An optimal chemoprophylaxis and treatment control for a spatiotemporal tuberculosis model

Abstract

In this work, we propose 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. Based on an existing model, we add the Laplacian term in each class to describe the spatial mobility of its individuals, which led us to a SEIR reaction-diffusion system. Then, controls with treatment and chemoprophylaxis are incorporated to reduce the latently infected (exposed) and actively infected individual populations to fight against the spread of the disease. Theoretically the existence, positivity and boundness of state systems have been proved, also the existence of controls has been shown, and a characterization of the controls in terms of the state functions and the adjoint functions has been provided. 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 both control strategies controls on treatment and chemoprophylaxis strategies are used simultaneously.