research article the dynamical behaviors in a stochastic...

Research ArticleThe Dynamical Behaviors in a Stochastic SIS Epidemic Modelwith Nonlinear Incidence

Ramziya Rifhat Qing Ge and Zhidong Teng

College of Mathematics and Systems Science Xinjiang University Urumqi 830046 China

Correspondence should be addressed to Zhidong Teng zhidong tengsinacom

Received 8 February 2016 Accepted 22 May 2016

Academic Editor Chuangyin Dang

Copyright copy 2016 Ramziya Rifhat et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A stochastic SIS-type epidemic model with general nonlinear incidence and disease-induced mortality is investigated It is provedthat the dynamical behaviors of the model are determined by a certain threshold value 119877

0 That is when 119877

0lt 1 and together with

an additional condition the disease is extinct with probability one and when

119877

0gt 1 the disease is permanent in the mean in

probability and when there is not disease-related death the disease oscillates stochastically about a positive number Furthermorewhen

119877

0gt 1 the model admits positive recurrence and a unique stationary distribution Particularly the effects of the intensities

of stochastic perturbation for the dynamical behaviors of the model are discussed in detail and the dynamical behaviors for thestochastic SIS epidemicmodel with standard incidence are established Finally the numerical simulations are presented to illustratethe proposed open problems

1 Introduction

Our real life is full of randomness and stochasticityThereforeusing stochastic dynamical models can gain more real ben-efits Particularly stochastic dynamical models can provideus with some additional degrees of realism in comparison totheir deterministic counterparts There are different possibleapproaches which result in different effects on the epidemicdynamical systems to include random perturbations in themodels In particular the following three approaches are seenmost often The first one is parameters perturbation thesecond one is the environmental noise that is proportional tothe variables and the last one is the robustness of the positiveequilibrium of the deterministic models

In recent years various types of stochastic epidemicdynamical models are established and investigated widelyThemain research subjects include the existence and unique-ness of positive solution with any positive initial value inprobability mean the persistence and extinction of the dis-ease in probability mean the asymptotical behaviors aroundthe disease-free equilibrium and the endemic equilibrium ofthe deterministic models and the existence of the stationarydistribution as well as ergodicity Many important results

have been established in many literatures for example [1ndash16]and the references cited therein Particularly for stochasticSI type epidemic models in [6] Gray et al constructed astochastic SIS epidemic model with constant population sizewhere the authors not only obtained the existence of theunique global positive solution with any positive initial valuebut also established the threshold value conditions that isthe disease dies out or persists Furthermore in the case ofthe persistence the authors also showed the existence of astationary distribution and finally computed the mean valueand variance of the stationary distribution

However from articles [1ndash16] and the references citedtherein we see that there are still many important problemswhich are not studied completely and impactfully For exam-ple see the following

(1) The stochastic epidemic models with general non-linear incidence are not investigated Up to nowonly some special cases of nonlinear incidence forexample saturated incidence rate are consideredBut we all know that the nonlinear incidence ratein the theory of mathematical epidemiology is veryimportant

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2016 Article ID 5218163 14 pageshttpdxdoiorg10115520165218163

2 Computational and Mathematical Methods in Medicine

(2) For the stochastic epidemic models with the standardincidence up to now we do not find any interestingresearches

(3) The conditions obtained on the existence of uniquestationary distribution are very rigorous Whetherthere is a unique stationary distribution onlywhen themodel is permanent in the mean with probability oneis still an open problem

Motivated by the above work in this paper we considerthe following deterministic SIS epidemic model with nonlin-ear incidence rate and disease-induced mortality

119889119878 (119905)

119889119905

= Λ minus 120573119891 (119878 (119905) 119868 (119905)) + 120574119868 (119905) minus 120583119878 (119905)

119889119868 (119905)

119889119905

= 120573119891 (119878 (119905) 119868 (119905)) minus (120583 + 120574 + 120572) 119868 (119905)

(1)

In model (1) 119878 and 119868 denote the susceptible and infectiousindividualsΛ denotes the recruitment rate of the susceptible120583 is the natural death rate of 119878 and 119868 120572 is the disease-relateddeath rate the transmission of the infection is governedby a nonlinear incidence rate 120573119891(119878 119868) where 120573 denotesthe transmission coefficient between compartments 119878 and 119868119891(119878 119868) is a continuously differentiable function of 119878 and 119868 and120574 denotes the per capita disease contact rate

Now we assume that the random effects of the envi-ronment make the transmission coefficient 120573 of disease indeterministic model (1) generate random disturbance Thatis 120573 rarr 120573 + 120590

119861(119905) where 119861(119905) is a one-dimensional standardBrownian motion defined on some probability space Thusmodel (1) will become into the following stochastic SISepidemic model with nonlinear incidence rate

119889119878 (119905) = [Λ minus 120573119891 (119878 (119905) 119868 (119905)) + 120574119868 (119905) minus 120583119878 (119905)] 119889119905

minus 120590119891 (119878 (119905) 119868 (119905)) 119889119861 (119905)

119889119868 (119905) = [120573119891 (119878 (119905) 119868 (119905)) minus (120583 + 120574 + 120572) 119868 (119905)] 119889119905

+ 120590119891 (119878 (119905) 119868 (119905)) 119889119861 (119905)

(2)

In this paper we investigate the dynamical behaviors ofmodel (2) By using the Lyapunov function method Itorsquosformula and the theory of stochastic analysis [17 18] we willestablish a series of new interesting criteria on the extinctionof the disease permanence in the mean of the model withprobability oneThe stochastic oscillation of the disease abouta positive number in the case where there is not disease-related death is also obtained Further we study the positiverecurrence and the existence of stationary distribution formodel (2) and a new criterion is established Particularlythe effects of the intensities of stochastic perturbation for thedynamical behaviors of the model are discussed in detailFor some special cases of nonlinear incidence 119891(119878 119868) forexample 119891(119878 119868) = 119878119868119873 (standard incidence) and 119891(119878 119868) =ℎ(119878)119892(119868) many idiographic criteria on the extinction per-manence and stationary distribution are established Lastlysome affirmative answers for the open problems which areproposed in this paper also are given by the numerical

examples (the numerical simulation method can be found in[19])

The organization of this paper is as follows In Section 2the preliminaries are given and some useful lemmas areintroduced In Section 3 the sufficient conditions are estab-lished which ensure that the disease dies out with probabilityone In Section 4 we establish the sufficient conditions whichensure that the disease in model (2) is permanent in themean with probability one and when there is not disease-related death the disease oscillates stochastically about apositive number In Section 5 the existence on the uniquestationary distribution of model (2) is proved In Section 6the numerical simulations are carried out to illustrate someopen problems Lastly a brief discussion is given in the endto conclude this work

2 Preliminaries

Denote 1198772+= (119909

1 119909

2) 119909

1gt 0 119909

2gt 0 119877

+0= [0infin) and

119877

+= (0infin) Throughout this paper we assume that model

(2) is defined on a complete probability space (Ω 119865119905

119905ge0 119875)

with a filtration 119865119905

119905ge0satisfying the usual conditions that is

119865

119905

119905ge0is right continuous and 119865

0contains all 119875-null sets

In model (2) 119878 and 119868 denote the susceptible and infectedfractions of the population respectively and 119873 = 119878 + 119868

is the total size of the population among whom the diseaseis spreading the parameters Λ 120583 120573 and 120574 are given asin model (1) the transmission of the infection is governedby a nonlinear incidence rate 120573119878119892(119868) 119861(119905) denotes one-dimensional standard Brownianmotion defined on the aboveprobability space and 120590 represents the intensity of theBrownian motion 119861(119905) Throughout this paper we alwaysassume the following

(H) 119891(119878 119868) is two-order continuously differentiable forany 119878 ge 0 119868 ge 0 and 119878 + 119868 gt 0 For each fixed 119868 ge 0119891(119878 119868) is increasing for 119878 gt 0 and for each fixed 119878 ge 0119891(119878 119868)119868 is decreasing for 119868 gt 0 119891(119878 0) = 119891(0 119868) = 0for any 119878 gt 0 and 119868 gt 0 and 120597119891(1198780 0)120597119868 gt 0 where119878

0= Λ120583

Particularly when 119891(119878 119868) = ℎ(119878)119892(119868) then assumption(H) becomes in the following form

(Hlowast) ℎ(119878) and 119892(119868) are continuously differentiable for 119878 ge 0and 119868 ge 0 ℎ(119878) is increasing for 119878 ge 0 and 119892(119868)119868 isdecreasing for 119868 gt 0

Remark 1 From (H) by simple calculating we can obtainthat for any 119878 gt 0 and 119868 gt 0 0 le 119891(119878 119868) le (120597119891(119878 0)120597119868)119868and for any 119878

2gt 119878

1gt 0 120597119891(119878

2 0)120597119868 ge 120597119891(119878

1 0)120597119868

Remark 2 When119891(119878 119868) = 119878119868119873 (standard incidence) where119873 = 119878+119868119891(119878 119868) = 119878119868(1+120596

1119868+120596

2119878) (Beddington-DeAngelis

incidence) with constants 1205961ge 0 and 120596

2ge 0 and 119891(119878 119868) =

119878119868(1 + 120596119868

2) with constant 120596 ge 0 then (H) is satisfied

Now we give the following result for function 119891(119878 119868)

Computational and Mathematical Methods in Medicine 3

Lemma 3 For any constants 119901 gt 119902 gt 0 let 119863 = (119878 119868) 119878 gt

0 119868 gt 0 119902 le 119878 + 119868 le 119901 Then

max(119878119868)isin119863

119891 (119878 119868)

119878

119891 (119878 119868)

119868

lt infin (3)

max(119878119868)isin119863

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119878

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

lt infin (4)

Theproof of Lemma 3 is simple In fact from (H) we have

lim119878rarr0

119891 (119878 119868)

119878

=

120597119891 (0 119868)

120597119878

lim119868rarr0

119891 (119878 119868)

119878

=

120597119891 (119878 0)

120597119868

(5)

Hence conclusion (3) holds Define the functions

119867(119878 119868) =

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2 119868 gt 0

1

2

120597

2119891 (119878 0)

120597119868

2 119868 = 0

(119878 119868) isin 119863

119866 (119878 119868) =

1

119868

120597119891 (119878 119868)

120597119868

119868 gt 0

120597

2119891 (119878 0)

120597119868120597119878

119868 = 0

(119878 119868) isin 119863

(6)

Using the LrsquoHospital principle from (H) we have

lim119868rarr0

(

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2) =

1

2

120597

2119891 (119878 0)

120597119868

2

lim119868rarr0

1

119868

120597119891 (119878 119868)

120597119878

=

120597

2119891 (119878 0)

120597119868120597119878

(7)

This shows that119867(119878 119868) and119866(119878 119868) are continuous for (119878 119868) isin119863 Therefore conclusion (4) also is true

Next on the existence of global positive solutions andthe ultimate boundedness of solutions for model (2) withprobability one we have the result as follows

Lemma 4 For any initial value (119878(0) 119868(0)) isin 119877

2

+ model

(2) has a unique solution (119878(119905) 119868(119905)) defined on 119905 isin 119877

+0

satisfying (119878(119905) 119868(119905)) isin 119877

2

+for all 119905 ge 0 with probability one

Furthermore when 120572 gt 0 then 119878

0le lim inf

119905rarrinfin119873(119905) le

lim sup119905rarrinfin

119873(119905) le 119878

0 and when 120572 = 0 then lim119905rarrinfin

119873(119905) =

119878

0 where119873(119905) = 119878(119905) + 119868(119905) and 1198780= Λ(120583 + 120572)

Lemma 4 can be proved by using the method which isgiven in [6] We hence omit it here

3 Extinction of the Disease

Define the constants

119877

0=

120573 (120597119891 (119878

0 0) 120597119868)

120583 + 120574 + 120572

119877

0= 119877

0minus

120590

2(120597119891 (119878

0 0) 120597119868)

2

2 (120583 + 120574 + 120572)

(8)

We have that 119877

0is the basic reproduction number of

deterministic model (1) On the extinction of the disease inprobability for model (2) we have the following result

Theorem5 Assume that one of the following conditions holds

(a) 1205902 le 120573(120597119891(1198780 0)120597119868) and 1198770lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)

Then disease 119868 in model (2) is extinct with probability oneThatis for any initial value (119878(0) 119868(0)) isin 1198772

+ solution (119878(119905) 119868(119905)) of

model (2) has lim119905rarrinfin

119868(119905) = 0 as

Proof By Lemma 4 we have (119878(119905) 119868(119905)) isin 1198772+as for all 119905 ge 0

and lim sup119905rarrinfin

(119878(119905)+119868(119905)) le 119878

0 For any 120578 gt 0 there is119879

0gt 0

such that 119878(119905) + 119868(119905) lt 119878

0+ 120578 for all 119905 ge 119879

0 Hence for any

119905 ge 119879

0

119891 (119878 (119905) 119868 (119905))

119868 (119905)

isin (0

120597119891 (119878

0+ 120578 0)

120597119868

] (9)

With Itorsquos formula (see [17 18]) we have

119889 log 119868 (119905) = [120573119891 (119878 (119905) 119868 (119905))

119868 (119905)

minus (120583 + 120574 + 120572)

minus

120590

2

2

(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

)

2

]119889119905 + 120590

sdot

119891 (119878 (119905) 119868 (119905))

119868 (119905)

119889119861 (119905)

(10)

Hence for any 120576 gt 0

log 119868 (119905)119905

le

log 119868 (0)119905

+

120573 + 120576

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

minus (120583 + 120574 + 120572)

minus

120590

2

2

1

119905

int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119904

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(11)

Define a function

119892 (119906) = (120573 + 120576) 119906 minus

120590

2

2

119906

2minus (120583 + 120574 + 120572)

(12)


When 120590 = 0 119892(119906) is monotone increasing for 119906 isin 119877

+ and

when 120590 gt 0 119892(119906) is monotone increasing for 119906 isin [0 (120573 +

120576)120590

2) and monotone decreasing for 119906 isin [(120573 + 120576)1205902infin)

If condition (a) holds then when 120590 = 0 from (9) wedirectly have

119892(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

) le 119892(

120597119891 (119878

0+ 120578 0)

120597119868

) forall119905 ge 119879

0

(13)

When 120590 gt 0 since 120597119891(1198780 0)120597119868 le 1205731205902 we can choose 120578 gt 0such that 120578 le 120576 and 120597119891(1198780 + 120578 0)120597119868 lt (120573 + 120576)120590

2 From (9)we also have inequality (13) Hence when 119905 ge 119879

0

log 119868 (119905)119905

le

log 119868 (0)119905

+

1

119905

int

119905

0

119892(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

) 119889119904

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le

log 119868 (0)119905

+

1

119905

int

1198790

0

119892(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

) 119889119904

+

1

119905

119892(

120597119891 (119878

0+ 120578 0)

120597119868

) (119905 minus 119879

0)

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(14)

By the large number theorem for martingales (see [17] orLemma A1 given in [9]) we obtain


log 119868 (119905)119905

le 119892(

120597119891 (119878

0+ 120578 0)

120597119868

) as (15)

From the arbitrariness of 120576 and 120578 we further obtain


log 119868 (119905)119905

le 120573

120597119891 (119878

0 0)

120597119868

minus

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus (120583 + 120574 + 120572)

= (120583 + 120574 + 120572) (

119877

0minus 1) lt 0 as

(16)

If condition (b) holds then since 120590 gt 0 119892(119906) hasmaximum value (120573 + 120576)221205902 minus (120583 + 120574 + 120572) at 119906 = (120573 + 120576)1205902and for any 119905 ge 0 we have

120573119892(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

) le

(120573 + 120576)

2

2120590

2minus (120583 + 120574 + 120572)

(17)

which implies

log 119868 (119905)119905

le

log 119868 (0)119905

+

(120573 + 120576)

2

2120590

2minus (120583 + 120574 + 120572)

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(18)

With the large number theorem formartingales and arbitrari-ness of 120576 we obtain


log 119868 (119905)119905

le

120573

2

2120590

2minus (120583 + 120574 + 120572) lt 0 as (19)

From (16) and (19) we finally have lim119905rarrinfin

119868(119905) = 0 as Thiscompletes the proof

Now we give a further discussion for conditions (a) and(b) of Theorem 5 by using the intensity 120590 of stochastic per-turbation and basic reproduction number119877

0of deterministic

model (1)When 119877

0le 1 then for any 120590 gt 0 119877

0lt 1 and it is easy

to prove that one of the conditions (a) and (b) of Theorem 5holdsTherefore for any 120590 gt 0 the conclusions ofTheorem 5hold Let 1 lt 119877

0le 2 From

119877

0= 1 we have

120590 ≜ 120590 =

radic2 (120583 + 120574 + 120572) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

(20)

Denote

120590

1=

120573

radic2 (120583 + 120574 + 120572)

120590

2= radic

120573

120597119891 (119878

0 0) 120597119868

(21)

Since 1205901le 120590

2 we easily prove that when 120590 gt 120590 one of the

conditions (a) and (b) ofTheorem 5 holds Therefore for any120590 gt 120590 the conclusions of Theorem 5 hold When 119877

0gt 2

we have 1205901gt 120590

2and 120590

1ge 120590 ge 120590

2 Hence condition (a) in

Theorem 5 does not hold We only can obtain that for any120590 gt 120590

1the conclusions of Theorem 5 hold Summarizing the

above discussions we have the following result as a corollaryof Theorem 5

Corollary 6 Assume that one of the following conditionsholds

(a) 1198770le 1 and 120590 gt 0

(b) 1 lt 1198770le 2 and 120590 gt 120590

(c) 1198770gt 2 and 120590 gt 120590

1

Then disease 119868 in model (2) is extinct with probability one

Corollary 7 Let 119891(119878 119868) = 119878119868119873 (standard incidence)Assume that one of the following conditions holds

(a) 1205902 le 120573 and 1198770= 120573(120583 + 120574 + 120572) minus 120590

22(120583 + 120574 + 120572) lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)



Corollary 8 Let 119891(119878 119868) = ℎ(119878)119892(119868) Assume that (Hlowast) holdsand one of the following conditions holds

(a) 1205902 le 120573ℎ(1198780)1198921015840(0) and 1198770= 120573ℎ(119878

0)119892

1015840(0)(120583+120574+120572)minus

120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 + 120574 + 120572) lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)


Remark 9 It is easy to see that in Theorem 5 the conditions119877

0gt 2 and 120590 le 120590 le 120590

1are not included Therefore

an interesting conjecture for model (2) is proposed that isif the above condition holds then the disease still dies outwith probability one In Section 6 we will give an affirmativeanswer by using the numerical simulations see Example 1

Remark 10 In the above discussions we see that case 1198770=

1 has not been considered An interesting open problemis whether when

119877

0= 1 the disease in model (2) also is

extinct with probability one A numerical example is givenin Section 6 see Example 2

4 Permanence of the Disease

On the permanence of the disease in the mean with probabil-ity one for model (2) we establish the following results

Theorem 11 If 1198770

gt 1 then disease 119868 in model (2) ispermanent in the mean with probability one That is there isa constant119898

119868gt 0 such that for any initial value (119878(0) 119868(0)) isin

119877

2

+ solution (119878(119905) 119868(119905)) of model (2) satisfies

lim inf119905rarrinfin

1

119905

int

119905

0

119868 (119904) 119889119904 ge 119898

119868119886119904 (22)

Proof From

119877

0gt 1 we choose a small enough constant 120576 gt 0

such that

120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572) minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

gt 0

(23)

By Lemma 4 it is clear that for any initial value(119878(0) 119868(0)) isin 119877

2



(1119905) int

119905

0119868(119904)119889119904 le 119878

0 and for above 120576 gt 0 there is119879

0gt 0 such that 119878

0minus 120576 le 119878(119905) + 119868(119905) le 119878

0+ 120576 as for all 119905 ge 119879

0

Denote the set 119863120576= (119878 119868) 119878

0minus 120576 le 119878 + 119868 le 119878

0+ 120576 Since

119889119873(119905) = (Λ minus 120583119873(119905) minus 120572119868(119905))119889119905 we obtain for any 119905 gt 1198790

int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904 = minus

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+

119873 (119879

0) minus 119873 (119905)

120583

(24)

From (10) for any 119905 ge 1198790

log 119868 (119905) = log 119868 (0) + 120573int119905

0

[

120597119891 (119878

0 0)

120597119868

+

119891 (119878 (119904) 119868 (119904))

119868 (119904)

minus

120597119891 (119878

0 0)

120597119868

] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119904

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(25)

Since 119891(119878 119868)119868 for 119878 gt 0 and 119868 gt 0 is continuously differen-tiable lim

119868rarr0(119891(119878 119868)119868) = 120597119891(119878 0)120597119868 exists for any 119878 gt 0

and set 119863120576is convex and connected by the Lagrange mean

value theorem when 119905 ge 1198790we have

119891 (119878 (119905) 119868 (119905))

119868 (119905)

minus

120597119891 (119878

0 0)

120597119868

= (

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868

minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119905)

+

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119905) minus 119878

0)

(26)

where (120585(119905) 120601(119905)) isin 119863120576 Let constants

119872

1= max(119878119868)isin119863120576

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119872

2= max(119878119868)isin119863120576

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119878

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(27)

From Lemma 3 we have 0 lt 11987211198722lt infin For any 119905 ge 119879

0 we

have

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868

minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

ge minus119872

1as

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119878

le 119872

2as

(28)

From (25) and Remark 1 we further have

log 119868 (119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

+ 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

+ 120573int

119905

1198790

[(

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868


minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119904) +

1

120601 (119905)

sdot

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119904) minus 119878

0)] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) ge log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 + 120573119872

2int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904

minus (120583 + 120574 + 120572) 119905 minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119905 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 minus 1205731198722

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879

0) minus 119873 (119905)) minus (120583 + 120574 + 120572) 119905 minus

1

2

sdot 120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = 119867 (119905) + 120579119905

minus 120579

0int

119905

0

119878 (119904) 119889119904

(29)

where

119867(119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

minus 120573

120597119891 (119878

0 0)

120597119868

119879

0

+ 120573(119872

1+119872

2

120583 + 120572

120583

)int

1198790

0

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879

0) minus 119873 (119905))

+ 120590int

119905

0

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

120579 = 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572)

minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

120579

0= 120573(119872

1+119872

2

120583 + 120572

120583

)

(30)

By the large number theorem for martingales and Lemma 4lim119905rarrinfin

(119867(119905)119905) = 0 as Therefore from Lemma 52 given in[16] we finally obtain lim inf

119905rarrinfin(1119905) int

119905

0119868(119904)119889119904 ge 120579120579

0as

This completes the proof

Remark 12 From (20) we have that 1198770gt 1 is equivalent to

120590 lt 120590 Therefore Theorem 11 also can be rewritten by usingintensity 120590 of stochastic perturbation in the following form if120590 lt 120590 then disease 119868 in model (2) is permanent in the meanwith probability one

Remark 13 Combining Corollary 6 and Remark 12 we canobtain that when 1 lt 119877

0le 2 number 120590 is a threshold value

When 0 lt 120590 lt 120590 the disease 119868 in model (2) is permanentin the mean and when 120590 gt 120590 the disease 119868 is extinct withprobability one However when 119877

0gt 2 then the alike results

are not established Therefore it yet is an interesting openproblem

Theorem 14 Susceptible 119878 in model (2) also is permanent inthe mean with probability oneThat is there is a constant119898

119878gt

0 such that for any initial value (119878(0) 119868(0)) isin 119877

2

+ solution

(119878(119905) 119868(119905)) of model (2) satisfies


1

119905

int

119905

0

119878 (119904) 119889119904 ge 119898

119878119886119904 (31)

Proof By Lemma 4 we easily see that for any initial value(119878(0) 119868(0)) isin 119877

2



(1119905) int

119905

0119878(119904)119889119904 le 119878

0 and for any small enoughconstant 120576 gt 0 there is 119879


0minus 120576 le 119878(119905) + 119868(119905) le

119878

0+120576 for all 119905 ge 119879

0 Hence by Lemma 3 when 119905 ge 119879

0we have

119891(119878(119905) 119868(119905)) le 119872

119878119878(119905) where119872

119878= max

119863120576119891(119878 119868)119878 lt infin

Integrating the first equation of model (2) we obtain for any119905 ge 119879

0

119878 (119905) minus 119878 (0)

119905

= Λ minus

1

119905

int

119905

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904) minus 120574119868 (119904)] 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

ge Λ minus

1

119905

int

1198790

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904)] 119889119904

minus

1

119905

int

119905

1198790

[120573119872

119878+ 120583] 119878 (119904) 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

(32)


Therefore with the large number theorem formartingales wefinally have


1

119905

int

119905

0

119878 (119904) 119889119904 ge

Λ

120573119872

119878+ 120583

(33)


As consequences of Theorems 11 and 14 we have thefollowing corollaries

Corollary 15 Let 119891(119878 119868) = 119878119868119873 (standard incidence) If

119877

0= (120573minus(12)120590

2)(120583+120574+120572) gt 1 thenmodel (2) is permanent

in the mean with probability one

Corollary 16 Let 119891(119878 119868) = ℎ(119878)119892(119868) Assume that (Hlowast) holdsand

119877

0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574 + 120572) minus 120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 +

120574 + 120572) gt 1 then model (2) is permanent in the mean withprobability one

We further have the result on the weak permanence ofmodel (2) in probability

Corollary 17 Assume that 1198770gt 1 Then there is a constant

120585 gt 0 such that for any initial value (119878(0) 119868(0)) isin 1198772+ solution



119868 (119905) ge 120585


119878 (119905) ge 120585

as

(34)

Now we discuss special case 120572 = 0 for model (2)that is there is not disease-related death in model (2) Wecan establish the following more precise results on the weakpermanence of the disease in probability compared to theconclusion given in Corollary 17

Theorem 18 Let 120572 = 0 in model (2) If 1198770gt 1 then for any

initial value (119878(0) 119868(0)) isin 1198772+ solution (119878(119905) 119868(119905)) of model (2)

satisfies


119868 (119905) ge 120585 119886119904 (35)


119868 (119905) le 120585 119886119904 (36)

where 120585 gt 0 satisfies the equation

119891 (119878

0minus 120585 120585)

120585

=

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(37)

Proof FromLemma4we know that lim119905rarrinfin

(119878(119905)+119868(119905)) = 119878

0Without loss of generality we assume that 119878(119905) + 119868(119905) equiv 1198780 forall 119905 ge 0 From (10) for any 119905 ge 0

log 119868 (119905) = log 119868 (0) + int119905

0

[

[

120573

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

minus (120583 + 120574) minus

120590

2

2

(

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

)

2

]

]

119889119904

+ int

119905

0

120590

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

(38)

Define a function 119906(119868) = 119891(1198780 minus 119868 119868)119868 Then for any 119905 ge 0

log 119868 (119905) = log 119868 (0) + int119905

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(39)

where function119892(119906) = 120573119906minus(12059022)1199062minus(120583+120574)With condition

119877

0gt 1 we have 119892(0) = minus(120583 + 120574) lt 0 and

119892(

120597119891 (119878

0 0)

120597119868

) = minus

120590

2

2

(

120597119891 (119878

0 0)

120597119868

)

2

+ 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574) gt 0

(40)

Hence 119892(119906) = 0 has a positive root 120578 in (0 120597119891(119878

0 0)120597119868)

which is

120578 =

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(41)

Since 119906(119868) is monotone decreasing for 119868 isin (0 1198780) 119906(1198780) = 0and

lim119868rarr0+

119906 (119868) = lim119868rarr0+

119891 (119878

0minus 119868 119868)

119868

=

120597119891 (119878

0 0)

120597119868

(42)

there is a unique 120585 isin (0 1198780) such that 119906(120585) = 119891(1198780minus120585 120585)120585 = 120578and 119892(119906(120585)) = 119892(120578) = 0

When 120590 gt 0 and 1205731205902 lt 120597119891(1198780 0)120597119868 since function 119892(119906)has maximum value 119892(1205731205902) at 119906 = 120573120590

2 and 119892(120573120590

2) gt

119892(120597119891(119878

0 0)120597119868) there is a unique 119868 such that 119906(119868) = 120573120590

2From 120578 isin (0 120597119891(119878

0 0)120597119868) and 119892(120578) = 0 we have 120578 lt 120573120590

2Hence 0 lt 119868 lt 120585 lt 1198780

From the above discussion we obtain that 119892(119906(119868)) gt 0

is strictly increasing on 119868 isin (0

119868) 119892(119906(119868)) gt 0 is strictlydecreasing on 119868 isin (119868 120585) and 119892(119906(119868)) lt 0 is strictly decreasingon 119868 isin (120585 1198780)

When 1205902 le 120573(120597119891(119878

0 0)120597119868) similarly to the above dis-

cussion we can obtain that 119892(119906(119868)) gt 0 is strictly decreasing


on 119868 isin (0 120585) and 119892(119906(119868)) lt 0 is strictly decreasing on 119868 isin

(120585 119878

0)

Now we firstly prove that (35) is true If it is not true thenthere is an enough small 120576

0isin (0 1) such that 119875(Ω

1) gt 120576

0

where Ω1= lim sup

119905rarrinfin119868(119905) lt 120585 Hence for every 120596 isin Ω

1

there is a constant 1198791= 119879

1(120596) ge 119879

0such that

119868 (119905) le 120585 minus 120576

0forall119905 ge 119879

1 (43)

With the above discussion we know that 119892(119906(119868(119905))) ge 119892(119906(120585minus120576

0)) gt 0 for all 119905 ge 119879

1 From (39) we further obtain for any

119905 ge 119879

1

log 119868 (119905) ge log 119868 (0) + int1198791

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 minus 120576

0)) (119905 minus 119879

1)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(44)

From the large number theorem for martingales we havelim inf

119905rarrinfin(log 119868(119905)119905) le 119892(119906(120585 minus 120576

0)) gt 0 which implies

119868(119905) rarr infin as 119905 rarr infin This leads to a contradiction with (43)Next we prove that (36) holds If it is not true then there

is an enough small 1205761isin (0 1) such that 119875(Ω

2) gt 120576

1 where

Ω

2= lim inf

119905rarrinfin119868(119905) gt 120585 Hence for every 120596 isin Ω

2 there is

119879

2= 119879

2(120596) ge 119879

0such that

119868 (119905) ge 120585 + 120576

1forall119905 ge 119879

2 (45)

With the above discussionwe have119892(119906(119868(119905))) le 119892(119906(120585+1205761)) lt

0 for all 119905 ge 1198792 Together with (39) we further obtain for any

119905 ge 119879

2

log 119868 (119905) = log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

1198792

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 + 120576

1)) (119905 minus 119879

2)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(46)

With the large number theorem for martingales we havelim sup

119905rarrinfin(log 119868(119905)119905) le 119892(119906(120585 + 120576

1)) lt 0 which implies

119868(119905) rarr 0 as 119905 rarr infin This leads to a contradiction with (45)This completes the proof

Remark 19 Theorem 18 indicates that if 1198770gt 1 and 120572 =

0 then any solution (119878(119905) 119868(119905)) of model (2) with initialvalue (119878(0) 119868(0)) isin 119877

2

+oscillates about a positive number

120585 Therefore an interesting open problem is whether there is

a more less positive 119898 than number 120585 such that any solution(119878(119905) 119868(119905)) of model (2) with initial value (119878(0) 119868(0)) isin 119877

2

+


119868(119905) ge 119898 as In Section 6 we will give anaffirmative answer by using the numerical simulations seeExample 3

From Theorem 18 we easily see that number 120585 willarise from the change when the noise intensity 120590 changesTherefore it is very interesting and important to discuss hownumber 120585 changes along with the change of 120590 We have thefollowing result

Theorem 20 Assume that 120572 = 0 in model (2) and

119877

0gt 1 Let number 120585 be given in Theorem 18 and 119877

0=

120573(120597119891(119878

0 0)120597119868)(120583 + 120574) Then one has the following

(a) 120585 as the function of 120590 is defined for

0 lt 120590 lt

radic2 (120583 + 120574) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

fl

(47)

(b) 120585 is monotone decreasing for 120590 isin (0 )(c) lim

120590rarr0120585 = 119868

lowast where (119878lowast 119868lowast) is the endemic equilib-rium of deterministic model (1)

(d) If 1 le 119877

0le 2 then lim

120590rarr120585 = 0 and if 119877

0gt 2 then

lim120590rarr

120585 = 120585

2 where 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(48)

Proof Since

119891 (119878

0minus 120585 120585)

120585

= 120578

(49)

by the inverse function theorem we obtain that 120585 as thefunction of 120578 is defined for 120578 isin (0 120597119891(1198780 0)120597119868) From

120578 =

120573 minusradic120573

2minus 2120590

2(120583 + 120574)

120590

2

(50)

we can obtain that 120578 isin (0 120597119891(119878

0 0)120597119868) when 0 lt 120590 lt

Therefore 120585 as a function of 120590 is defined for 0 lt 120590 lt Computing the derivative of 120578 with respect to 120590 we have

119889120578

119889120590

=

minus2120573

120590

3+

2 (120583 + 120574)

120590radic120573

2minus 2120590

2(120583 + 120574)

+

2radic120573

2minus 2120590

2(120583 + 120574)

120590

3

=

2120573

2minus 2120590

2(120583 + 120574) minus 2120573

radic120573

2minus 2120590

2(120583 + 120574)

120590

3radic120573

2minus 2120590

2(120583 + 120574)

(51)


Since

[2120573

2minus 2120590

2(120583 + 120574)]

2

minus (2120573radic120573

2minus 2120590

2(120583 + 120574))

2

= 4120590

4(120583 + 120574)

2gt 0

(52)

we have 119889120578119889120590 gt 0 From the definition of 120585 we easilysee that 120585 is monotone decreasing for 120578 From (49) and (H)we obtain that 119889120585119889120578 exists and is continuous for 120578 Since(120597120597120585)(119891(119878

0minus 120585 120585)120585) lt 0 we have 119889120585119889120578 lt 0 Therefore

119889120585119889120590 = (119889120585119889120578)(119889120578119889120590) lt 0 It follows that 120585 is monotone

decreasing as 120590 increases Thus both lim120590rarr0

120585 and lim120590rarr

120585

exist Let lim120590rarr0

120585 = 120585

1and lim

120590rarr120585 = 120585

2 We have

lim120590rarr0

120578 = lim120590rarr0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

=

120583 + 120574

120573

(53)

Hence lim120590rarr0

(119891(119878

0minus 120585 120585)120585) = lim

120590rarr0120578 = (120583 + 120574)120573 This

shows that 119891(1198780 minus 1205851 120585

1)120585

1= (120583 + 120574)120573 Let (119878lowast 119868lowast) be the

endemic equilibriumof deterministicmodel (1) thenwe have119891(119878

0minus119868

lowast 119868

lowast)119868

lowast= (120583+120574)120573 Hence 120585

1= 119868

lowast This shows thatlim120590rarr0

120585 = 119868

lowastOn the other hand we have

lim120590rarr

120578 =


2minus 2

2(120583 + 120574)

2=

(120597119891 (119878

0 0) 120597119868) (120573 (120597119891 (119878

0 0) 120597119868) minus

1003816

1003816

1003816

1003816

1003816

120573 (120597119891 (119878

0 0) 120597119868) minus 2 (120583 + 120574)

1003816

1003816

1003816

1003816

1003816

)

2 (120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574))

(54)

If 1 le 119877

0le 2 then from (54) we obtain lim

120590rarr120578 =

120597119891(119878

0 0)120597119868 Hence

lim120590rarr

119891 (S0 minus 120585 120585)120585

=

120597119891 (119878

0 0)

120597119868

(55)

This shows that lim120590rarr

120585 = 0 If 1198770gt 2 then we have from

(54)

lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(56)

which implies

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(57)

Therefore we have lim120590rarr

120585 = 120585

2 and 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

(119877

0minus 1)

(58)


Conclusion (b) of Theorem 20 shows that when 120572 = 0

in model (2) number 120585 monotonically decreases when 120590

increases in (0 ) and when 120590 = 0 120585 has a maximum value119868

lowast by Conclusion (c) Therefore 0 lt 120585 lt 119868

lowast when 120590 gt 0 If1 le 119877

0le 2 then when 120590 = 120585 has a minimum value 0 and

if 1198770gt 2 then when 120590 = 120585 has a minimum value 120585

2gt 0 by

Conclusion (d)It is clear that when in model (2) 120572 = 0 then = 120590 from

(20) On the other hand from Conclusion (c) of Corollary 7we see that if 119877

0gt 2 then when 120590 gt 120590

1 where 120590

1is given in

(21) we have lim119905rarrinfin

119868(119905) = 0 as for any solution (119878(119905) 119868(119905))

ofmodel (2)with initial value (119878(0) 119868(0)) isin 1198772+ which implies

that 120585 = 0 Therefore when 119877

0gt 2 we can propose an

interesting open problem whether there is a critical value120590

lowastisin ( 120590

1) such that when 120590 isin (0 120590lowast) we have the fact that

120585 is monotonically decreasing and 120585 gt 0 and when 120590 gt 120590lowast wehave 120585 = 0

Remark 21 When 1198770gt 2 then from (56) we obtain

lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

(59)

namely

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

=

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(60)

where (119878lowast 119868lowast) is the endemic equilibrium of deterministicmodel (1) Hence

119891 (119878

0minus 120585

2 120585

2)

120585

2

gt

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(61)

Consequently 0 lt 1205852lt 119868

lowast

Remark 22 When 119891(119878 119868) = 119878119868 we easily validate thatTheorems 20 and 24 degenerate into Theorems 51 and 54which are given in [19] respectively Therefore Theorems 18and 20 are the considerable extension ofTheorems 51 and 54in general nonlinear incidence cases respectively

Remark 23 For the case 120572 gt 0 in model (2) an interestingand important open problem is when

119877

0gt 1 whether we

also can establish similar results as Theorems 18 and 20Furthermore as an improvement of the results obtained in


Corollary 17 we also propose another open problem onlywhen

119877

0gt 1 we also can establish the permanence of the

disease with probability one that is there is a constant119898 gt 0

such that for any solution (119878(119905) 119868(119905)) of model (2) with initialvalue (119878(0) 119868(0)) isin 119877

2

+ one has lim

119905rarrinfin119868(119905) ge 119898 as In

Section 6 we will give an affirmative answer by using thenumerical simulations see Example 3

5 Stationary Distribution

FromTheorems 11 and 14 we obtain that when 1198770gt 1model

(2) is permanent in the mean with probability one Howeverwhen 119877

0gt 1model (2) also has a stationary distribution We

have an affirmative answer as follows

Theorem 24 If 1198770gt 1 then model (2) is positive recurrent

and has a unique stationary distribution

Proof Here the method given in the proof ofTheorem 51 in[17] is improved and developed By Lemma 4 and Remark 9we only need to give the proof in region Γ where Γ = (119878 119868) 119878 ge 0 119868 ge 0 119878

0le 119878 + 119868 le 119878

0 Let (119878(119905) 119868(119905)) be any solution

of model (1) with (119878(0) 119868(0)) isin Γ as for all 119905 ge 0 Let 119886 gt 0

be a large enough constant and let

119863 = (119878 119868) isin Γ

1

119886

lt 119878 lt 119878

0minus

1

119886

1

119886

lt 119868 lt 119878

0minus

1

119886

(62)

When (119878 119868) isin Γ 119863 then either 0 lt 119878 lt 1119886 or 0 lt 119868 lt 1119886The diffusion matrix for model (56) is

119860 (119878 119868) = (

120590

2119891

2(119878 119868) minus120590

2119891

2(119878 119868)

minus120590

2119891

2(119878 119868) 120590

2119891

2(119878 119868)

) (63)

For any (119878 119868) isin 119863 we have 12059021198912(119878 119868) ge 120590

2(119891(1119886 119878

0minus

1119886)(119886119878

0minus 1))

2Choose a Lyapunov function as follows

119881 (119878 119868) = Ψ

1(119868) + Ψ

2(119878 119868) + Ψ

3(119878) (64)

where

Ψ

1(119868) =

1

V119868

minusV

Ψ

2(119878 119868) =

1

V119868

minusV(119878

0minus 119878)

Ψ

3(119878) =

1

119878

(65)

and 0 lt V lt 1 is a constant Computing 119871Ψ1 by Remark 1 we

have

119871Ψ

1= minus119868

minus(V+1)(120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119868

minus(V+2)119891

2(119878 119868) le 119868

minusV(120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

)

+ 119868

minusV120573(

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

)

(66)

Applying the Lagrange mean value theorem we have

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

=

1

120601

120597119891 (120585 120601)

120597119878

(119878

0minus 119878)

+ (

119891 (120585 120601)

120601

2minus

1

120601

120597119891 (120585 120601)

120597119868

) 119868

le 119872

1(119878

0minus 119878) +119872

2119868 +119872

3119877

(67)

where (120585 120601) isin Γ and

119872

1= max(119878119868)isinΓ

1

119868

120597119891 (119878 119868)

120597119878

119872

2= max(119878119868)isinΓ

119891 (119878 119868)

119868

2minus

1

119868

120597119891 (119878 119868)

120597119868

(68)

By Lemma 3 we have 0 le 11987211198722lt infin We hence have

119871Ψ

1le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 120573119872

1(119878

0minus 119878) 119868

minusV+ 120573119872

2119868

1minusV

(69)

Computing 119871Ψ2 by Remark 1 we have

119871Ψ

2= minus

1

V119868

minusV(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) minus 119868

minus(V+1)(119878

0

minus 119878) (120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119891

2(119878 119868) 119868

minus(V+2)(119878

0minus 119878) minus

1

2

119868

minus(V+1)120590

2119891

2(119878 119868)

= minus

1

V119868

minusV(120583 (119878

0minus 119878) minus 120573119891 (119878 119868) + 120574119868)

minus 119868

minusV(119878

0minus 119878) (120573

119891 (119878 119868)

119868

minus (120583 + 120572 + 120574)) +

1

2

(1 + V)

sdot 120590

2(

119891 (119878 119868)

119868

)

2

119868

minusV(119878

0minus 119878) minus 120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusV


= 119868

minusV(119878

0minus 119878)(minus

120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)

Computing 119871Ψ3 we have

119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)

where by Lemma 3 1198720= max

Γ119891(119878 119868)119878 lt infin From the

above calculations we obtain that for any (119878 119868) isin Γ 119863

119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)

and when V gt 0 is small enough it follows that

120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)

we finally obtain that when 119886 gt 0 is large enough

119871119881 lt minus1 as forall (119878 119868) isin Γ 119863 (75)

FromTheorem 22 given in [10] we know that model (2) hasa unique stationary distribution 120585 such that

119875 lim119879rarrinfin

1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)


Remark 25 ComparingTheorem 24 withTheorem 62 givenin [19] we see thatTheorem 62 is extended and improved tothe general stochastic SIS epidemic model (2)

Remark 26 Since 1198770gt 1 is equivalent to 120590 lt 120590 we also have

that if 120590 lt 120590 then model (2) is positive recurrent and has aunique stationary distribution

Particularly for some special cases of nonlinear incidence119891(119878 119868) we have the following idiographic results on thestationary distribution as the consequences of Theorem 24


119877

0= (120573 minus (12)120590

2)(120583 + 120574 + 120572) gt 1 then model (2) is positive

recurrent and has a unique stationary distribution

Corollary 28 Let119891(119878 119868) = ℎ(119878)119892(119868) Assume that (Hlowast) holdsand 119877

0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+

120572) gt 1 then model (2) is positive recurrent and has a uniquestationary distribution

Combining Corollary 6 Theorem 11 Remark 12 Theo-rem 24 and Remark 26 we can finally establish the followingsummarization result by using intensity 120590 of stochastic per-turbation and basic reproduction number119877

0of deterministic

model (1)

Corollary 29 (a) Let 1198770le 1 Then for any 120590 gt 0 the disease

in model (2) is extinct with probability one(b) Let 1 lt 119877

0le 2 Then for any 0 lt 120590 lt 120590 model (2) is

permanent in the mean with probability one and has a uniquestationary distribution and for any 120590 gt 120590 the disease in model(2) is extinct with probability one


0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)

StochasticDeterministic

(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)

Figure 1 (a) is trajectories of the solution 119868(119905) with the initial value 119868(0) = 05 and (b) with the initial value 119868(0) = 006

(c) Let 1198770gt 2 Then for any 0 lt 120590 lt 120590 model (2) is

permanent in the mean with probability one and has a uniquestationary distribution and for any 120590 gt 120590

1 where 120590

1is given

in (20) the disease in model (2) is extinct with probability one

6 Numerical Simulations

In this section we analyze the stochastic behavior of model(2) by means of the numerical simulations in order to makereaders understand our results more better The numericalsimulation method can be found in [19] Throughout thefollowing numerical simulations we choose119891(119878 119868) = 119878119868(1+120596119868) where 120596 gt 0 is a constant The correspondingdiscretization system of model (2) is given as follows

119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)

where 120585119896(119896 = 1 2 ) are the Gaussian random variables

which follow standard normal distribution119873(0 1)

Example 1 In model (2) we choose Λ = 2000 120573 = 060 120583 =11 120574 = 13 120590 = 0075 and 120572 = 2

By computing we have 1198770= 4195 gt 2 119877

0= 06715 lt 1

120573119878

0minus 120590

2= minus00023 lt 0 and 1205902 minus 12057322(120583 + 120574) = minus00019 lt

0 which is the case of Remark 9 From the numerical

simulations we see that the disease will die out (see Figure 1)An affirmative answer is given for the open problemproposedin Remark 9

Example 2 In model (2) choose Λ = 2000 120573 = 09 120583 = 30120574 = 12 and 120590 = 009

By computing we have

119877

0= 1 From the numerical

simulations given in Figure 2 we know that the disease willdie outTherefore an affirmative answer is given for the openproblem proposed in Remark 10

Example 3 In model (2) choose Λ = 2000 120573 = 05 120583 = 30120574 = 20 120590 = 002 and 120572 = 2

We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037

The numerical simulations are found in Figure 3 We cansee that solution 119868(119905) of model (2) oscillates up and down at120585 which further show that the conclusions of Theorems 14and 18 are true At the same time this example also showsthat the disease in model (2) is permanent with probabilityone Therefore an affirmative answer is given for the openproblems proposed in Remarks 19 and 23

7 Discussion

In this paper we investigated a class of stochastic SIS epidemicmodels with nonlinear incidence rate which include thestandard incidence Beddington-DeAngelis incidence andnonlinear incidence ℎ(119878)119892(119868) A series of criteria in the prob-ability mean on the extinction of the disease the persistenceand permanence in themean of the disease and the existenceof the stationary distribution are established Furthermorethe numerical examples are carried out to illustrate theproposed open problems in this paper


Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)

DeterministicStochastic

(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)

StochasticDeterministic120585

(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)


It is easily seen that the research given in [6] for thestochastic SIS epidemic model with bilinear incidence isextended to the model with general nonlinear incidence anddisease-inducedmortality Particularly we see that stochasticSIS epidemic model with standard incidence is investigatedfor the first time

The researches given in this paper show that stochasticmodel (2) has more rich dynamical properties than thecorresponding deterministic model (1) Particularly stochas-tic model (2) has no endemic equilibrium Thus this canbring more difficulty for us to investigate model (2) but on

the other hand this also makes model (2) have more richresearchful subjects than deterministic model (1) We candiscuss not only the extinction persistence and permanencein the mean of disease in probability but also the existenceand uniqueness of stationary distribution the asymptoticalbehaviors of solutions of stochastic model (2) around theequilibrium of deterministic model (1) and so forth

In addition we easily see that when intensity 120590 gt 0 ofthe stochastic perturbation then 119877

0gt

119877

0 This shows that

when 119877

0gt 1 we still can have 119877

0lt 1 Therefore there is

a very interesting and important phenomenon that is for


deterministic model (1) the disease is permanent but for thecorresponding stochasticmodel (2) the disease is extinct withprobability one see Conclusion (c) of Corollary 29

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the Doctorial Subjects Foun-dation of The Ministry of Education of China (Grant no2013651110001) and the National Natural Science Foundationof China (Grants nos 11271312 11401512 and 11261056)

References

[1] E Beretta V Kolmanovskii and L Shaikhet ldquoStability of epi-demic model with time delays influenced by stochastic pertur-bationsrdquoMathematics and Computers in Simulation vol 45 no3-4 pp 269ndash277 1998

[2] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[3] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A Statistical Mechanics and ItsApplications vol 354 pp 111ndash126 2005

[4] N Dalal D Greenhalgh and X Mao ldquoA stochastic model forinternal HIV dynamicsrdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 1084ndash1101 2008

[5] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[6] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011

[7] Q Yang D Jiang N Shi and C Ji ldquoThe ergodicity and extin-ction of stochastically perturbed SIR and SEIR epidemicmodelswith saturated incidencerdquo Journal of Mathematical Analysis andApplications vol 388 no 1 pp 248ndash271 2012

[8] A Lahrouz L Omari and D Kioach ldquoGlobal analysis of adeterministic and stochastic nonlinear SIRS epidemic modelrdquoNonlinear Analysis Modelling and Control vol 16 no 1 pp 59ndash76 2011

[9] Y Zhao D Jiang and D OrsquoRegan ldquoThe extinction and persis-tence of the stochastic SIS epidemic model with vaccinationrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 4916ndash4927 2013

[10] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computation vol 233pp 10ndash19 2014

[11] A Lahrouz and L Omari ldquoExtinction and stationary distri-bution of a stochastic SIRS epidemic model with non-linearincidencerdquo StatisticsampProbability Letters vol 83 no 4 pp 960ndash968 2013

[12] Y Cai Y Kang M Banerjee and W Wang ldquoA stochastic SIRSepidemic model with infectious force under intervention stra-tegiesrdquo Journal of Differential Equations vol 259 no 12 pp7463ndash7502 2015

[13] Q Yang and X Mao ldquoStochastic dynamics of SIRS epidemicmodels with random perturbationrdquo Mathematical Biosciencesand Engineering vol 11 no 4 pp 1003ndash1025 2014

[14] A Lahrouz and A Settati ldquoQualitative study of a nonlinearstochastic SIRS epidemic systemrdquo Stochastic Analysis and Appli-cations vol 32 no 6 pp 992ndash1008 2014

[15] F Wang X Wang S Zhang and C Ding ldquoOn pulse vaccinestrategy in a periodic stochastic SIR epidemic modelrdquo ChaosSolitons amp Fractals vol 66 pp 127ndash135 2014

[16] C Ji and D Jiang ldquoThreshold behaviour of a stochastic SIRmodelrdquo Applied Mathematical Modelling vol 38 no 21-22 pp5067ndash5079 2014

[17] X Mao Stochastic Differential Equations and Applications Hor-wood Chichester UK 2nd edition 2008

[18] R Z Hasminskii Stochastic Stability of Differential Equations1980

[19] D J Higham ldquoAn algorithmic introduction to numerical simu-lation of stochastic differential equationsrdquo SIAMReview vol 43no 3 pp 525ndash546 2001

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


(2) For the stochastic epidemic models with the standardincidence up to now we do not find any interestingresearches

(3) The conditions obtained on the existence of uniquestationary distribution are very rigorous Whetherthere is a unique stationary distribution onlywhen themodel is permanent in the mean with probability oneis still an open problem

Motivated by the above work in this paper we considerthe following deterministic SIS epidemic model with nonlin-ear incidence rate and disease-induced mortality

119889119878 (119905)

119889119905

= Λ minus 120573119891 (119878 (119905) 119868 (119905)) + 120574119868 (119905) minus 120583119878 (119905)

119889119868 (119905)

119889119905

= 120573119891 (119878 (119905) 119868 (119905)) minus (120583 + 120574 + 120572) 119868 (119905)

(1)

In model (1) 119878 and 119868 denote the susceptible and infectiousindividualsΛ denotes the recruitment rate of the susceptible120583 is the natural death rate of 119878 and 119868 120572 is the disease-relateddeath rate the transmission of the infection is governedby a nonlinear incidence rate 120573119891(119878 119868) where 120573 denotesthe transmission coefficient between compartments 119878 and 119868119891(119878 119868) is a continuously differentiable function of 119878 and 119868 and120574 denotes the per capita disease contact rate

Now we assume that the random effects of the envi-ronment make the transmission coefficient 120573 of disease indeterministic model (1) generate random disturbance Thatis 120573 rarr 120573 + 120590

119861(119905) where 119861(119905) is a one-dimensional standardBrownian motion defined on some probability space Thusmodel (1) will become into the following stochastic SISepidemic model with nonlinear incidence rate

119889119878 (119905) = [Λ minus 120573119891 (119878 (119905) 119868 (119905)) + 120574119868 (119905) minus 120583119878 (119905)] 119889119905

minus 120590119891 (119878 (119905) 119868 (119905)) 119889119861 (119905)

119889119868 (119905) = [120573119891 (119878 (119905) 119868 (119905)) minus (120583 + 120574 + 120572) 119868 (119905)] 119889119905

+ 120590119891 (119878 (119905) 119868 (119905)) 119889119861 (119905)

(2)

In this paper we investigate the dynamical behaviors ofmodel (2) By using the Lyapunov function method Itorsquosformula and the theory of stochastic analysis [17 18] we willestablish a series of new interesting criteria on the extinctionof the disease permanence in the mean of the model withprobability oneThe stochastic oscillation of the disease abouta positive number in the case where there is not disease-related death is also obtained Further we study the positiverecurrence and the existence of stationary distribution formodel (2) and a new criterion is established Particularlythe effects of the intensities of stochastic perturbation for thedynamical behaviors of the model are discussed in detailFor some special cases of nonlinear incidence 119891(119878 119868) forexample 119891(119878 119868) = 119878119868119873 (standard incidence) and 119891(119878 119868) =ℎ(119878)119892(119868) many idiographic criteria on the extinction per-manence and stationary distribution are established Lastlysome affirmative answers for the open problems which areproposed in this paper also are given by the numerical

examples (the numerical simulation method can be found in[19])

The organization of this paper is as follows In Section 2the preliminaries are given and some useful lemmas areintroduced In Section 3 the sufficient conditions are estab-lished which ensure that the disease dies out with probabilityone In Section 4 we establish the sufficient conditions whichensure that the disease in model (2) is permanent in themean with probability one and when there is not disease-related death the disease oscillates stochastically about apositive number In Section 5 the existence on the uniquestationary distribution of model (2) is proved In Section 6the numerical simulations are carried out to illustrate someopen problems Lastly a brief discussion is given in the endto conclude this work

2 Preliminaries

Denote 1198772+= (119909

1 119909

2) 119909

1gt 0 119909

2gt 0 119877

+0= [0infin) and

119877

+= (0infin) Throughout this paper we assume that model

(2) is defined on a complete probability space (Ω 119865119905

119905ge0 119875)

with a filtration 119865119905

119905ge0satisfying the usual conditions that is

119865

119905

119905ge0is right continuous and 119865

0contains all 119875-null sets

In model (2) 119878 and 119868 denote the susceptible and infectedfractions of the population respectively and 119873 = 119878 + 119868

is the total size of the population among whom the diseaseis spreading the parameters Λ 120583 120573 and 120574 are given asin model (1) the transmission of the infection is governedby a nonlinear incidence rate 120573119878119892(119868) 119861(119905) denotes one-dimensional standard Brownianmotion defined on the aboveprobability space and 120590 represents the intensity of theBrownian motion 119861(119905) Throughout this paper we alwaysassume the following

(H) 119891(119878 119868) is two-order continuously differentiable forany 119878 ge 0 119868 ge 0 and 119878 + 119868 gt 0 For each fixed 119868 ge 0119891(119878 119868) is increasing for 119878 gt 0 and for each fixed 119878 ge 0119891(119878 119868)119868 is decreasing for 119868 gt 0 119891(119878 0) = 119891(0 119868) = 0for any 119878 gt 0 and 119868 gt 0 and 120597119891(1198780 0)120597119868 gt 0 where119878

0= Λ120583

Particularly when 119891(119878 119868) = ℎ(119878)119892(119868) then assumption(H) becomes in the following form

(Hlowast) ℎ(119878) and 119892(119868) are continuously differentiable for 119878 ge 0and 119868 ge 0 ℎ(119878) is increasing for 119878 ge 0 and 119892(119868)119868 isdecreasing for 119868 gt 0

Remark 1 From (H) by simple calculating we can obtainthat for any 119878 gt 0 and 119868 gt 0 0 le 119891(119878 119868) le (120597119891(119878 0)120597119868)119868and for any 119878

2gt 119878

1gt 0 120597119891(119878

2 0)120597119868 ge 120597119891(119878

1 0)120597119868

Remark 2 When119891(119878 119868) = 119878119868119873 (standard incidence) where119873 = 119878+119868119891(119878 119868) = 119878119868(1+120596

1119868+120596

2119878) (Beddington-DeAngelis

incidence) with constants 1205961ge 0 and 120596

2ge 0 and 119891(119878 119868) =

119878119868(1 + 120596119868

2) with constant 120596 ge 0 then (H) is satisfied

Now we give the following result for function 119891(119878 119868)



0 119868 gt 0 119902 le 119878 + 119868 le 119901 Then

max(119878119868)isin119863

119891 (119878 119868)

119878

119891 (119878 119868)

119868

lt infin (3)

max(119878119868)isin119863

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119878

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

lt infin (4)


lim119878rarr0

119891 (119878 119868)

119878

=

120597119891 (0 119868)

120597119878

lim119868rarr0

119891 (119878 119868)

119878

=

120597119891 (119878 0)

120597119868

(5)


119867(119878 119868) =

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2 119868 gt 0

1

2

120597

2119891 (119878 0)

120597119868

2 119868 = 0

(119878 119868) isin 119863

119866 (119878 119868) =

1

119868

120597119891 (119878 119868)

120597119868

119868 gt 0

120597

2119891 (119878 0)

120597119868120597119878

119868 = 0

(119878 119868) isin 119863

(6)


lim119868rarr0

(

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2) =

1

2

120597

2119891 (119878 0)

120597119868

2

lim119868rarr0

1

119868

120597119891 (119878 119868)

120597119878

=

120597

2119891 (119878 0)

120597119868120597119878

(7)




2

+ model


+0

satisfying (119878(119905) 119868(119905)) isin 119877

2



0le lim inf

119905rarrinfin119873(119905) le


119873(119905) le 119878


119873(119905) =

119878

0 where119873(119905) = 119878(119905) + 119868(119905) and 1198780= Λ(120583 + 120572)




119877

0=

120573 (120597119891 (119878

0 0) 120597119868)

120583 + 120574 + 120572

119877

0= 119877

0minus

120590

2(120597119891 (119878

0 0) 120597119868)

2

2 (120583 + 120574 + 120572)

(8)

We have that 119877




(a) 1205902 le 120573(120597119891(1198780 0)120597119868) and 1198770lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)


+ solution (119878(119905) 119868(119905)) of


119868(119905) = 0 as



(119878(119905)+119868(119905)) le 119878


0gt 0

such that 119878(119905) + 119868(119905) lt 119878

0+ 120578 for all 119905 ge 119879

0 Hence for any

119905 ge 119879

0

119891 (119878 (119905) 119868 (119905))

119868 (119905)

isin (0

120597119891 (119878

0+ 120578 0)

120597119868

] (9)


119889 log 119868 (119905) = [120573119891 (119878 (119905) 119868 (119905))

119868 (119905)

minus (120583 + 120574 + 120572)

minus

120590

2

2

(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

)

2

]119889119905 + 120590

sdot

119891 (119878 (119905) 119868 (119905))

119868 (119905)

119889119861 (119905)

(10)


log 119868 (119905)119905

le

log 119868 (0)119905

+

120573 + 120576

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

minus (120583 + 120574 + 120572)

minus

120590

2

2

1

119905

int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119904

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(11)

Define a function

119892 (119906) = (120573 + 120576) 119906 minus

120590

2

2

119906

2minus (120583 + 120574 + 120572)

(12)



+ and


120576)120590



119892(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

) le 119892(

120597119891 (119878

0+ 120578 0)

120597119868

) forall119905 ge 119879

0

(13)



0

log 119868 (119905)119905

le

log 119868 (0)119905

+

1

119905

int

119905

0

119892(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

) 119889119904

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le

log 119868 (0)119905

+

1

119905

int

1198790

0

119892(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

) 119889119904

+

1

119905

119892(

120597119891 (119878

0+ 120578 0)

120597119868

) (119905 minus 119879

0)

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(14)



log 119868 (119905)119905

le 119892(

120597119891 (119878

0+ 120578 0)

120597119868

) as (15)



log 119868 (119905)119905

le 120573

120597119891 (119878

0 0)

120597119868

minus

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus (120583 + 120574 + 120572)

= (120583 + 120574 + 120572) (

119877

0minus 1) lt 0 as

(16)


120573119892(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

) le

(120573 + 120576)

2

2120590

2minus (120583 + 120574 + 120572)

(17)

which implies

log 119868 (119905)119905

le

log 119868 (0)119905

+

(120573 + 120576)

2

2120590

2minus (120583 + 120574 + 120572)

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(18)



log 119868 (119905)119905

le

120573

2

2120590

2minus (120583 + 120574 + 120572) lt 0 as (19)




0of deterministic


0le 1 then for any 120590 gt 0 119877



0le 2 From

119877

0= 1 we have

120590 ≜ 120590 =

radic2 (120583 + 120574 + 120572) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

(20)

Denote

120590

1=

120573

radic2 (120583 + 120574 + 120572)

120590

2= radic

120573

120597119891 (119878

0 0) 120597119868

(21)

Since 1205901le 120590



0gt 2

we have 1205901gt 120590

2and 120590

1ge 120590 ge 120590






(a) 1198770le 1 and 120590 gt 0

(b) 1 lt 1198770le 2 and 120590 gt 120590

(c) 1198770gt 2 and 120590 gt 120590

1



(a) 1205902 le 120573 and 1198770= 120573(120583 + 120574 + 120572) minus 120590

22(120583 + 120574 + 120572) lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)




(a) 1205902 le 120573ℎ(1198780)1198921015840(0) and 1198770= 120573ℎ(119878

0)119892

1015840(0)(120583+120574+120572)minus

120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 + 120574 + 120572) lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)



0gt 2 and 120590 le 120590 le 120590





119877








119877

2



1

119905

int

119905

0

119868 (119904) 119889119904 ge 119898

119868119886119904 (22)

Proof From

119877


such that

120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572) minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

gt 0

(23)


2



(1119905) int

119905

0119868(119904)119889119904 le 119878



0minus 120576 le 119878(119905) + 119868(119905) le 119878

0+ 120576 as for all 119905 ge 119879

0

Denote the set 119863120576= (119878 119868) 119878

0minus 120576 le 119878 + 119868 le 119878

0+ 120576 Since


int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904 = minus

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+

119873 (119879

0) minus 119873 (119905)

120583

(24)

From (10) for any 119905 ge 1198790

log 119868 (119905) = log 119868 (0) + 120573int119905

0

[

120597119891 (119878

0 0)

120597119868

+

119891 (119878 (119904) 119868 (119904))

119868 (119904)

minus

120597119891 (119878

0 0)

120597119868

] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119904

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(25)





119891 (119878 (119905) 119868 (119905))

119868 (119905)

minus

120597119891 (119878

0 0)

120597119868

= (

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868

minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119905)

+

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119905) minus 119878

0)

(26)


119872

1= max(119878119868)isin119863120576

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119872

2= max(119878119868)isin119863120576

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119878

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(27)


0 we

have

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868

minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

ge minus119872

1as

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119878

le 119872

2as

(28)


log 119868 (119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

+ 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

+ 120573int

119905

1198790

[(

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868


minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119904) +

1

120601 (119905)

sdot

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119904) minus 119878

0)] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) ge log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 + 120573119872

2int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904

minus (120583 + 120574 + 120572) 119905 minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119905 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 minus 1205731198722

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879


1

2

sdot 120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = 119867 (119905) + 120579119905

minus 120579

0int

119905

0

119878 (119904) 119889119904

(29)

where

119867(119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

minus 120573

120597119891 (119878

0 0)

120597119868

119879

0

+ 120573(119872

1+119872

2

120583 + 120572

120583

)int

1198790

0

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879

0) minus 119873 (119905))

+ 120590int

119905

0

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

120579 = 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572)

minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

120579

0= 120573(119872

1+119872

2

120583 + 120572

120583

)

(30)




119905

0119868(119904)119889119904 ge 120579120579

0as










119878gt


2

+ solution



1

119905

int

119905

0

119878 (119904) 119889119904 ge 119898

119878119886119904 (31)


2



(1119905) int

119905

0119878(119904)119889119904 le 119878



0minus 120576 le 119878(119905) + 119868(119905) le

119878

0+120576 for all 119905 ge 119879


0we have

119891(119878(119905) 119868(119905)) le 119872

119878119878(119905) where119872

119878= max

119863120576119891(119878 119868)119878 lt infin


0

119878 (119905) minus 119878 (0)

119905

= Λ minus

1

119905

int

119905

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904) minus 120574119868 (119904)] 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

ge Λ minus

1

119905

int

1198790

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904)] 119889119904

minus

1

119905

int

119905

1198790

[120573119872

119878+ 120583] 119878 (119904) 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

(32)




1

119905

int

119905

0

119878 (119904) 119889119904 ge

Λ

120573119872

119878+ 120583

(33)




119877

0= (120573minus(12)120590




119877

0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574 + 120572) minus 120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 +







119868 (119905) ge 120585


119878 (119905) ge 120585

as

(34)




satisfies


119868 (119905) ge 120585 119886119904 (35)


119868 (119905) le 120585 119886119904 (36)


119891 (119878

0minus 120585 120585)

120585

=

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(37)


(119878(119905)+119868(119905)) = 119878


log 119868 (119905) = log 119868 (0) + int119905

0

[

[

120573

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

minus (120583 + 120574) minus

120590

2

2

(

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

)

2

]

]

119889119904

+ int

119905

0

120590

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

(38)


log 119868 (119905) = log 119868 (0) + int119905

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(39)


119877


119892(

120597119891 (119878

0 0)

120597119868

) = minus

120590

2

2

(

120597119891 (119878

0 0)

120597119868

)

2

+ 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574) gt 0

(40)


0 0)120597119868)

which is

120578 =

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(41)


lim119868rarr0+

119906 (119868) = lim119868rarr0+

119891 (119878

0minus 119868 119868)

119868

=

120597119891 (119878

0 0)

120597119868

(42)



2 and 119892(120573120590

2) gt

119892(120597119891(119878


2From 120578 isin (0 120597119891(119878

0 0)120597119868) and 119892(120578) = 0 we have 120578 lt 120573120590

2Hence 0 lt 119868 lt 120585 lt 1198780




When 1205902 le 120573(120597119891(119878





(120585 119878

0)



1) gt 120576

0

where Ω1= lim sup


1


1(120596) ge 119879

0such that

119868 (119905) le 120585 minus 120576

0forall119905 ge 119879

1 (43)


0)) gt 0 for all 119905 ge 119879


119905 ge 119879

1

log 119868 (119905) ge log 119868 (0) + int1198791

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 minus 120576

0)) (119905 minus 119879

1)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(44)






2) gt 120576

1 where

Ω

2= lim inf


2 there is

119879

2= 119879

2(120596) ge 119879

0such that

119868 (119905) ge 120585 + 120576

1forall119905 ge 119879

2 (45)



119905 ge 119879

2

log 119868 (119905) = log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

1198792

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 + 120576

1)) (119905 minus 119879

2)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(46)


119905rarrinfin(log 119868(119905)119905) le 119892(119906(120585 + 120576





2




2

+





119877


0=

120573(120597119891(119878



0 lt 120590 lt

radic2 (120583 + 120574) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

fl

(47)


120590rarr0120585 = 119868


(d) If 1 le 119877

0le 2 then lim

120590rarr120585 = 0 and if 119877

0gt 2 then

lim120590rarr

120585 = 120585

2 where 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(48)

Proof Since

119891 (119878

0minus 120585 120585)

120585

= 120578

(49)


120578 =


2minus 2120590

2(120583 + 120574)

120590

2

(50)


0 0)120597119868) when 0 lt 120590 lt


119889120578

119889120590

=

minus2120573

120590

3+

2 (120583 + 120574)

120590radic120573

2minus 2120590

2(120583 + 120574)

+

2radic120573

2minus 2120590

2(120583 + 120574)

120590

3

=

2120573

2minus 2120590

2(120583 + 120574) minus 2120573

radic120573

2minus 2120590

2(120583 + 120574)

120590

3radic120573

2minus 2120590

2(120583 + 120574)

(51)


Since

[2120573

2minus 2120590

2(120583 + 120574)]

2


2minus 2120590

2(120583 + 120574))

2

= 4120590

4(120583 + 120574)

2gt 0

(52)






120585


120585 = 120585

1and lim

120590rarr120585 = 120585

2 We have

lim120590rarr0

120578 = lim120590rarr0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

=

120583 + 120574

120573

(53)


(119891(119878

0minus 120585 120585)120585) = lim

120590rarr0120578 = (120583 + 120574)120573 This

shows that 119891(1198780 minus 1205851 120585

1)120585



0minus119868

lowast 119868

lowast)119868

lowast= (120583+120574)120573 Hence 120585

1= 119868


120585 = 119868


lim120590rarr

120578 =


2minus 2

2(120583 + 120574)

2=

(120597119891 (119878

0 0) 120597119868) (120573 (120597119891 (119878

0 0) 120597119868) minus

1003816

1003816

1003816

1003816

1003816

120573 (120597119891 (119878

0 0) 120597119868) minus 2 (120583 + 120574)

1003816

1003816

1003816

1003816

1003816

)

2 (120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574))

(54)

If 1 le 119877


120590rarr120578 =

120597119891(119878

0 0)120597119868 Hence

lim120590rarr

119891 (S0 minus 120585 120585)120585

=

120597119891 (119878

0 0)

120597119868

(55)



(54)

lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(56)

which implies

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(57)


120585 = 120585

2 and 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

(119877

0minus 1)

(58)









2gt 0 by



0gt 2 then when 120590 gt 120590

1 where 120590

1is given in







lowastisin ( 120590




lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

(59)

namely

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

=

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(60)


119891 (119878

0minus 120585

2 120585

2)

120585

2

gt

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(61)


lowast



119877

0gt 1 whether we




119877




2

+ one has lim











0le 119878 + 119868 le 119878




119863 = (119878 119868) isin Γ

1

119886

lt 119878 lt 119878

0minus

1

119886

1

119886

lt 119868 lt 119878

0minus

1

119886

(62)


119860 (119878 119868) = (

120590

2119891

2(119878 119868) minus120590

2119891

2(119878 119868)

minus120590

2119891

2(119878 119868) 120590

2119891

2(119878 119868)

) (63)


2(119891(1119886 119878

0minus

1119886)(119886119878

0minus 1))


119881 (119878 119868) = Ψ

1(119868) + Ψ

2(119878 119868) + Ψ

3(119878) (64)

where

Ψ

1(119868) =

1

V119868

minusV

Ψ

2(119878 119868) =

1

V119868

minusV(119878

0minus 119878)

Ψ

3(119878) =

1

119878

(65)


have

119871Ψ

1= minus119868

minus(V+1)(120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119868

minus(V+2)119891

2(119878 119868) le 119868

minusV(120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

)

+ 119868

minusV120573(

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

)

(66)


120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

=

1

120601

120597119891 (120585 120601)

120597119878

(119878

0minus 119878)

+ (

119891 (120585 120601)

120601

2minus

1

120601

120597119891 (120585 120601)

120597119868

) 119868

le 119872

1(119878

0minus 119878) +119872

2119868 +119872

3119877

(67)


119872

1= max(119878119868)isinΓ

1

119868

120597119891 (119878 119868)

120597119878

119872

2= max(119878119868)isinΓ

119891 (119878 119868)

119868

2minus

1

119868

120597119891 (119878 119868)

120597119868

(68)


119871Ψ

1le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 120573119872

1(119878

0minus 119878) 119868

minusV+ 120573119872

2119868

1minusV

(69)


119871Ψ

2= minus

1

V119868


minus(V+1)(119878

0

minus 119878) (120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119891

2(119878 119868) 119868

minus(V+2)(119878


1

2

119868

minus(V+1)120590

2119891

2(119878 119868)

= minus

1

V119868

minusV(120583 (119878

0minus 119878) minus 120573119891 (119878 119868) + 120574119868)

minus 119868

minusV(119878

0minus 119878) (120573

119891 (119878 119868)

119868

minus (120583 + 120572 + 120574)) +

1

2

(1 + V)

sdot 120590

2(

119891 (119878 119868)

119868

)

2

119868

minusV(119878

0minus 119878) minus 120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusV


= 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)


119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)




119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)


120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)





1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)







119877

0= (120573 minus (12)120590




0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+



0of deterministic

model (1)






0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















0 119868 gt 0 119902 le 119878 + 119868 le 119901 Then

max(119878119868)isin119863

119891 (119878 119868)

119878

119891 (119878 119868)

119868

lt infin (3)

max(119878119868)isin119863

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119878

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

lt infin (4)


lim119878rarr0

119891 (119878 119868)

119878

=

120597119891 (0 119868)

120597119878

lim119868rarr0

119891 (119878 119868)

119878

=

120597119891 (119878 0)

120597119868

(5)


119867(119878 119868) =

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2 119868 gt 0

1

2

120597

2119891 (119878 0)

120597119868

2 119868 = 0

(119878 119868) isin 119863

119866 (119878 119868) =

1

119868

120597119891 (119878 119868)

120597119868

119868 gt 0

120597

2119891 (119878 0)

120597119868120597119878

119868 = 0

(119878 119868) isin 119863

(6)


lim119868rarr0

(

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2) =

1

2

120597

2119891 (119878 0)

120597119868

2

lim119868rarr0

1

119868

120597119891 (119878 119868)

120597119878

=

120597

2119891 (119878 0)

120597119868120597119878

(7)




2

+ model


+0

satisfying (119878(119905) 119868(119905)) isin 119877

2



0le lim inf

119905rarrinfin119873(119905) le


119873(119905) le 119878


119873(119905) =

119878

0 where119873(119905) = 119878(119905) + 119868(119905) and 1198780= Λ(120583 + 120572)




119877

0=

120573 (120597119891 (119878

0 0) 120597119868)

120583 + 120574 + 120572

119877

0= 119877

0minus

120590

2(120597119891 (119878

0 0) 120597119868)

2

2 (120583 + 120574 + 120572)

(8)

We have that 119877




(a) 1205902 le 120573(120597119891(1198780 0)120597119868) and 1198770lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)


+ solution (119878(119905) 119868(119905)) of


119868(119905) = 0 as



(119878(119905)+119868(119905)) le 119878


0gt 0

such that 119878(119905) + 119868(119905) lt 119878

0+ 120578 for all 119905 ge 119879

0 Hence for any

119905 ge 119879

0

119891 (119878 (119905) 119868 (119905))

119868 (119905)

isin (0

120597119891 (119878

0+ 120578 0)

120597119868

] (9)


119889 log 119868 (119905) = [120573119891 (119878 (119905) 119868 (119905))

119868 (119905)

minus (120583 + 120574 + 120572)

minus

120590

2

2

(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

)

2

]119889119905 + 120590

sdot

119891 (119878 (119905) 119868 (119905))

119868 (119905)

119889119861 (119905)

(10)


log 119868 (119905)119905

le

log 119868 (0)119905

+

120573 + 120576

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

minus (120583 + 120574 + 120572)

minus

120590

2

2

1

119905

int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119904

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(11)

Define a function

119892 (119906) = (120573 + 120576) 119906 minus

120590

2

2

119906

2minus (120583 + 120574 + 120572)

(12)



+ and


120576)120590



119892(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

) le 119892(

120597119891 (119878

0+ 120578 0)

120597119868

) forall119905 ge 119879

0

(13)



0

log 119868 (119905)119905

le

log 119868 (0)119905

+

1

119905

int

119905

0

119892(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

) 119889119904

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le

log 119868 (0)119905

+

1

119905

int

1198790

0

119892(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

) 119889119904

+

1

119905

119892(

120597119891 (119878

0+ 120578 0)

120597119868

) (119905 minus 119879

0)

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(14)



log 119868 (119905)119905

le 119892(

120597119891 (119878

0+ 120578 0)

120597119868

) as (15)



log 119868 (119905)119905

le 120573

120597119891 (119878

0 0)

120597119868

minus

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus (120583 + 120574 + 120572)

= (120583 + 120574 + 120572) (

119877

0minus 1) lt 0 as

(16)


120573119892(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

) le

(120573 + 120576)

2

2120590

2minus (120583 + 120574 + 120572)

(17)

which implies

log 119868 (119905)119905

le

log 119868 (0)119905

+

(120573 + 120576)

2

2120590

2minus (120583 + 120574 + 120572)

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(18)



log 119868 (119905)119905

le

120573

2

2120590

2minus (120583 + 120574 + 120572) lt 0 as (19)




0of deterministic


0le 1 then for any 120590 gt 0 119877



0le 2 From

119877

0= 1 we have

120590 ≜ 120590 =

radic2 (120583 + 120574 + 120572) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

(20)

Denote

120590

1=

120573

radic2 (120583 + 120574 + 120572)

120590

2= radic

120573

120597119891 (119878

0 0) 120597119868

(21)

Since 1205901le 120590



0gt 2

we have 1205901gt 120590

2and 120590

1ge 120590 ge 120590






(a) 1198770le 1 and 120590 gt 0

(b) 1 lt 1198770le 2 and 120590 gt 120590

(c) 1198770gt 2 and 120590 gt 120590

1



(a) 1205902 le 120573 and 1198770= 120573(120583 + 120574 + 120572) minus 120590

22(120583 + 120574 + 120572) lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)




(a) 1205902 le 120573ℎ(1198780)1198921015840(0) and 1198770= 120573ℎ(119878

0)119892

1015840(0)(120583+120574+120572)minus

120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 + 120574 + 120572) lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)



0gt 2 and 120590 le 120590 le 120590





119877








119877

2



1

119905

int

119905

0

119868 (119904) 119889119904 ge 119898

119868119886119904 (22)

Proof From

119877


such that

120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572) minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

gt 0

(23)


2



(1119905) int

119905

0119868(119904)119889119904 le 119878



0minus 120576 le 119878(119905) + 119868(119905) le 119878

0+ 120576 as for all 119905 ge 119879

0

Denote the set 119863120576= (119878 119868) 119878

0minus 120576 le 119878 + 119868 le 119878

0+ 120576 Since


int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904 = minus

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+

119873 (119879

0) minus 119873 (119905)

120583

(24)

From (10) for any 119905 ge 1198790

log 119868 (119905) = log 119868 (0) + 120573int119905

0

[

120597119891 (119878

0 0)

120597119868

+

119891 (119878 (119904) 119868 (119904))

119868 (119904)

minus

120597119891 (119878

0 0)

120597119868

] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119904

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(25)





119891 (119878 (119905) 119868 (119905))

119868 (119905)

minus

120597119891 (119878

0 0)

120597119868

= (

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868

minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119905)

+

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119905) minus 119878

0)

(26)


119872

1= max(119878119868)isin119863120576

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119872

2= max(119878119868)isin119863120576

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119878

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(27)


0 we

have

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868

minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

ge minus119872

1as

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119878

le 119872

2as

(28)


log 119868 (119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

+ 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

+ 120573int

119905

1198790

[(

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868


minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119904) +

1

120601 (119905)

sdot

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119904) minus 119878

0)] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) ge log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 + 120573119872

2int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904

minus (120583 + 120574 + 120572) 119905 minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119905 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 minus 1205731198722

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879


1

2

sdot 120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = 119867 (119905) + 120579119905

minus 120579

0int

119905

0

119878 (119904) 119889119904

(29)

where

119867(119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

minus 120573

120597119891 (119878

0 0)

120597119868

119879

0

+ 120573(119872

1+119872

2

120583 + 120572

120583

)int

1198790

0

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879

0) minus 119873 (119905))

+ 120590int

119905

0

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

120579 = 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572)

minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

120579

0= 120573(119872

1+119872

2

120583 + 120572

120583

)

(30)




119905

0119868(119904)119889119904 ge 120579120579

0as










119878gt


2

+ solution



1

119905

int

119905

0

119878 (119904) 119889119904 ge 119898

119878119886119904 (31)


2



(1119905) int

119905

0119878(119904)119889119904 le 119878



0minus 120576 le 119878(119905) + 119868(119905) le

119878

0+120576 for all 119905 ge 119879


0we have

119891(119878(119905) 119868(119905)) le 119872

119878119878(119905) where119872

119878= max

119863120576119891(119878 119868)119878 lt infin


0

119878 (119905) minus 119878 (0)

119905

= Λ minus

1

119905

int

119905

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904) minus 120574119868 (119904)] 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

ge Λ minus

1

119905

int

1198790

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904)] 119889119904

minus

1

119905

int

119905

1198790

[120573119872

119878+ 120583] 119878 (119904) 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

(32)




1

119905

int

119905

0

119878 (119904) 119889119904 ge

Λ

120573119872

119878+ 120583

(33)




119877

0= (120573minus(12)120590




119877

0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574 + 120572) minus 120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 +







119868 (119905) ge 120585


119878 (119905) ge 120585

as

(34)




satisfies


119868 (119905) ge 120585 119886119904 (35)


119868 (119905) le 120585 119886119904 (36)


119891 (119878

0minus 120585 120585)

120585

=

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(37)


(119878(119905)+119868(119905)) = 119878


log 119868 (119905) = log 119868 (0) + int119905

0

[

[

120573

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

minus (120583 + 120574) minus

120590

2

2

(

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

)

2

]

]

119889119904

+ int

119905

0

120590

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

(38)


log 119868 (119905) = log 119868 (0) + int119905

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(39)


119877


119892(

120597119891 (119878

0 0)

120597119868

) = minus

120590

2

2

(

120597119891 (119878

0 0)

120597119868

)

2

+ 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574) gt 0

(40)


0 0)120597119868)

which is

120578 =

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(41)


lim119868rarr0+

119906 (119868) = lim119868rarr0+

119891 (119878

0minus 119868 119868)

119868

=

120597119891 (119878

0 0)

120597119868

(42)



2 and 119892(120573120590

2) gt

119892(120597119891(119878


2From 120578 isin (0 120597119891(119878

0 0)120597119868) and 119892(120578) = 0 we have 120578 lt 120573120590

2Hence 0 lt 119868 lt 120585 lt 1198780




When 1205902 le 120573(120597119891(119878





(120585 119878

0)



1) gt 120576

0

where Ω1= lim sup


1


1(120596) ge 119879

0such that

119868 (119905) le 120585 minus 120576

0forall119905 ge 119879

1 (43)


0)) gt 0 for all 119905 ge 119879


119905 ge 119879

1

log 119868 (119905) ge log 119868 (0) + int1198791

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 minus 120576

0)) (119905 minus 119879

1)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(44)






2) gt 120576

1 where

Ω

2= lim inf


2 there is

119879

2= 119879

2(120596) ge 119879

0such that

119868 (119905) ge 120585 + 120576

1forall119905 ge 119879

2 (45)



119905 ge 119879

2

log 119868 (119905) = log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

1198792

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 + 120576

1)) (119905 minus 119879

2)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(46)


119905rarrinfin(log 119868(119905)119905) le 119892(119906(120585 + 120576





2




2

+





119877


0=

120573(120597119891(119878



0 lt 120590 lt

radic2 (120583 + 120574) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

fl

(47)


120590rarr0120585 = 119868


(d) If 1 le 119877

0le 2 then lim

120590rarr120585 = 0 and if 119877

0gt 2 then

lim120590rarr

120585 = 120585

2 where 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(48)

Proof Since

119891 (119878

0minus 120585 120585)

120585

= 120578

(49)


120578 =


2minus 2120590

2(120583 + 120574)

120590

2

(50)


0 0)120597119868) when 0 lt 120590 lt


119889120578

119889120590

=

minus2120573

120590

3+

2 (120583 + 120574)

120590radic120573

2minus 2120590

2(120583 + 120574)

+

2radic120573

2minus 2120590

2(120583 + 120574)

120590

3

=

2120573

2minus 2120590

2(120583 + 120574) minus 2120573

radic120573

2minus 2120590

2(120583 + 120574)

120590

3radic120573

2minus 2120590

2(120583 + 120574)

(51)


Since

[2120573

2minus 2120590

2(120583 + 120574)]

2


2minus 2120590

2(120583 + 120574))

2

= 4120590

4(120583 + 120574)

2gt 0

(52)






120585


120585 = 120585

1and lim

120590rarr120585 = 120585

2 We have

lim120590rarr0

120578 = lim120590rarr0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

=

120583 + 120574

120573

(53)


(119891(119878

0minus 120585 120585)120585) = lim

120590rarr0120578 = (120583 + 120574)120573 This

shows that 119891(1198780 minus 1205851 120585

1)120585



0minus119868

lowast 119868

lowast)119868

lowast= (120583+120574)120573 Hence 120585

1= 119868


120585 = 119868


lim120590rarr

120578 =


2minus 2

2(120583 + 120574)

2=

(120597119891 (119878

0 0) 120597119868) (120573 (120597119891 (119878

0 0) 120597119868) minus

1003816

1003816

1003816

1003816

1003816

120573 (120597119891 (119878

0 0) 120597119868) minus 2 (120583 + 120574)

1003816

1003816

1003816

1003816

1003816

)

2 (120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574))

(54)

If 1 le 119877


120590rarr120578 =

120597119891(119878

0 0)120597119868 Hence

lim120590rarr

119891 (S0 minus 120585 120585)120585

=

120597119891 (119878

0 0)

120597119868

(55)



(54)

lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(56)

which implies

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(57)


120585 = 120585

2 and 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

(119877

0minus 1)

(58)









2gt 0 by



0gt 2 then when 120590 gt 120590

1 where 120590

1is given in







lowastisin ( 120590




lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

(59)

namely

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

=

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(60)


119891 (119878

0minus 120585

2 120585

2)

120585

2

gt

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(61)


lowast



119877

0gt 1 whether we




119877




2

+ one has lim











0le 119878 + 119868 le 119878




119863 = (119878 119868) isin Γ

1

119886

lt 119878 lt 119878

0minus

1

119886

1

119886

lt 119868 lt 119878

0minus

1

119886

(62)


119860 (119878 119868) = (

120590

2119891

2(119878 119868) minus120590

2119891

2(119878 119868)

minus120590

2119891

2(119878 119868) 120590

2119891

2(119878 119868)

) (63)


2(119891(1119886 119878

0minus

1119886)(119886119878

0minus 1))


119881 (119878 119868) = Ψ

1(119868) + Ψ

2(119878 119868) + Ψ

3(119878) (64)

where

Ψ

1(119868) =

1

V119868

minusV

Ψ

2(119878 119868) =

1

V119868

minusV(119878

0minus 119878)

Ψ

3(119878) =

1

119878

(65)


have

119871Ψ

1= minus119868

minus(V+1)(120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119868

minus(V+2)119891

2(119878 119868) le 119868

minusV(120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

)

+ 119868

minusV120573(

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

)

(66)


120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

=

1

120601

120597119891 (120585 120601)

120597119878

(119878

0minus 119878)

+ (

119891 (120585 120601)

120601

2minus

1

120601

120597119891 (120585 120601)

120597119868

) 119868

le 119872

1(119878

0minus 119878) +119872

2119868 +119872

3119877

(67)


119872

1= max(119878119868)isinΓ

1

119868

120597119891 (119878 119868)

120597119878

119872

2= max(119878119868)isinΓ

119891 (119878 119868)

119868

2minus

1

119868

120597119891 (119878 119868)

120597119868

(68)


119871Ψ

1le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 120573119872

1(119878

0minus 119878) 119868

minusV+ 120573119872

2119868

1minusV

(69)


119871Ψ

2= minus

1

V119868


minus(V+1)(119878

0

minus 119878) (120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119891

2(119878 119868) 119868

minus(V+2)(119878


1

2

119868

minus(V+1)120590

2119891

2(119878 119868)

= minus

1

V119868

minusV(120583 (119878

0minus 119878) minus 120573119891 (119878 119868) + 120574119868)

minus 119868

minusV(119878

0minus 119878) (120573

119891 (119878 119868)

119868

minus (120583 + 120572 + 120574)) +

1

2

(1 + V)

sdot 120590

2(

119891 (119878 119868)

119868

)

2

119868

minusV(119878

0minus 119878) minus 120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusV


= 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)


119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)




119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)


120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)





1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)







119877

0= (120573 minus (12)120590




0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+



0of deterministic

model (1)






0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















+ and


120576)120590



119892(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

) le 119892(

120597119891 (119878

0+ 120578 0)

120597119868

) forall119905 ge 119879

0

(13)



0

log 119868 (119905)119905

le

log 119868 (0)119905

+

1

119905

int

119905

0

119892(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

) 119889119904

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le

log 119868 (0)119905

+

1

119905

int

1198790

0

119892(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

) 119889119904

+

1

119905

119892(

120597119891 (119878

0+ 120578 0)

120597119868

) (119905 minus 119879

0)

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(14)



log 119868 (119905)119905

le 119892(

120597119891 (119878

0+ 120578 0)

120597119868

) as (15)



log 119868 (119905)119905

le 120573

120597119891 (119878

0 0)

120597119868

minus

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus (120583 + 120574 + 120572)

= (120583 + 120574 + 120572) (

119877

0minus 1) lt 0 as

(16)


120573119892(

119891 (119878 (119905) 119868 (119905))

119868 (119905)

) le

(120573 + 120576)

2

2120590

2minus (120583 + 120574 + 120572)

(17)

which implies

log 119868 (119905)119905

le

log 119868 (0)119905

+

(120573 + 120576)

2

2120590

2minus (120583 + 120574 + 120572)

+

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(18)



log 119868 (119905)119905

le

120573

2

2120590

2minus (120583 + 120574 + 120572) lt 0 as (19)




0of deterministic


0le 1 then for any 120590 gt 0 119877



0le 2 From

119877

0= 1 we have

120590 ≜ 120590 =

radic2 (120583 + 120574 + 120572) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

(20)

Denote

120590

1=

120573

radic2 (120583 + 120574 + 120572)

120590

2= radic

120573

120597119891 (119878

0 0) 120597119868

(21)

Since 1205901le 120590



0gt 2

we have 1205901gt 120590

2and 120590

1ge 120590 ge 120590






(a) 1198770le 1 and 120590 gt 0

(b) 1 lt 1198770le 2 and 120590 gt 120590

(c) 1198770gt 2 and 120590 gt 120590

1



(a) 1205902 le 120573 and 1198770= 120573(120583 + 120574 + 120572) minus 120590

22(120583 + 120574 + 120572) lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)




(a) 1205902 le 120573ℎ(1198780)1198921015840(0) and 1198770= 120573ℎ(119878

0)119892

1015840(0)(120583+120574+120572)minus

120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 + 120574 + 120572) lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)



0gt 2 and 120590 le 120590 le 120590





119877








119877

2



1

119905

int

119905

0

119868 (119904) 119889119904 ge 119898

119868119886119904 (22)

Proof From

119877


such that

120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572) minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

gt 0

(23)


2



(1119905) int

119905

0119868(119904)119889119904 le 119878



0minus 120576 le 119878(119905) + 119868(119905) le 119878

0+ 120576 as for all 119905 ge 119879

0

Denote the set 119863120576= (119878 119868) 119878

0minus 120576 le 119878 + 119868 le 119878

0+ 120576 Since


int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904 = minus

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+

119873 (119879

0) minus 119873 (119905)

120583

(24)

From (10) for any 119905 ge 1198790

log 119868 (119905) = log 119868 (0) + 120573int119905

0

[

120597119891 (119878

0 0)

120597119868

+

119891 (119878 (119904) 119868 (119904))

119868 (119904)

minus

120597119891 (119878

0 0)

120597119868

] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119904

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(25)





119891 (119878 (119905) 119868 (119905))

119868 (119905)

minus

120597119891 (119878

0 0)

120597119868

= (

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868

minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119905)

+

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119905) minus 119878

0)

(26)


119872

1= max(119878119868)isin119863120576

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119872

2= max(119878119868)isin119863120576

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119878

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(27)


0 we

have

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868

minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

ge minus119872

1as

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119878

le 119872

2as

(28)


log 119868 (119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

+ 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

+ 120573int

119905

1198790

[(

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868


minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119904) +

1

120601 (119905)

sdot

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119904) minus 119878

0)] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) ge log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 + 120573119872

2int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904

minus (120583 + 120574 + 120572) 119905 minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119905 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 minus 1205731198722

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879


1

2

sdot 120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = 119867 (119905) + 120579119905

minus 120579

0int

119905

0

119878 (119904) 119889119904

(29)

where

119867(119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

minus 120573

120597119891 (119878

0 0)

120597119868

119879

0

+ 120573(119872

1+119872

2

120583 + 120572

120583

)int

1198790

0

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879

0) minus 119873 (119905))

+ 120590int

119905

0

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

120579 = 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572)

minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

120579

0= 120573(119872

1+119872

2

120583 + 120572

120583

)

(30)




119905

0119868(119904)119889119904 ge 120579120579

0as










119878gt


2

+ solution



1

119905

int

119905

0

119878 (119904) 119889119904 ge 119898

119878119886119904 (31)


2



(1119905) int

119905

0119878(119904)119889119904 le 119878



0minus 120576 le 119878(119905) + 119868(119905) le

119878

0+120576 for all 119905 ge 119879


0we have

119891(119878(119905) 119868(119905)) le 119872

119878119878(119905) where119872

119878= max

119863120576119891(119878 119868)119878 lt infin


0

119878 (119905) minus 119878 (0)

119905

= Λ minus

1

119905

int

119905

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904) minus 120574119868 (119904)] 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

ge Λ minus

1

119905

int

1198790

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904)] 119889119904

minus

1

119905

int

119905

1198790

[120573119872

119878+ 120583] 119878 (119904) 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

(32)




1

119905

int

119905

0

119878 (119904) 119889119904 ge

Λ

120573119872

119878+ 120583

(33)




119877

0= (120573minus(12)120590




119877

0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574 + 120572) minus 120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 +







119868 (119905) ge 120585


119878 (119905) ge 120585

as

(34)




satisfies


119868 (119905) ge 120585 119886119904 (35)


119868 (119905) le 120585 119886119904 (36)


119891 (119878

0minus 120585 120585)

120585

=

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(37)


(119878(119905)+119868(119905)) = 119878


log 119868 (119905) = log 119868 (0) + int119905

0

[

[

120573

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

minus (120583 + 120574) minus

120590

2

2

(

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

)

2

]

]

119889119904

+ int

119905

0

120590

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

(38)


log 119868 (119905) = log 119868 (0) + int119905

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(39)


119877


119892(

120597119891 (119878

0 0)

120597119868

) = minus

120590

2

2

(

120597119891 (119878

0 0)

120597119868

)

2

+ 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574) gt 0

(40)


0 0)120597119868)

which is

120578 =

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(41)


lim119868rarr0+

119906 (119868) = lim119868rarr0+

119891 (119878

0minus 119868 119868)

119868

=

120597119891 (119878

0 0)

120597119868

(42)



2 and 119892(120573120590

2) gt

119892(120597119891(119878


2From 120578 isin (0 120597119891(119878

0 0)120597119868) and 119892(120578) = 0 we have 120578 lt 120573120590

2Hence 0 lt 119868 lt 120585 lt 1198780




When 1205902 le 120573(120597119891(119878





(120585 119878

0)



1) gt 120576

0

where Ω1= lim sup


1


1(120596) ge 119879

0such that

119868 (119905) le 120585 minus 120576

0forall119905 ge 119879

1 (43)


0)) gt 0 for all 119905 ge 119879


119905 ge 119879

1

log 119868 (119905) ge log 119868 (0) + int1198791

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 minus 120576

0)) (119905 minus 119879

1)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(44)






2) gt 120576

1 where

Ω

2= lim inf


2 there is

119879

2= 119879

2(120596) ge 119879

0such that

119868 (119905) ge 120585 + 120576

1forall119905 ge 119879

2 (45)



119905 ge 119879

2

log 119868 (119905) = log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

1198792

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 + 120576

1)) (119905 minus 119879

2)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(46)


119905rarrinfin(log 119868(119905)119905) le 119892(119906(120585 + 120576





2




2

+





119877


0=

120573(120597119891(119878



0 lt 120590 lt

radic2 (120583 + 120574) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

fl

(47)


120590rarr0120585 = 119868


(d) If 1 le 119877

0le 2 then lim

120590rarr120585 = 0 and if 119877

0gt 2 then

lim120590rarr

120585 = 120585

2 where 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(48)

Proof Since

119891 (119878

0minus 120585 120585)

120585

= 120578

(49)


120578 =


2minus 2120590

2(120583 + 120574)

120590

2

(50)


0 0)120597119868) when 0 lt 120590 lt


119889120578

119889120590

=

minus2120573

120590

3+

2 (120583 + 120574)

120590radic120573

2minus 2120590

2(120583 + 120574)

+

2radic120573

2minus 2120590

2(120583 + 120574)

120590

3

=

2120573

2minus 2120590

2(120583 + 120574) minus 2120573

radic120573

2minus 2120590

2(120583 + 120574)

120590

3radic120573

2minus 2120590

2(120583 + 120574)

(51)


Since

[2120573

2minus 2120590

2(120583 + 120574)]

2


2minus 2120590

2(120583 + 120574))

2

= 4120590

4(120583 + 120574)

2gt 0

(52)






120585


120585 = 120585

1and lim

120590rarr120585 = 120585

2 We have

lim120590rarr0

120578 = lim120590rarr0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

=

120583 + 120574

120573

(53)


(119891(119878

0minus 120585 120585)120585) = lim

120590rarr0120578 = (120583 + 120574)120573 This

shows that 119891(1198780 minus 1205851 120585

1)120585



0minus119868

lowast 119868

lowast)119868

lowast= (120583+120574)120573 Hence 120585

1= 119868


120585 = 119868


lim120590rarr

120578 =


2minus 2

2(120583 + 120574)

2=

(120597119891 (119878

0 0) 120597119868) (120573 (120597119891 (119878

0 0) 120597119868) minus

1003816

1003816

1003816

1003816

1003816

120573 (120597119891 (119878

0 0) 120597119868) minus 2 (120583 + 120574)

1003816

1003816

1003816

1003816

1003816

)

2 (120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574))

(54)

If 1 le 119877


120590rarr120578 =

120597119891(119878

0 0)120597119868 Hence

lim120590rarr

119891 (S0 minus 120585 120585)120585

=

120597119891 (119878

0 0)

120597119868

(55)



(54)

lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(56)

which implies

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(57)


120585 = 120585

2 and 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

(119877

0minus 1)

(58)









2gt 0 by



0gt 2 then when 120590 gt 120590

1 where 120590

1is given in







lowastisin ( 120590




lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

(59)

namely

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

=

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(60)


119891 (119878

0minus 120585

2 120585

2)

120585

2

gt

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(61)


lowast



119877

0gt 1 whether we




119877




2

+ one has lim











0le 119878 + 119868 le 119878




119863 = (119878 119868) isin Γ

1

119886

lt 119878 lt 119878

0minus

1

119886

1

119886

lt 119868 lt 119878

0minus

1

119886

(62)


119860 (119878 119868) = (

120590

2119891

2(119878 119868) minus120590

2119891

2(119878 119868)

minus120590

2119891

2(119878 119868) 120590

2119891

2(119878 119868)

) (63)


2(119891(1119886 119878

0minus

1119886)(119886119878

0minus 1))


119881 (119878 119868) = Ψ

1(119868) + Ψ

2(119878 119868) + Ψ

3(119878) (64)

where

Ψ

1(119868) =

1

V119868

minusV

Ψ

2(119878 119868) =

1

V119868

minusV(119878

0minus 119878)

Ψ

3(119878) =

1

119878

(65)


have

119871Ψ

1= minus119868

minus(V+1)(120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119868

minus(V+2)119891

2(119878 119868) le 119868

minusV(120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

)

+ 119868

minusV120573(

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

)

(66)


120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

=

1

120601

120597119891 (120585 120601)

120597119878

(119878

0minus 119878)

+ (

119891 (120585 120601)

120601

2minus

1

120601

120597119891 (120585 120601)

120597119868

) 119868

le 119872

1(119878

0minus 119878) +119872

2119868 +119872

3119877

(67)


119872

1= max(119878119868)isinΓ

1

119868

120597119891 (119878 119868)

120597119878

119872

2= max(119878119868)isinΓ

119891 (119878 119868)

119868

2minus

1

119868

120597119891 (119878 119868)

120597119868

(68)


119871Ψ

1le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 120573119872

1(119878

0minus 119878) 119868

minusV+ 120573119872

2119868

1minusV

(69)


119871Ψ

2= minus

1

V119868


minus(V+1)(119878

0

minus 119878) (120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119891

2(119878 119868) 119868

minus(V+2)(119878


1

2

119868

minus(V+1)120590

2119891

2(119878 119868)

= minus

1

V119868

minusV(120583 (119878

0minus 119878) minus 120573119891 (119878 119868) + 120574119868)

minus 119868

minusV(119878

0minus 119878) (120573

119891 (119878 119868)

119868

minus (120583 + 120572 + 120574)) +

1

2

(1 + V)

sdot 120590

2(

119891 (119878 119868)

119868

)

2

119868

minusV(119878

0minus 119878) minus 120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusV


= 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)


119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)




119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)


120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)





1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)







119877

0= (120573 minus (12)120590




0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+



0of deterministic

model (1)






0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















(a) 1205902 le 120573ℎ(1198780)1198921015840(0) and 1198770= 120573ℎ(119878

0)119892

1015840(0)(120583+120574+120572)minus

120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 + 120574 + 120572) lt 1

(b) 1205902 gt 12057322(120583 + 120574 + 120572)



0gt 2 and 120590 le 120590 le 120590





119877








119877

2



1

119905

int

119905

0

119868 (119904) 119889119904 ge 119898

119868119886119904 (22)

Proof From

119877


such that

120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572) minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

gt 0

(23)


2



(1119905) int

119905

0119868(119904)119889119904 le 119878



0minus 120576 le 119878(119905) + 119868(119905) le 119878

0+ 120576 as for all 119905 ge 119879

0

Denote the set 119863120576= (119878 119868) 119878

0minus 120576 le 119878 + 119868 le 119878

0+ 120576 Since


int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904 = minus

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+

119873 (119879

0) minus 119873 (119905)

120583

(24)

From (10) for any 119905 ge 1198790

log 119868 (119905) = log 119868 (0) + 120573int119905

0

[

120597119891 (119878

0 0)

120597119868

+

119891 (119878 (119904) 119868 (119904))

119868 (119904)

minus

120597119891 (119878

0 0)

120597119868

] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119904

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(25)





119891 (119878 (119905) 119868 (119905))

119868 (119905)

minus

120597119891 (119878

0 0)

120597119868

= (

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868

minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119905)

+

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119905) minus 119878

0)

(26)


119872

1= max(119878119868)isin119863120576

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119868

minus

119891 (119878 119868)

119868

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119872

2= max(119878119868)isin119863120576

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

119868

120597119891 (119878 119868)

120597119878

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(27)


0 we

have

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868

minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

ge minus119872

1as

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119878

le 119872

2as

(28)


log 119868 (119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

+ 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

+ 120573int

119905

1198790

[(

1

120601 (119905)

120597119891 (120585 (119905) 120601 (119905))

120597119868


minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119904) +

1

120601 (119905)

sdot

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119904) minus 119878

0)] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) ge log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 + 120573119872

2int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904

minus (120583 + 120574 + 120572) 119905 minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119905 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 minus 1205731198722

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879


1

2

sdot 120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = 119867 (119905) + 120579119905

minus 120579

0int

119905

0

119878 (119904) 119889119904

(29)

where

119867(119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

minus 120573

120597119891 (119878

0 0)

120597119868

119879

0

+ 120573(119872

1+119872

2

120583 + 120572

120583

)int

1198790

0

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879

0) minus 119873 (119905))

+ 120590int

119905

0

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

120579 = 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572)

minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

120579

0= 120573(119872

1+119872

2

120583 + 120572

120583

)

(30)




119905

0119868(119904)119889119904 ge 120579120579

0as










119878gt


2

+ solution



1

119905

int

119905

0

119878 (119904) 119889119904 ge 119898

119878119886119904 (31)


2



(1119905) int

119905

0119878(119904)119889119904 le 119878



0minus 120576 le 119878(119905) + 119868(119905) le

119878

0+120576 for all 119905 ge 119879


0we have

119891(119878(119905) 119868(119905)) le 119872

119878119878(119905) where119872

119878= max

119863120576119891(119878 119868)119878 lt infin


0

119878 (119905) minus 119878 (0)

119905

= Λ minus

1

119905

int

119905

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904) minus 120574119868 (119904)] 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

ge Λ minus

1

119905

int

1198790

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904)] 119889119904

minus

1

119905

int

119905

1198790

[120573119872

119878+ 120583] 119878 (119904) 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

(32)




1

119905

int

119905

0

119878 (119904) 119889119904 ge

Λ

120573119872

119878+ 120583

(33)




119877

0= (120573minus(12)120590




119877

0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574 + 120572) minus 120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 +







119868 (119905) ge 120585


119878 (119905) ge 120585

as

(34)




satisfies


119868 (119905) ge 120585 119886119904 (35)


119868 (119905) le 120585 119886119904 (36)


119891 (119878

0minus 120585 120585)

120585

=

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(37)


(119878(119905)+119868(119905)) = 119878


log 119868 (119905) = log 119868 (0) + int119905

0

[

[

120573

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

minus (120583 + 120574) minus

120590

2

2

(

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

)

2

]

]

119889119904

+ int

119905

0

120590

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

(38)


log 119868 (119905) = log 119868 (0) + int119905

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(39)


119877


119892(

120597119891 (119878

0 0)

120597119868

) = minus

120590

2

2

(

120597119891 (119878

0 0)

120597119868

)

2

+ 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574) gt 0

(40)


0 0)120597119868)

which is

120578 =

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(41)


lim119868rarr0+

119906 (119868) = lim119868rarr0+

119891 (119878

0minus 119868 119868)

119868

=

120597119891 (119878

0 0)

120597119868

(42)



2 and 119892(120573120590

2) gt

119892(120597119891(119878


2From 120578 isin (0 120597119891(119878

0 0)120597119868) and 119892(120578) = 0 we have 120578 lt 120573120590

2Hence 0 lt 119868 lt 120585 lt 1198780




When 1205902 le 120573(120597119891(119878





(120585 119878

0)



1) gt 120576

0

where Ω1= lim sup


1


1(120596) ge 119879

0such that

119868 (119905) le 120585 minus 120576

0forall119905 ge 119879

1 (43)


0)) gt 0 for all 119905 ge 119879


119905 ge 119879

1

log 119868 (119905) ge log 119868 (0) + int1198791

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 minus 120576

0)) (119905 minus 119879

1)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(44)






2) gt 120576

1 where

Ω

2= lim inf


2 there is

119879

2= 119879

2(120596) ge 119879

0such that

119868 (119905) ge 120585 + 120576

1forall119905 ge 119879

2 (45)



119905 ge 119879

2

log 119868 (119905) = log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

1198792

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 + 120576

1)) (119905 minus 119879

2)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(46)


119905rarrinfin(log 119868(119905)119905) le 119892(119906(120585 + 120576





2




2

+





119877


0=

120573(120597119891(119878



0 lt 120590 lt

radic2 (120583 + 120574) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

fl

(47)


120590rarr0120585 = 119868


(d) If 1 le 119877

0le 2 then lim

120590rarr120585 = 0 and if 119877

0gt 2 then

lim120590rarr

120585 = 120585

2 where 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(48)

Proof Since

119891 (119878

0minus 120585 120585)

120585

= 120578

(49)


120578 =


2minus 2120590

2(120583 + 120574)

120590

2

(50)


0 0)120597119868) when 0 lt 120590 lt


119889120578

119889120590

=

minus2120573

120590

3+

2 (120583 + 120574)

120590radic120573

2minus 2120590

2(120583 + 120574)

+

2radic120573

2minus 2120590

2(120583 + 120574)

120590

3

=

2120573

2minus 2120590

2(120583 + 120574) minus 2120573

radic120573

2minus 2120590

2(120583 + 120574)

120590

3radic120573

2minus 2120590

2(120583 + 120574)

(51)


Since

[2120573

2minus 2120590

2(120583 + 120574)]

2


2minus 2120590

2(120583 + 120574))

2

= 4120590

4(120583 + 120574)

2gt 0

(52)






120585


120585 = 120585

1and lim

120590rarr120585 = 120585

2 We have

lim120590rarr0

120578 = lim120590rarr0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

=

120583 + 120574

120573

(53)


(119891(119878

0minus 120585 120585)120585) = lim

120590rarr0120578 = (120583 + 120574)120573 This

shows that 119891(1198780 minus 1205851 120585

1)120585



0minus119868

lowast 119868

lowast)119868

lowast= (120583+120574)120573 Hence 120585

1= 119868


120585 = 119868


lim120590rarr

120578 =


2minus 2

2(120583 + 120574)

2=

(120597119891 (119878

0 0) 120597119868) (120573 (120597119891 (119878

0 0) 120597119868) minus

1003816

1003816

1003816

1003816

1003816

120573 (120597119891 (119878

0 0) 120597119868) minus 2 (120583 + 120574)

1003816

1003816

1003816

1003816

1003816

)

2 (120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574))

(54)

If 1 le 119877


120590rarr120578 =

120597119891(119878

0 0)120597119868 Hence

lim120590rarr

119891 (S0 minus 120585 120585)120585

=

120597119891 (119878

0 0)

120597119868

(55)



(54)

lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(56)

which implies

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(57)


120585 = 120585

2 and 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

(119877

0minus 1)

(58)









2gt 0 by



0gt 2 then when 120590 gt 120590

1 where 120590

1is given in







lowastisin ( 120590




lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

(59)

namely

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

=

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(60)


119891 (119878

0minus 120585

2 120585

2)

120585

2

gt

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(61)


lowast



119877

0gt 1 whether we




119877




2

+ one has lim











0le 119878 + 119868 le 119878




119863 = (119878 119868) isin Γ

1

119886

lt 119878 lt 119878

0minus

1

119886

1

119886

lt 119868 lt 119878

0minus

1

119886

(62)


119860 (119878 119868) = (

120590

2119891

2(119878 119868) minus120590

2119891

2(119878 119868)

minus120590

2119891

2(119878 119868) 120590

2119891

2(119878 119868)

) (63)


2(119891(1119886 119878

0minus

1119886)(119886119878

0minus 1))


119881 (119878 119868) = Ψ

1(119868) + Ψ

2(119878 119868) + Ψ

3(119878) (64)

where

Ψ

1(119868) =

1

V119868

minusV

Ψ

2(119878 119868) =

1

V119868

minusV(119878

0minus 119878)

Ψ

3(119878) =

1

119878

(65)


have

119871Ψ

1= minus119868

minus(V+1)(120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119868

minus(V+2)119891

2(119878 119868) le 119868

minusV(120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

)

+ 119868

minusV120573(

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

)

(66)


120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

=

1

120601

120597119891 (120585 120601)

120597119878

(119878

0minus 119878)

+ (

119891 (120585 120601)

120601

2minus

1

120601

120597119891 (120585 120601)

120597119868

) 119868

le 119872

1(119878

0minus 119878) +119872

2119868 +119872

3119877

(67)


119872

1= max(119878119868)isinΓ

1

119868

120597119891 (119878 119868)

120597119878

119872

2= max(119878119868)isinΓ

119891 (119878 119868)

119868

2minus

1

119868

120597119891 (119878 119868)

120597119868

(68)


119871Ψ

1le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 120573119872

1(119878

0minus 119878) 119868

minusV+ 120573119872

2119868

1minusV

(69)


119871Ψ

2= minus

1

V119868


minus(V+1)(119878

0

minus 119878) (120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119891

2(119878 119868) 119868

minus(V+2)(119878


1

2

119868

minus(V+1)120590

2119891

2(119878 119868)

= minus

1

V119868

minusV(120583 (119878

0minus 119878) minus 120573119891 (119878 119868) + 120574119868)

minus 119868

minusV(119878

0minus 119878) (120573

119891 (119878 119868)

119868

minus (120583 + 120572 + 120574)) +

1

2

(1 + V)

sdot 120590

2(

119891 (119878 119868)

119868

)

2

119868

minusV(119878

0minus 119878) minus 120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusV


= 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)


119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)




119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)


120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)





1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)







119877

0= (120573 minus (12)120590




0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+



0of deterministic

model (1)






0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















minus

119891 (120585 (119905) 120601 (119905))

120601

2(119905)

) 119868 (119904) +

1

120601 (119905)

sdot

120597119891 (120585 (119905) 120601 (119905))

120597119878

(119878 (119904) minus 119878

0)] 119889119904 minus (120583 + 120574

+ 120572) 119905 minus

1

2

120590

2int

119905

0

(

119891 (119878 (119904) 119868 (119904))

119868 (119904)

)

2

119889119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) ge log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 + 120573119872

2int

119905

1198790

(119878 (119904) minus 119878

0) 119889119904

minus (120583 + 120574 + 120572) 119905 minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = log 119868 (0)

+ 120573int

1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119905 + 120573

120597119891 (119878

0 0)

120597119868

(119905 minus 119879

0)

minus 120573119872

1int

119905

1198790

119868 (119904) 119889119904 minus 1205731198722

120583 + 120572

120583

int

119905

1198790

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879


1

2

sdot 120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

119905

+ 120590int

119905

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904) = 119867 (119905) + 120579119905

minus 120579

0int

119905

0

119878 (119904) 119889119904

(29)

where

119867(119905) = log 119868 (0) + 120573int1198790

0

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119904

minus 120573

120597119891 (119878

0 0)

120597119868

119879

0

+ 120573(119872

1+119872

2

120583 + 120572

120583

)int

1198790

0

119868 (119904) 119889119904

+ 120573119872

2

1

120583

(119873 (119879

0) minus 119873 (119905))

+ 120590int

119905

0

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

120579 = 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574 + 120572)

minus

1

2

120590

2(

120597119891 (119878

0+ 120576 0)

120597119868

)

2

120579

0= 120573(119872

1+119872

2

120583 + 120572

120583

)

(30)




119905

0119868(119904)119889119904 ge 120579120579

0as










119878gt


2

+ solution



1

119905

int

119905

0

119878 (119904) 119889119904 ge 119898

119878119886119904 (31)


2



(1119905) int

119905

0119878(119904)119889119904 le 119878



0minus 120576 le 119878(119905) + 119868(119905) le

119878

0+120576 for all 119905 ge 119879


0we have

119891(119878(119905) 119868(119905)) le 119872

119878119878(119905) where119872

119878= max

119863120576119891(119878 119868)119878 lt infin


0

119878 (119905) minus 119878 (0)

119905

= Λ minus

1

119905

int

119905

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904) minus 120574119868 (119904)] 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

ge Λ minus

1

119905

int

1198790

0

[120573119891 (119878 (119904) 119868 (119904)) + 120583119878 (119904)] 119889119904

minus

1

119905

int

119905

1198790

[120573119872

119878+ 120583] 119878 (119904) 119889119904

minus

120590

119905

int

119905

0

119891 (119878 (119904) 119868 (119904)) 119889119861 (119904)

(32)




1

119905

int

119905

0

119878 (119904) 119889119904 ge

Λ

120573119872

119878+ 120583

(33)




119877

0= (120573minus(12)120590




119877

0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574 + 120572) minus 120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 +







119868 (119905) ge 120585


119878 (119905) ge 120585

as

(34)




satisfies


119868 (119905) ge 120585 119886119904 (35)


119868 (119905) le 120585 119886119904 (36)


119891 (119878

0minus 120585 120585)

120585

=

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(37)


(119878(119905)+119868(119905)) = 119878


log 119868 (119905) = log 119868 (0) + int119905

0

[

[

120573

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

minus (120583 + 120574) minus

120590

2

2

(

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

)

2

]

]

119889119904

+ int

119905

0

120590

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

(38)


log 119868 (119905) = log 119868 (0) + int119905

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(39)


119877


119892(

120597119891 (119878

0 0)

120597119868

) = minus

120590

2

2

(

120597119891 (119878

0 0)

120597119868

)

2

+ 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574) gt 0

(40)


0 0)120597119868)

which is

120578 =

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(41)


lim119868rarr0+

119906 (119868) = lim119868rarr0+

119891 (119878

0minus 119868 119868)

119868

=

120597119891 (119878

0 0)

120597119868

(42)



2 and 119892(120573120590

2) gt

119892(120597119891(119878


2From 120578 isin (0 120597119891(119878

0 0)120597119868) and 119892(120578) = 0 we have 120578 lt 120573120590

2Hence 0 lt 119868 lt 120585 lt 1198780




When 1205902 le 120573(120597119891(119878





(120585 119878

0)



1) gt 120576

0

where Ω1= lim sup


1


1(120596) ge 119879

0such that

119868 (119905) le 120585 minus 120576

0forall119905 ge 119879

1 (43)


0)) gt 0 for all 119905 ge 119879


119905 ge 119879

1

log 119868 (119905) ge log 119868 (0) + int1198791

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 minus 120576

0)) (119905 minus 119879

1)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(44)






2) gt 120576

1 where

Ω

2= lim inf


2 there is

119879

2= 119879

2(120596) ge 119879

0such that

119868 (119905) ge 120585 + 120576

1forall119905 ge 119879

2 (45)



119905 ge 119879

2

log 119868 (119905) = log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

1198792

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 + 120576

1)) (119905 minus 119879

2)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(46)


119905rarrinfin(log 119868(119905)119905) le 119892(119906(120585 + 120576





2




2

+





119877


0=

120573(120597119891(119878



0 lt 120590 lt

radic2 (120583 + 120574) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

fl

(47)


120590rarr0120585 = 119868


(d) If 1 le 119877

0le 2 then lim

120590rarr120585 = 0 and if 119877

0gt 2 then

lim120590rarr

120585 = 120585

2 where 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(48)

Proof Since

119891 (119878

0minus 120585 120585)

120585

= 120578

(49)


120578 =


2minus 2120590

2(120583 + 120574)

120590

2

(50)


0 0)120597119868) when 0 lt 120590 lt


119889120578

119889120590

=

minus2120573

120590

3+

2 (120583 + 120574)

120590radic120573

2minus 2120590

2(120583 + 120574)

+

2radic120573

2minus 2120590

2(120583 + 120574)

120590

3

=

2120573

2minus 2120590

2(120583 + 120574) minus 2120573

radic120573

2minus 2120590

2(120583 + 120574)

120590

3radic120573

2minus 2120590

2(120583 + 120574)

(51)


Since

[2120573

2minus 2120590

2(120583 + 120574)]

2


2minus 2120590

2(120583 + 120574))

2

= 4120590

4(120583 + 120574)

2gt 0

(52)






120585


120585 = 120585

1and lim

120590rarr120585 = 120585

2 We have

lim120590rarr0

120578 = lim120590rarr0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

=

120583 + 120574

120573

(53)


(119891(119878

0minus 120585 120585)120585) = lim

120590rarr0120578 = (120583 + 120574)120573 This

shows that 119891(1198780 minus 1205851 120585

1)120585



0minus119868

lowast 119868

lowast)119868

lowast= (120583+120574)120573 Hence 120585

1= 119868


120585 = 119868


lim120590rarr

120578 =


2minus 2

2(120583 + 120574)

2=

(120597119891 (119878

0 0) 120597119868) (120573 (120597119891 (119878

0 0) 120597119868) minus

1003816

1003816

1003816

1003816

1003816

120573 (120597119891 (119878

0 0) 120597119868) minus 2 (120583 + 120574)

1003816

1003816

1003816

1003816

1003816

)

2 (120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574))

(54)

If 1 le 119877


120590rarr120578 =

120597119891(119878

0 0)120597119868 Hence

lim120590rarr

119891 (S0 minus 120585 120585)120585

=

120597119891 (119878

0 0)

120597119868

(55)



(54)

lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(56)

which implies

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(57)


120585 = 120585

2 and 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

(119877

0minus 1)

(58)









2gt 0 by



0gt 2 then when 120590 gt 120590

1 where 120590

1is given in







lowastisin ( 120590




lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

(59)

namely

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

=

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(60)


119891 (119878

0minus 120585

2 120585

2)

120585

2

gt

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(61)


lowast



119877

0gt 1 whether we




119877




2

+ one has lim











0le 119878 + 119868 le 119878




119863 = (119878 119868) isin Γ

1

119886

lt 119878 lt 119878

0minus

1

119886

1

119886

lt 119868 lt 119878

0minus

1

119886

(62)


119860 (119878 119868) = (

120590

2119891

2(119878 119868) minus120590

2119891

2(119878 119868)

minus120590

2119891

2(119878 119868) 120590

2119891

2(119878 119868)

) (63)


2(119891(1119886 119878

0minus

1119886)(119886119878

0minus 1))


119881 (119878 119868) = Ψ

1(119868) + Ψ

2(119878 119868) + Ψ

3(119878) (64)

where

Ψ

1(119868) =

1

V119868

minusV

Ψ

2(119878 119868) =

1

V119868

minusV(119878

0minus 119878)

Ψ

3(119878) =

1

119878

(65)


have

119871Ψ

1= minus119868

minus(V+1)(120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119868

minus(V+2)119891

2(119878 119868) le 119868

minusV(120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

)

+ 119868

minusV120573(

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

)

(66)


120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

=

1

120601

120597119891 (120585 120601)

120597119878

(119878

0minus 119878)

+ (

119891 (120585 120601)

120601

2minus

1

120601

120597119891 (120585 120601)

120597119868

) 119868

le 119872

1(119878

0minus 119878) +119872

2119868 +119872

3119877

(67)


119872

1= max(119878119868)isinΓ

1

119868

120597119891 (119878 119868)

120597119878

119872

2= max(119878119868)isinΓ

119891 (119878 119868)

119868

2minus

1

119868

120597119891 (119878 119868)

120597119868

(68)


119871Ψ

1le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 120573119872

1(119878

0minus 119878) 119868

minusV+ 120573119872

2119868

1minusV

(69)


119871Ψ

2= minus

1

V119868


minus(V+1)(119878

0

minus 119878) (120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119891

2(119878 119868) 119868

minus(V+2)(119878


1

2

119868

minus(V+1)120590

2119891

2(119878 119868)

= minus

1

V119868

minusV(120583 (119878

0minus 119878) minus 120573119891 (119878 119868) + 120574119868)

minus 119868

minusV(119878

0minus 119878) (120573

119891 (119878 119868)

119868

minus (120583 + 120572 + 120574)) +

1

2

(1 + V)

sdot 120590

2(

119891 (119878 119868)

119868

)

2

119868

minusV(119878

0minus 119878) minus 120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusV


= 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)


119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)




119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)


120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)





1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)







119877

0= (120573 minus (12)120590




0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+



0of deterministic

model (1)






0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















1

119905

int

119905

0

119878 (119904) 119889119904 ge

Λ

120573119872

119878+ 120583

(33)




119877

0= (120573minus(12)120590




119877

0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574 + 120572) minus 120590

2(ℎ(119878

0)119892

1015840(0))

22(120583 +







119868 (119905) ge 120585


119878 (119905) ge 120585

as

(34)




satisfies


119868 (119905) ge 120585 119886119904 (35)


119868 (119905) le 120585 119886119904 (36)


119891 (119878

0minus 120585 120585)

120585

=

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(37)


(119878(119905)+119868(119905)) = 119878


log 119868 (119905) = log 119868 (0) + int119905

0

[

[

120573

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

minus (120583 + 120574) minus

120590

2

2

(

119891 (119878

0minus 119868 (119904) 119868 (119904))

119868 (119904)

)

2

]

]

119889119904

+ int

119905

0

120590

119891 (119878 (119905) 119868 (119904))

119868 (119904)

119889119861 (119904)

(38)


log 119868 (119905) = log 119868 (0) + int119905

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(39)


119877


119892(

120597119891 (119878

0 0)

120597119868

) = minus

120590

2

2

(

120597119891 (119878

0 0)

120597119868

)

2

+ 120573

120597119891 (119878

0 0)

120597119868

minus (120583 + 120574) gt 0

(40)


0 0)120597119868)

which is

120578 =

120583 + 120574

120573

120590 = 0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

120590 gt 0

(41)


lim119868rarr0+

119906 (119868) = lim119868rarr0+

119891 (119878

0minus 119868 119868)

119868

=

120597119891 (119878

0 0)

120597119868

(42)



2 and 119892(120573120590

2) gt

119892(120597119891(119878


2From 120578 isin (0 120597119891(119878

0 0)120597119868) and 119892(120578) = 0 we have 120578 lt 120573120590

2Hence 0 lt 119868 lt 120585 lt 1198780




When 1205902 le 120573(120597119891(119878





(120585 119878

0)



1) gt 120576

0

where Ω1= lim sup


1


1(120596) ge 119879

0such that

119868 (119905) le 120585 minus 120576

0forall119905 ge 119879

1 (43)


0)) gt 0 for all 119905 ge 119879


119905 ge 119879

1

log 119868 (119905) ge log 119868 (0) + int1198791

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 minus 120576

0)) (119905 minus 119879

1)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(44)






2) gt 120576

1 where

Ω

2= lim inf


2 there is

119879

2= 119879

2(120596) ge 119879

0such that

119868 (119905) ge 120585 + 120576

1forall119905 ge 119879

2 (45)



119905 ge 119879

2

log 119868 (119905) = log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

1198792

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 + 120576

1)) (119905 minus 119879

2)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(46)


119905rarrinfin(log 119868(119905)119905) le 119892(119906(120585 + 120576





2




2

+





119877


0=

120573(120597119891(119878



0 lt 120590 lt

radic2 (120583 + 120574) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

fl

(47)


120590rarr0120585 = 119868


(d) If 1 le 119877

0le 2 then lim

120590rarr120585 = 0 and if 119877

0gt 2 then

lim120590rarr

120585 = 120585

2 where 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(48)

Proof Since

119891 (119878

0minus 120585 120585)

120585

= 120578

(49)


120578 =


2minus 2120590

2(120583 + 120574)

120590

2

(50)


0 0)120597119868) when 0 lt 120590 lt


119889120578

119889120590

=

minus2120573

120590

3+

2 (120583 + 120574)

120590radic120573

2minus 2120590

2(120583 + 120574)

+

2radic120573

2minus 2120590

2(120583 + 120574)

120590

3

=

2120573

2minus 2120590

2(120583 + 120574) minus 2120573

radic120573

2minus 2120590

2(120583 + 120574)

120590

3radic120573

2minus 2120590

2(120583 + 120574)

(51)


Since

[2120573

2minus 2120590

2(120583 + 120574)]

2


2minus 2120590

2(120583 + 120574))

2

= 4120590

4(120583 + 120574)

2gt 0

(52)






120585


120585 = 120585

1and lim

120590rarr120585 = 120585

2 We have

lim120590rarr0

120578 = lim120590rarr0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

=

120583 + 120574

120573

(53)


(119891(119878

0minus 120585 120585)120585) = lim

120590rarr0120578 = (120583 + 120574)120573 This

shows that 119891(1198780 minus 1205851 120585

1)120585



0minus119868

lowast 119868

lowast)119868

lowast= (120583+120574)120573 Hence 120585

1= 119868


120585 = 119868


lim120590rarr

120578 =


2minus 2

2(120583 + 120574)

2=

(120597119891 (119878

0 0) 120597119868) (120573 (120597119891 (119878

0 0) 120597119868) minus

1003816

1003816

1003816

1003816

1003816

120573 (120597119891 (119878

0 0) 120597119868) minus 2 (120583 + 120574)

1003816

1003816

1003816

1003816

1003816

)

2 (120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574))

(54)

If 1 le 119877


120590rarr120578 =

120597119891(119878

0 0)120597119868 Hence

lim120590rarr

119891 (S0 minus 120585 120585)120585

=

120597119891 (119878

0 0)

120597119868

(55)



(54)

lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(56)

which implies

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(57)


120585 = 120585

2 and 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

(119877

0minus 1)

(58)









2gt 0 by



0gt 2 then when 120590 gt 120590

1 where 120590

1is given in







lowastisin ( 120590




lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

(59)

namely

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

=

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(60)


119891 (119878

0minus 120585

2 120585

2)

120585

2

gt

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(61)


lowast



119877

0gt 1 whether we




119877




2

+ one has lim











0le 119878 + 119868 le 119878




119863 = (119878 119868) isin Γ

1

119886

lt 119878 lt 119878

0minus

1

119886

1

119886

lt 119868 lt 119878

0minus

1

119886

(62)


119860 (119878 119868) = (

120590

2119891

2(119878 119868) minus120590

2119891

2(119878 119868)

minus120590

2119891

2(119878 119868) 120590

2119891

2(119878 119868)

) (63)


2(119891(1119886 119878

0minus

1119886)(119886119878

0minus 1))


119881 (119878 119868) = Ψ

1(119868) + Ψ

2(119878 119868) + Ψ

3(119878) (64)

where

Ψ

1(119868) =

1

V119868

minusV

Ψ

2(119878 119868) =

1

V119868

minusV(119878

0minus 119878)

Ψ

3(119878) =

1

119878

(65)


have

119871Ψ

1= minus119868

minus(V+1)(120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119868

minus(V+2)119891

2(119878 119868) le 119868

minusV(120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

)

+ 119868

minusV120573(

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

)

(66)


120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

=

1

120601

120597119891 (120585 120601)

120597119878

(119878

0minus 119878)

+ (

119891 (120585 120601)

120601

2minus

1

120601

120597119891 (120585 120601)

120597119868

) 119868

le 119872

1(119878

0minus 119878) +119872

2119868 +119872

3119877

(67)


119872

1= max(119878119868)isinΓ

1

119868

120597119891 (119878 119868)

120597119878

119872

2= max(119878119868)isinΓ

119891 (119878 119868)

119868

2minus

1

119868

120597119891 (119878 119868)

120597119868

(68)


119871Ψ

1le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 120573119872

1(119878

0minus 119878) 119868

minusV+ 120573119872

2119868

1minusV

(69)


119871Ψ

2= minus

1

V119868


minus(V+1)(119878

0

minus 119878) (120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119891

2(119878 119868) 119868

minus(V+2)(119878


1

2

119868

minus(V+1)120590

2119891

2(119878 119868)

= minus

1

V119868

minusV(120583 (119878

0minus 119878) minus 120573119891 (119878 119868) + 120574119868)

minus 119868

minusV(119878

0minus 119878) (120573

119891 (119878 119868)

119868

minus (120583 + 120572 + 120574)) +

1

2

(1 + V)

sdot 120590

2(

119891 (119878 119868)

119868

)

2

119868

minusV(119878

0minus 119878) minus 120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusV


= 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)


119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)




119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)


120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)





1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)







119877

0= (120573 minus (12)120590




0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+



0of deterministic

model (1)






0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















(120585 119878

0)



1) gt 120576

0

where Ω1= lim sup


1


1(120596) ge 119879

0such that

119868 (119905) le 120585 minus 120576

0forall119905 ge 119879

1 (43)


0)) gt 0 for all 119905 ge 119879


119905 ge 119879

1

log 119868 (119905) ge log 119868 (0) + int1198791

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 minus 120576

0)) (119905 minus 119879

1)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(44)






2) gt 120576

1 where

Ω

2= lim inf


2 there is

119879

2= 119879

2(120596) ge 119879

0such that

119868 (119905) ge 120585 + 120576

1forall119905 ge 119879

2 (45)



119905 ge 119879

2

log 119868 (119905) = log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

1198792

119892 (119906 (119868 (119904))) 119889119904

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

le log 119868 (0) + int1198792

0

119892 (119906 (119868 (119904))) 119889119904

+ 119892 (119906 (120585 + 120576

1)) (119905 minus 119879

2)

+ int

119905

0

120590

119891 (119878 (119904) 119868 (119904))

119868 (119904)

119889119861 (119904)

(46)


119905rarrinfin(log 119868(119905)119905) le 119892(119906(120585 + 120576





2




2

+





119877


0=

120573(120597119891(119878



0 lt 120590 lt

radic2 (120583 + 120574) (119877

0minus 1)

120597119891 (119878

0 0) 120597119868

fl

(47)


120590rarr0120585 = 119868


(d) If 1 le 119877

0le 2 then lim

120590rarr120585 = 0 and if 119877

0gt 2 then

lim120590rarr

120585 = 120585

2 where 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(48)

Proof Since

119891 (119878

0minus 120585 120585)

120585

= 120578

(49)


120578 =


2minus 2120590

2(120583 + 120574)

120590

2

(50)


0 0)120597119868) when 0 lt 120590 lt


119889120578

119889120590

=

minus2120573

120590

3+

2 (120583 + 120574)

120590radic120573

2minus 2120590

2(120583 + 120574)

+

2radic120573

2minus 2120590

2(120583 + 120574)

120590

3

=

2120573

2minus 2120590

2(120583 + 120574) minus 2120573

radic120573

2minus 2120590

2(120583 + 120574)

120590

3radic120573

2minus 2120590

2(120583 + 120574)

(51)


Since

[2120573

2minus 2120590

2(120583 + 120574)]

2


2minus 2120590

2(120583 + 120574))

2

= 4120590

4(120583 + 120574)

2gt 0

(52)






120585


120585 = 120585

1and lim

120590rarr120585 = 120585

2 We have

lim120590rarr0

120578 = lim120590rarr0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

=

120583 + 120574

120573

(53)


(119891(119878

0minus 120585 120585)120585) = lim

120590rarr0120578 = (120583 + 120574)120573 This

shows that 119891(1198780 minus 1205851 120585

1)120585



0minus119868

lowast 119868

lowast)119868

lowast= (120583+120574)120573 Hence 120585

1= 119868


120585 = 119868


lim120590rarr

120578 =


2minus 2

2(120583 + 120574)

2=

(120597119891 (119878

0 0) 120597119868) (120573 (120597119891 (119878

0 0) 120597119868) minus

1003816

1003816

1003816

1003816

1003816

120573 (120597119891 (119878

0 0) 120597119868) minus 2 (120583 + 120574)

1003816

1003816

1003816

1003816

1003816

)

2 (120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574))

(54)

If 1 le 119877


120590rarr120578 =

120597119891(119878

0 0)120597119868 Hence

lim120590rarr

119891 (S0 minus 120585 120585)120585

=

120597119891 (119878

0 0)

120597119868

(55)



(54)

lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(56)

which implies

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(57)


120585 = 120585

2 and 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

(119877

0minus 1)

(58)









2gt 0 by



0gt 2 then when 120590 gt 120590

1 where 120590

1is given in







lowastisin ( 120590




lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

(59)

namely

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

=

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(60)


119891 (119878

0minus 120585

2 120585

2)

120585

2

gt

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(61)


lowast



119877

0gt 1 whether we




119877




2

+ one has lim











0le 119878 + 119868 le 119878




119863 = (119878 119868) isin Γ

1

119886

lt 119878 lt 119878

0minus

1

119886

1

119886

lt 119868 lt 119878

0minus

1

119886

(62)


119860 (119878 119868) = (

120590

2119891

2(119878 119868) minus120590

2119891

2(119878 119868)

minus120590

2119891

2(119878 119868) 120590

2119891

2(119878 119868)

) (63)


2(119891(1119886 119878

0minus

1119886)(119886119878

0minus 1))


119881 (119878 119868) = Ψ

1(119868) + Ψ

2(119878 119868) + Ψ

3(119878) (64)

where

Ψ

1(119868) =

1

V119868

minusV

Ψ

2(119878 119868) =

1

V119868

minusV(119878

0minus 119878)

Ψ

3(119878) =

1

119878

(65)


have

119871Ψ

1= minus119868

minus(V+1)(120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119868

minus(V+2)119891

2(119878 119868) le 119868

minusV(120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

)

+ 119868

minusV120573(

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

)

(66)


120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

=

1

120601

120597119891 (120585 120601)

120597119878

(119878

0minus 119878)

+ (

119891 (120585 120601)

120601

2minus

1

120601

120597119891 (120585 120601)

120597119868

) 119868

le 119872

1(119878

0minus 119878) +119872

2119868 +119872

3119877

(67)


119872

1= max(119878119868)isinΓ

1

119868

120597119891 (119878 119868)

120597119878

119872

2= max(119878119868)isinΓ

119891 (119878 119868)

119868

2minus

1

119868

120597119891 (119878 119868)

120597119868

(68)


119871Ψ

1le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 120573119872

1(119878

0minus 119878) 119868

minusV+ 120573119872

2119868

1minusV

(69)


119871Ψ

2= minus

1

V119868


minus(V+1)(119878

0

minus 119878) (120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119891

2(119878 119868) 119868

minus(V+2)(119878


1

2

119868

minus(V+1)120590

2119891

2(119878 119868)

= minus

1

V119868

minusV(120583 (119878

0minus 119878) minus 120573119891 (119878 119868) + 120574119868)

minus 119868

minusV(119878

0minus 119878) (120573

119891 (119878 119868)

119868

minus (120583 + 120572 + 120574)) +

1

2

(1 + V)

sdot 120590

2(

119891 (119878 119868)

119868

)

2

119868

minusV(119878

0minus 119878) minus 120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusV


= 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)


119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)




119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)


120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)





1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)







119877

0= (120573 minus (12)120590




0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+



0of deterministic

model (1)






0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















Since

[2120573

2minus 2120590

2(120583 + 120574)]

2


2minus 2120590

2(120583 + 120574))

2

= 4120590

4(120583 + 120574)

2gt 0

(52)






120585


120585 = 120585

1and lim

120590rarr120585 = 120585

2 We have

lim120590rarr0

120578 = lim120590rarr0

2 (120583 + 120574)

120573 +radic120573

2minus 2120590

2(120583 + 120574)

=

120583 + 120574

120573

(53)


(119891(119878

0minus 120585 120585)120585) = lim

120590rarr0120578 = (120583 + 120574)120573 This

shows that 119891(1198780 minus 1205851 120585

1)120585



0minus119868

lowast 119868

lowast)119868

lowast= (120583+120574)120573 Hence 120585

1= 119868


120585 = 119868


lim120590rarr

120578 =


2minus 2

2(120583 + 120574)

2=

(120597119891 (119878

0 0) 120597119868) (120573 (120597119891 (119878

0 0) 120597119868) minus

1003816

1003816

1003816

1003816

1003816

120573 (120597119891 (119878

0 0) 120597119868) minus 2 (120583 + 120574)

1003816

1003816

1003816

1003816

1003816

)

2 (120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574))

(54)

If 1 le 119877


120590rarr120578 =

120597119891(119878

0 0)120597119868 Hence

lim120590rarr

119891 (S0 minus 120585 120585)120585

=

120597119891 (119878

0 0)

120597119868

(55)



(54)

lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(56)

which implies

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

120597119891 (119878

0 0) 120597119868

119877

0minus 1

(57)


120585 = 120585

2 and 120585

2satisfies

119891 (119878

0minus 120585

2 120585

2)

120585

2

=

120597119891 (119878

0 0) 120597119868

(119877

0minus 1)

(58)









2gt 0 by



0gt 2 then when 120590 gt 120590

1 where 120590

1is given in







lowastisin ( 120590




lim120590rarr

120578 =

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

(59)

namely

lim120590rarr

119891 (119878

0minus 120585 120585)

120585

=

(120597119891 (119878

0 0) 120597119868) (120583 + 120574)

120573 (120597119891 (119878

0 0) 120597119868) minus (120583 + 120574)

gt

120583 + 120574

120573

=

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(60)


119891 (119878

0minus 120585

2 120585

2)

120585

2

gt

119891 (119878

0minus 119868

lowast 119868

lowast)

119868

lowast

(61)


lowast



119877

0gt 1 whether we




119877




2

+ one has lim











0le 119878 + 119868 le 119878




119863 = (119878 119868) isin Γ

1

119886

lt 119878 lt 119878

0minus

1

119886

1

119886

lt 119868 lt 119878

0minus

1

119886

(62)


119860 (119878 119868) = (

120590

2119891

2(119878 119868) minus120590

2119891

2(119878 119868)

minus120590

2119891

2(119878 119868) 120590

2119891

2(119878 119868)

) (63)


2(119891(1119886 119878

0minus

1119886)(119886119878

0minus 1))


119881 (119878 119868) = Ψ

1(119868) + Ψ

2(119878 119868) + Ψ

3(119878) (64)

where

Ψ

1(119868) =

1

V119868

minusV

Ψ

2(119878 119868) =

1

V119868

minusV(119878

0minus 119878)

Ψ

3(119878) =

1

119878

(65)


have

119871Ψ

1= minus119868

minus(V+1)(120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119868

minus(V+2)119891

2(119878 119868) le 119868

minusV(120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

)

+ 119868

minusV120573(

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

)

(66)


120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

=

1

120601

120597119891 (120585 120601)

120597119878

(119878

0minus 119878)

+ (

119891 (120585 120601)

120601

2minus

1

120601

120597119891 (120585 120601)

120597119868

) 119868

le 119872

1(119878

0minus 119878) +119872

2119868 +119872

3119877

(67)


119872

1= max(119878119868)isinΓ

1

119868

120597119891 (119878 119868)

120597119878

119872

2= max(119878119868)isinΓ

119891 (119878 119868)

119868

2minus

1

119868

120597119891 (119878 119868)

120597119868

(68)


119871Ψ

1le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 120573119872

1(119878

0minus 119878) 119868

minusV+ 120573119872

2119868

1minusV

(69)


119871Ψ

2= minus

1

V119868


minus(V+1)(119878

0

minus 119878) (120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119891

2(119878 119868) 119868

minus(V+2)(119878


1

2

119868

minus(V+1)120590

2119891

2(119878 119868)

= minus

1

V119868

minusV(120583 (119878

0minus 119878) minus 120573119891 (119878 119868) + 120574119868)

minus 119868

minusV(119878

0minus 119878) (120573

119891 (119878 119868)

119868

minus (120583 + 120572 + 120574)) +

1

2

(1 + V)

sdot 120590

2(

119891 (119878 119868)

119868

)

2

119868

minusV(119878

0minus 119878) minus 120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusV


= 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)


119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)




119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)


120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)





1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)







119877

0= (120573 minus (12)120590




0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+



0of deterministic

model (1)






0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119877




2

+ one has lim











0le 119878 + 119868 le 119878




119863 = (119878 119868) isin Γ

1

119886

lt 119878 lt 119878

0minus

1

119886

1

119886

lt 119868 lt 119878

0minus

1

119886

(62)


119860 (119878 119868) = (

120590

2119891

2(119878 119868) minus120590

2119891

2(119878 119868)

minus120590

2119891

2(119878 119868) 120590

2119891

2(119878 119868)

) (63)


2(119891(1119886 119878

0minus

1119886)(119886119878

0minus 1))


119881 (119878 119868) = Ψ

1(119868) + Ψ

2(119878 119868) + Ψ

3(119878) (64)

where

Ψ

1(119868) =

1

V119868

minusV

Ψ

2(119878 119868) =

1

V119868

minusV(119878

0minus 119878)

Ψ

3(119878) =

1

119878

(65)


have

119871Ψ

1= minus119868

minus(V+1)(120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119868

minus(V+2)119891

2(119878 119868) le 119868

minusV(120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

)

+ 119868

minusV120573(

120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

)

(66)


120597119891 (119878

0 0)

120597119868

minus

119891 (119878 119868)

119868

=

1

120601

120597119891 (120585 120601)

120597119878

(119878

0minus 119878)

+ (

119891 (120585 120601)

120601

2minus

1

120601

120597119891 (120585 120601)

120597119868

) 119868

le 119872

1(119878

0minus 119878) +119872

2119868 +119872

3119877

(67)


119872

1= max(119878119868)isinΓ

1

119868

120597119891 (119878 119868)

120597119878

119872

2= max(119878119868)isinΓ

119891 (119878 119868)

119868

2minus

1

119868

120597119891 (119878 119868)

120597119868

(68)


119871Ψ

1le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 120573119872

1(119878

0minus 119878) 119868

minusV+ 120573119872

2119868

1minusV

(69)


119871Ψ

2= minus

1

V119868


minus(V+1)(119878

0

minus 119878) (120573119891 (119878 119868) minus (120583 + 120572 + 120574) 119868) +

1

2

(1 + V)

sdot 120590

2119891

2(119878 119868) 119868

minus(V+2)(119878


1

2

119868

minus(V+1)120590

2119891

2(119878 119868)

= minus

1

V119868

minusV(120583 (119878

0minus 119878) minus 120573119891 (119878 119868) + 120574119868)

minus 119868

minusV(119878

0minus 119878) (120573

119891 (119878 119868)

119868

minus (120583 + 120572 + 120574)) +

1

2

(1 + V)

sdot 120590

2(

119891 (119878 119868)

119868

)

2

119868

minusV(119878

0minus 119878) minus 120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusV


= 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)


119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)




119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)


120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)





1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)







119877

0= (120573 minus (12)120590




0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+



0of deterministic

model (1)






0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















= 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574 minus 120573

119891 (119878 119868)

119868

+

1

2

(1 + V) 1205902 (119891 (119878 119868)

119868

)

2

) + 119868

1minusV(

120573

V119891 (119878 119868)

119868

minus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

) minus

120575

V119868

minusV+1le 119868

minusV(119878

0minus 119878)

sdot (minus

120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

) +

120573

V120597119891 (119878

0 0)

120597119868

sdot 119868

1minusVminus

1

2

120590

2(

119891 (119878 119868)

119868

)

2

119868

1minusVminus

120575

V119868

minusV+1

(70)


119871Ψ

3= minus

1

119878

2(Λ minus 120583119878 minus 120573119891 (119878 119868) + 120574119868) +

1

119878

3120590

2119891

2(119878 119868)

le minus

Λ

119878

2+

120583

119878

+ 120573

119891 (119878 119868)

119878

1

119878

+ 120590

2(

119891 (119878 119868)

119878

)

21

119878

minus

120574

119878

2119868 le minus

Λ

119878

2+

1

119878

(120583 + 120573119872

0+ 120590

2119872

2

0) minus

120574

119878

2119868

le minus

Λ

2119878

2+

1

2Λ

(120583 + 120573119872

0+ 120590

2119872

2

0)

2

minus

120574

119878

2119868

(71)




119871119881 le 119868

minusV(120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

) + 119868

minusV(119878


120583

V+ 120583 + 120572 + 120574

+

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1) + (119878

0)

1minusV

sdot (120573119872

2+

120573

V120597119891 (119878

0 0)

120597119868

) minus

Λ

2119878

2+

1

2120583

(120583 + 120573119872

0

+ 120590

2119872

2

0)

2

(72)

Since

120583 + 120572 + 120574 +

1

2

120590

2(

120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

(73)


120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

minus 120573

120597119891 (119878

0 0)

120597119868

lt 0

minus

120583

V+ 120583 + 120572 + 120574 +

1

2

(1 + V) 1205902(120597119891 (119878

0 0)

120597119868

)

2

+ 120573119872

1lt 0

(74)





1

119879

int

119879

0

(119878 (119905) 119868 (119905)) 119889119905 = int

Γ

(119878 119868) 120585 (119889 (119878 119868))

= 1

(76)







119877

0= (120573 minus (12)120590




0= 120573ℎ(119878

0)119892

1015840(0)(120583 + 120574+120572) minus120590

2(ℎ(119878

0)119892

1015840(0))

22(120583+ 120574+



0of deterministic

model (1)






0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 50 100 150 200 250 300minus05

0

05

1

15

2

Time T

I(t)


(a)

Time T0 50 100 150 200 250 300

minus02

0

02

04

06

08

1

12

14

16

18

I(t)


(b)




1 where 120590

1is given




119878

119896+1= 119878

119896+ [Λ minus

120573119878

119896119868

119896

1 + 120572119868

119896

+ 120574119868

119896minus 120583119878

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

119868

119896+1= 119868

119896+ [

120573119878

119896119868

119896

1 + 120572119868

119896

minus (120583 + 120574) 119868

119896]Δ119905

+

119878

119896119868

119896

1 + 120572119868

119896

[120590120585

119896radic

Δ119905 +

1

2

120590

2(120585

2

119896minus 1)Δ119905]

(77)





0= 06715 lt 1

120573119878

0minus 120590






119877




We have

119877

0= 1200 119877

0= 12500 and 120585 = 01037


7 Discussion



Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















Time T0 50 100 150 200

0

01

02

03

04

05

06

07I(t)


(a)

Time T


0 50 100 150 2000

01

02

03

04

05

06

07

08

I(t)

(b)


Time T0 50 100 150 200

0

005

01

015

02

025

03

035

04

045

05

I(t)


(a)

Time T0 50 100 150 200

0

005

01

015

02

025

03

035

I(t)


(b)






0gt

119877

0 This shows that

when 119877






Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















Competing Interests


Acknowledgments


References

























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















research article the dynamical behaviors in a stochastic...

Documents