Analysis of the impact of treatments on HIV/AIDS and Tuberculosis co-infected population under random perturbations

In this work, we study the impact of treatments at different stages of Human Immunodeficiency Virus (HIV) and Tuberculosis (TB) co-infection in a population under the influence of random perturbations. This is achieved by constructing a stochastic epidemic model describing the transmission and treatment of the diseases. The model is created with the assumption that transmission rates fluctuate rapidly compared to the evolution of the untreated diseases. The basic reproduction numbers corresponding to the population with HIV infection only (with n stages of infections and treatments), the population with tuberculosis infection only, and the overall population with co-infection (with n stages of infection/treatments) are derived in the presence and absence of noise perturbations. These are used to discuss the long term behavior of the population around a disease-free equilibrium and an endemic equilibrium, and to analyze the effect of noise and treatments on the system. We also showed conditions under which TB infected population dynamic undergoes backward bifurcation and give conditions for disease eradication in the entire population. Analysis shows that small perturbations to the disease-free equilibrium can initially grow under certain conditions, and the introduction of TB treatment is effective in eliminating the co-infection. Numerical simulations are presented for validation of our results using published parameters.