An application of discrete-time SEIR model to the COVID-19 spread

TL;DR

This paper applies a discrete-time SEIR model to analyze COVID-19 spread, demonstrating the model's convergence properties and discussing its effectiveness for Uzbekistan.

Contribution

It introduces a quadratic stochastic operator form of the SEIR model and proves its regularity and convergence, providing new insights into epidemic modeling.

Findings

The QSO has an uncountable set of fixed points on the boundary.

All trajectories of the system are convergent.

The model's efficiency is discussed for Uzbekistan.

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 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.