Competing species models with an infectious disease

Roberto A. Saenz; Herbert W. Hethcote

doi:10.3934/mbe.2006.3.219

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2006, Volume 3, Issue 1: 219-235. Doi: 10.3934/mbe.2006.3.219

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Competing species models with an infectious disease

Roberto A. Saenz^1, and
Herbert W. Hethcote^1,

1.
Applied Mathematical and Computational Sciences, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242

Received: December 31, 2004

Revised: March 31, 2005

Published: October 2005

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Abstract

The frequency-dependent (standard) form of the incidence is used for the transmission dynamics of an infectious disease in a competing species model. In the global analysis of the SIS model with the birth rate independent of the population size, a modified reproduction number $\mathbf{R}_1$ determines the asymptotic behavior, so that the disease dies out if $\mathbf{R}_1 \leq 1$ and approaches a globally attractive endemic equilibrium if $\mathbf{R}_1 > 1$. Because the disease- reduced reproduction and disease-related death rates are often different in two competing species, a shared disease can change the outcome of the competition. Models of SIR and SIRS type are also considered. A key result in all of these models with the frequency-dependent incidence is that the disease must either die out in both species or remain endemic in both species.

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Mathematics Subject Classification: 92D30, 92D40.

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