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Dynamics of a Two-Dimensional Discrete-Time SIS Model

We analyze a two-dimensional discrete-time SIS model with a non-constant total population. Our goal is to determine the interaction between the total population, the susceptible class and the infective class, and the implications this may have for the disease dynamics. Utilizing a constant recruitment rate in the susceptible class, it is possible to assume the existence of an asymptotic limiting equation which enables us to reduce the system of two-equations into a single, dynamically equivalent equation. In this case, we are able to demonstrate the global stability of the disease-free and the endemic equilibria when the basic reproductive number (R0) is less than one and greater than one, respectively. When we consider a non-constant recruitment rate, the total population bifurcates as we vary the birth rate and the death rate. Using computer simulations, we observe different behavior among the infective class and the total population, and possibly, the occurrence of a strange attractor.

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Jaime H. Barrera, Cornell University
Ariel Cintrón-Arias, Cornell University
Nicolas Davidenko, Harvard University
Lisa R. Denogean, Cornell University
Saúl Ramón Franco-Gonzalez, University of California-Irvine