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.
Citation: Roberto A. Saenz, Herbert W. Hethcote. Competing species models with an infectious disease[J]. Mathematical Biosciences and Engineering, 2006, 3(1): 219-235. doi: 10.3934/mbe.2006.3.219
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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