Hopf bifurcation in an age-structured SIR epidemic model

In this paper, we study the occurrence of a sustained periodic solution via the Hopf bifurcation in an age-structured SIR epidemic model. Under the assumption that the transmission rate depends on the age of infective individuals and the product of the transmission rate and the population age distribution is concentrated in a specific age, we reformulate the model into an integral equation of Fredholm type. We then define the basic reproduction number R-0 and show that the unique positive endemic equilibrium of the integral equation exists if and only if R-0 > 1. We derive a characteristic equation for the endemic equilibrium, and regarding the specific age as a bifurcation parameter, we obtain a sufficient condition for the occurrence of the Hopf bifurcation. Finally, we provide a numerical example that supports our theoretical result. (C) 2018 Elsevier Ltd. All rights reserved.