A time-varying SIS epidemic model with incidence rate depending on the susceptible and infective populations with eventual impulsive effects

This paper studies a time-varying SIS (i.e., containing susceptible and infected populations) propagation disease model exhibiting a nonlinear incidence rate and impulsive eventual culling of both populations so that the individuals recover with no immunity to the disease. The nonlinear incidence rate consists of two time-varying additive terms proportional to the susceptible and infected populations normalized to the total population. The disease transmission dynamics does not necessarily take into account the total population as a normalizing effect. In this sense, the considered model is a mixed pseudo-mass action (at the level of the nonlinear incidence rate) and true-mass action model (at the level of disease transmission). However, such a normalization may be considered though a change from the disease transmission function to a normalized on so that the whole model be of true-mass action type. The positivity and stability of both the impulse- free and impulsive under pulse culling variants of the model are investigated in this paper.