In this paper we introduce and analyze a non-standard discretized SIS epidemic model for a homogeneous population. The presented model is a discrete version of the continuous model known from literature and used by us for building a model for a heterogeneous population. Firstly, we discuss basic properties of the discrete system. In particular, boundedness of variables and positivity of solutions of the system are investigated. Then we focus on stability of stationary states. Results for the disease-free stationary state are depicted with the use of a basic reproduction number computed for the system. For this state we also manage to prove its global stability for a given condition. It transpires that the behavior of the disease-free state is the same as its behavior in the analogous continuous system. In case of the endemic stationary state, however, the results are presented with respect to a step size of discretization. Local stability of this state is guaranteed for a sufficiently small critical value of the step size. We also conduct numerical simulations confirming theoretical results about boundedness of variables and global stability of the disease-free state of the analyzed system. Furthermore, the simulations ascertain a possibility of appearance of Neimark-Sacker bifurcation for the endemic state. As a bifurcation parameter the step size of discretization is chosen. The simulations suggest the appearance of a supercritical bifurcation.
Citation: Marcin Choiński, Mariusz Bodzioch, Urszula Foryś. A non-standard discretized SIS model of epidemics[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 115-133. doi: 10.3934/mbe.2022006
In this paper we introduce and analyze a non-standard discretized SIS epidemic model for a homogeneous population. The presented model is a discrete version of the continuous model known from literature and used by us for building a model for a heterogeneous population. Firstly, we discuss basic properties of the discrete system. In particular, boundedness of variables and positivity of solutions of the system are investigated. Then we focus on stability of stationary states. Results for the disease-free stationary state are depicted with the use of a basic reproduction number computed for the system. For this state we also manage to prove its global stability for a given condition. It transpires that the behavior of the disease-free state is the same as its behavior in the analogous continuous system. In case of the endemic stationary state, however, the results are presented with respect to a step size of discretization. Local stability of this state is guaranteed for a sufficiently small critical value of the step size. We also conduct numerical simulations confirming theoretical results about boundedness of variables and global stability of the disease-free state of the analyzed system. Furthermore, the simulations ascertain a possibility of appearance of Neimark-Sacker bifurcation for the endemic state. As a bifurcation parameter the step size of discretization is chosen. The simulations suggest the appearance of a supercritical bifurcation.
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Marcin Choiński, Mariusz Bodzioch, Urszula Foryś. A non-standard discretized SIS model of epidemics[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 115-133. doi: 10.3934/mbe.2022006
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