Stability and Hopf Bifurcation for a delayed hand–foot–mouth disease model with continuous age-structure in the exposed class

Abstract

Hand–foot–mouth disease (HFMD) is a mild and highly contagious viral infectious disease common in young children, but anyone can get it. In order to reveal the transmission phenomena of HFMD, we formulate a HFMD model with age structure for latently infected individuals and atime delay. The time delay occurs during the transition from latent to infectious individuals. We reformulate the model as an abstract Cauchy problem and show the presence of equilibria. We specify the basic reproduction number ${ℛ}_0$ which determines the threshold dynamics of the HFMD model. For ${ℛ}_0<1$, the disease-free equilibrium ${\bar{E}}_0$ is globally asymptotically stable. For ${ℛ}_0>1$, we derive that the endemic equilibrium ${\bar{E}}_{\ast }$ is unstable, which is the criteria for the occurrence of Hopf bifurcation. Finally, some numerical simulations demonstrate the obtained theoretical results and shed light on the impact of time delay on the evolution of HFMD spread.