Dynamical Behaviour of an HIV/AIDS Epidemic Model

Abstract

In this paper we have developed a five compartmental HIV/AIDS epidemic model with two infectious stages before full-blown AIDS defined, i.e., asymptomatic phase and symptomatic phase. Boundedness and non-negativity analysis of the solutions, existence and stability analysis of the model at various equilibrium points are discussed thoroughly. Basic reproduction number (\(R_0\)) is calculated using next generation matrix method. It is found that the system is locally as well as globally asymptotically stable at disease free equilibrium \(E_0\) when \(R_0 1\), endemic equilibrium \(E^*\) exists and the system becomes locally asymptotically stable at \(E^*\) if Routh–Hurwitz criterion is satisfied. We also obtain necessary conditions for the global asymptotic stability of the system at \(E^*\) by geometrical approach. The effect of single discrete time delay on the model is also discussed. We have considered the time delay \(\tau \) as a time lag due to the development of the infection until signs or symptoms first appear. The length of delay preserving the stability is estimated using Nyquist criteria and existence conditions of the Hopf-bifurcation for the time delay are investigated by choosing the time delay \(\tau \) as a bifurcation parameter. Important analytic results are numerically verified using MATLAB, which shows the reliability of our model from the practical point of view.