Basic concepts for the Kermack and McKendrick model with static heterogeneity.

Abstract

In this paper, we consider the infection-age-dependent Kermack-McKendrick model, where host individuals are distributed in a continuous state space. To provide a mathematical foundation for the heterogeneous model, we develop a $L^1$ -framework to formulate basic epidemiological concepts. First, we show the mathematical well-posedness of the basic model under appropriate conditions allowing for unbounded structural variables in an unbounded domain. Next, we define the basic reproduction number and prove pandemic threshold results. We then present a systematic procedure to compute the effective reproduction number and the herd immunity threshold. Finally, we give some illustrative examples and concrete results by using the separable mixing assumption.