Asymptotic behavior of a discrete-time density-dependent SI epidemic model with constant recruitment.

Abstract

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.