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

KulenoviĆ, M. R. S.; NurkanoviĆ, M.; Yakubu, Abdul-Aziz

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