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Asymptotic behavior of a discrete-time density-dependent SI epidemic model with constant recruitment. (English) Zbl 1490.39028

Summary: We use the epidemic threshold parameter, \({{\mathcal{R}}}_0 \), and invariant rectangles to investigate the global asymptotic behavior of solutions of the density-dependent discrete-time SI epidemic model where the variables \(S_n\) and \(I_n\) represent the populations of susceptibles and infectives at time \(n = 0,1,\ldots \), respectively. The model features constant survival “probabilities” of susceptible and infective individuals and the constant recruitment per the unit time interval \([n, n+1]\) into the susceptible class. We compute the basic reproductive number, \({{\mathcal{R}}}_0 \), and use it to prove that independent of positive initial population sizes, \({{\mathcal{R}}}_0<1\) implies the unique disease-free equilibrium is globally stable and the infective population goes extinct. However, the unique endemic equilibrium is globally stable and the infective population persists whenever \({{\mathcal{R}}}_0>1\) and the constant survival probability of susceptible is either less than or equal than 1/3 or the constant recruitment is large enough.

MSC:

39A60 Applications of difference equations
39A30 Stability theory for difference equations
92D30 Epidemiology
92D25 Population dynamics (general)
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