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

We use the epidemic threshold parameter, , and invariant rectangles to investigate the global asymptotic behavior of solutions of the density-dependent discrete-time SI epidemic model where the variables and represent the populations of susceptibles and infectives at time , respectively. The model features constant survival "probabilities" of susceptible and infective individuals and the constant recruitment per the unit time interval into the susceptible class. We compute the basic reproductive number, , and use it to prove that independent of positive initial population sizes, 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 and the constant survival probability of susceptible is either less than or equal than 1/3 or the constant recruitment is large enough.