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On a modification of a discrete epidemic model. (English) Zbl 1197.39010
Summary: In this paper under some conditions on the constants $A,B\in 0,\infty$) we study the existence of positive solutions, the existence of a unique nonnegative equilibrium and the convergence of the positive solutions to the nonnegative equilibrium of the system of difference equations ${x}_{n+1}=\left(1-{y}_{n}-{y}_{n-1}\right)\left(1-{e}^{-A{y}_{n}}\right),\phantom{\rule{3.33333pt}{0ex}}{y}_{n+1}=\left(1-{x}_{n}-{x}_{n-1}\right)\left(1-{e}^{-B{x}_{n}}\right)$ where $A,B\in \left(0,\infty \right)$ and the initial values ${x}_{-1},{x}_{0},{y}_{-1},{y}_{0}$ are positive numbers which satisfy the relations ${x}_{0}+{x}_{1}<1,{y}_{0}+{y}_{1}<1·1-y0>\left(1-{x}_{0}-{x}_{-1}\right)\left(1-{e}^{B{x}_{0}},1-{x}_{0}>\left(1-{y}_{0}-{y}_{1}\right)\left(1-{e}^{-A{y}_{0}}\right)$.
##### MSC:
 39A30 Stability theory (difference equations) 92D30 Epidemiology
##### References:
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