A second order accurate approximation for fractional derivatives with singular and non-singular kernel applied to a HIV model

Abstract

In this manuscript we examine the CD$4^{+}$ T cells model of HIV infection under the consideration of two different fractional differentiation operators namely Caputo and Caputo-Fabrizio (CF). Moreover, the generalized HIV model is investigated by considering Reverse Transcriptase (RT) inhibitors as a drug treatment for HIV. The threshold values for the stability of the equilibrium point belonging to non-infected case are calculated for both models with and without treatment. For the numerical solutions of the studied model, we construct trapezoidal approximation schemes having second order accuracy for the approximation of fractional operators with singular and non-singular kernel. The stability and convergence of the proposed schemes are analyzed analytically. To illustrate the dynamics given by these two fractional operators, we perform numerical simulations of the HIV model for different biological scenarios with and without drug concentration. The studied biological cases are identified by considering different values of the parameters such as infection rate, growth rate of CD4${}^{+}$ T cells, clearance rate of virus particles and also the order of the fractional derivative.

Introduction

The fractional calculus provides new aspects in describing complicated dynamics of realistic systems having memory effects; nevertheless, understanding and anticipating such systems become always a serious research problem for scientists. This is due to the fact that the classic type of fractional derivatives have a singular kernel, and hence, they may not always be able to describe the non-locality of real-world dynamics properly. In order to characterize better the nonlocal systems, new fractional derivatives with nonsingular kernel have been suggested and applied for practical purposes. One of the best choice among the existing definitions is the Caputo–Fabrizio (CF) fractional operator based on the exponential function [1]. The main outcome of this new derivative is that it exhibits new asymptotic behaviors, which differ from the classic case of fractional operators. Some important extensions of this operator is reported on the existence of solutions for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations [2]. Another extension of CF was reported recently in [3].

An interesting model based on the movement of wave on the surface of shallow water using CF has been studied in [4]. Their results showed that the movement of waves portrays with control on the surface of shallow water with CF derivative. However, the models presented in the literatures were not able to portray and control the real movement of these waves. Unsteady flows of an incompressible Maxwell fluid through a circular tube using CF fractional derivatives are investigated in [5] and they revealed that the fluid velocity and the associated heat transfer modeled by fractional derivatives are quite distinct from those of the ordinary fluids. Further they have shown that the fluid velocity and the thermal performance in cylindrical tubes can be controlled by regulating the fractional derivative parameter. A fractional order epidemiological model of dengue fever is investigated within three types of operators known as the Caputo, the CF, and the ABC [6]. It has shown that the efficiency rates of the fractional-order operators are higher than that of the existing classical model by comparing the obtained solutions with real data of dengue outbreak.

The human immunodeficiency virus (HIV) gives basis to grow acquired immunodeficiency syndrome (AIDS), where it may take 10–15 years to develop AIDS after being infected by the virus. According to the statistics given by WHO (World Health Organization) the number of people who are newly infected with HIV is increasing as fast as the ones living with HIV currently. For HIV-1 infection the primary target is an activated CD4${}^{+}$ T-cell [7]. The availability of “target” cells is a significant control factor, i.e. cells that HIV is capable to transmit infection [8], [9]. CD4 T-cells are HIV preferential targets. Clinical data from a diverse range propose that availability of CD$4^{+}$ T-cells restricted virus replication. An infected CD4 T-cell makes multiple HIV copies and does not act upon its function in the human immune system. A shortage of CD4 T-cell count affect the working of the human immune system. Viral load is raised by stirring the immune system together with IL-2 [10]. A durable control of HIV-1 is caused by monotherapy through the anti-retroviral drug didanosine while it is unified by means of the immunosuppressive drug hydroxyurea [11]. De Boer et al. investigated anti CD4 therapy for AID based on mathematical models [8]. A stochastic HIV infection model of integer order with T-cell proliferation and CTL immune response is formulated in [12]. A deterministic model for the in-vivo dynamics of HIV is formulated and analysed in [13].

In recent years, different types of epidemic models of fractional differential equations have been proposed to study their dynamics. Fractional-order SIR, SIRS and SIS epidemic models are discussed in [14], [15]. A non-local interacting epidemic model is studied in [16]. The stability analysis for fractional order SEIR model is given in details by [17]. In [18], a fractional-order model of HIV-1 primary infection is analyzed. HIV infection of CD$4^{+}$ T-cells of fractional order is presented in [19]. In [20], the authors proposed a model for malaria transmission under control strategies using fractional operator. In [21] global asymptotic stability is obtained for one type of reaction diffusion equation describing the population with the logistic type of growth. An implicit numerical scheme for fractional order HIV model is proposed in [22] where a finite difference based approximation is used for Caputo derivative. Pinto et al. [23] examined the fractional order HIV model containing the population of the latent infected cells, macrophages and CTLs and further in [24] two transmission models and treatments are proposed. As a next step, the multi-order fractional derivatives are investigated in [25] for the dynamics of such models by using Bernstein operational matrices in an arbitrary interval to obtain the approximate analytical solution of the HIV infection model. Recently, the effect of time-varying drug exposure is investigated of the fractional order HIV infection model [26]. A fractional order SICA epidemiological model with constant recruitment rate, mass action incidence and variable population size, for HIV/AIDS transmission is analyzed in [27].

Due to the reason that many fractional-order differential equations do not possess exact analytic solutions, developing reliable numerical methods has a deep necessity and importance. In recent literature, various schemes have been suggested to solve the fractional-order differential equations and applied to several areas of science and engineering [28], [29], [30], [31], [32], [33]. The purpose of this manuscript is to construct an efficient numerical scheme to solve the HIV model using both Caputo and CF fractional operators as two different types of fractional differentiation operators having singular and non-singular kernels, respectively, in their definitions. For this purpose, we integrate both sides of the fractional system with respect to time t and utilize a second-order trapezoidal formula for the approximation of the corresponding integral. Therefore, the main outcome becomes the direct transformation of the fractional differential equation into the fractional integral equation, without a need of discretization of the time fractional derivative.

The plan of the manuscript is as follows: In Section 2, some necessary definitions are given together with the description of the integer-order model of interaction between target cells, T, productively infected T-cells, I, and HIV virus particles V. Then, in Section 3, the generalized model is newly formulated within Caputo and CF fractional operators and studied with and without RT inhibitor drug treatment. Moreover, the stability results for the non-infected equilibrium points of the system are presented for both cases. Section 4 is associated to the construction of the trapezoidal method to solve the fractional model with Caputo and CF operators, further the stability and convergence results of the proposed numerical schemes established. In Section 5, the dynamics of the HIV model given by different orders of the considered fractional derivatives are studied numerically. A set of biological cases that differ by the biological model parameters are simulated correspondingly to demonstrate the obtained numerical results of the generalized fractional HIV model. We conclude our paper in Section 6.