Download Dynamics of Infectious Diseases

Dynamics of Infectious Diseases
Using Lotka-Volterra equations?
VS
Predator
dy
 y ( c  dx )
dt
Prey
dx
 x(a  by )
dt
Full model
c
dN
Susceptible
d
a
Infectious
d
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
b
Removed
d
Reduced model (Classic KermackMcKendrick Model)
Susceptible
a
Infectious
b
Removed
where
a is the infection rate
b is the removal rate of infectives
dS
 aSI
dt
dI
 aSI  bI
dt
dR
 bI
dt
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
> 0 if S0 >
< 0 if S0 <
“THRESHOLD EFFECT”
b
a
b
a
dI
( aS  b) I

b

 1  ,  
dS
aSI
S
a
Integrating the equation,
I  S   ln S = 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
dS
  aSI *  bI
dt
dI
 aSI * bI
dt
b
a
Male
Susceptible, S
Infectious, I
Female
Susceptible, S*
Infectious, I*
a*
dS *
 a * S * I  b * I *
dt
where
a,a* is the infection rate
b,b* is the removal rate of infectives
b*
dI *
 a *S * I  b* I *
dt
• Since we have the condition S(t)+I(t)=N and
S*(t)+I*(t)=N*, we can simplify the equations to
dI
 aI * ( N  I )  bI
dt
dI *
 a * I ( N *  I *)  b * I *
dt
• Equating both equations to zero, we can obtain the
steady states
NN *   *
Is    N *
NN *   *
I *s   *  N
b
b*
  , * 
a
a*
AIDS (Autoimmune Deficiency Syndrome)
B

Natural Death
Susceptible X
c
Infectious Y
p

Natural Death
(1  p )

Natural
Death
AIDS A
d

Disease induced Death
Seropositive Z
(non-infectious)

Natural Death
dX
Y
 B  X  cX ,  
dt
N
dY
 cX  (   )Y
dt
dA
 p Y  ( d   ) A
dt
dZ
 (1  p )Y  Z
dt
N (t )  X (t )  Y (t )  Z (t )  A(t )