Simple criss-cross model of epidemic for heterogeneous populations

In this paper we consider a class of criss-cross models describing the dynamics of epidemic of infectious illness. We follow the ideas presented in the paper by Romaszko et al., where the authors described actions of active detecting of tuberculosis (TB) among homeless subpopulation in Warmian-Masurian province of Poland. However, the class of epidemiological models analyzed in this paper, because of their universalism, can be applied in modeling dynamics of epidemics of various kinds of illnesses. In the original model, the whole population is divided into subpopulations of non-homeless and homeless people. Each of the subpopulations consists of two groups – susceptible and infected individuals. We consider the division of the whole population into two subpopulations described by different model parameters. We focus on the analysis of the basic criss-cross model depending on the form of a function describing transmission of illness. We consider two types of this function – first, called standard incidence function, was used in the original work of Romaszko et al., and second, bilinear function. The most important property of this model is related to its Malthusian origin, and this property is independent of the transmission function. This means that in many cases the size of one of the subpopulations or the whole population grows boundlessly or the population goes to extinction. However, for the bilinear transmission function coexistence of both subpopulations is also possible. We also analyze the influence of active detection onto the model dynamics. The basic criss-cross model is fitted to the demographic data from Poland which then allows for making a short-term prediction on TB dynamics.