The co-dynamics of Malaria and Tuberculosis with optimal control strategies

Abstract

Malaria and Tuberculosis are both the severe and causing death diseases in the world. The
occurrence of TB and malaria as a coinfection is also an alarming threat to the human. Therefore, we
consider a mathematical model of the dynamics of malaria and tuberculosis coinfection and explore its
theoretical results. We formulate the model and obtain their basic properties. The stability of each model
is obtained and discussed. We show that at the disease free case each model is locally asymptotically
stable, when the basic reproduction number less than unity. Further, we analyze the phenomenon of
backward bifurcation for coinfection model. For the sub models, we present the local stability for the
disease free case whenever basic reproduction number less than 1. Further, an optimal control problem is
presented to investigate the dynamics of malaria and tuberculosis coinfection. The numerical results with
different scenarios are presented. The mathematical model with and without control problem are solved
numerically using the Runge-Kutta backward and forward scheme of order four. We show selection of
appropriate control as a control strategies for the disease eliminations. Every set of control has its importance
for elimination of infection in TB and malaria compartments but the results are more appropriate when
utilizing all the controls simultaneously at last strategies. The present problem of coinfection among TB
and malaria infection address the disease well and will reduce the disease burden in community with
suggestions of the inserted control profiles. The suggested controls profiles are highly relevant and will
best decrease the both the disease and its dual infection.