Threshold dynamics of difference equations for SEIR model with nonlinear incidence function and infinite delay

Soufiane Bentout; Salih Djilali; Sunil Kumar; Tarik Mohammed Touaoula

doi:10.1140/epjp/s13360-021-01466-0

Threshold dynamics of difference equations for SEIR model with nonlinear incidence function and infinite delay

Eur Phys J Plus. 2021;136(5):587. doi: 10.1140/epjp/s13360-021-01466-0. Epub 2021 May 27.

Authors

Soufiane Bentout^{1

2}, Salih Djilali^{1

3}, Sunil Kumar^{4

5}, Tarik Mohammed Touaoula^{1

6}

Affiliations

¹ Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, Université de Tlemcen, Tlemcen, Algeria.
² Department of Mathematics and Informatics, Belhadj Bouchaib University of Ain Temouchent, BP 284 RP, 46000 Ain Temouchent, Algeria.
³ Faculty of Exact and Computer Sciences, Mathematic Department, Hassiba Benbouali university, Chlef, Algeria.
⁴ Department of Mathematics, National Institute of Technology Jamshedpur, Jharkhand, 831014 India.
⁵ Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE.
⁶ Department of Mathematics, University of Tlemcen, 13000 Tlemcen, Algeria.

Abstract

In this research, we explore the global conduct of age-structured SEIR system with nonlinear incidence functional (NIF), where a threshold behavior is obtained. More precisely, we will analyze the investigated model differently, where we will rewrite it as a difference equations with infinite delay by the help of the characteristic method. Using standard conditions on the nonlinear incidence functional that can fit with a vast class of a well-known incidence functionals, we investigated the global asymptotic stability (GAS) of the disease-free equilibrium (DFE) using a Lyapunov functional (LF) for $R_{0} \leq 1$ . The total trajectory method is utilized for avoiding proving the local behavior of equilibria. Further, in the case $R_{0} > 1$ we achieved the persistence of the infection and the GAS of the endemic equilibrium state (EE) using the weakly $ρ$ -persistence theory, where a proper LF is obtained. The achieved results are checked numerically using graphical representations.