Dynamics of acquired immunity boosted by exposure to infection

A mathematical model is presented to describe the dynamics of immunity which can be boosted by reexposure to infection. Immunity is assumed to last until a specified interval of time elapses without an exposure. This assumption is incorporated into a compartmental model as a differential-delay equation. When the model is applied to malaria epidemiology, the prevalence of disease among adults is greatest at intermediate rates of infection. Observed age-prevalence curves have shapes similar to those of the predicted curves. Immunity in an individual is formulated in terms of a stochastic process, and an expression for the average duration of immunity is obtained. The average duration of immunity increases with the rate of exposure, but the presence of mortality (or other kinds of removal) shortens the average duration observed, in analogy with the theoryof competing risks.