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Abstract
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It is well known that dynamical systems are very useful tools to
study the viral disease such as HIV, HBV, HCV, Ebola and
Influenza. This paper deals with the model
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\begin{equation}\label{1.3}
\begin{split}
& \frac{dT(t)}{dt}=s-dT+rT(1-\frac{T}{T_{M}})-\frac{b_{1} TV}{1+aV}-b_{2}TI,\\
& \frac{dI(t)}{dt}=\frac{b_{1}TV}{1+aV}+b_{2}TI-\delta I,\\
& \frac{dV(t)}{dt}=hI-lV,
\end{split}
\end{equation}
which is a mathematical model of the
cell-to-cell and the cell-free spread of HIV with both linear and
nonlinear functional responses and logistic target cell growth.
The reproduction number of each mode of transmission has been
calculated and their sum has been considered as the basic
reproduction number. Based on the values of the reproduction
number, the local and global stability of the rest points have
been investigated. Choosing a suitable bifurcation parameter, some
qualifications for the occurrence of Hopf bifurcation around the parameter
have also been obtained. Moreover, numerical simulations are presented
to support the analytical results. Finally, to study the effect of
the drug on the disease process, some control conditions are
determined. Since two modes of transmission and both linear and
nonlinear functional responses have been included in this
manuscript, our obtained results are generalization of those in
the literature. Moreover, the results have been obtained
with weaker assumptions in comparison with the previous ones.
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