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

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 $$R_0\le 1$$ . 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 $$\rho $$ -persistence theory, where a proper LF is obtained. The achieved results are checked numerically using graphical representations.