The role of synaptic transmission in a HIV model with memory.

A fractional order model for HIV with two transmission modes and treatment is proposed.Global stability of the disease-free equilibrium is analyzed.Cell-to-cell transmission has a strong impact in the value of the reproduction number.Increasing drug efficacy decreases the infectiousness of the disease. We propose a mathematical model with memory for the dynamics of HIV epidemics, where two transmission modes, cell-to-cell and virus-to-cell, and drug resistance are considered. Systems with memory, or fractional order systems, have largely been applied to the modeling of several real life phenomena. Here, we consider a fractional model where the order of the non-integer derivative takes values in the interval 0.5, 1.0. We prove the local and global stability of the disease-free equilibrium. We study the role of the cell-to-cell transmission probability on the dynamics of the model, and on the value of the reproduction number, R0, for distinct values of the fractional order derivative, α. Moreover, we show evidence of an improvement of HIV infected patients quality of life, due to the increase of the drug efficacy. In the end, important inferences are drawn.