Eradication and Control
Let R be the effective reproductive rate of a microparasite:
Criterion for eradication:
R 1
Immunization Programmes
If we immunized a proportion, p, of the susceptible population,
the effective reproductive rate is at most:
R  R0 (1  p)
Where R0 is the basic reproductive rate.
Clearly, if the following is true, then the
eradication criterion is achieved.
R  R0 (1  p)  1
Eradication and Control
Solving p in terms of R0 , we obtain p  1  (1/ R0 )
Where the critical p value, pc  1  (1/ R0 )
In Terms of A and L
If we subsitute the relationship R0  L / A, we obtain the following:
p  1  ( A / L) and pc  1  ( A / L)
Where A is the average age of infection and L is the expected life span.
Immunization and new equilibria



We can modify the SIR cohort model for
immunization programme.
If we assume that fraction p of new borns are
successfully immunized for an infection, we
would only have to change the initial values of
X and Z.
X(0)=(1-p)N(0) and Z(0)=pN(0)
SIR with Immunization Programme
Solving the new system of equations, we obtain:
X (a )  (1- p) N (0)l ( a) exp(  ' a), where  ' is
the new force of infection.
We can now find the new R 0 by finding the new
equilibrium susceptible proportion x*.
SIR with Immunization Programme
Since x*=  X ( a ) da /  N ( a )da
For Type I Survival:
(1  p )(1  e   ' L )
x* 
'L
For Type II Survival:
(1  p ) 
x* 
 ' 
SIR with Immunization Programme
Since R 0 =1/x*
For Type I Survival:
R0 
'L
 ' L
(1  p )(1  e )
For Type II Survival:
 ' 
R0 
(1  p ) 
SIR with Immunization Programme
We can estimate  ' by first estimating R0 , then solving  '
in terms of R0 using the previous equations.
For Type II Survival,
1
 '   R0 (1   p)
R0
Subsituting 1-1/ R0  pc ,
 '   R0 ( pc  p)
Age Specific Immunization
If we take the general case of immunization at age b,
we can derive these formulas for R 0 :
Type I:
R0 
 'L
1  pe   'b  (1  p)e   ' L
Type II:
1  ( '/  )
R0 
1  pe  (  '   )b
Average Age of Infection
For the new average age of infection, we simply
take the first moment of the new lambda*X(a).
We obtain:
1  (1   ' L)e  ' L
A' 
 '(1  e  ' L )
Type I
1
A
A' 

 '  1  p
Type II
Average Age of Infection
Programme Specific Criteria

Before, we assumed immunization for the
entire population.

What is our prediction for the age specific
immunization program?

We can obtain that information by taking the
limit of the infection force to 0.
Programme Specific Criteria
For Type I:
1  (1/ R0 ) L  A
pc 

1  (b / L) L  b
For Type II:
pc  [1  (1/ R0 )]e
b/ L
 [1  ( A / L)]e
b/ L
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Immunization control