Abstract
In this paper, we have studied the global dynamics of HIV model with two transmission paths: direct transmission through cells-to-cells contact and indirect transmission through virus. We have derived a four dimensional mathematical model including uninfected $CD_4^{+T}$ cells, infected $CD4^+{T}$ cells, virus and the CTL immune response cells. The nonnegativity and boundedness property of the solutions the proposed mathematical system have been analysed, and the basic reproduction ratio $R_0$ has been derived with the help of next generation matrix method. We also discussed the local and global stability with respect to the basic reproduction ratio of both disease-free and interior equilibrium points under certain conditions. Through numerical simulations, we have validated the all analytical findings. We have established that the disease-free equilibrium is globally stable for $R_0 1$ whenever exists. It is also observed that cells-to-cells transmission rate is more effective compare to virus-to-cells infection rate.
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