Abstract

The purpose of this paper is to propose a variable fractional-order model with a constant time delay of the coinfection of HIV/AIDS and malaria. The proposed model describes the interaction between HIV/AIDS and malaria. This model is presented by using variable fractional-order derivative which is an extension of the constant fractional-order derivative to explain a certain pattern in the development of infection of several patients. The presented model has been solved numerically via the predictor-corrector scheme. The local and global stability conditions of the disease-free equilibrium are investigated. Also, numerical simulations are presented for different variable fractional-order derivatives in Caputo sense.

1. Introduction

The human immunodeficiency virus (HIV) and malaria are considered among the most challenging global public health issues in the last few decades. HIV and malaria are life-threatening diseases which have similar geographic distributions [1]. They cause millions of deaths every year in several areas especially in Africa, Asia, and Latin America. In 2017, HIV killed about one million people [2] while malaria killed roughly 435 000 people worldwide [3]. HIV can be transmitted through certain body fluids while malaria is transmitted through bites of infected mosquitoes.

HIV is considered as one of the most deadly infectious diseases which strikes the human immune system and destroy the CD4+ cells. AIDS is the last stage of HIV which occurs when the CD4+ cells of the human body count drops below 200 cells/mm [4]. In this stage, the immune system cannot defend the body against the attacks by several opportunistic diseases. On the other hand, if malaria parasite invades the bloodstream, then, it destroys red blood cells. So, malaria infection may be developed to anemia or cerebral malaria, which can cause disabilities and death [5]. The coinfection of HIV and malaria has become endemic in several developing countries. World health organization (WHO) reports indicating that more than two million people die every year because of the malaria/AIDS coinfection [6]. The interaction between HIV and malaria in Sub-Saharan Africa has become among the major public health problems [7] and has resulted in many economic disasters [1] by negatively affecting the contribution of the labor force to the national economy.

Recently, increasing research efforts have been made to obtain an effective vaccine to halt the progression and transmission of malaria. Vaccination target is to reduce the rate of human infection, the severity of the disease [810], and the parasite’s transmission to mosquitoes. Clinical trials in Africa proved that a malaria vaccine is partially protective [11].

From mathematicians’ perspective, mathematical models are significant tools that help us to understand the current state and the future progress of infectious diseases in human networks in order to control and prevent such diseases. Several mathematical models have been presented to study the prevalence and the coinfection of HIV and malaria, but most of such models are integer or constant fractional-order models [1222]. This paper is devoted to propose a delay variable fractional-order model for the coinfection of HIV/AIDS and malaria. In this model, a discrete time delay is incorporated in the variables of active humans who are infected by malaria and the coinfected humans while a discrete time delay is incorporated in the variable of the infectious mosquitoes. After a time , susceptible people become infected by malaria while exposed individuals become infectious after the same time. On the other hand, mosquitoes become infectious after time . Introducing such a time delay to the proposed model is essential to characterize the time needed to start in vaccination and treatments processes. The merits of the proposed model are clear from putting in the time delay with the variable fractional-order derivative which is an extension of the constant fractional-order in the same model. Hence, using the proposed variable fractional-order model with time delay gives a better understanding of the interaction between malaria and HIV. To the best of our knowledge, the presented model is the first variable fractional-order model with a time delay which describes the prevalence and interactions between HIV and malaria. In this model, the integer order derivative is used to distinguish the short memory of systems, while the variable fractional-order derivative is utilized to characterize the variable memory of systems.

This paper is organized as follows. In Section 2, some preliminaries of fractional calculus and the algorithm of the predictor-corrector method are presented while Section 3 describes the proposed model. In Section 4, the disease-free equilibrium and stability are presented. The basic reproduction number is computed in Section 5. Section 6 is devoted to the numerical results and discussions. Our conclusion is illustrated in Section 7.

2. Preliminaries

2.1. Fractional Calculus

The fractional calculus is considered as a mathematical tool for characterizing memory of biological and epidemiological systems. The classical integer order derivative can be used to describe the short memory of the dynamical systems, while fractional-order derivative has the merit of describing the long memory of dynamical systems. The variable fractional-order derivative is an extension of the constant fractional-order derivative and has been introduced in several scientific fields [2325]. Also, it is a powerful tool to characterize memory that varies from point to point. Furthermore, the variable fractional-order derivative can be applied to describe the variable memory of dynamical systems [26].

In this section, we present some basic definitions of constant/variable fractional-order derivatives as follows.

Definition 1 (Riemann–Liouville derivatives of fractional-order ). Let be a bounded and continuous function; then Riemann–Liouville fractional-order derivative of is defined as follows [27].
(i) Left Riemann–Liouville derivative of fractional-order is defined by(ii) Right Riemann–Liouville derivative of fractional-order is defined by

Definition 2 (Caputo derivatives of fractional-order ). Let be a bounded and continuous function; then the Caputo fractional-order derivative of is defined as follows [27].
(i) Left Caputo derivative of fractional-order is defined by(ii) Right Caputo derivative of fractional-order is defined by

Definition 3 (Riemann–Liouville derivatives of variable fractional-order ). Let be a bounded and continuous function; then Riemann–Liouville fractional-order derivative of is defined as follows [27].
(i) Left Riemann–Liouville derivative of variable fractional-order is defined by(ii) Right Riemann–Liouville derivative of fractional-order is defined by

Definition 4 (Caputo derivatives of variable fractional-order ). Let be a bounded and continuous function; then the Caputo fractional-order derivative of is defined as follows [27].
(i) Left Caputo derivative of fractional-order is defined by(ii) Right Caputo derivative of fractional-order is defined by

2.2. Predictor-Corrector Method

There are many techniques for solving a delay variable fractional-order models such as finite difference [28], Hermite wavelet [29], and Adams-Bashforth-Morton [30] methods. In this section, we state a predictor-corrector method for solving a delay variable fractional-order model [31].

Let where , , and is a smooth function. Suppose a uniform grid , where and are integers such that and .

The predictor approximation is defined by whereThe corrector approximation is defined by where

3. The Model

The proposed variable fractional-order model with a constant delay in this paper is based on the constant fractional delay model proposed in [32]. This model consists of 12 compartments, as follows:where the population of mosquitoes as follows:where are the infectious mosquitoes and are the susceptible mosquitoes.

And the population of human is divided into the following classes: are the susceptible individuals are the individuals vaccinated against malaria are the individuals infected with malaria are individuals infected and vaccinated against malaria are the coinfected individuals showing no symptoms of AIDS are the individuals asymptomatically infected with HIV/AIDS are the HIV infected individuals showing symptoms of AIDS are the coinfected individuals showing symptoms of AIDS

Besides, all human are subject to natural death, occurring at a rate . Susceptible individuals get in the human population at a rate . The parameter p is the proportion of individuals successfully vaccinated, where is the proportion getting in the class and is the proportion getting in the class . Susceptible individuals enter the class after some time . The rate of infection by malaria parasite of susceptible individuals is given bywhere is the proportion of individuals in the community and models the efficacy of adopted strategies for individuals protection. is the rate of female mosquitoes’ bites. The value of the probability that a bite of an infectious mosquito leads to the infection of a susceptible human is . The efficacy of the preerythrocytic vaccine is given by . Vaccinated individuals may become susceptible at a rate . The rate of infection with HIV/AIDS of susceptible individuals is :where is the infectiousness of individuals in the AIDS stage of HIV infection. is the effective contact rate for HIV infection. Infectiousness to malaria of coinfected individuals showing symptoms of AIDS is .

Parameter models the effect of the preerythrocytic vaccine in the raising of the recovery. Parameter models the effect of the preerythrocytic vaccine in the decreasing of mortality due to disease. The rate of recovery of individuals infected with malaria and going to the susceptible class is . The rate of death of individuals infected with malaria is . models the decrease in sexual activity due to malaria disease. is the rate of recovery of the coinfected individuals showing no symptoms of AIDS from malaria. refers to the increased malaria mortality of individuals coinfected with HIV. indicates the rise in HIV mortality due to the coinfection with malaria. is the rate of development of to AIDS. The rate of death from AIDS is . The rate of natural death of is . is the assumed rise in susceptibility to malaria as a result of HIV infection. The rate of recovery of from malaria is . is the rise in susceptibility to malaria of individuals of . defines those coinfected individuals develop to AIDS faster than those infected only with HIV.

The rate of natural death of mosquitoes is . The rate of infection by the Anopheles parasite of susceptible mosquitoes is given by where defines the decreasing of transmission from vaccinated humans to susceptible mosquitoes. The probability that a mosquito’s bite in a malaria infective human tends to infection of the mosquito is . The exposed mosquitoes turn infectious after time . The rate of increasing mortality due to the presence of the parasite in the body is . In other words, all mosquitoes are subjected to a natural death, at a rate of . It is assumed that the infectious mosquitoes are subjected to death rate because of the presence of the parasite in their bodies at a rate and that they do not recover before they die [32].

4. The Disease-Free Equilibrium and Stability

The equilibrium point of a dynamical system is a solution that does not change with time.

To obtain the disease-free equilibrium of model (14), letThen the disease-free equilibrium is The stability of disease-free equilibrium is defined by a sign of real part of eigenvalues of the Jacobian matrix evaluated at disease-free equilibrium. The Jacobian matrix is the matrix of partial derivatives of the right-hand side with respect to state variables.

The Jacobian matrix of model (14) around the disease-free equilibrium iswhereThe eigenvalues of the Jacobian matrix areThe remaining five eigenvalues are obtained from the following matrix:whereThat matrix M has the characteristic equationwhere

Using the results in [33], the disease-free equilibrium is locally asymptotically stable if the Routh-Hurwitz determinants , , , , aresatisfying , , , and . These conditions are the needed sufficient conditions to verify for .

We can put system (14) in the following form:Let , , ; then is continuous with respect to and globally Lipschitz continuous with respect to , , and in the following norm: that is,for some Lipschitz constants , , and , and , . So satisfies the standard conditions for the existence and uniqueness of solutions [34].

Also, let be an equilibrium point of system (33). To determine the local stability of the system (33) we can use the indirect method of Lyapunov which uses the linearization of a system [35].

The linearization of the system (33) is where , , and are matrices evaluated at the disease-free equilibrium (essentially a Jacobian matrix for each time delay) [36].

It follows that, for each fixed t, the remainder is

And the remainder tends to zero as , , tend to zero. But, the remainder may not tend to zero uniformly. So we need a stronger condition which is

If (37) holds, then system (35) is the linearization of the system (33). Once the linearization exits, its stability defines the local stability of the original nonlinear system.

Let , , be bounded. If is a uniformly asymptotically stable equilibrium point of system (35) then is a locally uniformly asymptotically stable equilibrium point of system (33).

5. The Basic Reproduction Number

In epidemiology, the basic reproduction number is defined as the number of secondary infections due to a single infection in a totally susceptible population. It is useful since it decides if or not an infectious disease can spread through a population. When , the infection will be able to spread in a population. But if , the infection will disappear. For , there was, at least, one stable endemic equilibrium [32]. In some cases, the basic reproduction number is not enough to predict the spread of epidemics because bifurcation may occur.

The basic reproduction number of the model (14) is shown in [32]where is the basic reproduction number of malaria model and is the basic reproduction of HIV model as follows:

Theorem 5 (see [43]). If , then the disease-free equilibrium is globally asymptotically stable in .
= {(, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , such that the solution of the system (14) is positive}.

Proof. From the previous section according to Routh-Hurwitz conditions defined by (31) must be greater than zero so we will rewrite in terms of and after some manipulation as follows:Thus if and so the disease-free equilibrium is globally asymptotically stable in .

6. Numerical Results and Discussions

Applying the predictor-corrector method to solve model (14) with initial conditions,And the values of parameters are shown in Table 1

We investigate the model behavior in two cases. Firstly, the variable fractional-order is . Secondly, the variable fractional-order is a periodic function .

In Figure 1, we show the effect of the parameter which is the susceptibility to malaria of individuals showing symptoms of AIDS. It is shown that when increases; the number of HIV infected individuals showing symptoms of AIDS decreases. Besides, when we use the variable fractional-order means the memory of the model is described as a periodic function; hence the behavior of the model is also periodic. Also, when we use the variable fractional-order means the memory in the model is described by a decreasing function so the model behavior is slower with time as in Figures 2 and 3.

In Figure 4, we show the effect of the parameter which is HIV mortality due to the coinfection with malaria. It is shown that when increases; it leads to decreasing of new cases of malaria. Besides, when we use the variable fractional-order means the memory of the model is described as a periodic function; hence the behavior of the model is also periodic. Also, when we use the variable fractional-order means the memory in the model is described by a decreasing function so the model behavior is slower with time as in Figures 5 and 6.

The presented numerical results indicate that the proposed delay variable fractional-order model is a generalization of the constant fractional-order model with a time delay which has been presented in [32].

7. Conclusion

A delay variable fractional-order model for the coinfection of HIV/AIDS and malaria which includes malaria vaccination and personal protection strategies is proposed in this paper. Also, the basic reproduction number and stability of the disease-free equilibrium have been studied. The numerical results showed the impact of changing the parameters values such as and on the number of the infected individuals with malaria/HIV, coinfected individuals, and infectious mosquitoes as well. The variable fractional-order derivative in the proposed model is used to distinguish the effect of the memory that changes over time on the disease progression of distinct patients. In Our future work, comparisons between the numerical results and real data will be held in order to examine the numerical simulation results at different variable fractional-order

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.