*Fractional Calculus—Short Recap*

Fractional Calculus has been a hot research topic in the last few decades. Researchers from distinct scientific areas, theoretical and applied, have studied fractional order models to obtain a deeper understanding of real world phenomena [4,12–17]. Fractional order models are characterized by a 'memory' property, which brings additional information to analyze the systems' dynamical behaviors.

The classical definitions for a derivative of fractional (non-integer) order are the Caputo (C), the Riemann-Liouville (RL) and the Grünwald-Letnikov (GL). Let (0, *t*) be the interval, instead of (*a*, *t*), for simplification. The function *y*(*τ*) is smooth in every interval (0, *t*), *t* ≤ *T*. The RL definition reads:

$$D\_{RL}^{\mathfrak{a}}y(t) = \begin{cases} \frac{1}{\Gamma(m-\mathfrak{a})} \frac{d^m}{dt^m} \int\_0^t \frac{y(\tau)}{(t-\tau)^{n+1-m}}, & m-1 \le n < m\\ \frac{d^m y(t)}{dt^m} & n = m \end{cases}$$

where Γ is the Euler Gamma function. The Caputo definition is given by:

$$D\_{\mathbb{C}}^{a}y(t) = \begin{cases} \frac{1}{\Gamma(m-a)} \int\_{0}^{t} \frac{y^{m}(\tau)}{(t-\tau)^{a+1-m}} & m-1 \le a < m\\ \frac{d^{m}y(t)}{dt^{m}} & n = m \end{cases}$$

The GL definition is based on finite differences and is equivalent to the RL formula:

$$D\_{GL}^{\alpha}y(t) = \lim\_{h \to 0} \quad h^{-\alpha} \sum\_{k=0}^{n} (-1)^{k} \frac{\Gamma(\alpha+1)}{k! \Gamma(\alpha-k+1)} y(x-kh), \ nh = x$$

The memory effect in biology/epidemiology/immunology is extremely important, thus the appearance of fractional order models in the study of patterns arising in these models comes as a natural generalization of the integer order models [18–21]. In [20], the authors generalize an integer order model for HIV dynamics to include a fractional order derivative. In Arafa et al. [18] the authors generalize an integer order model for HIV dynamics to include a fractional order derivative. They conclude that the fractional order model provides a better fit to real data from 10 patients than the integer order model. Pinto [4] studies the role of the latent reservoir in the persistence of the latent reservoir and of the plasma viremia in a fractional-order (FO) model for HIV infection. The model assumes that (i) the latently infected cells may undergo bystander proliferation, without active viral production, (ii) the latent cell activation rate decreases with time on ART, and (iii) the productively infected cells' death rate is a function of the infected cell density. The model clarifies the role of the latent reservoir in the persistence of the latent reservoir and of the plasma virus. The non-integer order derivative is associated with distinct velocities in the dynamics of the latent reservoir and of plasma virus. In [12], the authors study the effect of the HIV viral load in a coinfection fractional order model for HIV and HCV (hepatitis C virus) coinfection. HIV has a significant impact on the burden of the coinfection. Moreover, the order of the fractional derivative may pave the way to a better understanding of the individuals' compliance to treatment, the distinct responses of the immune system. The non-integer order derivative adds another degree of freedom to the model. In what concerns drug diffusion in tissues, there are some interesting results in the literature. In [22], the authors propose non-integer order (fractional order) models to represent anomalous diffusion, memory effects and power-law clearance rates, typical of drug uptake and diffusion in a case-study of a drug used for cancer therapy. They conclude that fractional models avoid unbounded accumulation of drugs, seen in the integer order approach, and help to prevent life-threatening side-effects on patients. In 2017 [23], the authors provide a review on pharmacokinetic models and propose their generalizations to fractional orders. The new models account for tissue trapping as well as short- and long-time recirculating effects. The benefits from such approach are twofold: (i) a better understanding of secondary effects on patients under treatment; and (ii) avoidance of unbounded drug accumulation.

With the aforementioned ideas in mind, we outline the paper as follows. In Section 2, we describe the proposed model. We follow with the computation of the reproductive number and the stability of the disease free equilibrium in Section 3. Then, in Section 4, we prove the global stability of the disease free equilibrium. The model is simulated and the corresponding results are discussed in Section 5. Finally, in Section 6, we conclude this work.
