AN OPTIMAL CONTROL MODEL FOR TUBERCULOSIS-DIABETES DYNAMICS

It is known that tuberculosis and diabetes mellitus, apart from posing individual threat to human health, also pair to form a dangerous alliance with increased risk. Taking into consideration the crucial aspects of this alliance, in this paper, we propose a compartmental model for TB, using a non-linear system of ordinary differential equations, with distinct classes for non-diabetes and diabetes at various stages of TB infection. Positivity analysis of the model along with computation of the basic reproduction number $R_0$ is also carried out. We then formulate an associated optimal control problem with screening/treatment rates at the latent TB stage as control variables and seeking to minimize the number of active TB infected individuals, both at diabetes and non-diabetes levels. Using Pontryagin’s Maximum Principle, we derive expressions for optimal control inputs. Illustrative numerical simulations of this control problem show how the optimal controls which are applied initially at higher rates, and then at a steady rate continuously, help to keep the proportions of the latent and active TB infected in the entire adult population at negligible levels. Our illustrations, though partly based on hypothetical data, underline the necessity and effectiveness of routine TB checks and treatment at the latent TB infected stage in order to combat the progression of TB which is fueled through its alliance with diabetes. We also illustrate how the proposed optimal control model can be extended to the non-autonomous case where some parameters are time-varying.