Chapter 13 TIME DELAYS IN EPIDEMIC MODELS Modeling and Numerical Considerations

Continuous time deterministic epidemic models are traditionally formulated as systems of ordinary differential equations for the numbers of individuals in various disease states, with the sojourn time in a state being exponentially distributed. Time delays are introduced to model constant sojourn times in a state, for example, the infective or immune state. Models then become delay-differential and/or integral equations. For a review of some epidemic models with delay see van den Driessche [228]. More generally, an arbitrarily distributed sojourn time in a state, for example, the infective or immune state, is used by some authors (see [69] and the references therein). When introduced in an explicit way, time delays may change the qualitative behavior of a model; for example, an epidemic model with generalized logistic dynamics can have periodic solutions when the time in the infective stage is constant [112]. Qualitative differences that arise from including time delay in an explicit way in models that include vertical transmission are explored in [38, Chapter 4]. In population biology, a maturation time delay is used to explain observed oscillations [192]. This delay is included in an epidemic model by Cooke, van den Driessche and