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Journal of Applied Nonlinear Dynamics 12(2) (2023) 327--337 | DOI:10.5890/JAND.2023.06.010

U. A. Rozikov$^{1,2,3}$, S. K. Shoyimardonov$^1$

$^1$ V.I. Romanovskiy Institute of Mathematics of Uzbek Academy of Sciences, Tashkent, Uzbekistan

$^2$ AKFA University, National Park Street, Tashkent, Uzbekistan

$^3$ Faculty of Mathematics, National University of Uzbekistan, 100174, Tashkent, Uzbekistan

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Abstract

The Susceptible-Exposed-Infectious-Recovered (SEIR) model is applied in several countries to ascertain the spread of the coronavirus disease 2019 (COVID-19). We consider discrete-time SEIR epidemic model in a closed system which does not account for births or deaths, total population size under consideration is constant. This dynamical system is generated by a non-linear evolution operator depending on four parameters. Under some conditions on parameters we reduce the evolution operator to a quadratic stochastic operator (QSO) which maps 3-dimensional simplex to itself. We show that the QSO has uncountable set of fixed points (all laying on the boundary of the simplex). It is shown that all trajectories of the dynamical system (generated by the QSO) of the SEIR model are convergent (i.e. the QSO is regular). Moreover, we discuss the efficiency of the model for Uzbekistan.

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