Periodically forced discrete-time SIS epidemic model with diseaseinduced mortality

John E. Franke; Abdul-Aziz Yakubu

doi:10.3934/mbe.2011.8.385

Article Contents

2011, Volume 8, Issue 2: 385-408. Doi: 10.3934/mbe.2011.8.385

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Periodically forced discrete-time SIS epidemic model with disease induced mortality

John E. Franke^1, and
Abdul-Aziz Yakubu^2,

1.
Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205
2.
Department of Mathematics, Howard University, Washington, DC 20059

Received: January 31, 2010

Revised: July 31, 2010

Published: March 2011

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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.

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Mathematics Subject Classification: Primary: 37G15, 37G35; Secondary: 39A11, 92B05.

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