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.