dynamical analysis of a delayed hiv virus dynamic...
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Research ArticleDynamical Analysis of a Delayed HIV Virus Dynamic Model withCell-to-Cell Transmission and Apoptosis of Bystander Cells
Tongqian Zhang 12 Junling Wang1 Yi Song 1 and Zhichao Jiang 3
1College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China2State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and theMinistry of Science and Technology Shandong University of Science and Technology Qingdao 266590 China3School of Liberal Arts and Sciences North China Institute of Aerospace Engineering Langfang 065000 China
Correspondence should be addressed to Yi Song songyi_sysinacom
Received 20 November 2019 Revised 25 April 2020 Accepted 20 July 2020 Published 17 August 2020
Academic Editor Zhile Yang
Copyright copy 2020 Tongqian Zhang et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
In this paper a delayed viral dynamical model that considers two different transmission methods of the virus and apoptosis ofbystander cells is proposed and investigated e basic reproductive number R0 of the model is derived Based on the basicreproductive number we prove that the disease-free equilibrium E0 is globally asymptotically stable for R0 lt 1 by constructingsuitable Lyapunov functional For R0 gt 1 by regarding the time delay as bifurcation parameter the existence of local Hopfbifurcation is investigated e results show that time delay can change the stability of endemic equilibrium and cause periodicoscillations Finally we give some numerical simulations to illustrate the theoretical findings
1 Introduction and Model Formulation
AIDS is a serious immune-mediated disorder caused byhuman immunodeficiency virus (HIV) infection Contin-uous high-level HIV replication leads to a chronic fatalinfection By destroying important cells in the human im-mune system such as helper Tcells (especially CD4+Tcells)macrophages and dendritic cells HIV destroys or damagesthe functions of various immune cells in the immune systemresulting in a wide range of immune abnormalities [1] Withthe development of infection the human immune systembegins to weaken which leads to the extreme decline of thebodyrsquos resistance leading to a variety of opportunistic in-fections such as herpes zoster tuberculosis Pneumocystiscarinii pneumonia toxoplasmosis microsporidia and so on[2 3]
Since its first discovery in the United States in 1981AIDS has spread very rapidly By 2018 AIDS has been foundin almost every country in the world According to the latestUNAIDS report approximately 379 million peopleworldwide have been infected with HIV by 2018
approximately 233 million people are receiving anti-retroviral treatment nearly 770000 people have died ofAIDS-related diseases [4] AIDS has become one of the mostintractable infectious diseases in todayrsquos society posing ahuge threat to the survival and further development ofmankind Especially with the globalization of human ac-tivities the increasing frequency of peoplersquos communicationand the continuous evolution of HIV virus pose newchallenges to the prevention and treatment of AIDS Due toseveral unique characteristics of HIV replication andtransmission such as two transmission mechanisms of thevirus and the lack of proofreading mechanisms effectivevaccines and specific medicines that can completely cureAIDS have not been developed until now [5] Only pro-phylactic blockers have been developed that can be usedbefore and after the onset of high-risk behavior and thisantiretroviral drug can only delay the progression of thedisease erefore a more in-depth study of the HIV virus isvery important
Mathematical modeling plays an important role in thestudy of biological systems Mathematical models can
HindawiComplexityVolume 2020 Article ID 2313102 12 pageshttpsdoiorg10115520202313102
provide a more typical more refined and more quantitativedescription of complex systems [6ndash18] Due to the uncer-tainty of clinical measurement the low quality of samplesand the high risk of repeated trials the use of mathematicalmodels to describe the infection process qualitatively andquantitatively to study the mechanism of AIDS infectionand the treatment of AIDS is of great significance to thecontrol of HIV infection [19ndash21]
In 1996 Nowak et al [22] proposed a virus infectionmodel of ordinary differential equation describing the re-lationship among infected cells uninfected cells and freeviruses as follows
_x(t) s minus dx(t) minus βx(t)v(t)
_y(t) βx(t)v(t) minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎨
⎪⎪⎩(1)
where x(t) y(t) and v(t) are the concentrations of unin-fected CD4+T cells infected CD4+T cells and the free virusparticles respectively s is the rate at which new uninfectedCD4+T cells are produced and d is the natural mortality ofuninfected CD4+T cells p is the mortality of infectedCD4+T cells infected with virus or immune system β is theprobability of uninfected CD4+T cells infected with the freevirus k is the rate of CD4+Tcells producing free virus and u
is the rate at which the bodyrsquos immune system clears the
virus e authors derived the basic reproductive numberR0 βλkadu and proved that when R0 lt 1 the infection willnot spread and the system will return to the uninfected stateand when R0 gt 1 the number of infected cells will increasewhile the number of uninfected cells will decrease and thesystem will converge to the disease-free equilibrium In themodel the term βxv is the rate that infected cells are pro-duced by the contact of uninfected cells and free virususually this transmission method is called cell-free virusspread Based on model (1) large mathematical modelsconsidering cell-free virus transmission and differentfunctional responses have been proposed and analyzed forexample βxv(x + y) in [23] βxv(1 + bv) in [24 25]βxv(1 + ax + bv) in [26] βxv(1 + ax + bv + abxv) in [27]and general nonlinear functional response [28ndash30]
Perelson and Nelson [31] Wang and Li [32] and Huet al [33] extended model (1) by introducing logistic growthfor susceptible CD4+ T cells However many researchstudies have shown that virus can be transmitted directlyfrom cell to cell by virological synapses ie cell-to-celltransmission [34ndash41] en recently Lai and Zou [42]proposed a viral dynamical model that combines both cell-free virus transmission and cell-to-cell transmission asfollows
_x(t) rx(t) 1 minusx(t) + αy(t)
xM
1113888 1113889 minus β1x(t)v(t) minus β2x(t)y(t)
_y(t) β1x(t)v(t) + β2x(t)y(t) minus dyy(t)
_v(t) cy(t) minus dvv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(2)
where r is the target cell growth rate limited by the carryingcapacity of the target cell xM and the constant α indicates therestriction of infected cells that are usually applied to thegrowth of cells of target cells α≫ 1 ey found that if Rgt 1then the infection will persist and the Hopf bifurcation willoccur within a certain range of parameters where R is thebasic reproductive number
What is interesting is that there are some differencesbetween the above models about the decrease of uninfectedcells e decrease of uninfected cells x in model (1) is due tothe natural death or the infection by free virus while inmodel (2) the decrease of x is caused by the density-de-pendent mortality or the infection by free virus or the in-fection by infected cells However recent research studieshave shown that apoptosis is an important factor of thedecrease of uninfected cells [43] and HIV can produceapoptosis signals that induce apoptosis in uninfectedCD4+T cells [44 45] en by taking into account the timedelay in HIV virus from HIV infection to produce new viralparticles [34] and apoptosis of bystander cells Cheng et al
[46] proposed a virus model for HIV infection with discretedelay as follows
_x(t) s minus dx(t) minus cx(t)y(t) minus βx(t)v(t)
_y(t) eminus λτβx(t minus τ)v(t minus τ) minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎨
⎪⎪⎩(3)
where c is the apoptotic rate of uninfected cells induced byinfected CD4+Tcells and eminus λτ represents the probability ofsurvival of infected cells during the incubation periodfrom virus particles entering cells to releasing new virusparticles ey proved that when the basic reproductivenumber R0 lt 1 the infection-free equilibrium is globallyasymptotically stable and when R0 gt 1 the infectedequilibrium is locally asymptotically stable and the systemis uniformly persistent Guo and Ma [47] extended model(3) by using a general functional response and obtainedthe global properties of the model Ji et al [48] expandedmodel (3) by considering a full logistic growth of unin-fected cells
2 Complexity
en based on model (3) and motivated by [32 47 48]considering the time delay in HIV virus from HIV infectionto produce new viral particles and the time delay in the cell-
to-cell transmission and the apoptosis effect of uninfectedCD4+T cells we propose a delayed viral model as follows
_x(t) s + rx(t) 1 minusx(t) + y(t)
T1113888 1113889 minus dx(t) minus βx(t)v(t) minus αx(t)y(t) minus cx(t)y(t)
_y(t) δx(t minus τ)v(t minus τ) + ηx(t minus τ)y(t minus τ) minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(4)
where x y and v are the concentrations of uninfected cellsinfected cells and free virus s is the rate that new uninfectedcells are produced r is the intrinsic growth rate and T is themaximum level of CD4+T cells d p and u are the naturalmortalities of uninfected cells infected cells and free virusrespectively α is the probability of healthy cells being infectedby infected cells δ represents the survival rate of infected cellsbefore effective infection c is the apoptotic rate of uninfectedcells induced by infected CD4+Tcells and η is the survival rateof healthy cells before effective infection by infected cells
e remainder of the paper is organized as follows InSection 2 we will give the initial conditions of model (4) andsome preliminary properties In Section 3 we will discuss thestability of the equilibria In Section 4 we illustrate the mainresults by some numerical simulations Finally we give abrief conclusion and topics for further research in the future
2 Basic Reproductive Number andthe Equilibria
Let C C([minus τ 0] R3) be the Banach space of continuousfunctions mapping from interval [minus τ 0] to R3 equipped withthe sup-norme initial condition of the model (4) is givenas follows
x(θ) ϕ1(θ) y(θ) ϕ2(θ) v(θ) ϕ3(θ) (θ isin [minus τ 0])
(5)
where ϕ (ϕ1ϕ2 ϕ3) isin C such that ϕi(θ) ge 0(θ isin [minus τ 0]
i 1 2 3) About the boundedness and non-negativity of thesolutions of model (4) by using the method in [49] we canprove the following lemma
Lemma 1 6e solution (x(t) y(t) v(t)) of model (4) withinitial condition (5) is existent unique and non-negative on[0 +infin) and also ultimately bounded Moreover
lim supt⟶+infin
x(t) ≔ x0
lim supt⟶+infin
(x(t) + y(t + τ)) ≔s + rx0
d1
lim supt⟶+infin
v(t) ≔k s + rx0( 1113857
d1u
(6)
where d1 min d p1113864 1113865 and x0 T2r(r minus d+
(d minus r)2 + 4rsT1113969
)
e basic reproductive number of model (4) isR0 ((δk + ηu)pu)x0 en model (4) always has a virus-free equilibrium E0 (x0 0 0) where x0 T2r(r minus d+
(d minus r)2 + 4rsT1113969
) and if R0 gt 1 there exists a unique epi-demic equilibrium Elowast (xlowast ylowast vlowast) where
xlowast
pu
δk + ηu
ylowast
s + xlowast r minus rxlowastT( ) minus d( )
xlowast((rT) +(βku) + α + c)
vlowast
kylowast
u
(7)
3 Stability for E0 and Elowast
In this section we will discuss the stability forE0 and Elowast withthe increase of R0 We start with R0 lt 1
Theorem 1 For model (4) if R0 lt 1 E0 is globally asymp-totically stable for any time delay τ gt 0
Proof We consider the linearization system of model (4) atE0 as follows
_x(t) r minus d minus2rx0
T1113874 1113875x(t) minus
r
T+ α + c1113874 1113875x0y(t) minus βx0v(t)
_y(t) ηx0 minus p( 1113857y(t minus τ) + δx0v(t minus τ)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
(8)
e characteristic equation for E0 is given as
λ minus r + d +2r
Tx01113874 1113875 λ + p minus ηx0e
minus λτ1113872 1113873(λ + u) minus kδx0e
minus λτ1113960 1113961 0
(9)It is easy to see that (9) has a root
Complexity 3
λ1 r minus d minus2r
Tx0 minus
(d minus r)2 minus4rs
T
1113970
lt 0 (10)
en we only need to discuss the distribution of theroots of
λ + p minus ηx0eminus λτ
1113872 1113873(λ + u) minus kδx0eminus λτ
0 (11)
Obviously equation (11) can be written as
λ2 +(p + u)λ minus ηx0eminus λτλ + pu minus (ηu + kδ)x0e
minus λτ 0
(12)
When R0 lt 1 pu minus (ηu + kδ)x0 gt 0 therefore λ 0 isnot a root of (12) If (12) has a pair of pure imaginary rootλ iω(ωgt 0) for any τ gt 0 substituting it into (12) we canget that
iω(p + u) + pu minus ω21113872 1113873 i ωηx0 cos(ωτ) minus x0(ηu + kδ)sin(ωτ)1113858 1113859
+ ωηx0 sin(ωτ) + x0(ηu + kδ)cos(ωτ)1113858 1113859
(13)
and then separating its real and imaginary parts one gets
ω(p + u) ωηx0 cos(ωτ) minus x0(ηu + kδ)sin(ωτ)
pu minus ω2 ωηx0 sin(ωτ) + x0(ηu + kδ)cos(ωτ)1113896 (14)
Simplifying the above formula and letting κ ω2 wehave
f(κ) κ2 + p2
+ u2
minus x20η
21113872 1113873κ + p
2u2
minus x20(ηu + kδ)
2 0
(15)
On the one hand it is easy to get that
p2u2
minus x20(ηu + kδ)
2 p
2u2 1 minus R
201113872 1113873gt 0 (16)
On the other hand
p2
+ u2
minus x20η
2
1R20
minus 11113888 1113889x20y
2+
1R20
δ2k2
u2 +2δky
u1113888 1113889x
20 + u
2 gt 0
(17)
where R0 ((δk + ηu)pu)x0 lt 1 is usedClearly it indicates that f(κ)gt 0 which is in conflict
with f(κ) 0 is shows that all the roots of (8) havenegative real parts for any time delay τ ge 0 erefore thedisease-free equilibrium E0 is locally asymptotically stablefor any time delay τ ge 0
Next let us prove that if R0 lt 1 E0 is globally attractivefor any time delay τ ge 0 Define
G ϕ ϕ1 ϕ2ϕ3( 1113857 isin C1113868111386811138681113868 x0 ge ϕ1 ge 0ϕ2 ge 0ϕ3 ge 01113966 1113967
(18)
It is easy to show thatG attracts all solutions of model (4)and is also positively invariant with respect to model (4) IfR0 lt 1 let us choose a functional W(ϕ) on G as follows
W(ϕ) mϕ2(0) + ϕ3(0) + n 11139460
minus τϕ2(θ)dθ + ε1113946
0
minus τϕ3(θ)dθ
(19)
where m k(p minus ηx0) n mηx0 kηx0(p minus ηx0) n+
k mp and εgt 0 is a constant to be chosen later SinceR0 lt 1 we have m ngt 0 It can be found that W(ϕ) iscontinuous on the subset G in C From the invariance of Gfor any ϕ isin G the solution (x(t) y(t) v(t)) of model (4)satisfies x(t)le x0 for any tge 0 It follows from (4) that
_W(ϕ) m δϕ1(minus τ)ϕ3(minus τ) + ηϕ1(minus τ)ϕ2(minus τ)1113858 1113859 minus mpϕ2(0) + kϕ2(0) minus uϕ3(0)
+ n ϕ2(0) minus ϕ2(minus τ)( 1113857 + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
lem δϕ1(minus τ)ϕ3(minus τ) + ηx0ϕ2(minus τ)1113858 1113859 minus mpϕ2(0) + kϕ2(0) minus uϕ3(0)
+ n ϕ2(0) minus ϕ2(minus τ)( 1113857 + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
mδ ϕ1(minus τ)ϕ3(minus τ)( 1113857 + nϕ2(0) + kϕ2(0) minus mpϕ2(0) minus uϕ3(0)
+ ε ϕ3(0) minus ϕ3(minus τ)( 1113857
mδϕ1(minus τ)ϕ3(minus τ) minus uϕ3(0) + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
ϕ3(minus τ) mδϕ1(minus τ) minus ε( 1113857 + ϕ3(0)(ε minus u)
leϕ3(minus τ) mδx0 minus ε( 1113857 + ϕ3(0)(ε minus u)
(20)
SinceR0 lt 1 it is possible that we can choose the parameterεgt 0 such that mδx0 lt εlt u us it has_W(ϕ)le ϕ3(0)(ε minus u)le 0 for any ϕ isin G All these manifestthat W(ϕ) is a Lyapunov functional on the subset G By using
LyapunovndashLaSalle invariance principle the disease-free equi-librium E0 of model (4) is globally asymptotically stable
Next let us analyze the stability of the epidemic equi-librium Elowast And the linearized system of model (4) at Elowast is
4 Complexity
_x(t) minuss
xlowast+
rxlowast
T1113888 1113889x(t) minus
r
T+ α + c1113874 1113875xlowasty(t) minus βxlowastv(t)
_y(t) δ vlowastx(t minus τ) + xlowastv(t minus τ)( ) + η ylowastx(t minus τ) + xlowasty(t minus τ)( 1113857 minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(21)
e characteristic equation for Elowast is given as
λ +s
xlowast+ r
xlowast
T1113888 1113889 λ + p minus ηx
lowaste
minus λτ1113872 1113873(λ + u) minus kδx
lowaste
minus λτ1113960 1113961
+ eminus λτ δv
lowast+ ηylowast
( 1113857 xlowast r
T+ α + c1113874 1113875(λ + u) + kβx
lowast1113876 1113877 0
(22)
On the other hand (22) is equivalent to
P(λ) + Q(λ)eminus λτ
0 (23)
where
P(λ) λ3 + a1λ2
+ a2λ + a3 (24)
Q(λ) b1λ2
+ b2λ + b3 (25)
with
a1 p + u + B
a2 pu + pB + Bu
a3 puB
b1 minus I b2 GE minus BI minus uI minus kH
b3 GEu + GkF minus kBH minus BIu
G(Eu + kF) minus B(kH + Iu)
(26)
B s
xlowast+ r
xlowast
T
E xlowast r
T+ α + c1113874 1113875
F βxlowast
G δvlowast
+ ηylowast
H δxlowast
I ηxlowast
(27)
When τ 0 (12) reduces to
λ3 + a1 + b1( 1113857λ2 + a2 + b2( 1113857λ + a3 + b3 0 (28)
Notice that
a1 + b1 p + u + B minus I
gtp + u minus I
p + u minus ηxlowast
kpδ + kuδ + ηu2
kδ + ηugt 0
a2 + b2 B(p + u minus I) + GE + pu minus uI minus kH
B(p + u minus I) + GEgt 0
a3 + b3 puB + G(Eu + kF) minus B(kH + Iu)
G(Eu + kF)gt 0
H1 a1 + b1 gt 0
H2 a1 + b1( 1113857 a2 + b2( 1113857 minus a3 + b3( 1113857
B(p + u minus I)2
+ GE(p + u minus I) + B2(p + u minus I)
+ G(EB minus Eu minus kF)
(29)
erefore if R0 gt 1 and H2 gt 0 hold by theRouthndashHurwitz criterion the epidemic equilibrium Elowast islocally asymptotically stable for any time delay τ 0
Next let us investigate the stability of Elowast when τ gt 0Since a3 + b3 puB + G(Eu + kF) minus B(kH + Iu) G
(Eu + kF)gt 0 τ 0 is not the root of (23) For τ gt 0 weassume that (23) has the pure imaginary root λ iω(ωgt 0)and substituting it into (23) we can get that
minus iω3minus a1ω
2+ ia2ω + a3 + minus b1ω
2+ ib2ω + b31113872 1113873
middot (cosωτ minus i sinωτ) 0(30)
and then separating its real and imaginary parts we get
ω3minus a2ω b1ω
2minus b31113872 1113873sin(ωτ) + b2ω cos(ωτ)
a1ω2
minus a3 b3 minus b1ω2
1113872 1113873cos(ωτ) + b2ω sin(ωτ)(31)
Clearly it is equivalent to
ω6+ a
21 minus b
21 minus 2a21113872 1113873ω4
+ a22 minus b
22 minus 2b1b31113872 1113873ω2
+ a23 minus b
231113872 1113873 0
(32)
Let c1 a21 minus b21 minus 2a2 c2 a2
2 minus b22 minus 2a1a3 minus 2b1b3
and c3 a23 minus b23 (22) can be written as
Complexity 5
ω6+ c1ω
4+ c2ω
2+ c3 0 (33)
Let ϖ ω2 (33) is equivalent to
ϖ3 + c1ϖ2
+ c2ϖ + c3 0 (34)
Define
g(ϖ) ϖ3 + c1ϖ2
+ c2ϖ + c3 (35)
and thus
gprime(ϖ) 3ϖ2 + 2c1ϖ + c2 (36)
We consider that
3ϖ2 + 2c1ϖ + c2 0 (37)
and it has two real roots given as ϖ1 (minus c1 +Δ
radic)3 and
ϖ2 (minus c1 minusΔ
radic)3 where Δ c21 minus 3c2 Next we get the
following lemma from [50]
Lemma 2 For polynomial (34) the following conclusions aregiven
(i) If c3 lt 0 (34) has at least one positive root(ii) If c3 ge 0 and Δlt 0 (34) has no real root(iii) If c3 ge 0 and Δgt 0 and if and only ifϖ1 (minus c1 +
Δ
radic)3gt 0 and g(ϖ1)le 0 (34) has real
roots
Assume that g(ϖ) 0 has positive real roots thereforewe can suppose that (34) has k(1le kle 3) positive real rootsdenoted as ϖ1ϖ2 and ϖ3 Moreover (33) has positive realroots ωk
ϖk
radic From (31) we can get that
cos(ωτ) b2 minus a1b1( 1113857ω4 + a1b3 + a3b1 minus a2b2( 1113857ω2 minus a3b3
b22ω2 + b3 minus b1ω2( 11138572
(38)
In addition we get the corresponding τ(n)k gt 0 such that
(33) has pure imaginary λ iωk where
τ(n)k
1ωk
arccosb2 minus a1b1( 1113857ω4
k + a1b3 + a3b1 minus a2b2( 1113857ω2k minus a3b3
b22ω2k + b3 minus b1ω2
k1113872 11138732
⎛⎜⎝ ⎞⎟⎠ + 2nπ⎧⎪⎨
⎪⎩
⎫⎪⎬
⎪⎭ (39)
in which k 1 2 3 n 0 1 2
Define
τlowast minkisin[123]
τ(0)k1113966 1113967 (40)
Differentiating the two sides of (15) with respect to τ itfollows that
3λ2 + 2a1λ + a21113872 1113873dλdτ
+ 2b1λ + b2( 1113857eminus λτdλ
dτ
minus b1λ2
+ b2λ + b31113872 1113873eminus λτ λ + τ
dλdτ
1113888 1113889 0
(41)
and therefore
dλdτ
1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 + 2a1ωki
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873i
+2b1ωki + b2
minus b2ω2k + b3ωk minus b1ω3
k1113872 1113873iminus
τωki
(42)
end(Res(λ))
dτ1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 a2ω2k minus ω4
k( 1113857 + 2a1ωk a1ω3k minus a3ωk( 1113857
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873
+2b1ωk b3ωk minus b1ω3
k( 1113857 minus b22ω2k
b22ω4k + b3ωk minus b1ω3
k1113872 11138732
(43)
From (31) we have
ω3minus a2ω1113872 1113873
2+ a1ω minus a3( 1113857
2 b
22ω
2+ b1ω
2minus b31113872 1113873
2 (44)
Hence
d(Res(λ))
dτ1113888 11138891113888 1113889
minus 1
λiωk
3ϖ3k + 2c1ϖ2k + c2ϖk
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
ωkgprime ϖk( 1113857
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
(45)
Since ωk gt 0 this is enough to demonstrate thatRe(dλ(τ)dτ)|ττ(n)
k
and gprime(ϖk) have the same sign Com-bining Lemma 2 with (45) we have the followingconclusions
Theorem 2 If R0 gt 1 the following results hold
(i) If c3 ge 0 and Δle 0 then the epidemic equilibriumElowast (xlowast ylowast vlowast) is locally asymptotically stable
(ii) If c3 ge 0 or c3 lt 0 and Δgt 0 then the epidemicequilibrium Elowast (xlowast ylowast vlowast) is asymptotically sta-ble when τ isin [0 τlowast) and unstable when τ gt τlowast
(iii) If the conditions of (ii) are all satisfied andgprime(ϖk)ne 0 then model (1) undergoes a Hopf bifur-cation at Elowast when τ gt τ(n)
k (n 0 1 2 )
6 Complexity
4 Numerical Simulations
In this section we give some numerical simulations tovalidate our main results in Section 3 e basic parametersare set as follows [51] s 2 r 10 T 10 d
002 β 0000027 α 0000024 c 0001 δ 0002 η
0002 p 05 k 10 and u 05 Direct calculations withMaple 14 show thatR 08548lt 1 and E0 (101765 0 0)Figure 1 shows that virus-free equilibrium E0 of the system isstable
In the following we change T to 100 and direct cal-culations show system (4) has two equilibria ieE0 (101765 0 0) and E1 (119048 881959 17639171)
(R 84gt 1) We easily get
a3 03396
b3 18997
g(0) c3 a23 minus b
23 minus 34937lt 0
(46)
en equation (31) has one positive root u 3165equation (28) has one positive rootϖ0 1779 and the Hopfbifurcation value is τ0 05304 First let τ 052lt τ0 andselect three different initial values I1 (1 001 1700)I2 (1 80 001) and I3 (1 80 1700) respectively Nu-merical simulations show that the solutions of systemconverge to the same equilibrium respectively (see Fig-ures 2 3 and 4) In the biological sense tiny presence ofinfected cells or virus can lead to the disease transmissionNext let τ 054gt τ0 Figure 5 shows that periodic
x
0
2
4
6
8
10
12
40002000 50001000 60000 3000t
(a)
y
80
70
60
50
40
30
20
10
020001000 3000 4000 5000 60000
t
(b)
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 60000t
(c)
80 60 10840 6420 2
2004006008001000120014001600
(d)
Figure 1 Illustration of basic behavior of solutions of system (4) with initial condition I3 where R 08548lt 1 (a) time series of x (t)(b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 7
x
0
10
20
30
40
50
60
70
80
90
100
2000 4000 6000 8000 10000 12000 140000t
(a)
y
100004000 140000 6000 80002000 12000t
0
20
40
60
80
100
120
140
(b)
v
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
v
120100 8080y
6060x4040 20 20
1100
1200
1300
1400
1500
1600
1700
1800
(d)
Figure 2 Illustration of basic behavior of solutions of system (4) with τ 052 where R 84gt 1 (the initial condition is (1 001 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
1000 2000 3000 4000 5000 6000 7000 8000 90000t
0
10
20
30
40
50
60
(a)
y
1000 4000 6000 900070002000 80003000 50000t
40
50
60
70
80
90
100
(b)
Figure 3 Continued
8 Complexity
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 6000 7000 8000 90000t
(c)
v
901080 20
70x30
y
60 405050
50010001500
(d)
Figure 3 Illustration of basic behavior of solutions of system (4) with τ 052 whereR 84gt 1 (the initial condition is (1 80 001)) (a)time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
2000 4000 6000 8000 10000 12000 140000t
0
5
10
15
20
25
30
35
40
(a)
y
50
60
70
80
90
100
110
120
2000 4000 6000 8000 10000 12000 140000t
(b)
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
100 3080 201060
145015001550160016501700175018001850
v
y x
(d)
Figure 4 Illustration of basic behavior of solutions of system (4) with τ 052lt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 9
oscillations occur for τ gt τ0 en system (4) undergoes aHopf bifurcation at Elowast when τ τ0
5 Conclusion
In this paper we improved a delayed HIV virus dynamicmodel by introducing cell-to-cell transmission and apo-ptosis of bystander cells We derived the basic reproductivenumber R0 lt 1 and based on the basic reproductive numberthe stability of the equilibria of the system was investigatede mathematical analysis showed that if R0 lt 1 then thedisease-free equilibrium of system (4) is globally asymp-totically stable and for R0 gt 1 the system can produce Hopfbifurcation with the delay changing In biology HIV cannotsuccessfully invade healthy cells and will be cleared by theimmune system finally if R0 lt 1 and if R0 gt 1 uninfectedcells infected cells and free virus can coexist as periodicoscillations under certain conditions
However it is very necessary to consider antiretroviraltherapy in model (4) since it can make the model morerealistic and many scholars have noticed this topic andachieved excellent results for example Wang et al [52ndash54]
have studied the virus dynamics model with antiretroviraltherapy eir research shows that antiretroviral therapy canincrease the number of uninfected T cells and decrease thenumber of infectious virions and is very effective for thetreatment of AIDS patients In present model we do notconsider this factor and we will leave this for future research
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
Tongqian Zhang was supported by the Shandong ProvincialNatural Science Foundation of China (no ZR2019MA003)and Scientific Research Foundation of Shandong Universityof Science and Technology for Recruited Talents Zhichao
x
times104151 35 405 3252 450
t
0
5
10
15
20
25
30
35
40
45
(a)
times104
y
41505 2 25 3 450 1 35t
50
60
70
80
90
100
110
120
130
(b)
times104
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
151 35 405 3252 450t
(c)
120
v
40100y 30
x80 201060
145015001550160016501700175018001850
(d)
Figure 5 Illustration of basic behavior of solutions of system (4) with τ 054gt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
10 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
[1] R Weiss ldquoHow does HIV cause AIDSrdquo Science vol 260no 5112 pp 1273ndash1279 1993
[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
provide a more typical more refined and more quantitativedescription of complex systems [6ndash18] Due to the uncer-tainty of clinical measurement the low quality of samplesand the high risk of repeated trials the use of mathematicalmodels to describe the infection process qualitatively andquantitatively to study the mechanism of AIDS infectionand the treatment of AIDS is of great significance to thecontrol of HIV infection [19ndash21]
In 1996 Nowak et al [22] proposed a virus infectionmodel of ordinary differential equation describing the re-lationship among infected cells uninfected cells and freeviruses as follows
_x(t) s minus dx(t) minus βx(t)v(t)
_y(t) βx(t)v(t) minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎨
⎪⎪⎩(1)
where x(t) y(t) and v(t) are the concentrations of unin-fected CD4+T cells infected CD4+T cells and the free virusparticles respectively s is the rate at which new uninfectedCD4+T cells are produced and d is the natural mortality ofuninfected CD4+T cells p is the mortality of infectedCD4+T cells infected with virus or immune system β is theprobability of uninfected CD4+T cells infected with the freevirus k is the rate of CD4+Tcells producing free virus and u
is the rate at which the bodyrsquos immune system clears the
virus e authors derived the basic reproductive numberR0 βλkadu and proved that when R0 lt 1 the infection willnot spread and the system will return to the uninfected stateand when R0 gt 1 the number of infected cells will increasewhile the number of uninfected cells will decrease and thesystem will converge to the disease-free equilibrium In themodel the term βxv is the rate that infected cells are pro-duced by the contact of uninfected cells and free virususually this transmission method is called cell-free virusspread Based on model (1) large mathematical modelsconsidering cell-free virus transmission and differentfunctional responses have been proposed and analyzed forexample βxv(x + y) in [23] βxv(1 + bv) in [24 25]βxv(1 + ax + bv) in [26] βxv(1 + ax + bv + abxv) in [27]and general nonlinear functional response [28ndash30]
Perelson and Nelson [31] Wang and Li [32] and Huet al [33] extended model (1) by introducing logistic growthfor susceptible CD4+ T cells However many researchstudies have shown that virus can be transmitted directlyfrom cell to cell by virological synapses ie cell-to-celltransmission [34ndash41] en recently Lai and Zou [42]proposed a viral dynamical model that combines both cell-free virus transmission and cell-to-cell transmission asfollows
_x(t) rx(t) 1 minusx(t) + αy(t)
xM
1113888 1113889 minus β1x(t)v(t) minus β2x(t)y(t)
_y(t) β1x(t)v(t) + β2x(t)y(t) minus dyy(t)
_v(t) cy(t) minus dvv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(2)
where r is the target cell growth rate limited by the carryingcapacity of the target cell xM and the constant α indicates therestriction of infected cells that are usually applied to thegrowth of cells of target cells α≫ 1 ey found that if Rgt 1then the infection will persist and the Hopf bifurcation willoccur within a certain range of parameters where R is thebasic reproductive number
What is interesting is that there are some differencesbetween the above models about the decrease of uninfectedcells e decrease of uninfected cells x in model (1) is due tothe natural death or the infection by free virus while inmodel (2) the decrease of x is caused by the density-de-pendent mortality or the infection by free virus or the in-fection by infected cells However recent research studieshave shown that apoptosis is an important factor of thedecrease of uninfected cells [43] and HIV can produceapoptosis signals that induce apoptosis in uninfectedCD4+T cells [44 45] en by taking into account the timedelay in HIV virus from HIV infection to produce new viralparticles [34] and apoptosis of bystander cells Cheng et al
[46] proposed a virus model for HIV infection with discretedelay as follows
_x(t) s minus dx(t) minus cx(t)y(t) minus βx(t)v(t)
_y(t) eminus λτβx(t minus τ)v(t minus τ) minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎨
⎪⎪⎩(3)
where c is the apoptotic rate of uninfected cells induced byinfected CD4+Tcells and eminus λτ represents the probability ofsurvival of infected cells during the incubation periodfrom virus particles entering cells to releasing new virusparticles ey proved that when the basic reproductivenumber R0 lt 1 the infection-free equilibrium is globallyasymptotically stable and when R0 gt 1 the infectedequilibrium is locally asymptotically stable and the systemis uniformly persistent Guo and Ma [47] extended model(3) by using a general functional response and obtainedthe global properties of the model Ji et al [48] expandedmodel (3) by considering a full logistic growth of unin-fected cells
2 Complexity
en based on model (3) and motivated by [32 47 48]considering the time delay in HIV virus from HIV infectionto produce new viral particles and the time delay in the cell-
to-cell transmission and the apoptosis effect of uninfectedCD4+T cells we propose a delayed viral model as follows
_x(t) s + rx(t) 1 minusx(t) + y(t)
T1113888 1113889 minus dx(t) minus βx(t)v(t) minus αx(t)y(t) minus cx(t)y(t)
_y(t) δx(t minus τ)v(t minus τ) + ηx(t minus τ)y(t minus τ) minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(4)
where x y and v are the concentrations of uninfected cellsinfected cells and free virus s is the rate that new uninfectedcells are produced r is the intrinsic growth rate and T is themaximum level of CD4+T cells d p and u are the naturalmortalities of uninfected cells infected cells and free virusrespectively α is the probability of healthy cells being infectedby infected cells δ represents the survival rate of infected cellsbefore effective infection c is the apoptotic rate of uninfectedcells induced by infected CD4+Tcells and η is the survival rateof healthy cells before effective infection by infected cells
e remainder of the paper is organized as follows InSection 2 we will give the initial conditions of model (4) andsome preliminary properties In Section 3 we will discuss thestability of the equilibria In Section 4 we illustrate the mainresults by some numerical simulations Finally we give abrief conclusion and topics for further research in the future
2 Basic Reproductive Number andthe Equilibria
Let C C([minus τ 0] R3) be the Banach space of continuousfunctions mapping from interval [minus τ 0] to R3 equipped withthe sup-norme initial condition of the model (4) is givenas follows
x(θ) ϕ1(θ) y(θ) ϕ2(θ) v(θ) ϕ3(θ) (θ isin [minus τ 0])
(5)
where ϕ (ϕ1ϕ2 ϕ3) isin C such that ϕi(θ) ge 0(θ isin [minus τ 0]
i 1 2 3) About the boundedness and non-negativity of thesolutions of model (4) by using the method in [49] we canprove the following lemma
Lemma 1 6e solution (x(t) y(t) v(t)) of model (4) withinitial condition (5) is existent unique and non-negative on[0 +infin) and also ultimately bounded Moreover
lim supt⟶+infin
x(t) ≔ x0
lim supt⟶+infin
(x(t) + y(t + τ)) ≔s + rx0
d1
lim supt⟶+infin
v(t) ≔k s + rx0( 1113857
d1u
(6)
where d1 min d p1113864 1113865 and x0 T2r(r minus d+
(d minus r)2 + 4rsT1113969
)
e basic reproductive number of model (4) isR0 ((δk + ηu)pu)x0 en model (4) always has a virus-free equilibrium E0 (x0 0 0) where x0 T2r(r minus d+
(d minus r)2 + 4rsT1113969
) and if R0 gt 1 there exists a unique epi-demic equilibrium Elowast (xlowast ylowast vlowast) where
xlowast
pu
δk + ηu
ylowast
s + xlowast r minus rxlowastT( ) minus d( )
xlowast((rT) +(βku) + α + c)
vlowast
kylowast
u
(7)
3 Stability for E0 and Elowast
In this section we will discuss the stability forE0 and Elowast withthe increase of R0 We start with R0 lt 1
Theorem 1 For model (4) if R0 lt 1 E0 is globally asymp-totically stable for any time delay τ gt 0
Proof We consider the linearization system of model (4) atE0 as follows
_x(t) r minus d minus2rx0
T1113874 1113875x(t) minus
r
T+ α + c1113874 1113875x0y(t) minus βx0v(t)
_y(t) ηx0 minus p( 1113857y(t minus τ) + δx0v(t minus τ)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
(8)
e characteristic equation for E0 is given as
λ minus r + d +2r
Tx01113874 1113875 λ + p minus ηx0e
minus λτ1113872 1113873(λ + u) minus kδx0e
minus λτ1113960 1113961 0
(9)It is easy to see that (9) has a root
Complexity 3
λ1 r minus d minus2r
Tx0 minus
(d minus r)2 minus4rs
T
1113970
lt 0 (10)
en we only need to discuss the distribution of theroots of
λ + p minus ηx0eminus λτ
1113872 1113873(λ + u) minus kδx0eminus λτ
0 (11)
Obviously equation (11) can be written as
λ2 +(p + u)λ minus ηx0eminus λτλ + pu minus (ηu + kδ)x0e
minus λτ 0
(12)
When R0 lt 1 pu minus (ηu + kδ)x0 gt 0 therefore λ 0 isnot a root of (12) If (12) has a pair of pure imaginary rootλ iω(ωgt 0) for any τ gt 0 substituting it into (12) we canget that
iω(p + u) + pu minus ω21113872 1113873 i ωηx0 cos(ωτ) minus x0(ηu + kδ)sin(ωτ)1113858 1113859
+ ωηx0 sin(ωτ) + x0(ηu + kδ)cos(ωτ)1113858 1113859
(13)
and then separating its real and imaginary parts one gets
ω(p + u) ωηx0 cos(ωτ) minus x0(ηu + kδ)sin(ωτ)
pu minus ω2 ωηx0 sin(ωτ) + x0(ηu + kδ)cos(ωτ)1113896 (14)
Simplifying the above formula and letting κ ω2 wehave
f(κ) κ2 + p2
+ u2
minus x20η
21113872 1113873κ + p
2u2
minus x20(ηu + kδ)
2 0
(15)
On the one hand it is easy to get that
p2u2
minus x20(ηu + kδ)
2 p
2u2 1 minus R
201113872 1113873gt 0 (16)
On the other hand
p2
+ u2
minus x20η
2
1R20
minus 11113888 1113889x20y
2+
1R20
δ2k2
u2 +2δky
u1113888 1113889x
20 + u
2 gt 0
(17)
where R0 ((δk + ηu)pu)x0 lt 1 is usedClearly it indicates that f(κ)gt 0 which is in conflict
with f(κ) 0 is shows that all the roots of (8) havenegative real parts for any time delay τ ge 0 erefore thedisease-free equilibrium E0 is locally asymptotically stablefor any time delay τ ge 0
Next let us prove that if R0 lt 1 E0 is globally attractivefor any time delay τ ge 0 Define
G ϕ ϕ1 ϕ2ϕ3( 1113857 isin C1113868111386811138681113868 x0 ge ϕ1 ge 0ϕ2 ge 0ϕ3 ge 01113966 1113967
(18)
It is easy to show thatG attracts all solutions of model (4)and is also positively invariant with respect to model (4) IfR0 lt 1 let us choose a functional W(ϕ) on G as follows
W(ϕ) mϕ2(0) + ϕ3(0) + n 11139460
minus τϕ2(θ)dθ + ε1113946
0
minus τϕ3(θ)dθ
(19)
where m k(p minus ηx0) n mηx0 kηx0(p minus ηx0) n+
k mp and εgt 0 is a constant to be chosen later SinceR0 lt 1 we have m ngt 0 It can be found that W(ϕ) iscontinuous on the subset G in C From the invariance of Gfor any ϕ isin G the solution (x(t) y(t) v(t)) of model (4)satisfies x(t)le x0 for any tge 0 It follows from (4) that
_W(ϕ) m δϕ1(minus τ)ϕ3(minus τ) + ηϕ1(minus τ)ϕ2(minus τ)1113858 1113859 minus mpϕ2(0) + kϕ2(0) minus uϕ3(0)
+ n ϕ2(0) minus ϕ2(minus τ)( 1113857 + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
lem δϕ1(minus τ)ϕ3(minus τ) + ηx0ϕ2(minus τ)1113858 1113859 minus mpϕ2(0) + kϕ2(0) minus uϕ3(0)
+ n ϕ2(0) minus ϕ2(minus τ)( 1113857 + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
mδ ϕ1(minus τ)ϕ3(minus τ)( 1113857 + nϕ2(0) + kϕ2(0) minus mpϕ2(0) minus uϕ3(0)
+ ε ϕ3(0) minus ϕ3(minus τ)( 1113857
mδϕ1(minus τ)ϕ3(minus τ) minus uϕ3(0) + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
ϕ3(minus τ) mδϕ1(minus τ) minus ε( 1113857 + ϕ3(0)(ε minus u)
leϕ3(minus τ) mδx0 minus ε( 1113857 + ϕ3(0)(ε minus u)
(20)
SinceR0 lt 1 it is possible that we can choose the parameterεgt 0 such that mδx0 lt εlt u us it has_W(ϕ)le ϕ3(0)(ε minus u)le 0 for any ϕ isin G All these manifestthat W(ϕ) is a Lyapunov functional on the subset G By using
LyapunovndashLaSalle invariance principle the disease-free equi-librium E0 of model (4) is globally asymptotically stable
Next let us analyze the stability of the epidemic equi-librium Elowast And the linearized system of model (4) at Elowast is
4 Complexity
_x(t) minuss
xlowast+
rxlowast
T1113888 1113889x(t) minus
r
T+ α + c1113874 1113875xlowasty(t) minus βxlowastv(t)
_y(t) δ vlowastx(t minus τ) + xlowastv(t minus τ)( ) + η ylowastx(t minus τ) + xlowasty(t minus τ)( 1113857 minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(21)
e characteristic equation for Elowast is given as
λ +s
xlowast+ r
xlowast
T1113888 1113889 λ + p minus ηx
lowaste
minus λτ1113872 1113873(λ + u) minus kδx
lowaste
minus λτ1113960 1113961
+ eminus λτ δv
lowast+ ηylowast
( 1113857 xlowast r
T+ α + c1113874 1113875(λ + u) + kβx
lowast1113876 1113877 0
(22)
On the other hand (22) is equivalent to
P(λ) + Q(λ)eminus λτ
0 (23)
where
P(λ) λ3 + a1λ2
+ a2λ + a3 (24)
Q(λ) b1λ2
+ b2λ + b3 (25)
with
a1 p + u + B
a2 pu + pB + Bu
a3 puB
b1 minus I b2 GE minus BI minus uI minus kH
b3 GEu + GkF minus kBH minus BIu
G(Eu + kF) minus B(kH + Iu)
(26)
B s
xlowast+ r
xlowast
T
E xlowast r
T+ α + c1113874 1113875
F βxlowast
G δvlowast
+ ηylowast
H δxlowast
I ηxlowast
(27)
When τ 0 (12) reduces to
λ3 + a1 + b1( 1113857λ2 + a2 + b2( 1113857λ + a3 + b3 0 (28)
Notice that
a1 + b1 p + u + B minus I
gtp + u minus I
p + u minus ηxlowast
kpδ + kuδ + ηu2
kδ + ηugt 0
a2 + b2 B(p + u minus I) + GE + pu minus uI minus kH
B(p + u minus I) + GEgt 0
a3 + b3 puB + G(Eu + kF) minus B(kH + Iu)
G(Eu + kF)gt 0
H1 a1 + b1 gt 0
H2 a1 + b1( 1113857 a2 + b2( 1113857 minus a3 + b3( 1113857
B(p + u minus I)2
+ GE(p + u minus I) + B2(p + u minus I)
+ G(EB minus Eu minus kF)
(29)
erefore if R0 gt 1 and H2 gt 0 hold by theRouthndashHurwitz criterion the epidemic equilibrium Elowast islocally asymptotically stable for any time delay τ 0
Next let us investigate the stability of Elowast when τ gt 0Since a3 + b3 puB + G(Eu + kF) minus B(kH + Iu) G
(Eu + kF)gt 0 τ 0 is not the root of (23) For τ gt 0 weassume that (23) has the pure imaginary root λ iω(ωgt 0)and substituting it into (23) we can get that
minus iω3minus a1ω
2+ ia2ω + a3 + minus b1ω
2+ ib2ω + b31113872 1113873
middot (cosωτ minus i sinωτ) 0(30)
and then separating its real and imaginary parts we get
ω3minus a2ω b1ω
2minus b31113872 1113873sin(ωτ) + b2ω cos(ωτ)
a1ω2
minus a3 b3 minus b1ω2
1113872 1113873cos(ωτ) + b2ω sin(ωτ)(31)
Clearly it is equivalent to
ω6+ a
21 minus b
21 minus 2a21113872 1113873ω4
+ a22 minus b
22 minus 2b1b31113872 1113873ω2
+ a23 minus b
231113872 1113873 0
(32)
Let c1 a21 minus b21 minus 2a2 c2 a2
2 minus b22 minus 2a1a3 minus 2b1b3
and c3 a23 minus b23 (22) can be written as
Complexity 5
ω6+ c1ω
4+ c2ω
2+ c3 0 (33)
Let ϖ ω2 (33) is equivalent to
ϖ3 + c1ϖ2
+ c2ϖ + c3 0 (34)
Define
g(ϖ) ϖ3 + c1ϖ2
+ c2ϖ + c3 (35)
and thus
gprime(ϖ) 3ϖ2 + 2c1ϖ + c2 (36)
We consider that
3ϖ2 + 2c1ϖ + c2 0 (37)
and it has two real roots given as ϖ1 (minus c1 +Δ
radic)3 and
ϖ2 (minus c1 minusΔ
radic)3 where Δ c21 minus 3c2 Next we get the
following lemma from [50]
Lemma 2 For polynomial (34) the following conclusions aregiven
(i) If c3 lt 0 (34) has at least one positive root(ii) If c3 ge 0 and Δlt 0 (34) has no real root(iii) If c3 ge 0 and Δgt 0 and if and only ifϖ1 (minus c1 +
Δ
radic)3gt 0 and g(ϖ1)le 0 (34) has real
roots
Assume that g(ϖ) 0 has positive real roots thereforewe can suppose that (34) has k(1le kle 3) positive real rootsdenoted as ϖ1ϖ2 and ϖ3 Moreover (33) has positive realroots ωk
ϖk
radic From (31) we can get that
cos(ωτ) b2 minus a1b1( 1113857ω4 + a1b3 + a3b1 minus a2b2( 1113857ω2 minus a3b3
b22ω2 + b3 minus b1ω2( 11138572
(38)
In addition we get the corresponding τ(n)k gt 0 such that
(33) has pure imaginary λ iωk where
τ(n)k
1ωk
arccosb2 minus a1b1( 1113857ω4
k + a1b3 + a3b1 minus a2b2( 1113857ω2k minus a3b3
b22ω2k + b3 minus b1ω2
k1113872 11138732
⎛⎜⎝ ⎞⎟⎠ + 2nπ⎧⎪⎨
⎪⎩
⎫⎪⎬
⎪⎭ (39)
in which k 1 2 3 n 0 1 2
Define
τlowast minkisin[123]
τ(0)k1113966 1113967 (40)
Differentiating the two sides of (15) with respect to τ itfollows that
3λ2 + 2a1λ + a21113872 1113873dλdτ
+ 2b1λ + b2( 1113857eminus λτdλ
dτ
minus b1λ2
+ b2λ + b31113872 1113873eminus λτ λ + τ
dλdτ
1113888 1113889 0
(41)
and therefore
dλdτ
1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 + 2a1ωki
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873i
+2b1ωki + b2
minus b2ω2k + b3ωk minus b1ω3
k1113872 1113873iminus
τωki
(42)
end(Res(λ))
dτ1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 a2ω2k minus ω4
k( 1113857 + 2a1ωk a1ω3k minus a3ωk( 1113857
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873
+2b1ωk b3ωk minus b1ω3
k( 1113857 minus b22ω2k
b22ω4k + b3ωk minus b1ω3
k1113872 11138732
(43)
From (31) we have
ω3minus a2ω1113872 1113873
2+ a1ω minus a3( 1113857
2 b
22ω
2+ b1ω
2minus b31113872 1113873
2 (44)
Hence
d(Res(λ))
dτ1113888 11138891113888 1113889
minus 1
λiωk
3ϖ3k + 2c1ϖ2k + c2ϖk
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
ωkgprime ϖk( 1113857
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
(45)
Since ωk gt 0 this is enough to demonstrate thatRe(dλ(τ)dτ)|ττ(n)
k
and gprime(ϖk) have the same sign Com-bining Lemma 2 with (45) we have the followingconclusions
Theorem 2 If R0 gt 1 the following results hold
(i) If c3 ge 0 and Δle 0 then the epidemic equilibriumElowast (xlowast ylowast vlowast) is locally asymptotically stable
(ii) If c3 ge 0 or c3 lt 0 and Δgt 0 then the epidemicequilibrium Elowast (xlowast ylowast vlowast) is asymptotically sta-ble when τ isin [0 τlowast) and unstable when τ gt τlowast
(iii) If the conditions of (ii) are all satisfied andgprime(ϖk)ne 0 then model (1) undergoes a Hopf bifur-cation at Elowast when τ gt τ(n)
k (n 0 1 2 )
6 Complexity
4 Numerical Simulations
In this section we give some numerical simulations tovalidate our main results in Section 3 e basic parametersare set as follows [51] s 2 r 10 T 10 d
002 β 0000027 α 0000024 c 0001 δ 0002 η
0002 p 05 k 10 and u 05 Direct calculations withMaple 14 show thatR 08548lt 1 and E0 (101765 0 0)Figure 1 shows that virus-free equilibrium E0 of the system isstable
In the following we change T to 100 and direct cal-culations show system (4) has two equilibria ieE0 (101765 0 0) and E1 (119048 881959 17639171)
(R 84gt 1) We easily get
a3 03396
b3 18997
g(0) c3 a23 minus b
23 minus 34937lt 0
(46)
en equation (31) has one positive root u 3165equation (28) has one positive rootϖ0 1779 and the Hopfbifurcation value is τ0 05304 First let τ 052lt τ0 andselect three different initial values I1 (1 001 1700)I2 (1 80 001) and I3 (1 80 1700) respectively Nu-merical simulations show that the solutions of systemconverge to the same equilibrium respectively (see Fig-ures 2 3 and 4) In the biological sense tiny presence ofinfected cells or virus can lead to the disease transmissionNext let τ 054gt τ0 Figure 5 shows that periodic
x
0
2
4
6
8
10
12
40002000 50001000 60000 3000t
(a)
y
80
70
60
50
40
30
20
10
020001000 3000 4000 5000 60000
t
(b)
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 60000t
(c)
80 60 10840 6420 2
2004006008001000120014001600
(d)
Figure 1 Illustration of basic behavior of solutions of system (4) with initial condition I3 where R 08548lt 1 (a) time series of x (t)(b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 7
x
0
10
20
30
40
50
60
70
80
90
100
2000 4000 6000 8000 10000 12000 140000t
(a)
y
100004000 140000 6000 80002000 12000t
0
20
40
60
80
100
120
140
(b)
v
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
v
120100 8080y
6060x4040 20 20
1100
1200
1300
1400
1500
1600
1700
1800
(d)
Figure 2 Illustration of basic behavior of solutions of system (4) with τ 052 where R 84gt 1 (the initial condition is (1 001 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
1000 2000 3000 4000 5000 6000 7000 8000 90000t
0
10
20
30
40
50
60
(a)
y
1000 4000 6000 900070002000 80003000 50000t
40
50
60
70
80
90
100
(b)
Figure 3 Continued
8 Complexity
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 6000 7000 8000 90000t
(c)
v
901080 20
70x30
y
60 405050
50010001500
(d)
Figure 3 Illustration of basic behavior of solutions of system (4) with τ 052 whereR 84gt 1 (the initial condition is (1 80 001)) (a)time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
2000 4000 6000 8000 10000 12000 140000t
0
5
10
15
20
25
30
35
40
(a)
y
50
60
70
80
90
100
110
120
2000 4000 6000 8000 10000 12000 140000t
(b)
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
100 3080 201060
145015001550160016501700175018001850
v
y x
(d)
Figure 4 Illustration of basic behavior of solutions of system (4) with τ 052lt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 9
oscillations occur for τ gt τ0 en system (4) undergoes aHopf bifurcation at Elowast when τ τ0
5 Conclusion
In this paper we improved a delayed HIV virus dynamicmodel by introducing cell-to-cell transmission and apo-ptosis of bystander cells We derived the basic reproductivenumber R0 lt 1 and based on the basic reproductive numberthe stability of the equilibria of the system was investigatede mathematical analysis showed that if R0 lt 1 then thedisease-free equilibrium of system (4) is globally asymp-totically stable and for R0 gt 1 the system can produce Hopfbifurcation with the delay changing In biology HIV cannotsuccessfully invade healthy cells and will be cleared by theimmune system finally if R0 lt 1 and if R0 gt 1 uninfectedcells infected cells and free virus can coexist as periodicoscillations under certain conditions
However it is very necessary to consider antiretroviraltherapy in model (4) since it can make the model morerealistic and many scholars have noticed this topic andachieved excellent results for example Wang et al [52ndash54]
have studied the virus dynamics model with antiretroviraltherapy eir research shows that antiretroviral therapy canincrease the number of uninfected T cells and decrease thenumber of infectious virions and is very effective for thetreatment of AIDS patients In present model we do notconsider this factor and we will leave this for future research
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
Tongqian Zhang was supported by the Shandong ProvincialNatural Science Foundation of China (no ZR2019MA003)and Scientific Research Foundation of Shandong Universityof Science and Technology for Recruited Talents Zhichao
x
times104151 35 405 3252 450
t
0
5
10
15
20
25
30
35
40
45
(a)
times104
y
41505 2 25 3 450 1 35t
50
60
70
80
90
100
110
120
130
(b)
times104
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
151 35 405 3252 450t
(c)
120
v
40100y 30
x80 201060
145015001550160016501700175018001850
(d)
Figure 5 Illustration of basic behavior of solutions of system (4) with τ 054gt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
10 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
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[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
en based on model (3) and motivated by [32 47 48]considering the time delay in HIV virus from HIV infectionto produce new viral particles and the time delay in the cell-
to-cell transmission and the apoptosis effect of uninfectedCD4+T cells we propose a delayed viral model as follows
_x(t) s + rx(t) 1 minusx(t) + y(t)
T1113888 1113889 minus dx(t) minus βx(t)v(t) minus αx(t)y(t) minus cx(t)y(t)
_y(t) δx(t minus τ)v(t minus τ) + ηx(t minus τ)y(t minus τ) minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(4)
where x y and v are the concentrations of uninfected cellsinfected cells and free virus s is the rate that new uninfectedcells are produced r is the intrinsic growth rate and T is themaximum level of CD4+T cells d p and u are the naturalmortalities of uninfected cells infected cells and free virusrespectively α is the probability of healthy cells being infectedby infected cells δ represents the survival rate of infected cellsbefore effective infection c is the apoptotic rate of uninfectedcells induced by infected CD4+Tcells and η is the survival rateof healthy cells before effective infection by infected cells
e remainder of the paper is organized as follows InSection 2 we will give the initial conditions of model (4) andsome preliminary properties In Section 3 we will discuss thestability of the equilibria In Section 4 we illustrate the mainresults by some numerical simulations Finally we give abrief conclusion and topics for further research in the future
2 Basic Reproductive Number andthe Equilibria
Let C C([minus τ 0] R3) be the Banach space of continuousfunctions mapping from interval [minus τ 0] to R3 equipped withthe sup-norme initial condition of the model (4) is givenas follows
x(θ) ϕ1(θ) y(θ) ϕ2(θ) v(θ) ϕ3(θ) (θ isin [minus τ 0])
(5)
where ϕ (ϕ1ϕ2 ϕ3) isin C such that ϕi(θ) ge 0(θ isin [minus τ 0]
i 1 2 3) About the boundedness and non-negativity of thesolutions of model (4) by using the method in [49] we canprove the following lemma
Lemma 1 6e solution (x(t) y(t) v(t)) of model (4) withinitial condition (5) is existent unique and non-negative on[0 +infin) and also ultimately bounded Moreover
lim supt⟶+infin
x(t) ≔ x0
lim supt⟶+infin
(x(t) + y(t + τ)) ≔s + rx0
d1
lim supt⟶+infin
v(t) ≔k s + rx0( 1113857
d1u
(6)
where d1 min d p1113864 1113865 and x0 T2r(r minus d+
(d minus r)2 + 4rsT1113969
)
e basic reproductive number of model (4) isR0 ((δk + ηu)pu)x0 en model (4) always has a virus-free equilibrium E0 (x0 0 0) where x0 T2r(r minus d+
(d minus r)2 + 4rsT1113969
) and if R0 gt 1 there exists a unique epi-demic equilibrium Elowast (xlowast ylowast vlowast) where
xlowast
pu
δk + ηu
ylowast
s + xlowast r minus rxlowastT( ) minus d( )
xlowast((rT) +(βku) + α + c)
vlowast
kylowast
u
(7)
3 Stability for E0 and Elowast
In this section we will discuss the stability forE0 and Elowast withthe increase of R0 We start with R0 lt 1
Theorem 1 For model (4) if R0 lt 1 E0 is globally asymp-totically stable for any time delay τ gt 0
Proof We consider the linearization system of model (4) atE0 as follows
_x(t) r minus d minus2rx0
T1113874 1113875x(t) minus
r
T+ α + c1113874 1113875x0y(t) minus βx0v(t)
_y(t) ηx0 minus p( 1113857y(t minus τ) + δx0v(t minus τ)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
(8)
e characteristic equation for E0 is given as
λ minus r + d +2r
Tx01113874 1113875 λ + p minus ηx0e
minus λτ1113872 1113873(λ + u) minus kδx0e
minus λτ1113960 1113961 0
(9)It is easy to see that (9) has a root
Complexity 3
λ1 r minus d minus2r
Tx0 minus
(d minus r)2 minus4rs
T
1113970
lt 0 (10)
en we only need to discuss the distribution of theroots of
λ + p minus ηx0eminus λτ
1113872 1113873(λ + u) minus kδx0eminus λτ
0 (11)
Obviously equation (11) can be written as
λ2 +(p + u)λ minus ηx0eminus λτλ + pu minus (ηu + kδ)x0e
minus λτ 0
(12)
When R0 lt 1 pu minus (ηu + kδ)x0 gt 0 therefore λ 0 isnot a root of (12) If (12) has a pair of pure imaginary rootλ iω(ωgt 0) for any τ gt 0 substituting it into (12) we canget that
iω(p + u) + pu minus ω21113872 1113873 i ωηx0 cos(ωτ) minus x0(ηu + kδ)sin(ωτ)1113858 1113859
+ ωηx0 sin(ωτ) + x0(ηu + kδ)cos(ωτ)1113858 1113859
(13)
and then separating its real and imaginary parts one gets
ω(p + u) ωηx0 cos(ωτ) minus x0(ηu + kδ)sin(ωτ)
pu minus ω2 ωηx0 sin(ωτ) + x0(ηu + kδ)cos(ωτ)1113896 (14)
Simplifying the above formula and letting κ ω2 wehave
f(κ) κ2 + p2
+ u2
minus x20η
21113872 1113873κ + p
2u2
minus x20(ηu + kδ)
2 0
(15)
On the one hand it is easy to get that
p2u2
minus x20(ηu + kδ)
2 p
2u2 1 minus R
201113872 1113873gt 0 (16)
On the other hand
p2
+ u2
minus x20η
2
1R20
minus 11113888 1113889x20y
2+
1R20
δ2k2
u2 +2δky
u1113888 1113889x
20 + u
2 gt 0
(17)
where R0 ((δk + ηu)pu)x0 lt 1 is usedClearly it indicates that f(κ)gt 0 which is in conflict
with f(κ) 0 is shows that all the roots of (8) havenegative real parts for any time delay τ ge 0 erefore thedisease-free equilibrium E0 is locally asymptotically stablefor any time delay τ ge 0
Next let us prove that if R0 lt 1 E0 is globally attractivefor any time delay τ ge 0 Define
G ϕ ϕ1 ϕ2ϕ3( 1113857 isin C1113868111386811138681113868 x0 ge ϕ1 ge 0ϕ2 ge 0ϕ3 ge 01113966 1113967
(18)
It is easy to show thatG attracts all solutions of model (4)and is also positively invariant with respect to model (4) IfR0 lt 1 let us choose a functional W(ϕ) on G as follows
W(ϕ) mϕ2(0) + ϕ3(0) + n 11139460
minus τϕ2(θ)dθ + ε1113946
0
minus τϕ3(θ)dθ
(19)
where m k(p minus ηx0) n mηx0 kηx0(p minus ηx0) n+
k mp and εgt 0 is a constant to be chosen later SinceR0 lt 1 we have m ngt 0 It can be found that W(ϕ) iscontinuous on the subset G in C From the invariance of Gfor any ϕ isin G the solution (x(t) y(t) v(t)) of model (4)satisfies x(t)le x0 for any tge 0 It follows from (4) that
_W(ϕ) m δϕ1(minus τ)ϕ3(minus τ) + ηϕ1(minus τ)ϕ2(minus τ)1113858 1113859 minus mpϕ2(0) + kϕ2(0) minus uϕ3(0)
+ n ϕ2(0) minus ϕ2(minus τ)( 1113857 + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
lem δϕ1(minus τ)ϕ3(minus τ) + ηx0ϕ2(minus τ)1113858 1113859 minus mpϕ2(0) + kϕ2(0) minus uϕ3(0)
+ n ϕ2(0) minus ϕ2(minus τ)( 1113857 + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
mδ ϕ1(minus τ)ϕ3(minus τ)( 1113857 + nϕ2(0) + kϕ2(0) minus mpϕ2(0) minus uϕ3(0)
+ ε ϕ3(0) minus ϕ3(minus τ)( 1113857
mδϕ1(minus τ)ϕ3(minus τ) minus uϕ3(0) + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
ϕ3(minus τ) mδϕ1(minus τ) minus ε( 1113857 + ϕ3(0)(ε minus u)
leϕ3(minus τ) mδx0 minus ε( 1113857 + ϕ3(0)(ε minus u)
(20)
SinceR0 lt 1 it is possible that we can choose the parameterεgt 0 such that mδx0 lt εlt u us it has_W(ϕ)le ϕ3(0)(ε minus u)le 0 for any ϕ isin G All these manifestthat W(ϕ) is a Lyapunov functional on the subset G By using
LyapunovndashLaSalle invariance principle the disease-free equi-librium E0 of model (4) is globally asymptotically stable
Next let us analyze the stability of the epidemic equi-librium Elowast And the linearized system of model (4) at Elowast is
4 Complexity
_x(t) minuss
xlowast+
rxlowast
T1113888 1113889x(t) minus
r
T+ α + c1113874 1113875xlowasty(t) minus βxlowastv(t)
_y(t) δ vlowastx(t minus τ) + xlowastv(t minus τ)( ) + η ylowastx(t minus τ) + xlowasty(t minus τ)( 1113857 minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(21)
e characteristic equation for Elowast is given as
λ +s
xlowast+ r
xlowast
T1113888 1113889 λ + p minus ηx
lowaste
minus λτ1113872 1113873(λ + u) minus kδx
lowaste
minus λτ1113960 1113961
+ eminus λτ δv
lowast+ ηylowast
( 1113857 xlowast r
T+ α + c1113874 1113875(λ + u) + kβx
lowast1113876 1113877 0
(22)
On the other hand (22) is equivalent to
P(λ) + Q(λ)eminus λτ
0 (23)
where
P(λ) λ3 + a1λ2
+ a2λ + a3 (24)
Q(λ) b1λ2
+ b2λ + b3 (25)
with
a1 p + u + B
a2 pu + pB + Bu
a3 puB
b1 minus I b2 GE minus BI minus uI minus kH
b3 GEu + GkF minus kBH minus BIu
G(Eu + kF) minus B(kH + Iu)
(26)
B s
xlowast+ r
xlowast
T
E xlowast r
T+ α + c1113874 1113875
F βxlowast
G δvlowast
+ ηylowast
H δxlowast
I ηxlowast
(27)
When τ 0 (12) reduces to
λ3 + a1 + b1( 1113857λ2 + a2 + b2( 1113857λ + a3 + b3 0 (28)
Notice that
a1 + b1 p + u + B minus I
gtp + u minus I
p + u minus ηxlowast
kpδ + kuδ + ηu2
kδ + ηugt 0
a2 + b2 B(p + u minus I) + GE + pu minus uI minus kH
B(p + u minus I) + GEgt 0
a3 + b3 puB + G(Eu + kF) minus B(kH + Iu)
G(Eu + kF)gt 0
H1 a1 + b1 gt 0
H2 a1 + b1( 1113857 a2 + b2( 1113857 minus a3 + b3( 1113857
B(p + u minus I)2
+ GE(p + u minus I) + B2(p + u minus I)
+ G(EB minus Eu minus kF)
(29)
erefore if R0 gt 1 and H2 gt 0 hold by theRouthndashHurwitz criterion the epidemic equilibrium Elowast islocally asymptotically stable for any time delay τ 0
Next let us investigate the stability of Elowast when τ gt 0Since a3 + b3 puB + G(Eu + kF) minus B(kH + Iu) G
(Eu + kF)gt 0 τ 0 is not the root of (23) For τ gt 0 weassume that (23) has the pure imaginary root λ iω(ωgt 0)and substituting it into (23) we can get that
minus iω3minus a1ω
2+ ia2ω + a3 + minus b1ω
2+ ib2ω + b31113872 1113873
middot (cosωτ minus i sinωτ) 0(30)
and then separating its real and imaginary parts we get
ω3minus a2ω b1ω
2minus b31113872 1113873sin(ωτ) + b2ω cos(ωτ)
a1ω2
minus a3 b3 minus b1ω2
1113872 1113873cos(ωτ) + b2ω sin(ωτ)(31)
Clearly it is equivalent to
ω6+ a
21 minus b
21 minus 2a21113872 1113873ω4
+ a22 minus b
22 minus 2b1b31113872 1113873ω2
+ a23 minus b
231113872 1113873 0
(32)
Let c1 a21 minus b21 minus 2a2 c2 a2
2 minus b22 minus 2a1a3 minus 2b1b3
and c3 a23 minus b23 (22) can be written as
Complexity 5
ω6+ c1ω
4+ c2ω
2+ c3 0 (33)
Let ϖ ω2 (33) is equivalent to
ϖ3 + c1ϖ2
+ c2ϖ + c3 0 (34)
Define
g(ϖ) ϖ3 + c1ϖ2
+ c2ϖ + c3 (35)
and thus
gprime(ϖ) 3ϖ2 + 2c1ϖ + c2 (36)
We consider that
3ϖ2 + 2c1ϖ + c2 0 (37)
and it has two real roots given as ϖ1 (minus c1 +Δ
radic)3 and
ϖ2 (minus c1 minusΔ
radic)3 where Δ c21 minus 3c2 Next we get the
following lemma from [50]
Lemma 2 For polynomial (34) the following conclusions aregiven
(i) If c3 lt 0 (34) has at least one positive root(ii) If c3 ge 0 and Δlt 0 (34) has no real root(iii) If c3 ge 0 and Δgt 0 and if and only ifϖ1 (minus c1 +
Δ
radic)3gt 0 and g(ϖ1)le 0 (34) has real
roots
Assume that g(ϖ) 0 has positive real roots thereforewe can suppose that (34) has k(1le kle 3) positive real rootsdenoted as ϖ1ϖ2 and ϖ3 Moreover (33) has positive realroots ωk
ϖk
radic From (31) we can get that
cos(ωτ) b2 minus a1b1( 1113857ω4 + a1b3 + a3b1 minus a2b2( 1113857ω2 minus a3b3
b22ω2 + b3 minus b1ω2( 11138572
(38)
In addition we get the corresponding τ(n)k gt 0 such that
(33) has pure imaginary λ iωk where
τ(n)k
1ωk
arccosb2 minus a1b1( 1113857ω4
k + a1b3 + a3b1 minus a2b2( 1113857ω2k minus a3b3
b22ω2k + b3 minus b1ω2
k1113872 11138732
⎛⎜⎝ ⎞⎟⎠ + 2nπ⎧⎪⎨
⎪⎩
⎫⎪⎬
⎪⎭ (39)
in which k 1 2 3 n 0 1 2
Define
τlowast minkisin[123]
τ(0)k1113966 1113967 (40)
Differentiating the two sides of (15) with respect to τ itfollows that
3λ2 + 2a1λ + a21113872 1113873dλdτ
+ 2b1λ + b2( 1113857eminus λτdλ
dτ
minus b1λ2
+ b2λ + b31113872 1113873eminus λτ λ + τ
dλdτ
1113888 1113889 0
(41)
and therefore
dλdτ
1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 + 2a1ωki
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873i
+2b1ωki + b2
minus b2ω2k + b3ωk minus b1ω3
k1113872 1113873iminus
τωki
(42)
end(Res(λ))
dτ1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 a2ω2k minus ω4
k( 1113857 + 2a1ωk a1ω3k minus a3ωk( 1113857
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873
+2b1ωk b3ωk minus b1ω3
k( 1113857 minus b22ω2k
b22ω4k + b3ωk minus b1ω3
k1113872 11138732
(43)
From (31) we have
ω3minus a2ω1113872 1113873
2+ a1ω minus a3( 1113857
2 b
22ω
2+ b1ω
2minus b31113872 1113873
2 (44)
Hence
d(Res(λ))
dτ1113888 11138891113888 1113889
minus 1
λiωk
3ϖ3k + 2c1ϖ2k + c2ϖk
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
ωkgprime ϖk( 1113857
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
(45)
Since ωk gt 0 this is enough to demonstrate thatRe(dλ(τ)dτ)|ττ(n)
k
and gprime(ϖk) have the same sign Com-bining Lemma 2 with (45) we have the followingconclusions
Theorem 2 If R0 gt 1 the following results hold
(i) If c3 ge 0 and Δle 0 then the epidemic equilibriumElowast (xlowast ylowast vlowast) is locally asymptotically stable
(ii) If c3 ge 0 or c3 lt 0 and Δgt 0 then the epidemicequilibrium Elowast (xlowast ylowast vlowast) is asymptotically sta-ble when τ isin [0 τlowast) and unstable when τ gt τlowast
(iii) If the conditions of (ii) are all satisfied andgprime(ϖk)ne 0 then model (1) undergoes a Hopf bifur-cation at Elowast when τ gt τ(n)
k (n 0 1 2 )
6 Complexity
4 Numerical Simulations
In this section we give some numerical simulations tovalidate our main results in Section 3 e basic parametersare set as follows [51] s 2 r 10 T 10 d
002 β 0000027 α 0000024 c 0001 δ 0002 η
0002 p 05 k 10 and u 05 Direct calculations withMaple 14 show thatR 08548lt 1 and E0 (101765 0 0)Figure 1 shows that virus-free equilibrium E0 of the system isstable
In the following we change T to 100 and direct cal-culations show system (4) has two equilibria ieE0 (101765 0 0) and E1 (119048 881959 17639171)
(R 84gt 1) We easily get
a3 03396
b3 18997
g(0) c3 a23 minus b
23 minus 34937lt 0
(46)
en equation (31) has one positive root u 3165equation (28) has one positive rootϖ0 1779 and the Hopfbifurcation value is τ0 05304 First let τ 052lt τ0 andselect three different initial values I1 (1 001 1700)I2 (1 80 001) and I3 (1 80 1700) respectively Nu-merical simulations show that the solutions of systemconverge to the same equilibrium respectively (see Fig-ures 2 3 and 4) In the biological sense tiny presence ofinfected cells or virus can lead to the disease transmissionNext let τ 054gt τ0 Figure 5 shows that periodic
x
0
2
4
6
8
10
12
40002000 50001000 60000 3000t
(a)
y
80
70
60
50
40
30
20
10
020001000 3000 4000 5000 60000
t
(b)
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 60000t
(c)
80 60 10840 6420 2
2004006008001000120014001600
(d)
Figure 1 Illustration of basic behavior of solutions of system (4) with initial condition I3 where R 08548lt 1 (a) time series of x (t)(b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 7
x
0
10
20
30
40
50
60
70
80
90
100
2000 4000 6000 8000 10000 12000 140000t
(a)
y
100004000 140000 6000 80002000 12000t
0
20
40
60
80
100
120
140
(b)
v
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
v
120100 8080y
6060x4040 20 20
1100
1200
1300
1400
1500
1600
1700
1800
(d)
Figure 2 Illustration of basic behavior of solutions of system (4) with τ 052 where R 84gt 1 (the initial condition is (1 001 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
1000 2000 3000 4000 5000 6000 7000 8000 90000t
0
10
20
30
40
50
60
(a)
y
1000 4000 6000 900070002000 80003000 50000t
40
50
60
70
80
90
100
(b)
Figure 3 Continued
8 Complexity
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 6000 7000 8000 90000t
(c)
v
901080 20
70x30
y
60 405050
50010001500
(d)
Figure 3 Illustration of basic behavior of solutions of system (4) with τ 052 whereR 84gt 1 (the initial condition is (1 80 001)) (a)time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
2000 4000 6000 8000 10000 12000 140000t
0
5
10
15
20
25
30
35
40
(a)
y
50
60
70
80
90
100
110
120
2000 4000 6000 8000 10000 12000 140000t
(b)
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
100 3080 201060
145015001550160016501700175018001850
v
y x
(d)
Figure 4 Illustration of basic behavior of solutions of system (4) with τ 052lt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 9
oscillations occur for τ gt τ0 en system (4) undergoes aHopf bifurcation at Elowast when τ τ0
5 Conclusion
In this paper we improved a delayed HIV virus dynamicmodel by introducing cell-to-cell transmission and apo-ptosis of bystander cells We derived the basic reproductivenumber R0 lt 1 and based on the basic reproductive numberthe stability of the equilibria of the system was investigatede mathematical analysis showed that if R0 lt 1 then thedisease-free equilibrium of system (4) is globally asymp-totically stable and for R0 gt 1 the system can produce Hopfbifurcation with the delay changing In biology HIV cannotsuccessfully invade healthy cells and will be cleared by theimmune system finally if R0 lt 1 and if R0 gt 1 uninfectedcells infected cells and free virus can coexist as periodicoscillations under certain conditions
However it is very necessary to consider antiretroviraltherapy in model (4) since it can make the model morerealistic and many scholars have noticed this topic andachieved excellent results for example Wang et al [52ndash54]
have studied the virus dynamics model with antiretroviraltherapy eir research shows that antiretroviral therapy canincrease the number of uninfected T cells and decrease thenumber of infectious virions and is very effective for thetreatment of AIDS patients In present model we do notconsider this factor and we will leave this for future research
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
Tongqian Zhang was supported by the Shandong ProvincialNatural Science Foundation of China (no ZR2019MA003)and Scientific Research Foundation of Shandong Universityof Science and Technology for Recruited Talents Zhichao
x
times104151 35 405 3252 450
t
0
5
10
15
20
25
30
35
40
45
(a)
times104
y
41505 2 25 3 450 1 35t
50
60
70
80
90
100
110
120
130
(b)
times104
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
151 35 405 3252 450t
(c)
120
v
40100y 30
x80 201060
145015001550160016501700175018001850
(d)
Figure 5 Illustration of basic behavior of solutions of system (4) with τ 054gt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
10 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
[1] R Weiss ldquoHow does HIV cause AIDSrdquo Science vol 260no 5112 pp 1273ndash1279 1993
[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
λ1 r minus d minus2r
Tx0 minus
(d minus r)2 minus4rs
T
1113970
lt 0 (10)
en we only need to discuss the distribution of theroots of
λ + p minus ηx0eminus λτ
1113872 1113873(λ + u) minus kδx0eminus λτ
0 (11)
Obviously equation (11) can be written as
λ2 +(p + u)λ minus ηx0eminus λτλ + pu minus (ηu + kδ)x0e
minus λτ 0
(12)
When R0 lt 1 pu minus (ηu + kδ)x0 gt 0 therefore λ 0 isnot a root of (12) If (12) has a pair of pure imaginary rootλ iω(ωgt 0) for any τ gt 0 substituting it into (12) we canget that
iω(p + u) + pu minus ω21113872 1113873 i ωηx0 cos(ωτ) minus x0(ηu + kδ)sin(ωτ)1113858 1113859
+ ωηx0 sin(ωτ) + x0(ηu + kδ)cos(ωτ)1113858 1113859
(13)
and then separating its real and imaginary parts one gets
ω(p + u) ωηx0 cos(ωτ) minus x0(ηu + kδ)sin(ωτ)
pu minus ω2 ωηx0 sin(ωτ) + x0(ηu + kδ)cos(ωτ)1113896 (14)
Simplifying the above formula and letting κ ω2 wehave
f(κ) κ2 + p2
+ u2
minus x20η
21113872 1113873κ + p
2u2
minus x20(ηu + kδ)
2 0
(15)
On the one hand it is easy to get that
p2u2
minus x20(ηu + kδ)
2 p
2u2 1 minus R
201113872 1113873gt 0 (16)
On the other hand
p2
+ u2
minus x20η
2
1R20
minus 11113888 1113889x20y
2+
1R20
δ2k2
u2 +2δky
u1113888 1113889x
20 + u
2 gt 0
(17)
where R0 ((δk + ηu)pu)x0 lt 1 is usedClearly it indicates that f(κ)gt 0 which is in conflict
with f(κ) 0 is shows that all the roots of (8) havenegative real parts for any time delay τ ge 0 erefore thedisease-free equilibrium E0 is locally asymptotically stablefor any time delay τ ge 0
Next let us prove that if R0 lt 1 E0 is globally attractivefor any time delay τ ge 0 Define
G ϕ ϕ1 ϕ2ϕ3( 1113857 isin C1113868111386811138681113868 x0 ge ϕ1 ge 0ϕ2 ge 0ϕ3 ge 01113966 1113967
(18)
It is easy to show thatG attracts all solutions of model (4)and is also positively invariant with respect to model (4) IfR0 lt 1 let us choose a functional W(ϕ) on G as follows
W(ϕ) mϕ2(0) + ϕ3(0) + n 11139460
minus τϕ2(θ)dθ + ε1113946
0
minus τϕ3(θ)dθ
(19)
where m k(p minus ηx0) n mηx0 kηx0(p minus ηx0) n+
k mp and εgt 0 is a constant to be chosen later SinceR0 lt 1 we have m ngt 0 It can be found that W(ϕ) iscontinuous on the subset G in C From the invariance of Gfor any ϕ isin G the solution (x(t) y(t) v(t)) of model (4)satisfies x(t)le x0 for any tge 0 It follows from (4) that
_W(ϕ) m δϕ1(minus τ)ϕ3(minus τ) + ηϕ1(minus τ)ϕ2(minus τ)1113858 1113859 minus mpϕ2(0) + kϕ2(0) minus uϕ3(0)
+ n ϕ2(0) minus ϕ2(minus τ)( 1113857 + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
lem δϕ1(minus τ)ϕ3(minus τ) + ηx0ϕ2(minus τ)1113858 1113859 minus mpϕ2(0) + kϕ2(0) minus uϕ3(0)
+ n ϕ2(0) minus ϕ2(minus τ)( 1113857 + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
mδ ϕ1(minus τ)ϕ3(minus τ)( 1113857 + nϕ2(0) + kϕ2(0) minus mpϕ2(0) minus uϕ3(0)
+ ε ϕ3(0) minus ϕ3(minus τ)( 1113857
mδϕ1(minus τ)ϕ3(minus τ) minus uϕ3(0) + ε ϕ3(0) minus ϕ3(minus τ)( 1113857
ϕ3(minus τ) mδϕ1(minus τ) minus ε( 1113857 + ϕ3(0)(ε minus u)
leϕ3(minus τ) mδx0 minus ε( 1113857 + ϕ3(0)(ε minus u)
(20)
SinceR0 lt 1 it is possible that we can choose the parameterεgt 0 such that mδx0 lt εlt u us it has_W(ϕ)le ϕ3(0)(ε minus u)le 0 for any ϕ isin G All these manifestthat W(ϕ) is a Lyapunov functional on the subset G By using
LyapunovndashLaSalle invariance principle the disease-free equi-librium E0 of model (4) is globally asymptotically stable
Next let us analyze the stability of the epidemic equi-librium Elowast And the linearized system of model (4) at Elowast is
4 Complexity
_x(t) minuss
xlowast+
rxlowast
T1113888 1113889x(t) minus
r
T+ α + c1113874 1113875xlowasty(t) minus βxlowastv(t)
_y(t) δ vlowastx(t minus τ) + xlowastv(t minus τ)( ) + η ylowastx(t minus τ) + xlowasty(t minus τ)( 1113857 minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(21)
e characteristic equation for Elowast is given as
λ +s
xlowast+ r
xlowast
T1113888 1113889 λ + p minus ηx
lowaste
minus λτ1113872 1113873(λ + u) minus kδx
lowaste
minus λτ1113960 1113961
+ eminus λτ δv
lowast+ ηylowast
( 1113857 xlowast r
T+ α + c1113874 1113875(λ + u) + kβx
lowast1113876 1113877 0
(22)
On the other hand (22) is equivalent to
P(λ) + Q(λ)eminus λτ
0 (23)
where
P(λ) λ3 + a1λ2
+ a2λ + a3 (24)
Q(λ) b1λ2
+ b2λ + b3 (25)
with
a1 p + u + B
a2 pu + pB + Bu
a3 puB
b1 minus I b2 GE minus BI minus uI minus kH
b3 GEu + GkF minus kBH minus BIu
G(Eu + kF) minus B(kH + Iu)
(26)
B s
xlowast+ r
xlowast
T
E xlowast r
T+ α + c1113874 1113875
F βxlowast
G δvlowast
+ ηylowast
H δxlowast
I ηxlowast
(27)
When τ 0 (12) reduces to
λ3 + a1 + b1( 1113857λ2 + a2 + b2( 1113857λ + a3 + b3 0 (28)
Notice that
a1 + b1 p + u + B minus I
gtp + u minus I
p + u minus ηxlowast
kpδ + kuδ + ηu2
kδ + ηugt 0
a2 + b2 B(p + u minus I) + GE + pu minus uI minus kH
B(p + u minus I) + GEgt 0
a3 + b3 puB + G(Eu + kF) minus B(kH + Iu)
G(Eu + kF)gt 0
H1 a1 + b1 gt 0
H2 a1 + b1( 1113857 a2 + b2( 1113857 minus a3 + b3( 1113857
B(p + u minus I)2
+ GE(p + u minus I) + B2(p + u minus I)
+ G(EB minus Eu minus kF)
(29)
erefore if R0 gt 1 and H2 gt 0 hold by theRouthndashHurwitz criterion the epidemic equilibrium Elowast islocally asymptotically stable for any time delay τ 0
Next let us investigate the stability of Elowast when τ gt 0Since a3 + b3 puB + G(Eu + kF) minus B(kH + Iu) G
(Eu + kF)gt 0 τ 0 is not the root of (23) For τ gt 0 weassume that (23) has the pure imaginary root λ iω(ωgt 0)and substituting it into (23) we can get that
minus iω3minus a1ω
2+ ia2ω + a3 + minus b1ω
2+ ib2ω + b31113872 1113873
middot (cosωτ minus i sinωτ) 0(30)
and then separating its real and imaginary parts we get
ω3minus a2ω b1ω
2minus b31113872 1113873sin(ωτ) + b2ω cos(ωτ)
a1ω2
minus a3 b3 minus b1ω2
1113872 1113873cos(ωτ) + b2ω sin(ωτ)(31)
Clearly it is equivalent to
ω6+ a
21 minus b
21 minus 2a21113872 1113873ω4
+ a22 minus b
22 minus 2b1b31113872 1113873ω2
+ a23 minus b
231113872 1113873 0
(32)
Let c1 a21 minus b21 minus 2a2 c2 a2
2 minus b22 minus 2a1a3 minus 2b1b3
and c3 a23 minus b23 (22) can be written as
Complexity 5
ω6+ c1ω
4+ c2ω
2+ c3 0 (33)
Let ϖ ω2 (33) is equivalent to
ϖ3 + c1ϖ2
+ c2ϖ + c3 0 (34)
Define
g(ϖ) ϖ3 + c1ϖ2
+ c2ϖ + c3 (35)
and thus
gprime(ϖ) 3ϖ2 + 2c1ϖ + c2 (36)
We consider that
3ϖ2 + 2c1ϖ + c2 0 (37)
and it has two real roots given as ϖ1 (minus c1 +Δ
radic)3 and
ϖ2 (minus c1 minusΔ
radic)3 where Δ c21 minus 3c2 Next we get the
following lemma from [50]
Lemma 2 For polynomial (34) the following conclusions aregiven
(i) If c3 lt 0 (34) has at least one positive root(ii) If c3 ge 0 and Δlt 0 (34) has no real root(iii) If c3 ge 0 and Δgt 0 and if and only ifϖ1 (minus c1 +
Δ
radic)3gt 0 and g(ϖ1)le 0 (34) has real
roots
Assume that g(ϖ) 0 has positive real roots thereforewe can suppose that (34) has k(1le kle 3) positive real rootsdenoted as ϖ1ϖ2 and ϖ3 Moreover (33) has positive realroots ωk
ϖk
radic From (31) we can get that
cos(ωτ) b2 minus a1b1( 1113857ω4 + a1b3 + a3b1 minus a2b2( 1113857ω2 minus a3b3
b22ω2 + b3 minus b1ω2( 11138572
(38)
In addition we get the corresponding τ(n)k gt 0 such that
(33) has pure imaginary λ iωk where
τ(n)k
1ωk
arccosb2 minus a1b1( 1113857ω4
k + a1b3 + a3b1 minus a2b2( 1113857ω2k minus a3b3
b22ω2k + b3 minus b1ω2
k1113872 11138732
⎛⎜⎝ ⎞⎟⎠ + 2nπ⎧⎪⎨
⎪⎩
⎫⎪⎬
⎪⎭ (39)
in which k 1 2 3 n 0 1 2
Define
τlowast minkisin[123]
τ(0)k1113966 1113967 (40)
Differentiating the two sides of (15) with respect to τ itfollows that
3λ2 + 2a1λ + a21113872 1113873dλdτ
+ 2b1λ + b2( 1113857eminus λτdλ
dτ
minus b1λ2
+ b2λ + b31113872 1113873eminus λτ λ + τ
dλdτ
1113888 1113889 0
(41)
and therefore
dλdτ
1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 + 2a1ωki
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873i
+2b1ωki + b2
minus b2ω2k + b3ωk minus b1ω3
k1113872 1113873iminus
τωki
(42)
end(Res(λ))
dτ1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 a2ω2k minus ω4
k( 1113857 + 2a1ωk a1ω3k minus a3ωk( 1113857
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873
+2b1ωk b3ωk minus b1ω3
k( 1113857 minus b22ω2k
b22ω4k + b3ωk minus b1ω3
k1113872 11138732
(43)
From (31) we have
ω3minus a2ω1113872 1113873
2+ a1ω minus a3( 1113857
2 b
22ω
2+ b1ω
2minus b31113872 1113873
2 (44)
Hence
d(Res(λ))
dτ1113888 11138891113888 1113889
minus 1
λiωk
3ϖ3k + 2c1ϖ2k + c2ϖk
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
ωkgprime ϖk( 1113857
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
(45)
Since ωk gt 0 this is enough to demonstrate thatRe(dλ(τ)dτ)|ττ(n)
k
and gprime(ϖk) have the same sign Com-bining Lemma 2 with (45) we have the followingconclusions
Theorem 2 If R0 gt 1 the following results hold
(i) If c3 ge 0 and Δle 0 then the epidemic equilibriumElowast (xlowast ylowast vlowast) is locally asymptotically stable
(ii) If c3 ge 0 or c3 lt 0 and Δgt 0 then the epidemicequilibrium Elowast (xlowast ylowast vlowast) is asymptotically sta-ble when τ isin [0 τlowast) and unstable when τ gt τlowast
(iii) If the conditions of (ii) are all satisfied andgprime(ϖk)ne 0 then model (1) undergoes a Hopf bifur-cation at Elowast when τ gt τ(n)
k (n 0 1 2 )
6 Complexity
4 Numerical Simulations
In this section we give some numerical simulations tovalidate our main results in Section 3 e basic parametersare set as follows [51] s 2 r 10 T 10 d
002 β 0000027 α 0000024 c 0001 δ 0002 η
0002 p 05 k 10 and u 05 Direct calculations withMaple 14 show thatR 08548lt 1 and E0 (101765 0 0)Figure 1 shows that virus-free equilibrium E0 of the system isstable
In the following we change T to 100 and direct cal-culations show system (4) has two equilibria ieE0 (101765 0 0) and E1 (119048 881959 17639171)
(R 84gt 1) We easily get
a3 03396
b3 18997
g(0) c3 a23 minus b
23 minus 34937lt 0
(46)
en equation (31) has one positive root u 3165equation (28) has one positive rootϖ0 1779 and the Hopfbifurcation value is τ0 05304 First let τ 052lt τ0 andselect three different initial values I1 (1 001 1700)I2 (1 80 001) and I3 (1 80 1700) respectively Nu-merical simulations show that the solutions of systemconverge to the same equilibrium respectively (see Fig-ures 2 3 and 4) In the biological sense tiny presence ofinfected cells or virus can lead to the disease transmissionNext let τ 054gt τ0 Figure 5 shows that periodic
x
0
2
4
6
8
10
12
40002000 50001000 60000 3000t
(a)
y
80
70
60
50
40
30
20
10
020001000 3000 4000 5000 60000
t
(b)
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 60000t
(c)
80 60 10840 6420 2
2004006008001000120014001600
(d)
Figure 1 Illustration of basic behavior of solutions of system (4) with initial condition I3 where R 08548lt 1 (a) time series of x (t)(b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 7
x
0
10
20
30
40
50
60
70
80
90
100
2000 4000 6000 8000 10000 12000 140000t
(a)
y
100004000 140000 6000 80002000 12000t
0
20
40
60
80
100
120
140
(b)
v
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
v
120100 8080y
6060x4040 20 20
1100
1200
1300
1400
1500
1600
1700
1800
(d)
Figure 2 Illustration of basic behavior of solutions of system (4) with τ 052 where R 84gt 1 (the initial condition is (1 001 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
1000 2000 3000 4000 5000 6000 7000 8000 90000t
0
10
20
30
40
50
60
(a)
y
1000 4000 6000 900070002000 80003000 50000t
40
50
60
70
80
90
100
(b)
Figure 3 Continued
8 Complexity
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 6000 7000 8000 90000t
(c)
v
901080 20
70x30
y
60 405050
50010001500
(d)
Figure 3 Illustration of basic behavior of solutions of system (4) with τ 052 whereR 84gt 1 (the initial condition is (1 80 001)) (a)time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
2000 4000 6000 8000 10000 12000 140000t
0
5
10
15
20
25
30
35
40
(a)
y
50
60
70
80
90
100
110
120
2000 4000 6000 8000 10000 12000 140000t
(b)
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
100 3080 201060
145015001550160016501700175018001850
v
y x
(d)
Figure 4 Illustration of basic behavior of solutions of system (4) with τ 052lt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 9
oscillations occur for τ gt τ0 en system (4) undergoes aHopf bifurcation at Elowast when τ τ0
5 Conclusion
In this paper we improved a delayed HIV virus dynamicmodel by introducing cell-to-cell transmission and apo-ptosis of bystander cells We derived the basic reproductivenumber R0 lt 1 and based on the basic reproductive numberthe stability of the equilibria of the system was investigatede mathematical analysis showed that if R0 lt 1 then thedisease-free equilibrium of system (4) is globally asymp-totically stable and for R0 gt 1 the system can produce Hopfbifurcation with the delay changing In biology HIV cannotsuccessfully invade healthy cells and will be cleared by theimmune system finally if R0 lt 1 and if R0 gt 1 uninfectedcells infected cells and free virus can coexist as periodicoscillations under certain conditions
However it is very necessary to consider antiretroviraltherapy in model (4) since it can make the model morerealistic and many scholars have noticed this topic andachieved excellent results for example Wang et al [52ndash54]
have studied the virus dynamics model with antiretroviraltherapy eir research shows that antiretroviral therapy canincrease the number of uninfected T cells and decrease thenumber of infectious virions and is very effective for thetreatment of AIDS patients In present model we do notconsider this factor and we will leave this for future research
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
Tongqian Zhang was supported by the Shandong ProvincialNatural Science Foundation of China (no ZR2019MA003)and Scientific Research Foundation of Shandong Universityof Science and Technology for Recruited Talents Zhichao
x
times104151 35 405 3252 450
t
0
5
10
15
20
25
30
35
40
45
(a)
times104
y
41505 2 25 3 450 1 35t
50
60
70
80
90
100
110
120
130
(b)
times104
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
151 35 405 3252 450t
(c)
120
v
40100y 30
x80 201060
145015001550160016501700175018001850
(d)
Figure 5 Illustration of basic behavior of solutions of system (4) with τ 054gt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
10 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
[1] R Weiss ldquoHow does HIV cause AIDSrdquo Science vol 260no 5112 pp 1273ndash1279 1993
[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
_x(t) minuss
xlowast+
rxlowast
T1113888 1113889x(t) minus
r
T+ α + c1113874 1113875xlowasty(t) minus βxlowastv(t)
_y(t) δ vlowastx(t minus τ) + xlowastv(t minus τ)( ) + η ylowastx(t minus τ) + xlowasty(t minus τ)( 1113857 minus py(t)
_v(t) ky(t) minus uv(t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(21)
e characteristic equation for Elowast is given as
λ +s
xlowast+ r
xlowast
T1113888 1113889 λ + p minus ηx
lowaste
minus λτ1113872 1113873(λ + u) minus kδx
lowaste
minus λτ1113960 1113961
+ eminus λτ δv
lowast+ ηylowast
( 1113857 xlowast r
T+ α + c1113874 1113875(λ + u) + kβx
lowast1113876 1113877 0
(22)
On the other hand (22) is equivalent to
P(λ) + Q(λ)eminus λτ
0 (23)
where
P(λ) λ3 + a1λ2
+ a2λ + a3 (24)
Q(λ) b1λ2
+ b2λ + b3 (25)
with
a1 p + u + B
a2 pu + pB + Bu
a3 puB
b1 minus I b2 GE minus BI minus uI minus kH
b3 GEu + GkF minus kBH minus BIu
G(Eu + kF) minus B(kH + Iu)
(26)
B s
xlowast+ r
xlowast
T
E xlowast r
T+ α + c1113874 1113875
F βxlowast
G δvlowast
+ ηylowast
H δxlowast
I ηxlowast
(27)
When τ 0 (12) reduces to
λ3 + a1 + b1( 1113857λ2 + a2 + b2( 1113857λ + a3 + b3 0 (28)
Notice that
a1 + b1 p + u + B minus I
gtp + u minus I
p + u minus ηxlowast
kpδ + kuδ + ηu2
kδ + ηugt 0
a2 + b2 B(p + u minus I) + GE + pu minus uI minus kH
B(p + u minus I) + GEgt 0
a3 + b3 puB + G(Eu + kF) minus B(kH + Iu)
G(Eu + kF)gt 0
H1 a1 + b1 gt 0
H2 a1 + b1( 1113857 a2 + b2( 1113857 minus a3 + b3( 1113857
B(p + u minus I)2
+ GE(p + u minus I) + B2(p + u minus I)
+ G(EB minus Eu minus kF)
(29)
erefore if R0 gt 1 and H2 gt 0 hold by theRouthndashHurwitz criterion the epidemic equilibrium Elowast islocally asymptotically stable for any time delay τ 0
Next let us investigate the stability of Elowast when τ gt 0Since a3 + b3 puB + G(Eu + kF) minus B(kH + Iu) G
(Eu + kF)gt 0 τ 0 is not the root of (23) For τ gt 0 weassume that (23) has the pure imaginary root λ iω(ωgt 0)and substituting it into (23) we can get that
minus iω3minus a1ω
2+ ia2ω + a3 + minus b1ω
2+ ib2ω + b31113872 1113873
middot (cosωτ minus i sinωτ) 0(30)
and then separating its real and imaginary parts we get
ω3minus a2ω b1ω
2minus b31113872 1113873sin(ωτ) + b2ω cos(ωτ)
a1ω2
minus a3 b3 minus b1ω2
1113872 1113873cos(ωτ) + b2ω sin(ωτ)(31)
Clearly it is equivalent to
ω6+ a
21 minus b
21 minus 2a21113872 1113873ω4
+ a22 minus b
22 minus 2b1b31113872 1113873ω2
+ a23 minus b
231113872 1113873 0
(32)
Let c1 a21 minus b21 minus 2a2 c2 a2
2 minus b22 minus 2a1a3 minus 2b1b3
and c3 a23 minus b23 (22) can be written as
Complexity 5
ω6+ c1ω
4+ c2ω
2+ c3 0 (33)
Let ϖ ω2 (33) is equivalent to
ϖ3 + c1ϖ2
+ c2ϖ + c3 0 (34)
Define
g(ϖ) ϖ3 + c1ϖ2
+ c2ϖ + c3 (35)
and thus
gprime(ϖ) 3ϖ2 + 2c1ϖ + c2 (36)
We consider that
3ϖ2 + 2c1ϖ + c2 0 (37)
and it has two real roots given as ϖ1 (minus c1 +Δ
radic)3 and
ϖ2 (minus c1 minusΔ
radic)3 where Δ c21 minus 3c2 Next we get the
following lemma from [50]
Lemma 2 For polynomial (34) the following conclusions aregiven
(i) If c3 lt 0 (34) has at least one positive root(ii) If c3 ge 0 and Δlt 0 (34) has no real root(iii) If c3 ge 0 and Δgt 0 and if and only ifϖ1 (minus c1 +
Δ
radic)3gt 0 and g(ϖ1)le 0 (34) has real
roots
Assume that g(ϖ) 0 has positive real roots thereforewe can suppose that (34) has k(1le kle 3) positive real rootsdenoted as ϖ1ϖ2 and ϖ3 Moreover (33) has positive realroots ωk
ϖk
radic From (31) we can get that
cos(ωτ) b2 minus a1b1( 1113857ω4 + a1b3 + a3b1 minus a2b2( 1113857ω2 minus a3b3
b22ω2 + b3 minus b1ω2( 11138572
(38)
In addition we get the corresponding τ(n)k gt 0 such that
(33) has pure imaginary λ iωk where
τ(n)k
1ωk
arccosb2 minus a1b1( 1113857ω4
k + a1b3 + a3b1 minus a2b2( 1113857ω2k minus a3b3
b22ω2k + b3 minus b1ω2
k1113872 11138732
⎛⎜⎝ ⎞⎟⎠ + 2nπ⎧⎪⎨
⎪⎩
⎫⎪⎬
⎪⎭ (39)
in which k 1 2 3 n 0 1 2
Define
τlowast minkisin[123]
τ(0)k1113966 1113967 (40)
Differentiating the two sides of (15) with respect to τ itfollows that
3λ2 + 2a1λ + a21113872 1113873dλdτ
+ 2b1λ + b2( 1113857eminus λτdλ
dτ
minus b1λ2
+ b2λ + b31113872 1113873eminus λτ λ + τ
dλdτ
1113888 1113889 0
(41)
and therefore
dλdτ
1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 + 2a1ωki
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873i
+2b1ωki + b2
minus b2ω2k + b3ωk minus b1ω3
k1113872 1113873iminus
τωki
(42)
end(Res(λ))
dτ1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 a2ω2k minus ω4
k( 1113857 + 2a1ωk a1ω3k minus a3ωk( 1113857
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873
+2b1ωk b3ωk minus b1ω3
k( 1113857 minus b22ω2k
b22ω4k + b3ωk minus b1ω3
k1113872 11138732
(43)
From (31) we have
ω3minus a2ω1113872 1113873
2+ a1ω minus a3( 1113857
2 b
22ω
2+ b1ω
2minus b31113872 1113873
2 (44)
Hence
d(Res(λ))
dτ1113888 11138891113888 1113889
minus 1
λiωk
3ϖ3k + 2c1ϖ2k + c2ϖk
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
ωkgprime ϖk( 1113857
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
(45)
Since ωk gt 0 this is enough to demonstrate thatRe(dλ(τ)dτ)|ττ(n)
k
and gprime(ϖk) have the same sign Com-bining Lemma 2 with (45) we have the followingconclusions
Theorem 2 If R0 gt 1 the following results hold
(i) If c3 ge 0 and Δle 0 then the epidemic equilibriumElowast (xlowast ylowast vlowast) is locally asymptotically stable
(ii) If c3 ge 0 or c3 lt 0 and Δgt 0 then the epidemicequilibrium Elowast (xlowast ylowast vlowast) is asymptotically sta-ble when τ isin [0 τlowast) and unstable when τ gt τlowast
(iii) If the conditions of (ii) are all satisfied andgprime(ϖk)ne 0 then model (1) undergoes a Hopf bifur-cation at Elowast when τ gt τ(n)
k (n 0 1 2 )
6 Complexity
4 Numerical Simulations
In this section we give some numerical simulations tovalidate our main results in Section 3 e basic parametersare set as follows [51] s 2 r 10 T 10 d
002 β 0000027 α 0000024 c 0001 δ 0002 η
0002 p 05 k 10 and u 05 Direct calculations withMaple 14 show thatR 08548lt 1 and E0 (101765 0 0)Figure 1 shows that virus-free equilibrium E0 of the system isstable
In the following we change T to 100 and direct cal-culations show system (4) has two equilibria ieE0 (101765 0 0) and E1 (119048 881959 17639171)
(R 84gt 1) We easily get
a3 03396
b3 18997
g(0) c3 a23 minus b
23 minus 34937lt 0
(46)
en equation (31) has one positive root u 3165equation (28) has one positive rootϖ0 1779 and the Hopfbifurcation value is τ0 05304 First let τ 052lt τ0 andselect three different initial values I1 (1 001 1700)I2 (1 80 001) and I3 (1 80 1700) respectively Nu-merical simulations show that the solutions of systemconverge to the same equilibrium respectively (see Fig-ures 2 3 and 4) In the biological sense tiny presence ofinfected cells or virus can lead to the disease transmissionNext let τ 054gt τ0 Figure 5 shows that periodic
x
0
2
4
6
8
10
12
40002000 50001000 60000 3000t
(a)
y
80
70
60
50
40
30
20
10
020001000 3000 4000 5000 60000
t
(b)
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 60000t
(c)
80 60 10840 6420 2
2004006008001000120014001600
(d)
Figure 1 Illustration of basic behavior of solutions of system (4) with initial condition I3 where R 08548lt 1 (a) time series of x (t)(b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 7
x
0
10
20
30
40
50
60
70
80
90
100
2000 4000 6000 8000 10000 12000 140000t
(a)
y
100004000 140000 6000 80002000 12000t
0
20
40
60
80
100
120
140
(b)
v
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
v
120100 8080y
6060x4040 20 20
1100
1200
1300
1400
1500
1600
1700
1800
(d)
Figure 2 Illustration of basic behavior of solutions of system (4) with τ 052 where R 84gt 1 (the initial condition is (1 001 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
1000 2000 3000 4000 5000 6000 7000 8000 90000t
0
10
20
30
40
50
60
(a)
y
1000 4000 6000 900070002000 80003000 50000t
40
50
60
70
80
90
100
(b)
Figure 3 Continued
8 Complexity
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 6000 7000 8000 90000t
(c)
v
901080 20
70x30
y
60 405050
50010001500
(d)
Figure 3 Illustration of basic behavior of solutions of system (4) with τ 052 whereR 84gt 1 (the initial condition is (1 80 001)) (a)time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
2000 4000 6000 8000 10000 12000 140000t
0
5
10
15
20
25
30
35
40
(a)
y
50
60
70
80
90
100
110
120
2000 4000 6000 8000 10000 12000 140000t
(b)
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
100 3080 201060
145015001550160016501700175018001850
v
y x
(d)
Figure 4 Illustration of basic behavior of solutions of system (4) with τ 052lt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 9
oscillations occur for τ gt τ0 en system (4) undergoes aHopf bifurcation at Elowast when τ τ0
5 Conclusion
In this paper we improved a delayed HIV virus dynamicmodel by introducing cell-to-cell transmission and apo-ptosis of bystander cells We derived the basic reproductivenumber R0 lt 1 and based on the basic reproductive numberthe stability of the equilibria of the system was investigatede mathematical analysis showed that if R0 lt 1 then thedisease-free equilibrium of system (4) is globally asymp-totically stable and for R0 gt 1 the system can produce Hopfbifurcation with the delay changing In biology HIV cannotsuccessfully invade healthy cells and will be cleared by theimmune system finally if R0 lt 1 and if R0 gt 1 uninfectedcells infected cells and free virus can coexist as periodicoscillations under certain conditions
However it is very necessary to consider antiretroviraltherapy in model (4) since it can make the model morerealistic and many scholars have noticed this topic andachieved excellent results for example Wang et al [52ndash54]
have studied the virus dynamics model with antiretroviraltherapy eir research shows that antiretroviral therapy canincrease the number of uninfected T cells and decrease thenumber of infectious virions and is very effective for thetreatment of AIDS patients In present model we do notconsider this factor and we will leave this for future research
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
Tongqian Zhang was supported by the Shandong ProvincialNatural Science Foundation of China (no ZR2019MA003)and Scientific Research Foundation of Shandong Universityof Science and Technology for Recruited Talents Zhichao
x
times104151 35 405 3252 450
t
0
5
10
15
20
25
30
35
40
45
(a)
times104
y
41505 2 25 3 450 1 35t
50
60
70
80
90
100
110
120
130
(b)
times104
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
151 35 405 3252 450t
(c)
120
v
40100y 30
x80 201060
145015001550160016501700175018001850
(d)
Figure 5 Illustration of basic behavior of solutions of system (4) with τ 054gt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
10 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
[1] R Weiss ldquoHow does HIV cause AIDSrdquo Science vol 260no 5112 pp 1273ndash1279 1993
[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
ω6+ c1ω
4+ c2ω
2+ c3 0 (33)
Let ϖ ω2 (33) is equivalent to
ϖ3 + c1ϖ2
+ c2ϖ + c3 0 (34)
Define
g(ϖ) ϖ3 + c1ϖ2
+ c2ϖ + c3 (35)
and thus
gprime(ϖ) 3ϖ2 + 2c1ϖ + c2 (36)
We consider that
3ϖ2 + 2c1ϖ + c2 0 (37)
and it has two real roots given as ϖ1 (minus c1 +Δ
radic)3 and
ϖ2 (minus c1 minusΔ
radic)3 where Δ c21 minus 3c2 Next we get the
following lemma from [50]
Lemma 2 For polynomial (34) the following conclusions aregiven
(i) If c3 lt 0 (34) has at least one positive root(ii) If c3 ge 0 and Δlt 0 (34) has no real root(iii) If c3 ge 0 and Δgt 0 and if and only ifϖ1 (minus c1 +
Δ
radic)3gt 0 and g(ϖ1)le 0 (34) has real
roots
Assume that g(ϖ) 0 has positive real roots thereforewe can suppose that (34) has k(1le kle 3) positive real rootsdenoted as ϖ1ϖ2 and ϖ3 Moreover (33) has positive realroots ωk
ϖk
radic From (31) we can get that
cos(ωτ) b2 minus a1b1( 1113857ω4 + a1b3 + a3b1 minus a2b2( 1113857ω2 minus a3b3
b22ω2 + b3 minus b1ω2( 11138572
(38)
In addition we get the corresponding τ(n)k gt 0 such that
(33) has pure imaginary λ iωk where
τ(n)k
1ωk
arccosb2 minus a1b1( 1113857ω4
k + a1b3 + a3b1 minus a2b2( 1113857ω2k minus a3b3
b22ω2k + b3 minus b1ω2
k1113872 11138732
⎛⎜⎝ ⎞⎟⎠ + 2nπ⎧⎪⎨
⎪⎩
⎫⎪⎬
⎪⎭ (39)
in which k 1 2 3 n 0 1 2
Define
τlowast minkisin[123]
τ(0)k1113966 1113967 (40)
Differentiating the two sides of (15) with respect to τ itfollows that
3λ2 + 2a1λ + a21113872 1113873dλdτ
+ 2b1λ + b2( 1113857eminus λτdλ
dτ
minus b1λ2
+ b2λ + b31113872 1113873eminus λτ λ + τ
dλdτ
1113888 1113889 0
(41)
and therefore
dλdτ
1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 + 2a1ωki
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873i
+2b1ωki + b2
minus b2ω2k + b3ωk minus b1ω3
k1113872 1113873iminus
τωki
(42)
end(Res(λ))
dτ1113888 1113889
minus 1
λiωk
a2 minus 3ω2
k( 1113857 a2ω2k minus ω4
k( 1113857 + 2a1ωk a1ω3k minus a3ωk( 1113857
a2ω2k minus ω4
k1113872 1113873 + a1ω3k minus a3ωk1113872 1113873
+2b1ωk b3ωk minus b1ω3
k( 1113857 minus b22ω2k
b22ω4k + b3ωk minus b1ω3
k1113872 11138732
(43)
From (31) we have
ω3minus a2ω1113872 1113873
2+ a1ω minus a3( 1113857
2 b
22ω
2+ b1ω
2minus b31113872 1113873
2 (44)
Hence
d(Res(λ))
dτ1113888 11138891113888 1113889
minus 1
λiωk
3ϖ3k + 2c1ϖ2k + c2ϖk
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
ωkgprime ϖk( 1113857
ω2k b22ω2
k + b1ω2k minus b31113872 1113873
21113876 1113877
(45)
Since ωk gt 0 this is enough to demonstrate thatRe(dλ(τ)dτ)|ττ(n)
k
and gprime(ϖk) have the same sign Com-bining Lemma 2 with (45) we have the followingconclusions
Theorem 2 If R0 gt 1 the following results hold
(i) If c3 ge 0 and Δle 0 then the epidemic equilibriumElowast (xlowast ylowast vlowast) is locally asymptotically stable
(ii) If c3 ge 0 or c3 lt 0 and Δgt 0 then the epidemicequilibrium Elowast (xlowast ylowast vlowast) is asymptotically sta-ble when τ isin [0 τlowast) and unstable when τ gt τlowast
(iii) If the conditions of (ii) are all satisfied andgprime(ϖk)ne 0 then model (1) undergoes a Hopf bifur-cation at Elowast when τ gt τ(n)
k (n 0 1 2 )
6 Complexity
4 Numerical Simulations
In this section we give some numerical simulations tovalidate our main results in Section 3 e basic parametersare set as follows [51] s 2 r 10 T 10 d
002 β 0000027 α 0000024 c 0001 δ 0002 η
0002 p 05 k 10 and u 05 Direct calculations withMaple 14 show thatR 08548lt 1 and E0 (101765 0 0)Figure 1 shows that virus-free equilibrium E0 of the system isstable
In the following we change T to 100 and direct cal-culations show system (4) has two equilibria ieE0 (101765 0 0) and E1 (119048 881959 17639171)
(R 84gt 1) We easily get
a3 03396
b3 18997
g(0) c3 a23 minus b
23 minus 34937lt 0
(46)
en equation (31) has one positive root u 3165equation (28) has one positive rootϖ0 1779 and the Hopfbifurcation value is τ0 05304 First let τ 052lt τ0 andselect three different initial values I1 (1 001 1700)I2 (1 80 001) and I3 (1 80 1700) respectively Nu-merical simulations show that the solutions of systemconverge to the same equilibrium respectively (see Fig-ures 2 3 and 4) In the biological sense tiny presence ofinfected cells or virus can lead to the disease transmissionNext let τ 054gt τ0 Figure 5 shows that periodic
x
0
2
4
6
8
10
12
40002000 50001000 60000 3000t
(a)
y
80
70
60
50
40
30
20
10
020001000 3000 4000 5000 60000
t
(b)
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 60000t
(c)
80 60 10840 6420 2
2004006008001000120014001600
(d)
Figure 1 Illustration of basic behavior of solutions of system (4) with initial condition I3 where R 08548lt 1 (a) time series of x (t)(b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 7
x
0
10
20
30
40
50
60
70
80
90
100
2000 4000 6000 8000 10000 12000 140000t
(a)
y
100004000 140000 6000 80002000 12000t
0
20
40
60
80
100
120
140
(b)
v
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
v
120100 8080y
6060x4040 20 20
1100
1200
1300
1400
1500
1600
1700
1800
(d)
Figure 2 Illustration of basic behavior of solutions of system (4) with τ 052 where R 84gt 1 (the initial condition is (1 001 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
1000 2000 3000 4000 5000 6000 7000 8000 90000t
0
10
20
30
40
50
60
(a)
y
1000 4000 6000 900070002000 80003000 50000t
40
50
60
70
80
90
100
(b)
Figure 3 Continued
8 Complexity
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 6000 7000 8000 90000t
(c)
v
901080 20
70x30
y
60 405050
50010001500
(d)
Figure 3 Illustration of basic behavior of solutions of system (4) with τ 052 whereR 84gt 1 (the initial condition is (1 80 001)) (a)time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
2000 4000 6000 8000 10000 12000 140000t
0
5
10
15
20
25
30
35
40
(a)
y
50
60
70
80
90
100
110
120
2000 4000 6000 8000 10000 12000 140000t
(b)
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
100 3080 201060
145015001550160016501700175018001850
v
y x
(d)
Figure 4 Illustration of basic behavior of solutions of system (4) with τ 052lt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 9
oscillations occur for τ gt τ0 en system (4) undergoes aHopf bifurcation at Elowast when τ τ0
5 Conclusion
In this paper we improved a delayed HIV virus dynamicmodel by introducing cell-to-cell transmission and apo-ptosis of bystander cells We derived the basic reproductivenumber R0 lt 1 and based on the basic reproductive numberthe stability of the equilibria of the system was investigatede mathematical analysis showed that if R0 lt 1 then thedisease-free equilibrium of system (4) is globally asymp-totically stable and for R0 gt 1 the system can produce Hopfbifurcation with the delay changing In biology HIV cannotsuccessfully invade healthy cells and will be cleared by theimmune system finally if R0 lt 1 and if R0 gt 1 uninfectedcells infected cells and free virus can coexist as periodicoscillations under certain conditions
However it is very necessary to consider antiretroviraltherapy in model (4) since it can make the model morerealistic and many scholars have noticed this topic andachieved excellent results for example Wang et al [52ndash54]
have studied the virus dynamics model with antiretroviraltherapy eir research shows that antiretroviral therapy canincrease the number of uninfected T cells and decrease thenumber of infectious virions and is very effective for thetreatment of AIDS patients In present model we do notconsider this factor and we will leave this for future research
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
Tongqian Zhang was supported by the Shandong ProvincialNatural Science Foundation of China (no ZR2019MA003)and Scientific Research Foundation of Shandong Universityof Science and Technology for Recruited Talents Zhichao
x
times104151 35 405 3252 450
t
0
5
10
15
20
25
30
35
40
45
(a)
times104
y
41505 2 25 3 450 1 35t
50
60
70
80
90
100
110
120
130
(b)
times104
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
151 35 405 3252 450t
(c)
120
v
40100y 30
x80 201060
145015001550160016501700175018001850
(d)
Figure 5 Illustration of basic behavior of solutions of system (4) with τ 054gt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
10 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
[1] R Weiss ldquoHow does HIV cause AIDSrdquo Science vol 260no 5112 pp 1273ndash1279 1993
[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
4 Numerical Simulations
In this section we give some numerical simulations tovalidate our main results in Section 3 e basic parametersare set as follows [51] s 2 r 10 T 10 d
002 β 0000027 α 0000024 c 0001 δ 0002 η
0002 p 05 k 10 and u 05 Direct calculations withMaple 14 show thatR 08548lt 1 and E0 (101765 0 0)Figure 1 shows that virus-free equilibrium E0 of the system isstable
In the following we change T to 100 and direct cal-culations show system (4) has two equilibria ieE0 (101765 0 0) and E1 (119048 881959 17639171)
(R 84gt 1) We easily get
a3 03396
b3 18997
g(0) c3 a23 minus b
23 minus 34937lt 0
(46)
en equation (31) has one positive root u 3165equation (28) has one positive rootϖ0 1779 and the Hopfbifurcation value is τ0 05304 First let τ 052lt τ0 andselect three different initial values I1 (1 001 1700)I2 (1 80 001) and I3 (1 80 1700) respectively Nu-merical simulations show that the solutions of systemconverge to the same equilibrium respectively (see Fig-ures 2 3 and 4) In the biological sense tiny presence ofinfected cells or virus can lead to the disease transmissionNext let τ 054gt τ0 Figure 5 shows that periodic
x
0
2
4
6
8
10
12
40002000 50001000 60000 3000t
(a)
y
80
70
60
50
40
30
20
10
020001000 3000 4000 5000 60000
t
(b)
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 60000t
(c)
80 60 10840 6420 2
2004006008001000120014001600
(d)
Figure 1 Illustration of basic behavior of solutions of system (4) with initial condition I3 where R 08548lt 1 (a) time series of x (t)(b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 7
x
0
10
20
30
40
50
60
70
80
90
100
2000 4000 6000 8000 10000 12000 140000t
(a)
y
100004000 140000 6000 80002000 12000t
0
20
40
60
80
100
120
140
(b)
v
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
v
120100 8080y
6060x4040 20 20
1100
1200
1300
1400
1500
1600
1700
1800
(d)
Figure 2 Illustration of basic behavior of solutions of system (4) with τ 052 where R 84gt 1 (the initial condition is (1 001 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
1000 2000 3000 4000 5000 6000 7000 8000 90000t
0
10
20
30
40
50
60
(a)
y
1000 4000 6000 900070002000 80003000 50000t
40
50
60
70
80
90
100
(b)
Figure 3 Continued
8 Complexity
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 6000 7000 8000 90000t
(c)
v
901080 20
70x30
y
60 405050
50010001500
(d)
Figure 3 Illustration of basic behavior of solutions of system (4) with τ 052 whereR 84gt 1 (the initial condition is (1 80 001)) (a)time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
2000 4000 6000 8000 10000 12000 140000t
0
5
10
15
20
25
30
35
40
(a)
y
50
60
70
80
90
100
110
120
2000 4000 6000 8000 10000 12000 140000t
(b)
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
100 3080 201060
145015001550160016501700175018001850
v
y x
(d)
Figure 4 Illustration of basic behavior of solutions of system (4) with τ 052lt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 9
oscillations occur for τ gt τ0 en system (4) undergoes aHopf bifurcation at Elowast when τ τ0
5 Conclusion
In this paper we improved a delayed HIV virus dynamicmodel by introducing cell-to-cell transmission and apo-ptosis of bystander cells We derived the basic reproductivenumber R0 lt 1 and based on the basic reproductive numberthe stability of the equilibria of the system was investigatede mathematical analysis showed that if R0 lt 1 then thedisease-free equilibrium of system (4) is globally asymp-totically stable and for R0 gt 1 the system can produce Hopfbifurcation with the delay changing In biology HIV cannotsuccessfully invade healthy cells and will be cleared by theimmune system finally if R0 lt 1 and if R0 gt 1 uninfectedcells infected cells and free virus can coexist as periodicoscillations under certain conditions
However it is very necessary to consider antiretroviraltherapy in model (4) since it can make the model morerealistic and many scholars have noticed this topic andachieved excellent results for example Wang et al [52ndash54]
have studied the virus dynamics model with antiretroviraltherapy eir research shows that antiretroviral therapy canincrease the number of uninfected T cells and decrease thenumber of infectious virions and is very effective for thetreatment of AIDS patients In present model we do notconsider this factor and we will leave this for future research
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
Tongqian Zhang was supported by the Shandong ProvincialNatural Science Foundation of China (no ZR2019MA003)and Scientific Research Foundation of Shandong Universityof Science and Technology for Recruited Talents Zhichao
x
times104151 35 405 3252 450
t
0
5
10
15
20
25
30
35
40
45
(a)
times104
y
41505 2 25 3 450 1 35t
50
60
70
80
90
100
110
120
130
(b)
times104
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
151 35 405 3252 450t
(c)
120
v
40100y 30
x80 201060
145015001550160016501700175018001850
(d)
Figure 5 Illustration of basic behavior of solutions of system (4) with τ 054gt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
10 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
[1] R Weiss ldquoHow does HIV cause AIDSrdquo Science vol 260no 5112 pp 1273ndash1279 1993
[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
x
0
10
20
30
40
50
60
70
80
90
100
2000 4000 6000 8000 10000 12000 140000t
(a)
y
100004000 140000 6000 80002000 12000t
0
20
40
60
80
100
120
140
(b)
v
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
v
120100 8080y
6060x4040 20 20
1100
1200
1300
1400
1500
1600
1700
1800
(d)
Figure 2 Illustration of basic behavior of solutions of system (4) with τ 052 where R 84gt 1 (the initial condition is (1 001 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
1000 2000 3000 4000 5000 6000 7000 8000 90000t
0
10
20
30
40
50
60
(a)
y
1000 4000 6000 900070002000 80003000 50000t
40
50
60
70
80
90
100
(b)
Figure 3 Continued
8 Complexity
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 6000 7000 8000 90000t
(c)
v
901080 20
70x30
y
60 405050
50010001500
(d)
Figure 3 Illustration of basic behavior of solutions of system (4) with τ 052 whereR 84gt 1 (the initial condition is (1 80 001)) (a)time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
2000 4000 6000 8000 10000 12000 140000t
0
5
10
15
20
25
30
35
40
(a)
y
50
60
70
80
90
100
110
120
2000 4000 6000 8000 10000 12000 140000t
(b)
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
100 3080 201060
145015001550160016501700175018001850
v
y x
(d)
Figure 4 Illustration of basic behavior of solutions of system (4) with τ 052lt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 9
oscillations occur for τ gt τ0 en system (4) undergoes aHopf bifurcation at Elowast when τ τ0
5 Conclusion
In this paper we improved a delayed HIV virus dynamicmodel by introducing cell-to-cell transmission and apo-ptosis of bystander cells We derived the basic reproductivenumber R0 lt 1 and based on the basic reproductive numberthe stability of the equilibria of the system was investigatede mathematical analysis showed that if R0 lt 1 then thedisease-free equilibrium of system (4) is globally asymp-totically stable and for R0 gt 1 the system can produce Hopfbifurcation with the delay changing In biology HIV cannotsuccessfully invade healthy cells and will be cleared by theimmune system finally if R0 lt 1 and if R0 gt 1 uninfectedcells infected cells and free virus can coexist as periodicoscillations under certain conditions
However it is very necessary to consider antiretroviraltherapy in model (4) since it can make the model morerealistic and many scholars have noticed this topic andachieved excellent results for example Wang et al [52ndash54]
have studied the virus dynamics model with antiretroviraltherapy eir research shows that antiretroviral therapy canincrease the number of uninfected T cells and decrease thenumber of infectious virions and is very effective for thetreatment of AIDS patients In present model we do notconsider this factor and we will leave this for future research
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
Tongqian Zhang was supported by the Shandong ProvincialNatural Science Foundation of China (no ZR2019MA003)and Scientific Research Foundation of Shandong Universityof Science and Technology for Recruited Talents Zhichao
x
times104151 35 405 3252 450
t
0
5
10
15
20
25
30
35
40
45
(a)
times104
y
41505 2 25 3 450 1 35t
50
60
70
80
90
100
110
120
130
(b)
times104
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
151 35 405 3252 450t
(c)
120
v
40100y 30
x80 201060
145015001550160016501700175018001850
(d)
Figure 5 Illustration of basic behavior of solutions of system (4) with τ 054gt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
10 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
[1] R Weiss ldquoHow does HIV cause AIDSrdquo Science vol 260no 5112 pp 1273ndash1279 1993
[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
v
0
200
400
600
800
1000
1200
1400
1600
1800
1000 2000 3000 4000 5000 6000 7000 8000 90000t
(c)
v
901080 20
70x30
y
60 405050
50010001500
(d)
Figure 3 Illustration of basic behavior of solutions of system (4) with τ 052 whereR 84gt 1 (the initial condition is (1 80 001)) (a)time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
x
2000 4000 6000 8000 10000 12000 140000t
0
5
10
15
20
25
30
35
40
(a)
y
50
60
70
80
90
100
110
120
2000 4000 6000 8000 10000 12000 140000t
(b)
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
2000 4000 6000 8000 10000 12000 140000t
(c)
100 3080 201060
145015001550160016501700175018001850
v
y x
(d)
Figure 4 Illustration of basic behavior of solutions of system (4) with τ 052lt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
Complexity 9
oscillations occur for τ gt τ0 en system (4) undergoes aHopf bifurcation at Elowast when τ τ0
5 Conclusion
In this paper we improved a delayed HIV virus dynamicmodel by introducing cell-to-cell transmission and apo-ptosis of bystander cells We derived the basic reproductivenumber R0 lt 1 and based on the basic reproductive numberthe stability of the equilibria of the system was investigatede mathematical analysis showed that if R0 lt 1 then thedisease-free equilibrium of system (4) is globally asymp-totically stable and for R0 gt 1 the system can produce Hopfbifurcation with the delay changing In biology HIV cannotsuccessfully invade healthy cells and will be cleared by theimmune system finally if R0 lt 1 and if R0 gt 1 uninfectedcells infected cells and free virus can coexist as periodicoscillations under certain conditions
However it is very necessary to consider antiretroviraltherapy in model (4) since it can make the model morerealistic and many scholars have noticed this topic andachieved excellent results for example Wang et al [52ndash54]
have studied the virus dynamics model with antiretroviraltherapy eir research shows that antiretroviral therapy canincrease the number of uninfected T cells and decrease thenumber of infectious virions and is very effective for thetreatment of AIDS patients In present model we do notconsider this factor and we will leave this for future research
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
Tongqian Zhang was supported by the Shandong ProvincialNatural Science Foundation of China (no ZR2019MA003)and Scientific Research Foundation of Shandong Universityof Science and Technology for Recruited Talents Zhichao
x
times104151 35 405 3252 450
t
0
5
10
15
20
25
30
35
40
45
(a)
times104
y
41505 2 25 3 450 1 35t
50
60
70
80
90
100
110
120
130
(b)
times104
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
151 35 405 3252 450t
(c)
120
v
40100y 30
x80 201060
145015001550160016501700175018001850
(d)
Figure 5 Illustration of basic behavior of solutions of system (4) with τ 054gt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
10 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
[1] R Weiss ldquoHow does HIV cause AIDSrdquo Science vol 260no 5112 pp 1273ndash1279 1993
[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
oscillations occur for τ gt τ0 en system (4) undergoes aHopf bifurcation at Elowast when τ τ0
5 Conclusion
In this paper we improved a delayed HIV virus dynamicmodel by introducing cell-to-cell transmission and apo-ptosis of bystander cells We derived the basic reproductivenumber R0 lt 1 and based on the basic reproductive numberthe stability of the equilibria of the system was investigatede mathematical analysis showed that if R0 lt 1 then thedisease-free equilibrium of system (4) is globally asymp-totically stable and for R0 gt 1 the system can produce Hopfbifurcation with the delay changing In biology HIV cannotsuccessfully invade healthy cells and will be cleared by theimmune system finally if R0 lt 1 and if R0 gt 1 uninfectedcells infected cells and free virus can coexist as periodicoscillations under certain conditions
However it is very necessary to consider antiretroviraltherapy in model (4) since it can make the model morerealistic and many scholars have noticed this topic andachieved excellent results for example Wang et al [52ndash54]
have studied the virus dynamics model with antiretroviraltherapy eir research shows that antiretroviral therapy canincrease the number of uninfected T cells and decrease thenumber of infectious virions and is very effective for thetreatment of AIDS patients In present model we do notconsider this factor and we will leave this for future research
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
Tongqian Zhang was supported by the Shandong ProvincialNatural Science Foundation of China (no ZR2019MA003)and Scientific Research Foundation of Shandong Universityof Science and Technology for Recruited Talents Zhichao
x
times104151 35 405 3252 450
t
0
5
10
15
20
25
30
35
40
45
(a)
times104
y
41505 2 25 3 450 1 35t
50
60
70
80
90
100
110
120
130
(b)
times104
v
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
151 35 405 3252 450t
(c)
120
v
40100y 30
x80 201060
145015001550160016501700175018001850
(d)
Figure 5 Illustration of basic behavior of solutions of system (4) with τ 054gt τ0 whereR 84gt 1 (the initial condition is (1 80 1700))(a) time series of x (t) (b) time series of y (t) (c) time series of v (t) and (d) phase trajectories of x (t) y (t) and v (t)
10 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
[1] R Weiss ldquoHow does HIV cause AIDSrdquo Science vol 260no 5112 pp 1273ndash1279 1993
[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
Jiang was supported by the National Natural ScienceFoundation of China (no 11801014) Natural ScienceFoundation of Hebei Province (no A2018409004) HebeiProvince University Discipline Top Talent Selection andTraining Program (SLRC2019020) and 2020 Talent TrainingProject of Hebei Province
References
[1] R Weiss ldquoHow does HIV cause AIDSrdquo Science vol 260no 5112 pp 1273ndash1279 1993
[2] D A Jabs ldquoOcular manifestations of HIV infectionrdquoTransactions of the American Ophthalmological Societyvol 93 pp 623ndash683 1995
[3] A Sandhu and A K Samra ldquoOpportunistic infections anddisease implications in HIVAIDSrdquo International Journal ofPharmaceutical Science Invention vol 2 pp 47ndash54 2013
[4] UNAIDS ldquoe joint programme on AIDSrdquo Technical Reporthttpswwwunaidsorgen UNAIDS Geneva Switzerland2018
[5] N L Letvin ldquoProgress and obstacles in the development of anAIDS vaccinerdquo Nature Reviews Immunology vol 6 no 12pp 930ndash939 2006
[6] Y Wang M Lu and J Liu ldquoGlobal stability of a delayed virusmodel with latent infection and Beddington-DeAngelis in-fection functionrdquo Applied Mathematics Letters vol 107Article ID 106463 2020
[7] M Shen Y Xiao and L Rong ldquoGlobal stability of an in-fection-age structured HIV-1 model linking within-host andbetween-host dynamicsrdquo Mathematical Biosciences vol 263pp 37ndash50 2015
[8] S Iwami S Nakaoka and Y Takeuchi ldquoMathematicalanalysis of a HIV model with frequency dependence and viraldiversityrdquo Mathematical Biosciences and Engineering MBEvol 5 no 3 pp 457ndash476 2008
[9] T Guo Z Qiu and L Rong ldquoA within-host drug resistancemodel with continuous state-dependent viral strainsrdquoAppliedMathematics Letters vol 104 Article ID 106223 2020
[10] W Wang W Ma and Z Feng ldquoDynamics of reaction-dif-fusion equations for modeling CD4+ T cells decline withgeneral infection mechanism and distinct dispersal ratesrdquoNonlinear Analysis Real World Applications vol 51 ArticleID 102976 2020
[11] T Loudon S Pankavich and S Pankavich ldquoMathematicalanalysis and dynamic active subspaces for a long term modelof HIVrdquo Mathematical Biosciences and Engineering vol 14no 3 pp 709ndash733 2017
[12] O M Otunuga ldquoGlobal stability for a 2n+ 1 dimensionalHIVAIDS epidemic model with treatmentsrdquo MathematicalBiosciences vol 299 pp 138ndash152 2018
[13] Y Wang T Zhao T Zhao and J Liu ldquoViral dynamics of anHIV stochastic model with cell-to-cell infection CTL immuneresponse and distributed delaysrdquo Mathematical Biosciencesand Engineering vol 16 no 6 pp 7126ndash7154 2019
[14] T Feng Z Qiu X Meng and L Rong ldquoAnalysis of a sto-chastic HIV-1 infection model with degenerate diffusionrdquoAppliedMathematics and Computation vol 348 pp 437ndash4552019
[15] XWang Q Ge and Y Chen ldquoreshold dynamics of an HIVinfection model with two distinct cell subsetsrdquo AppliedMathematics Letters vol 103 p 106242 2020
[16] Y Liu and X Zou ldquoMathematical modeling of HIV-likeparticle assembly in vitrordquoMathematical Biosciences vol 288pp 46ndash51 2017
[17] N Bobko J P Zubelli and J Zubelli ldquoA singularly perturbedHIV model with treatment and antigenic variationrdquo Math-ematical Biosciences and Engineering vol 12 no 1 pp 1ndash212015
[18] T Guo Z Qiu and L Rong ldquoModeling the role of macro-phages in HIV persistence during antiretroviral therapyrdquoJournal of Mathematical Biology vol 81 no 1 pp 369ndash4022020
[19] H C Tuckwell and F Y M Wan ldquoFirst passage time todetection in stochastic population dynamical models for HIV-1rdquo Applied Mathematics Letters vol 13 no 5 pp 79ndash832000
[20] W Wang W Ma and Z Feng ldquoComplex dynamics of a timeperiodic nonlocal and time-delayed model of reaction-dif-fusion equations for modeling CD4+ T cells declinerdquo Journalof Computational and Applied Mathematics vol 367 ArticleID 112430 2020
[21] W Wang X Wang and Z Feng ldquoTime periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patientsrdquo Nonlinear Analysis Real World Applica-tions vol 57 Article ID 103184 2021
[22] M A Nowak and C R M Bangham ldquoPopulation dynamicsof immune responses to persistent virusesrdquo Science vol 272no 5258 pp 74ndash79 1996
[23] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[24] D Li andWMa ldquoAsymptotic properties of a HIV-1 infectionmodel with time delayrdquo Journal of Mathematical Analysis andApplications vol 335 no 1 pp 683ndash691 2007
[25] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal of MathematicalAnalysis and Applications vol 329 no 1 pp 281ndash297 2007
[26] G Huang W Ma and Y Takeuchi ldquoGlobal properties forvirus dynamics model with Beddington-DeAngelis functionalresponserdquo Applied Mathematics Letters vol 22 no 11pp 1690ndash1693 2009
[27] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withCrowley-Martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011
[28] K Hattaf and N Yousfi ldquoGlobal stability of a virus dynamicsmodel with cure rate and absorptionrdquo Journal of the EgyptianMathematical Society vol 22 no 3 pp 386ndash389 2014
[29] Y Tian and X Liu ldquoGlobal dynamics of a virus dynamicalmodel with general incidence rate and cure raterdquo Non-linear Analysis Real World Applications vol 16 pp 17ndash262014
[30] F Li S Zhang and X Meng ldquoDynamics analysis and nu-merical simulations of a delayed stochastic epidemic modelsubject to a general response functionrdquo Computational andApplied Mathematics vol 38 p 95 2019
[31] A S Perelson and P W Nelson ldquoMathematical analysis ofHIV-1 dynamics in vivordquo SIAM Review vol 41 no 1pp 3ndash44 1999
[32] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006
[33] Z Hu J Zhang H Wang W Ma and F Liao ldquoDynamicsanalysis of a delayed viral infection model with logistic growthand immune impairmentrdquo Applied Mathematical Modellingvol 38 no 2 pp 524ndash534 2014
[34] R V Culshaw S Ruan and GWebb ldquoAmathematical modelof cell-to-cell spread of HIV-1 that includes a time delayrdquo
Complexity 11
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity
Journal of Mathematical Biology vol 46 no 5 pp 425ndash4442003
[35] D Mcdonald and T J Hope ldquoRecruitment of HIV and itsreceptors to dendritic cell-T cell junctionsrdquo Science vol 300no 5623 pp 1295ndash1297 2003
[36] Q Sattentau ldquoAvoiding the void cell-to-cell spread of humanvirusesrdquo Nature Reviews Microbiology vol 6 no 11pp 815ndash826 2008
[37] D M Phillips ldquoe role of cell-to-cell transmission in HIVinfectionrdquo Aids vol 8 no 6 pp 719ndash732 1994
[38] C Jolly ldquoCell-to-cell transmission of retroviruses innateimmunity and interferon-induced restriction factorsrdquo Vi-rology vol 411 no 2 pp 251ndash259 2011
[39] C R M Bangham ldquoe immune control and cell-to-cellspread of human T-lymphotropic virus type 1rdquo Journal ofGeneral Virology vol 84 no 12 pp 3177ndash3189 2003
[40] S Krantic C Gimenez and C Rabourdin-Combe ldquoCell-to-cell contact via measles virus haemagglutinin-CD46 inter-action triggers CD46 downregulationrdquo Journal of GeneralVirology vol 76 no 11 pp 2793ndash2800 1995
[41] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 2020 Article ID 1 2020
[42] X Lai and X Zou ldquoModeling cell-to-cell spread of HIV-1 withlogistic target cell growthrdquo Journal of Mathematical Analysisand Applications vol 426 no 1 pp 563ndash584 2015
[43] M F Cotton D N Ikle E L Rapaport et al ldquoApoptosis ofCD4+ and CD8+ T cells isolated immediately ex vivo cor-relates with disease severity in human immunodeficiencyvirus type 1 infectionrdquo Pediatric Research vol 42 no 5pp 656ndash664 1997
[44] T H Finkel G Tudor-Williams N K Banda et al ldquoApo-ptosis occurs predominantly in bystander cells and not inproductively infected cells of HIV- and SIV-infected lymphnodesrdquo Nature Medicine vol 1 no 2 pp 129ndash134 1995
[45] N Selliah and T H Finkel ldquoBiochemical mechanisms of HIVinduced T cell apoptosisrdquo Cell Death amp Differentiation vol 8no 2 pp 127ndash136 2001
[46] W Cheng W Ma and S Guo ldquoA class of virus dynamicmodel with inhibitory effect on the growth of uninfectedT cells caused by infected T cells and its stability analysisrdquoCommunications on Pure and Applied Analysis vol 15 no 3pp 795ndash806 2016
[47] S Guo and W Ma ldquoGlobal behavior of delay differentialequations model of HIV infection with apoptosisrdquo Discrete ampContinuous Dynamical SystemsndashB vol 21 no 1 pp 103ndash1192016
[48] Y Ji W Ma and K Song ldquoModeling inhibitory effect on thegrowth of uninfected Tcells caused by infected Tcells stabilityand hopf bifurcationrdquo Computational and MathematicalMethods in Medicine vol 2018 Article ID 317689 10 pages2018
[49] S Hews S Eikenberry J D Nagy and Y Kuang ldquoRichdynamics of a hepatitis B viral infection model with logistichepatocyte growthrdquo Journal of Mathematical Biology vol 60no 4 pp 573ndash590 2010
[50] S Ruan and JWei ldquoOn the zeros of a third degree exponentialpolynomial with applications to a delayed model for thecontrol of testosterone secretionrdquo Mathematical Medicineand Biology vol 18 no 1 pp 41ndash52 2001
[51] K Zhuang and H Zhu ldquoStability and bifurcation analysis foran improved HIV model with time delay and cure raterdquo
Wseas Transactions on Mathematics vol 12 pp 860ndash8692013
[52] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed hiv pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009
[53] Y Wang Y Zhou F Brauer and J M Heffernan ldquoViraldynamics model with CTL immune response incorporatingantiretroviral therapyrdquo Journal of Mathematical Biologyvol 67 no 4 pp 901ndash934 2013
[54] Y Wang J Liu and L Liu ldquoViral dynamics of an HIV modelwith latent infection incorporating antiretroviral therapyrdquoAdvances in Difference Equations vol 2016 no 1 p 225 2016
12 Complexity