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Abstract
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It is well known that dynamical systems are very useful tools to study viral diseases such as HIV,
HBV, HCV, Ebola and Influenza. This paper focuses on 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 stabilities of the rest points are investigated. Choosing
a suitable bifurcation parameter, some conditions for the occurrence of Hopf bifurcation are
also 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 a generalization of those in the
literature. Moreover, the results are obtained with weaker assumptions in comparison with the
previous ones.
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