An inherently discrete–time $ SIS $ model based on the mass action law for a heterogeneous population

Abstract

In this paper, we introduce and analyze a discrete–time model of an epidemic spread in a heterogeneous population. As the heterogeneous population, we define a population in which we have two groups which differ in a risk of getting infected: a low–risk group and a high–risk group. We construct our model without discretization of its continuous–time counterpart, which is not a common approach. We indicate functions that reflect the probability of remaining healthy, which are based on the mass action law. Additionally, we discuss the existence and local stability of the stability states that appear in the system. Moreover, we provide conditions for their global stability. Some of the results are expressed with the use of the basic reproduction number $ \mathcal{R}_0 $. The novelty of our paper lies in assuming different values of every coefficient that describe a given process in each subpopulation. Thanks to that, we obtain the pure population's heterogeneity. Our results are in a line with expectations – the disease free stationary state is locally stable for $ \mathcal{R}_0 < 1 $ and loses its stability after crossing $ \mathcal{R}_0 = 1 $. We supplement our results with a numerical simulation that concerns the real case of a tuberculosis epidemic in Poland.

Keywords

• discrete–time systems
• $ sis $ model
• local stability
• global stability
• population heterogeneity
• dynamical systems