Basic concepts for the Kermack and McKendrick model with static heterogeneity

In this paper, we consider the infection-age-dependent Kermack--McKendrick model in which 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 the unbounded parameters with non-compact domain. Next we define the basic reproduction number and prove the pandemic threshold theorem. 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.