A contiunous-time $ SIS $ criss-cross model of co-infection in a heterogeneous population

In this paper, we introduce and analyze a contiunous-time model of co-infection dynamics in a heterogeneous population consisting of two subpopulations that differ in the risk of getting infected by individuals with two diseases. We assume that each parameter reflecting a given process for each subpopulation has different values, which makes the population completely heterogeneous. Such complexity and the population heterogeneity make our paper unique, reflecting co-infection dynamics. Moreover, we establish an epidemic spread for each disease not only in a sole subpopulation but also with criss-cross transmission, meaning between different subpopulations. The proposed system has a disease-free stationary state and two states reflecting the presence of one disease. We indicate conditions for their existence and local stability. The conditions for the local stability for states reflecting one disease have a complicated form, so we strengthened them so that they are more transparent. Investigation on the existence of a postulated endemic state corresponding to both disease's presence leads to a complex analysis, which is why we only provide an insight on this issue. Here, we also provide the basic reproduction number of our model and investigate properties of this number. The system has a universal structure; as such, it can be applied to investigate co-infection of different infectious diseases.