We use a periodically forced SIS epidemic model with disease induced
mortality to study the combined effects of seasonal trends and death on the
extinction and persistence of discretely reproducing populations. We
introduce the epidemic threshold parameter, $R_0$, for
predicting disease dynamics in periodic environments. Typically, $R_0<1$
implies disease extinction. However, in the presence of disease
induced mortality, we extend the results of Franke and Yakubu to periodic
environments and show that a small number of infectives can drive an
otherwise persistent population with $R_0>1$ to extinction.
Furthermore, we obtain conditions for the persistence of the total
population. In addition, we use the Beverton-Holt recruitment function to
show that the infective population exhibits period-doubling bifurcations
route to chaos where the disease-free susceptible population lives on a
2-cycle (non-chaotic) attractor.
Citation: John E. Franke, Abdul-Aziz Yakubu. Periodically forced discrete-time SIS epidemic model with diseaseinduced mortality[J]. Mathematical Biosciences and Engineering, 2011, 8(2): 385-408. doi: 10.3934/mbe.2011.8.385
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Abstract
We use a periodically forced SIS epidemic model with disease induced
mortality to study the combined effects of seasonal trends and death on the
extinction and persistence of discretely reproducing populations. We
introduce the epidemic threshold parameter, $R_0$, for
predicting disease dynamics in periodic environments. Typically, $R_0<1$
implies disease extinction. However, in the presence of disease
induced mortality, we extend the results of Franke and Yakubu to periodic
environments and show that a small number of infectives can drive an
otherwise persistent population with $R_0>1$ to extinction.
Furthermore, we obtain conditions for the persistence of the total
population. In addition, we use the Beverton-Holt recruitment function to
show that the infective population exhibits period-doubling bifurcations
route to chaos where the disease-free susceptible population lives on a
2-cycle (non-chaotic) attractor.