Abstract

In this paper, a mathematical fractional order Hepatitis C virus (HCV) spread model is presented for an analytical and numerical study. The model is a fractional order extension of the classical model. The paper includes the existence, singularity, Hyers-Ulam stability, and numerical solutions. Our numerical results are based on the Lagrange polynomial interpolation. We observe that the model of fractional order has the same behavior of the solutions as the integer order existing model.

1. Introduction

Hepatitis C is a kind of viral maladies caused by the Hepatitis C virus (HCV), which mostly damages the liver. People generally have minor or no symptoms when they first become infected. Black urine, Fever, yellow-tinged skin, and abdominal pain are all symptoms that might occur. The virus remains in the liver, in roughly 75 percent to 85 percent of patients who are infected. Initially, in the period of a chronic infection, there are usually no symptoms. However, it frequently develops to cirrhosis over time. Hepatitis C, on the other hand, can sometimes lead to liver cancer, blood cancer, and liver failure [1]. The most common way for HCV to spread is by blood-to-blood contact, which is related to injectable usage of drugs, improperly cleaned equipment for medical care, needle stick injuries in health care, and transfusions. The most typical reason for liver transplantation is Hepatitis C, even though virus generally returns after the procedure. Hepatitis C infects an estimated 71 million people (1 percent of the global population) in 2015. Low- and middle-income nations bear the brunt of the health burden, with Africa and Central and East Asia having the greatest rates of prevalence. In 2015, Hepatitis C caused around 167,000 liver cancer deaths and 326,000 cirrhosis deaths. 15th Hepatitis C is a disease that can be transmitted from one person to another. Hepatitis C was first identified in the 1970s, it was thought to be a kind of non-A non-B Hepatitis, and its presence was confirmed in 1989. Only humans and chimps are infected with Hepatitis C; for more details, see [2, 3].

In natural and physical sciences, mathematical and computational tools have been used to investigate phenomena at many scales, ranging from the global human population to individual atoms within a biomolecule. The relevant modeling methodologies span time spans ranging from years to picoseconds, region to region of interest (impacts ranging from evolutionary to atomic), and importance. This exploration will go over some of the most common and useful approaches in mathematics and computing. Differential equations, statistical models, dynamical systems, and game theoretic models are all examples of mathematical models; we refer to [4–10]. These and other types of models can be mixed and matched, resulting in a single model that has a diverse set of abstract structures. Logic models can be used in mathematical models in general. In many instances, the quality of a scientific topic is determined by how well theoretical mathematical models accord with the results of repeated experiments. As better theories are discovered, a lack of concordance between mathematical models that are theoretical and experimental findings frequently leads to significant advancements. Many mathematical models were formulated for the Hepatitis C diseases to understand the dynamics of the diseases and control the spreading of diseases; we refer to [11–14].

Fractional calculus is a discipline of mathematics that investigates the various ways in which the differentiation operator can be defined in terms of real or complex number powers. Because of their ability to include notion of nonlocal operators used to incorporate more complicated natural phenomena into mathematical equations, differential equations have attracted many scholars from practically all disciplines of science, technology, and engineering in recent years. The exponential decay law, the power law, and the extended Mittag-Leffler law were recommended as three dominants in fractional calculus. The kernel Mittag-Leffler function was shown to be more broadly applicable than the power law and exponential decay functions; both Riemann-Liouville and Caputo-Fabrizio are special examples of the Atangana-Baleanu fractional operator; we refer to [15–18]. Many biological models have been studied on fractional operators; Atangana and Alqahtani [19] considered a mathematical model river blindness as in Caputo sense and beta operators. Stability and numerical solutions were obtained for the fractional order model. Gómez-Aguilar et al. [20] examined a cancer model in three dimensions in the sense of Caputo-Fabrizio-Caputo and the novel fractional derivative with Mittag-Leffler kernel. Special solutions were obtained by an iterative process that used the Laplace transform rule, Sumudu-Picard integration approach, and the Adams-Moulton approach. Shah and Bushnaq [21] evaluated an endemic infection model in the fractional sense. Numerical solutions were obtained for the proposed model by combining the Laplace transform with the Adomian decomposition approach. Arfan et al. [22] studied semianalytical solutions for a fractional order COVID-19 model under Caputo derivative; for more details, see [23–26].

Inspired from the above literature, in this paper, we consider a Hepatitis C model in the sense of a fractional order derivative. Furthermore, we investigate the existence and uniqueness of the fractional order model with the help of a fixed point theorem and stability analysis of the fractional order Hepatitis C model. Finally, numerical simulations of the solutions are demonstrated and compared with classical derivatives by using different values of fractional order and parameters. This paper is organized as follows; Section 1: introduction of the paper; Section 2: framework of the model; Section 3: preliminaries; Section 4: existence of solutions; Section 5: numerical data fitting; Section 6: conclusion.

2. Framework of the Model

In this section, we discuss an integer order Hepatitis C model. The population is split into four classes based on their size. denotes class of susceptible, class of acutely infected, class of persistently infected, and class of treatment for infection:

With the initial conditions

where is the rate of natural death, is force of infection, is the rate of susceptibility of recovered, is the rate of recovery from acute infection, is the rate of progression to chronic infection, is the rate of death due to acute infection, is the rate of treatment failure of chronically infected, is the rate of recovery from chronic infection, is the rate of treatment of chronically infected, and is the rate of treatment cure. For more details on the existence of infection, endemic equilibrium, reproduction numbers, and stability of endemic equilibrium, see [11–13].

Definition 1 (see [17, 27]). On the basis of Mittag-Leffler kernel and for , the -fractional differential operator is given as where denote a weighted function which satisfied the main property :

Definition 2 (see [17, 27]). On the basis of Mittag-Leffler kernel and , the -fractional derivative is defined as

Definition 3 (see [17, 27]). Let ; the -integral is given

Lemma 4 (see [17]). Let function , then the AB fractional integral and derivative satisfy the following special character of Newton-Leibniz formula:

3. Existence Criteria

Let , where , for , with a norm defined by . Then, clearly, is Banach’s space. Let us consider system (1) in the sense of fractional order operator:

By employing Definition (3) to (8), we have

For simplicity in the above Equation (9), we introduce for given below:

For proving our results, we consider the following assumption . For the below continuous functions , such that there exist three constant , , such that the below hold:

Theorem 5. , for , satisfy Lipschitz condition if for defined in (17).
Consider for , below For , we have implies implies Thus, from (19)(–)(22), we have that for satisfying the Lipschitz condition. And this completes the proof. Assuming that , then we have For the iterative scheme of the -fractional order HCV model (8), define

Theorem 6. The -fractional order HCV model (8) has a solution if We define the function By the help of (29) and (30), we have Similarly, which ensure that , as for which completes the proof.