Simple Models for the Transmission of Microparasites Between Host Populations Living on non Coincident Spatial Domains
Résumé
The goal of this chapter is to provide a simple deterministic mathe- matical approach to modeling the transmission of microparasites between two host populations living on distinct spatial domains. We shall consider two prototypi- cal situations: (1), a vector borne disease and, (2), an environmentally transmitted disease. In our models direct horizontal criss{cross transmission from infectious in- dividuals of one population to susceptibles of the other one does not occur. Instead parasite transmission takes place either through indirect criss{cross contacts be- tween infective vectors and susceptible individuals and vice{versa in case (1), and through indirect contacts between susceptible hosts and the contaminated part of the environment and vice{versa in case (2). We shall also assume the microparasite is benign in one of the host populations, a reservoir, that is it has no impact on demography and dispersal of individuals. Next we assume it is lethal to the second population. In applications we have in mind the second population is human while the ¯rst one is an animal { avian or rodent { population. Simple mathematical deter- ministic models with spatio{temporal heterogeneities are developed, ranging from basic systems of ODEs for unstructured populations to Reaction{Di®usion mod- els for spatially structured populations to handle heterogeneous environments and populations living in distinct habitats. Besides showing the resulting mathematical problems are well{posed we analyze the existence and stability of endemic states. Under some circumstances, persistence thresholds are given.