Abstract
This paper presents a co-infection mathematical model of COVID-19 and TB to study their synergistic dynamics. We first investigated the single infection models of each disease and then the co-infection dynamics of the two diseases. Indeed, we calculated the basic reproduction number of each model, and then we studied the existence and the stability of the equilibrium points. We subsequently proved that the TB only infection model and the co-infection model exhibit backward and forward bifurcations. In addition, we performed a sensitivity analysis of the basic reproduction numbers to determine which parameters influence them the most. Furthermore, we applied Pontryagin’s maximum principle to our co-infection model to assess the impact of the use of an imperfect vaccine for COVID-19, taken as an optimal control strategy. Finally, we presented the results of the numerical simulations to support the theoretical findings.