American Institute of Mathematical Sciences

Advanced
2006, 3(1): 219-235. doi: 10.3934/mbe.2006.3.219

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, United States, United States

Received  January 2005 Revised  April 2005 Published  November 2005

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.
Keywords: coexistence, mathematical models, epidemiology, dynamics, competing species, ecology, infectious disease, reproduction number..
Mathematics Subject Classification: 92D30, 92D4.
Citation: Roberto A. Saenz, Herbert W. Hethcote. Competing species models with an infectious disease. Mathematical Biosciences & Engineering, 2006, 3 (1) : 219-235. doi: 10.3934/mbe.2006.3.219
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