#### Abstract

Hepatitis B is one of the infectious diseases among other contagious diseases. In this paper, we analyze and discuss the time dynamics of hepatitis B under the effect of various infectious periods. We propose a model keeping in view the role of acute and chronic infections stages and analyze the temporal dynamics. To do this, first we develop the model and use the concept of fractional theory to fractionalize. We also study the qualitative analysis of both the integer and fractional order models. For the fractionalizing purpose, the Caputo–Fabrizio (CF) operator is utilized. We prove also, the existence and uniqueness of the solution to the considered fractional-order model to show that the epidemic problem is feasible and well possessed. For this, we use the fixed point theory and its applications. Moreover, we prove that the considered model possesses a positive as well as bounded solution. We calculate the basic reproductive number to find the equilibria of the model to execute the stabilities and to investigate that the proposed fractional model is stable asymptotically. We support our analytical findings with the use of some graphical visualization.

#### 1. Introduction

The hepatitis B virus harms the liver. The structure of the host cells changes due to the hepatitis B virus (HBV). Hepatitis B virus targets the chief functional cells of the liver (hepatocytes) which perform an astonishing number of metabolic, secretory functions, and endocrime. Hepatitis B has different stages: acute and chronic. The subjection of the virus to someone for up to six months refers to the acute stage. The immunity system in this stage can finish the contamination, but sometime lead to a serious phase and cause long life illness. Then, it is called the chronic stage. If a person had a hepatitis B infection more than six months, then the person has chronic illness. In the chronic phase, the history of the acute phase vanishes. Hepatitis dissipates liver and finally failure of liver occurs and causes liver cancer [1]. The different sources for transferring this virus are the use of razors, (from blades sharing, toothbrushes etc.) vaginal, and semen, etc., [2–5]. Another source is vertical transmission. In this source, a child has hepatitis B from his infected mother of this virus [6]. Throughout the world, the infected population is in million. Out of these, about 93% belong to China [7, 8]. The effective to prevent people from this virus is Vaccination, to immunize from the HBV and almost adopt permeant immunity [9, 10].

For infectious diseases, mathematical modeling is a huge field that has wealthy literature. In order to learn about the dynamics of infectious virus diseases and provide the control mechanism, mathematical modeling plays a notable role in it [11]. The researchers got attention to hepatitis B because it is one of the main causes of death and developed various models (see [12–16]). Anderson et al. presented the simplest model to explore the effect of moving on the transferring of hepatitis B virus [17]. In the United Kingdom, Williams et al. examined and presented the dynamics of hepatitis B [18]. Moreover, to control hepatitis B in New Zeeland, a predicted mechanism model has been presented by Medley et al. [19]. Also in China, to evaluate the effect of the vaccination program, a model is presented with the effect of age by Zhao et al. [20]. Bakare et al. used SIR epidemic model and suggested the analysis of control [21]. Studying with control strategies, many epidemic models are presented by Kamyad et al. [22], respectively. Onyango et al. developed a model to evaluate the multiple endemic solution [23]. Also, Zhang et al. in Zinjiang investigated the dynamics of hepatitis B virus [24]. Presently, various models have been forecasted in [25–27], and in [28] to study the influences (effect) of various parameters on the disease and present suitable control strategy to the get rid of the required infection. Fractional calculus is an emerging area of applied sciences and got the attention of many researchers. It could be noted that models with fractional order provide more interesting results as compared to integer-order models [29–32]. Moreover, the analysis of fractional calculus is also used to find the inherited properties of different physical phenomena as observed in the field of science and technology [33, 34]. There are many examples of classical models which provides less accurateness, especially in forecasting the prediction of the future dynamic for systems, while instead of this, the fractional order is more useful [35, 36]. Recently, Khan et al. [37] formulated a model and investigated the dynamics of hepatitis B virus transmission. However, it could be noted that the model could be further improved by including some more complexity keeping in view the disease characteristics. Therefore, we will formulate the model and fractionalize to present more accurate dynamics of the disease.

The literature reveals that hepatitis B has many infectious periods as well as various transmission routes. Both the acute and chronic individuals are significant. However, the acute are very significant because they have no symptoms while transmit the disease to others. We formulate the model by extending the work studied by Khan et al. [37]. Nevertheless, the work proposed in the reported study is a great contribution to the field of mathematical epidemiology; however, it could be improved further by incorporating some more interesting parameters. We develop the model with some new features according to the hepatitis B characteristic to study the dynamics of the disease. We then fractionalize using the fractional theory because these models are more accurate than classical order models. Once to develop the model, we prove its existence with uniqueness and with the application of fixed point theory. Furthermore, the boundedness as well as positivity of positive solutions are discussed to show that the epidemic problem is biologically as well as mathematically feasible. We figure out the basic reproductive quantity to calculate the model equilibria and perform the stability analysis. We analyze that the considered model is stable. We use various approaches for the stability of the considered model with fractional order. Particularly, the result of the mean value theorem, linearization, and Barbalat’s Lemma to discuss the local and global properties of the problem is under consideration. The theocratical work is investigated with the help of some graphical results to verify and differentiate between integer and noninteger order.

#### 2. Model Formulation

We present the formulation of the problem in this section. Formulating the model with new features and parameters that were ignored in the work is proposed in [37]. Particularly, the authors do not consider the spreading from various infectious periods of acute and chronic, while the disease transmits from both types of infected individuals. Furthermore, the probability based transmission is more appropriate whenever there are more than one infection periods while in the work of Khan et al., the disease transmission rate was not taken to be probabilistic. Therefore, we keep the characteristics of hepatitis B in view and so the various compartmental sizes are population are assumed to be susceptible acute, chronic, and recovered. These groups of population are symbolized by , , , and , respectively. The following constraints are defined as follows:(i)All the epidemic variables and constants are non-negative involved in the epidemic process(ii)Two infectious periods are taken i.e., acute and chronic, while the transmission is also taken from both which was not taken in the above-reported work(iii)Vaccination of susceptible individuals will lead to recovery because the vaccine provides at least twenty-five years of immunity(iv)The spreading of the disease is taken from both acute and chronically infected individuals and further taken that lead to the portion of acute with probability while to the chronic with probability(v)The natural recovery of the acute population is taken to be with probability while those goes to the chronic stage are taken with probability(vi)Newborns are susceptible and the disease-induced death rate is taken in the chronic compartment

In the light of assumptions, we present the dynamics as follows:with initial population sizeswhere is the rate of newborn and assumed to be susceptible, while and are the diseases and reduced transmission rates of the disease. is the vaccination and is the death rate, while is the rate of moving i.e., from acute to chronic phase. is the occurance of the death rate from the disease of hepatitis B virus, while and are the recovery rates from chronic and acute stages, respectively.

#### 3. Preliminaries

In this section, we discuss the basic concepts of the fractional theory which will be helpful in obtaining our results.

*Definition 1. *[38–40] Suppose is a function such that , if and , , the Caputo–Fabrizio and Caputo fractional order derivative are define asandIn the above (3) and (4), *C* stands for Caputo and *CF* stands for the Caputo–Fabrizio, where and denote the normalization function, with .

*Definition 2. *[38–40] If and , both change with time , then the definition of Riemann–Liouville integral having order is given asand the Caputo–Fabrizio–Caputo (CF) integral having order in Caputo–Fabrizio–Caputo (CF) is given byWe modify the equations of the model described by system (1) to the related noninteger order (, ) using the Caputo–Fabrizio–Caputo (CF) operator. We use fractional derivatives instead of the ordinary derivatives, and for keeping the equal dimension of the equations on both sides, we take the power of every parameter as

#### 4. Existence and Uniqueness

In this section, we discuss the unique analysis of and the existence of the proposed fractional-order epidemiological model (7). For proving the uniqueness and existence of the solution, we use some necessary concepts of fixed point theory. So we transform the given model into the integral equation to analyze. Thus,

For the above system, the *CF*-fractional integral application gives

Let , , , and are the kernels and is defined by

Theorem 1. *The kernels , , , and given above satisfy the conditions of Lipschitz axioms.*

*Proof. *Suppose that and , and , and , and are the kernels for the two function, respectively , , , and , soBy using the application of Cauchy’s inequality, we obtain the system given below:Recursively, we obtainThe difference between consecutive terms with majorizing the norm application implieswhereFollowing the work of Qureshi et al., [41], the kernels satisfy the Lipschitz conditions, so

Theorem 2. *The solution of the proposed model (7) exists.*

*Proof. *By applying the equation (11) with iterative scheme, we obtainWe investigate that the (17) are the solutions of the model (7) consequently by making the substitutionswhere , , , and show the rest of the terms of the series; then,Applying Lipschitz axiom and taking norm on both sides, thenApplying as tends , we getwhich conclude that the reported model as given by (7) posses a solution.

Theorem 3. *The solution of the model given by equation (7) is unique.*

*Proof. *On contrary, we suppose that the reported model as described in (7) also is ; then,By majorizing, one may deriveBy applying Theorems 1 and 2, one may getThe inequalities as reported by the above (24) hold for each value of ; thus, we getHere, we discuss bounded-ness and the positivity of the proposed model (7) to prove the well-posed of the proposed problem. Furthermore, for the dynamics of the proposed problem, we state that a specific region that is positively invariant. To do this, the following Lemmas have been explored.

Lemma 1. *As are the solutions of the proposed model (7) and suppose that the model having non-negative initial conditions, then are non-negative solutions for every .*

*Proof. *For the proposed system given by (7), we take a general order fractional model asHere, denotes the order and represents the operator under consideration. Therefore, (26) takes the formAs contained in and . By following [42], we conclude that for every non-negative , the solutions are positive.

Lemma 2. *Let denote the region, in which the dynamics of the proposed model (7) is positively invariant; then,*

*Proof. *As denotes the total consider population, therefore, we haveThe solution for the (29) is given byHere, the Mittag–Leffler function is denoted by with . The above (30) whenever time increases without bound, then . Thus, if , then for every , while if , then contained in . So, we concluded that the dynamics of the reported model can be analyzed in the feasible portion .

#### 5. Steady State

The model as presented by (7) studied for equilibria such as endemic and disease free. Let be the state of disease free of the reported model; for studding point, the population is taken to be noninfectious. Therefore, the reported system given by (7) has a disease-free point , where , and . Now, to find the basic reproductive number, suppose , then system (7) giveswhere

Thus, the spectral radius of , i.e., , is the basic reproductive number.

Suppose is the endemic equilibrium and let , , . and , then the endemic equilibrium becomes

Thus, concerning to the local and global dynamics of the reported system, we find the dynamics results given below:

Theorem 4. *The disease-free point of the reported model described by (7) is stable locally asymptotically, if where endemic equilibrium is stable locally asymptotically if .*

*Proof. *Following [41], the linearizable description of the reported hepatitis B-virus model is given by (7) around guide to a matrix that is underOne eigenvalue is which is negative. For the remaining, we haveHere, we need to show that and , soandwhich implies thatClearly, we examine that holds if ; thus, we obtain the result that the equilibrium point of the reported model is stable locally asymptotically with condition .

In the same way, it can be proven that the endemic equilibrium of the given model is locally asymptotically stable. Concerning global dynamics of the fractional-order model, we obtain the below results.

Theorem 5. *The disease-free point of the reported model given by (7) is stable globally asymptotically if , where then the endemic point is stable globally asymptotically if .*

*Proof. *To study the global analysis of the epidemic model, we use the same methodology reported in [43]. Let us assume that and let it has a finite limit if varies without bound, so equation first of the reported model (7) becomesSince, for and for any , , so by following Theorem 3.1 in [44], it then leads towhere , and . Consequently, we getIn a same way, the of , and can be proved. Furthermore, let us assumeand