**2. The Model**

The model consists of seven classes: the healthy and susceptible CD4<sup>+</sup> T cells, *T*, the healthy and non-susceptible CD4<sup>+</sup> T cells, *TR*, the latently infected CD4<sup>+</sup> T cells, *L*, the infected and infectious CD4<sup>+</sup> T cells, *I*, the infected and non-infectious CD4<sup>+</sup> T cells, *IR*, the HIV virus, *V*, and the drug concentration in the plasma, *R*.

CD4<sup>+</sup> T cells are produced with rate *λ* and die with rate *μ*. These cells are infected by HIV and by infected CD4<sup>+</sup> T cells at rates *β*<sup>1</sup> and *β*2, respectively. The healthy T cells are inhibited by drug at rate *q*. A fraction, *η*, of infected CD4<sup>+</sup> T cells becomes latently infected. The latently infected CD4<sup>+</sup> T cells become productively infected at a rate *aL* and die with a rate *μL*. The infected CD4<sup>+</sup> T cells die with rate *a* and are inhibited by drug at rate *p*. The virus are produced by infected CD4<sup>+</sup> T cells at rate *k* and cleared at rate *c*. The dynamics of the drugs is as follows. For simplicity, we postulate that after taking the drug, the cell, *TR*, inhibits infection until it dies. We further assume that drugs are taken at times *t* = *tk*, and their effect is instantaneous. The latter results in a system of impulsive differential equations, with condition Δ*R* = Δ*Rk*, where Δ*Rk* is the dosage. For *t* = *tk*, the solutions are continuous and obey system (1). The drug, *R*, is cleared at rate *g*.

The nonlinear system of fractional differential equations describing the model is given by:

$$\begin{array}{rcl} \frac{d^aT}{dt^a} &=& \lambda^a - \mu^a T - \beta\_1^a T V - \beta\_2^a T I - q^a T R \\\\ \frac{d^aL}{dt^a} &=& \eta \beta\_1^a T V + \eta \beta\_2^a T I - a\_L^a L - \mu\_L^a L \\\\ \frac{d^aI}{dt^a} &=& (1 - \eta) \beta\_1^a T V + (1 - \eta) \beta\_2^a T I + a\_L^a L - a^a I - p^a I R \\\\ \frac{d^aV}{dt^a} &=& k^a I - c^a V \\\\ \frac{d^aT\_R}{dt^a} &=& q^a T R - d^a T\_R \\\\ \frac{d^aI\_R}{dt^a} &=& p^a I R - d^a I\_R \\\\ \frac{d^aR}{dt^a} &=& R\_k^a - g^a R \end{array} \tag{1}$$

where the parameter *α* ∈ (0, 1] is the order of the fractional derivative. The fractional derivative of the proposed model is used in the Caputo sense.
