Modelling the role of opportunistic diseases in coinfection

In this paper, we formulate a model for evaluating the effects of an opportunistic disease affecting only those individuals already infected by a primary disease. The opportunistic disease act on a faster time scale and it is represented by an SIS epidemic model with frequency-dependent transmission. The primary disease is governed by an SIS epidemic model with density-dependent transmission, and we consider two different recovery cases. The first one assumes a constant recovery rate whereas the second one takes into account limited treatment resources by means of a saturating treatment rate. No demographics is included in these models. Our results indicate that misunderstanding the role of the opportunistic disease may lead to wrong estimates of the overall potential amount of infected individuals. In the case of constant recovery rate, an expression measuring this discrepancy is derived, as well as conditions on the opportunistic disease imposing a coinfection endemic state on a primary disease otherwise tending to disappear. The case of saturating treatment rate adds the phenomenon of backward bifurcation, which fosters the presence of endemic coinfection and greater levels of infected individuals. Nevertheless, there are specific situations where increasing the opportunistic disease basic reproduction number helps to eradicate both diseases.