Eradication and ControlEradication and Control

1R

Let R be the effective reproductive rate of a microparasite:Let R be the effective reproductive rate of a microparasite:

Criterion for eradication:Criterion for eradication:

Immunization ProgrammesImmunization Programmes

0 (1 )R R p

If we immunized a proportion, p, of the susceptible population, If we immunized a proportion, p, of the susceptible population,

the effective reproductive rate is at most:the effective reproductive rate is at most:

Where R0 is the basic reproductive rate.

0 (1 ) 1R R p

Clearly, if the following is true, then the eradication criterion is achieved.

Eradication and ControlEradication and Control0 0Solving p in terms of , we obtain 1 (1/ )R p R

0Where the critical p value, 1 (1/ )cp R

In Terms of A and LIn Terms of A and L

0If we subsitute the relationship / , we obtain the following:

1 ( / ) and 1 ( / )

Where A is the average age of infection and L is the expected life span.

c

R L A

p A L p A L

Immunization and new equilibriaImmunization and new equilibria

We can modify the SIR cohort model for We can modify the SIR cohort model for immunization programme.immunization programme.

If we assume that fraction p of new borns are If we assume that fraction p of new borns are successfully immunized for an infection, we successfully immunized for an infection, we would only have to change the initial values of would only have to change the initial values of X and Z.X and Z.

X(0)=(1-p)N(0) and Z(0)=pN(0)X(0)=(1-p)N(0) and Z(0)=pN(0)

SIR with Immunization ProgrammeSIR with Immunization Programme

0

Solving the new system of equations, we obtain:

( ) (1- ) (0) ( ) exp( ' ), where ' is

the new force of infection.

We can now find the new R by finding the new

equilibrium susceptible proportion x*.

X a p N l a a

'

Since x*= ( ) / ( )

For Type I Survival:

(1 )(1 )*

'For Type II Survival:

(1 )*

'

L

X a da N a da

p ex

L

px

SIR with Immunization ProgrammeSIR with Immunization Programme

0

0 '

0

Since R =1/x*

For Type I Survival:

'

(1 )(1 )

For Type II Survival:

'

(1 )

L

LR

p e

Rp

SIR with Immunization ProgrammeSIR with Immunization Programme

SIR with Immunization ProgrammeSIR with Immunization Programme

0

0

00

0

0

We can estimate ' by first estimating , then solving '

in terms of using the previous equations.

For Type II Survival,

1' (1 )

Subsituting 1-1/ ,

' ( )c

c

R

R

R pR

R p

R p p

Age Specific ImmunizationAge Specific Immunization

0

0 ' '

0 ( ' )

If we take the general case of immunization at age b,

we can derive these formulas for R :

Type I:

'

1 (1 )

Type II:

1 ( '/ )

1

b L

b

LR

pe p e

Rpe

Average Age of InfectionAverage Age of Infection

'

'

1 (1 ' )'

'(1 )

L

L

L eA

e

For the new average age of infection, we simplyFor the new average age of infection, we simply

take the first moment of the new lambda*X(a).take the first moment of the new lambda*X(a).

We obtain:We obtain:

1'

' 1

AA

p

Type I Type II

Average Age of InfectionAverage Age of Infection

Programme Specific CriteriaProgramme Specific Criteria

Before, we assumed immunization for the Before, we assumed immunization for the entire population.entire population.

What is our prediction for the age specific What is our prediction for the age specific immunization program?immunization program?

We can obtain that information by taking the We can obtain that information by taking the limit of the infection force to 0.limit of the infection force to 0.

Programme Specific CriteriaProgramme Specific Criteria

0

/ /0

For Type I:

1 (1/ )

1 ( / )

For Type II:

[1 (1/ )] [1 ( / )]

c

b L b Lc

R L Ap

b L L b

p R e A L e