Periodic solutions for an “SVIQR” epidemic model in a seasonal environment with general incidence rate

Seasonality is repetitive in the ecological, biological and human systems. Seasonal factors affect both pathogen survival in the environment and host behavior. In this study, we considered a five-dimensional system of ordinary differential equations modeling an epidemic in a seasonal environment with a general incidence rate. We started by studying the autonomous system by investigating the global stability of steady states. Later, we proved the existence, uniqueness, positivity and boundedness of a periodic orbit in a non-autonomous system. We demonstrate that the global dynamics are determined using the basic reproduction number ${{ℛ}}_0$ which is defined by the spectral radius of a linear integral operator. We showed that if ${{ℛ}}_0<1$, then the disease-free periodic solution is globally asymptotically stable and if ${{ℛ}}_0>1$, then the trajectories converge to a limit cycle reflecting the persistence of the disease. Finally, we present a numerical investigation that support our results.