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# Simple criss-cross model of epidemic for heterogeneous populations

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## Abstract

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.

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... In our previous papers (see [17,18]) we analyzed continuous SIS models differing in a type of inflow into subpopulations. Basing on the system from [17], we constructed the discrete model with the use of the EEM. ...
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To review the transmission models of tuberculosis in heterogeneous population. The data used in this review were adopted mainly from the studies of models of tuberculosis reported from 1995 to 2006. Relevant literature on transmission models of tuberculosis in heterogeneous populations are referenced. Casual/random factors and genetic factors are the main reasons for epidemics of tuberculosis in recent years. Mass public transport is playing the primary role in casually close contact which can facilitate the transmission of tuberculosis. Genetic susceptibility not only varies endemic prevalence levels, but also drastically alters the effects of treatment for tuberculosis patients. Detailed studies further exhibit that casual contact and genetic factor are responsible for over 30% - 40% of the total new cases in recent years. The prevalence of tuberculosis could double (from 33% to 60%) if a genetically susceptible phenotype is present in only 30% of the population. And some challenges have emerged along with these exciting results. Casual/random contact, public transport and genetic susceptibility are responsible for most new tuberculosis cases and a wide variation in endemic tuberculosis levels between regions. Hence, the transmission model of tuberculosis in a heterogeneous population can provide more clues to underlying mechanism of tuberculosis transmission than in a homogeneous population. However, many challenges remain for us in understanding transmission of disease.
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