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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