Bifurcation analysis and optimal control of an infection age-structured epidemic model with vaccination and treatment

Abstract

This paper addresses the problem of bifurcation analysis and optimal control of an age-structured epidemic model. We first develop and analyze a deterministic epidemiological model for the transmission of an infectious disease described by a susceptible–vaccinated–infected (SVI). The model takes into account the age structure of the infection, imperfect vaccination and treatment of infected people who become susceptible again. The model without intervention strategies is completely analyzed. We compute the disease-free equilibrium and derive the basic reproduction number [Formula: see text]. The analysis of the model reveals the existence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated basic reproduction number is less than unity. In this case, controlling this infectious disease remains extremely difficult. Therefore, based on this continuous model, a control is formulated and solved as an optimal control theory problem, indicating how control terms on vaccination and treatment should be introduced in the population to minimize the number of infected individuals as well as the cost of applying controls. We establish the first-order necessary conditions for optimality and characterize optimal controls. The theory is supported by numerical simulations, which further suggest that the combination of intervention measures can significantly reduce the transmission dynamics of infectious diseases by keeping the population of infected individuals relatively low.