In this paper, a nonlinear population model for HIV-TB co-infection has been proposed. The model is incorporated with the effect of early and late initiation of HIV treatment in co-infectives already on TB treatment, on the occurrence of Immune Reconstitution Inflammatory syndrome (IRIS). A 15-dimensional (15D) mathematical model has been developed in this study. We begin with considering constant treatment rates and thereafter, proceed to time-dependent treatment rates for co-infectives as control parameters. The basic reproduction number, a threshold quantity, corresponding to each HIV and TB sub-model has been computed in case of constant controls. With constant values of control parameters, mathematical analysis shows the existence and local stability of the disease-free equilibrium point and the endemic equilibrium point for the model. Together with time-dependent parameters, an optimal control problem is introduced and solved using Pontryagin’s maximum principle with an objective to minimize the number of infectives and disease induced deaths along with the cost of treatment. Numerical simulations are performed to examine the effect of reproduction numbers on control profiles and to identify, the ideal combination of treatment strategies which provides minimum burden on a society. Numerical results imply that if both HIV and TB are endemic in the population, then in order to bring in minimum burden from the co-infection, optimal control efforts must be enforced rather than constant treatment rate.
Fast iterative solver for the optimal control of time‐dependent PDEs with Crank–Nicolson discretization in time
