Global phase portrait of a SIS model

In the qualitative theory of ordinary differential equations, we can find many papers whose objective is the classification of all the possible topological phase portraits of a given family of differential system. Most of the studies rely on systems with real parameters and the study consists of outlining their phase portraits by finding out some conditions on the parameters. Here, we studied a susceptible-infected-susceptible (SIS) model described by the differential system $\dot{x}=-bxy-mx+cy+mk$, $\dot{y}=bxy-(m+c)y$, where $b$, $c$, $k$, $m$ are real parameters with $b \neq 0$, $m \neq 0$ (see Brauer (2002). Such system describes an infectious disease from which infected people recover with immunity against reinfection. The integrability of such system has already been studied by Nucci and Leach (2004) and Llibre and Valls (2008). We found out two different topological classes of phase portraits.