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Simple Models for the Transmission of Microparasites Between Host Populations Living on Noncoincident Spatial Domains

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Structured Population Models in Biology and Epidemiology

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 1936))

The goal of this chapter is to provide a simple mathematical approach to modeling the transmission of microparasites between two host populations living on distinct spatial domains. We shall consider two prototypical situations (1), a vector borne disease and, (2), an environmentally transmitted disease. In our models direct horizontal criss-cross transmission from infectious individuals of one population to susceptibles of the other one does not occur. Instead parasite transmission takes place either through indirect criss-cross contacts between 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 first one is an animal – avian or rodent – population. Simple mathematical deterministic models with spatio-temporal heterogeneities are developed, ranging from basic systems of ODEs for unstructured populations to Reaction-Diffusion models 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.

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Authors and Affiliations

  1. Departments of Engineering Technology and Mathematics, University of Houston, Houston, Texas, 77204-3476, USA

    W. -E. Fitzgibbon

  2. UMR CNRS 5466 MAB & INRIA Futurs Anubis, case 26, Université Victor Segalen Bordeaux 2, 146 rue Léo Saignat, F 33076, Bordeaux Cedex, France

    M. Langlais

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  1. W. -E. Fitzgibbon
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  2. M. Langlais
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Editors and Affiliations

  1. University of Le Havre, 25 rue philippe Lebon, 76058, Le Havre, France

    Pierre Magal

  2. Department of Mathematics, University of Miami, Coral Gables, FL, 33124-4250, USA

    Shigui Ruan

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Fitzgibbon, W.E., Langlais, M. (2008). Simple Models for the Transmission of Microparasites Between Host Populations Living on Noncoincident Spatial Domains. In: Magal, P., Ruan, S. (eds) Structured Population Models in Biology and Epidemiology. Lecture Notes in Mathematics, vol 1936. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78273-5_3

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