Muhammad Abdullahi Yau - Lorestan university Nigeria Lecturer, Department of Mathematics, Nasarawa State University Keffi, Nigeria.

We consider a mathematical model of epidemic spread  in which the  population  is partitioned  into five compartments of susceptible S(t), Infected I(t), Removed R(t), Prevented U(t) and the Controlled W(t). We assume each of the compartments comprises of cohorts of individuals which are  identical with respect to the disease status. We derive five systems of equations to represent each of the subpopulations. The general stability of the disease free equilibrium (DFE) and the endemic equilibrium states of the linearized model are established using the linear stability theory and  the Routh-Hurwitz conditions are established and analyzed  in the domain of interest. We find that the DFE is locally asymptotically stable when the infected individuals received ART and use the condom but the endemic state is unstable to initial perturbations.  Also, we derive an expression for the basic  reproduction number  using the next generation matrix approach and  find that for  Ro<۱ the DFE is stable but for Ro>۱ is unstable.

Stability, Threshold, epidemic model, HIV, AIDS