Some bounds on estimates for reproductive ratios derived from the age-specific force of infection

In this paper we shall look at estimation of reproductive ratios for common childhood infections such as chickenpox, measles, mumps, and hepatitis A with and without a vaccination program. The paper starts with a survey of previous work in this area. We suppose that we are given data in the form of an age-related serological profile with a given vaccination program. This is used to estimate the reproductive ratio and evaluate vaccination campaigns. The effect of different mixing patterns, such as homogeneous mixing, assortative mixing, proportional mixing, and symmetric mixing are discussed. R phi denotes the reproductive ratio when a steady-state vaccination campaign phi is used. Assortative mixing maximizes the reproductive ratio R phi. A mixing pattern which minimizes R phi and a lower bound for R phi for the important symmetric mixing case are found. The most usual situation is that we are given the age-serological profile with no vaccination so that we have bounds for the basic reproductive ratio R0. These results are illustrated with an application to vaccination against hepatitis A in Bulgaria. Numerical evaluations of the effect of different elimination vaccination strategies are examined.