Global dynamics and optimal control of an age-structure Malaria transmission model with vaccination and relapse

Abstract

Malaria is a typical mosquito borne disease and one of the oldest human diseases, which is still one of the most serious threats to public health. In the process of prevention and control of this disease, the effectiveness of the vaccine and the relapse of disease are problems that have to be faced. We propose, in this paper, two-class-age structure dynamic model to describe the transmission of Malaria between mosquitoes and humans, where the age of vaccine and the age of recovered are introduced to discuss their impact on the transmission and control of this disease. The exact expression of the basic reproduction number is derived. Furthermore, the existence and local stability of the disease-free steady state and the endemic steady state are obtained, which are completely determined by the basic reproduction number. By applying the fluctuation lemma and the suitable Lyapunov functional, the global dynamics of steady states are investigated. That is, if the basic reproduction number is less than 1, the disease-free steady state is globally asymptotically stable, and the disease dies out; if the basic reproduction number is greater than 1, the endemic steady state is globally asymptotically stable, and the disease becomes endemic. In addition, an optimal control problem relative to this model is explored. By the application of the Gateaux derivative rule, we analyze the existence of optimal control and obtain necessary conditions for the control problem. Finally, some numerical simulations are carried out to explain the main theoretical results.