Dynamics of delay epidemic model with periodic transmission rate

Abstract

We introduce novel epidemic models with single and two strains described by systems of delay differential equations with a periodic time-dependent disease transmission rate, and based on the number of newly infected individuals. Transitions between infected, recovered, and returning to susceptible compartments due to waning immunity are determined by the accompanying time delays. Positiveness, existence and uniqueness of solutions are demonstrated with the help of fixed point theory. Reducing delay differential equations to integral equations facilitates determining the analytical estimation of the equilibrium solutions. When there are two strains, they compete with each other, and the strain with a larger basic reproduction number dominates in the population. However, both strains coexist, and the magnitudes of epidemic outbreaks are governed by the basic reproduction numbers. The results of this work are verified through comparison with seasonal influenza data.