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