![Page 1: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/1.jpg)
Dynamics of Infectious Diseases
![Page 2: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/2.jpg)
Using Lotka-Volterra equations?
Predator Prey
VS
)( byaxdt
dx)( dxcy
dt
dy
![Page 3: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/3.jpg)
Full model
SusceptibleSusceptible Infectious Removeda b
c
dN
ddd
where a is the infection rate b is the removal rate of infectives c is the rate of individuals losing immunity d is the mortality rate
![Page 4: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/4.jpg)
Reduced model (Classic Kermack-McKendrick Model)
SusceptibleSusceptible Infectious Removeda b
where a is the infection rate b is the removal rate of infectives
aSIdt
dS bIaSI
dt
dI bI
dt
dR
![Page 5: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/5.jpg)
}> 0 if S0 >
< 0 if S0 <a
b
a
b
“THRESHOLD EFFECT”
S(t) +I (t) + R(t) = N
We can set the initial conditions as
S(0)=S0 > 0 , I(0) =I0 > 0 , R(0) =0
![Page 6: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/6.jpg)
a
b
SaSI
IbaS
dS
dI
,1)(
Integrating the equation,
SSI ln = constant
= I0 + S0 – ρ ln S0
![Page 7: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/7.jpg)
• b is the removal rate from the infective class and is measured in unit (1/time)
• Thus, the reciprocal (1/b) is the average period of infectivity.
• is the fraction of population that comes into contact with an infective individual during the period of infectiveness
• The fraction is also known as infection’s contact rate, or intrinsic reproductive rate of disease.
![Page 8: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/8.jpg)
R0 is the basic reproduction rate of the infection, that is the number of infections produced by one primary infection in a whole susceptible population.
![Page 9: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/9.jpg)
Modelling venereal disease
Susceptible, SSusceptible, S Infectious, I
a
b
where a,a* is the infection rate b,b* is the removal rate of infectives
Susceptible, S*Susceptible, S* Infectious, I*
b*
a*Female
Male
bIaSIdt
dS *
*****
IbISadt
dS
bIaSIdt
dI *
*****
IbISadt
dI
![Page 10: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/10.jpg)
• Since we have the condition S(t)+I(t)=N and S*(t)+I*(t)=N*, we can simplify the equations to
• Equating both equations to zero, we can obtain the steady states
bIINaIdt
dI )(* ***)*(*
*IbINIa
dt
dI
*
**
N
NNI s
N
NNI s
*
***
*
**,
a
b
a
b
![Page 11: Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS](https://reader035.vdocuments.us/reader035/viewer/2022062423/5697c0051a28abf838cc4ce1/html5/thumbnails/11.jpg)
AIDS (Autoimmune Deficiency Syndrome)
c
Susceptible X
Infectious Y
Natural Death
AIDS A Seropositive Z(non-infectious)
Disease induced Death Natural Death
Natural Death
Natural Death
B
)()()()()(
)1(
)(
)(
,
tAtZtYtXtN
ZYpdt
dZ
AdYpdt
dA
YcXdt
dYN
YcXXB
dt
dX
p )1( p
d