Optimal control problem of a tuberculosis model with spatial dynamics

Abstract

Tuberculosis is an ancient contagious disease, and it causes more deaths worldwide than any other infectious diseases. Based on the fact that it is spread from person to person through the air, the tuberculosis can emerge in one region and spread to its neighbors in unprecedented durations. we propose here a Susceptible-Exposed-Infected-Recovered (SEIR) spatiotemporal model that characterizes the dynamics of tuberculosis disease by taking into consideration the spatial heterogeneity; in order to provide a realistic description of this disease. Then, controls on treatment, chemoprophylaxis are incorporated to reduce the latently infected (exposed) and actively infected individual populations to fight against the spread of the disease. Theoretically, we have proved the existence of optimal controls, and we have given a characterization of controls in terms of states and adjoint functions based on a discrete version of Pontryagin’s maximum principle. To illustrate the effectiveness of our theoretical results, we give numerical simulations for several scenarios. Our results indicate that the control effect is effective if controls on treatment and chemoprophylaxis strategies are used simultaneously.