A dynamical model for HIV–TB co-infection

A nonlinear mathematical model is proposed and analyzed to study the dynamics of HIV, TB and co-infection. The basic reproduction number for each of the two diseases (HIV and TB) is obtained. Four equilibrium points are found, out of which one is disease free equilibrium while the remaining three equilibrium points are such that they are having at least one of the diseases HIV and / or TB is present. When the basic reproduction number of the two diseases is less than one then the disease free equilibrium point is stable. This provides a threshold for the control of diseases. When TB free equilibrium point exists then it is locally stable if the basic reproduction number of HIV is more than that of TB. A condition for local stability of HIV free equilibrium point is also obtained. There exist some situations in which bi-stability is possible, that is, both of the diseases persist. The conditions for global stability of these points are also established. The conditions for existence of endemic equilibrium point are explored and it was found that it remains unstable whenever the endemic equilibrium point exists. The instability of endemic equilibrium point is used to interpret that the co-infection is not long lasting.