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

R
0
(1
 p )
Where R0 is the basic reproductive rate.
Clearly, if the following is true, then the eradication criterion is achieved.
R

R
0
(1
 p


Eradication and Control
Solving p in terms of R
0
, we obtain p
 
R
0
)
Where the critical p value, p c
 
R
0
)

In Terms of A and L
If we subsitute the relationship R
0

/ , we obtain the following: p
 
A L p c
 
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:
( )
 p N l a
  a
 the new force of infection.
We can now find the new R by finding the new
0 equilibrium susceptible proportion x*.

SIR with Immunization Programme
Since x*=

( ) /

( )
For Type I Survival: x *

(1
 p )(1
 e
 
' L
)

' L
For Type II Survival: x *

(1
 p )


SIR with Immunization Programme
Since R =1/x*
0
For Type I Survival:
R
0


' L
(1
 p )(1
 e
 
' L
)
For Type II Survival:
R
0

(1
 p )


SIR with Immunization Programme

R
0
 in terms of R
0
using the previous equations.
For Type II Survival,

'
 
R
0
(1

1

R
0 p )
Subsituting 1-1/

'
 
R p
0
( c
 p )
R
0
 p c
,

Age Specific Immunization
If we take the general case of immunization at age b, we can derive these formulas for R :
0
Type I:
R
0

1
 pe
 
' b

' L
 
)
 
' L
Type II:
R
0

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:
A '

  
' )

 
' L
'(1
 e
 
' L
)
Type I
A '

1
Type II

1

A p

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: p c


R
0
) L

A
1 ( / )


For Type II: p c
 
R
0
)] e
 