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.