Modelling, Analysis and Optimal Control to Co-Dynamics of HIV/AIDS-TB Diseases in Homogeneous Population

Abstract

In this paper, an optimal control mathematical model of HIV/AIDS and TB co-infection with vaccination and relapse is developed and analysed by dividing the total human population under consideration into five compartments, namely, susceptible (S), TB-infected (T ), HIV-infected (H), vaccinated (V ) and AIDS-infected (A). We analysed the steady states behaviour of the dynamical system representing the co-infection transmission dynamics of HIV/AIDS and TB epidemic. The mathematical model possesses four equilibrium points such as disease free, HIV/AIDS infection free, TB infection free and vaccination free. The stability of aforesaid cases is also investigated. A threshold parameter reproduction number R₀ is computed and if R₀ < 1 the disease dies out and it becomes endemic if R₀ > 1.
It is also found that the co-infection period also influences the transmission patterns of diseases. Some important theorems and results are proved. Optimal control solutions are provided to predict the efficacy of vaccination and control strategies. The sensitivity analysis has also been facilitated to carry out the effects of certain key parameters on the diseases co-dynamics. It is found that administration of appropriate vaccine at proper time could be more effective in controlling the co-infection. The relapse factor is also considered in the model where the vaccination fails.

Key words and phrases. HIV/AIDS-TB co-infection, relapse, vaccination.

2010 Mathematics Subject Classification. 92BXX