Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS.
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Transcript of Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS.
Dynamics of Infectious Diseases
Using Lotka-Volterra equations?
Predator Prey
VS
)( byaxdt
dx)( dxcy
dt
dy
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
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
}> 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
a
b
SaSI
IbaS
dS
dI
,1)(
Integrating the equation,
SSI ln = constant
= I0 + S0 – ρ ln S0
• 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.
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.
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
• 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
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