Improved uniform persistence for partially diffusive models of infectious diseases: cases of avian influenza and Ebola virus disease

Author:

Covington Ryan1,Patton Samuel1,Walker Elliott1,Yamazaki Kazuo2

Affiliation:

1. Texas Tech University, Department of Mathematics and Statistics, Lubbock, TX, 79409-1042, USA

2. University of Nebraska, Lincoln, Department of Mathematics, 203 Avery Hall, PO Box, 880130, Lincoln, NE 68588-0130, USA

Abstract

<abstract><p>Past works on partially diffusive models of diseases typically rely on a strong assumption regarding the initial data of their infection-related compartments in order to demonstrate uniform persistence in the case that the basic reproduction number $ \mathcal{R}_0 $ is above 1. Such a model for avian influenza was proposed, and its uniform persistence was proven for the case $ \mathcal{R}_0 &gt; 1 $ when all of the infected bird population, recovered bird population and virus concentration in water do not initially vanish. Similarly, a work regarding a model of the Ebola virus disease required that the infected human population does not initially vanish to show an analogous result. We introduce a modification on the standard method of proving uniform persistence, extending both of these results by weakening their respective assumptions to requiring that only one (rather than all) infection-related compartment is initially non-vanishing. That is, we show that, given $ \mathcal{R}_0 &gt; 1 $, if either the infected bird population or the viral concentration are initially nonzero anywhere in the case of avian influenza, or if any of the infected human population, viral concentration or population of deceased individuals who are under care are initially nonzero anywhere in the case of the Ebola virus disease, then their respective models predict uniform persistence. The difficulty which we overcome here is the lack of diffusion, and hence the inability to apply the minimum principle, in the equations of the avian influenza virus concentration in water and of the population of the individuals deceased due to the Ebola virus disease who are still in the process of caring.</p></abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

Applied Mathematics,Computational Mathematics,General Agricultural and Biological Sciences,Modeling and Simulation,General Medicine

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