# Modeling of Hepatitis B Virus Transmission with Fractional Analysis

Authors: Muhammad Khan; Tahir Khan; Imtiaz Ahmad; Zahir Shah; Alamzeb Khan
Journal: Mathematical Problems in Engineering
 (2022)

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## 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.


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## Body

## 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 Wt, Xt, Yt, and Zt, 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 withr probability while to the chronic with 1−r probability(v)
The natural recovery of the acute population is taken to be with probabilitys while those goes to the chronic stage are taken with 1−s probability(vi)
Newborns are susceptible and the disease-induced death rate is taken in the chronic compartmentIn the light of assumptions, we present the dynamics as follows:(1)dWtdt=Π−αWtXt−αγ0WtYt−ν+μ0Wt,dXtdt=rαWtXt+αγ0WtYt−μ0+β+γ1Xt,dYtdt=1−rαWtXt+αγ0WtYt+sβXt−μ0+γ2+μ1Yt,dZtdt=1−sβXt+νWt+γ2Yt−μ0Zt,with initial population sizes(2)W0,Z0>0,X0,Y0≥0,where Π is the rate of newborn and assumed to be susceptible, while α and γ0 are the diseases and reduced transmission rates of the disease. ν is the vaccination and μ0 is the death rate, while β is the rate of moving i.e., from acute to chronic phase. μ1 is the occurance of the death rate from the disease of hepatitis B virus, while γ2 and γ1 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 ℏt is a function such that ϕ∈H10,T,T>0, if σ>0 and n−1<σ<n, n∈ℕ, the Caputo–Fabrizio and Caputo fractional order derivative σ are define as(3)YD0,tσℏt1Γn−σ∫0tt−xn−σ−1ℏnxdx,and(4)CFD0,tσℏt=Kσ1−σ∫0tℏ′yexpσy−t1−σdy.
In the above (3) and (4), C stands for Caputo and CF stands for the Caputo–Fabrizio, where t>0 and Kσ denote the normalization function, with K1=0andK00.Definition 2.
[38–40] If ℏt and 0<σ<1, both change with time t, then the definition of Riemann–Liouville integral having order σ is given as(5)RLJ0,tσℏt=1Γσ∫0tt−yσ−1ℏydy,and the Caputo–Fabrizio–Caputo (CF) integral having order σ in Caputo–Fabrizio–Caputo (CF) is given by(6)CFJ0,tσℏt=21−σℏt2−σKσ+2σ2−σKσ∫0tℏydy.
We modify the equations of the model described by system (1) to the related noninteger order (σ, 0<σ<1) 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(7)CFD0,tσWt=Πσ−ασWtXt−ασγ0σWtYt−μ0σ+νσWt,CFD0,tσXt=rασWtXt+ασγ0σWtYt−μ0σ+βσ+γ1σXt,CFD0,tσYt=1−rασWtXt+ασγ0σWtYt+sβσXt−μ0σ+μ1σ+γ2σYt,CFD0,tσZt=1−sβσXt+νσWt+γ2σYt−μ0σZt.


## 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,(8)Wt=W0+CFJ0,tσΠσ−ασWtXt−ασγ0σWtYt−μ0σ+νσWt,Xt=X0+CFJ0,tσrασWtXt+ασγ0σWtYtt−μ0σ+βσ+γ1σXt,Yt=Y0+CFJ0,tσ1−rασWtXt+ασγ0σWtYt+sβσXt−μ0σ+μ1σ+γ2σYt,Zt=Z0+CFJ0,tσ1−sβσXt+νσWt+γ2σYt−μ0σZt.For the above system, theCF-fractional integral application gives(9)Wt=W0+21−σKσ2−σΠσ−ασWtXt−ασγ0σWtYt−μ0σ+νσWt+2σKσ2−σ∫0tΠσ−ασWyXy−ασγ0σWyYy−μ0σ+νσWydy,Xt=X0+21−σKσ2−σrασWtXt+ασγ0σWtYt−μ0σ+βσ+γ1σXt+2σKσ2−σ∫0trασWyXy+ασγ0σWyYy−μ0σ+βσ+γ1σXydy,Yt=Y0+21−σKσ2−σ1−rασWtXt+ασγ0σWtYt+sβσXt−μ0σ+μ1σ+γ2σYt+2σ2−σKσ∫0t1−rασWyXy+ασγ0σWyYy+sβσXy−μ0σ+μ1σ+γ2σYydy,Zt=Z0+21−σKσ2−σ1−sβσXt+νσWt+γ2σYt−μ0σZt+2σKσ2−σ∫0t1−sβσXy+νσWy+γ2σYy−μ0σZydy.Letℓ1, ℓ2, ℓ3, and ℓ4 are the kernels and is defined by(10)ℓ1Wt,t=Πσ−ασWtXt−ασγ0σWtYt−μ0σ+νσWt,ℓ2Xt,t=rασWtXt+ασγ0σWtYt−μ0σ+βσ+γ1σXt,ℓ3Yt,t=1−rασWtXt+ασγ0σWtYt+sβσXt−μ0σ+μ1σ+γ2σYt,ℓ4Zt,t=1−sβσXt+νσWt+γ2σYt−μ0σZt.Theorem 1.
The kernelsℓ1, ℓ2, ℓ3, and ℓ4 given above satisfy the conditions of Lipschitz axioms.Proof.
Suppose thatW1 and W, X1 and X, Y1 and Y, Z1 and Z are the kernels for the two function, respectively ℓ1, ℓ2, ℓ3, and ℓ4, so(11)ℓ1Wt,t−ℓ1W1t,t=Πσ−ασW−W1Xt−ασγ0σW−W1Yt−μ0σ+νσW−W1,ℓ2Xt,t−ℓ2X1t,t=rασWtX−X1+ασγ0σWtYt−μ0σ+βσ+γ1σX−X1,ℓ3Yt,t−ℓ3Y1t,t=1−rασWtXt+ασγ0σWtY−Y1+sβσXt−μ0σ+μ1σ+γ2σY−Y1,ℓ4Zt,t−ℓ4Z1t,t=1−sβσXt+νσWt+γ2σYt−μ0σZ−Z1.
By using the application of Cauchy’s inequality, we obtain the system given below:(12)ℓ1Wt,t−ℓ1W1t,t=Πσ−ασW−W1Xt−ασγ0σW−W1Yt−μ0σ+νσW−W1,ℓ2Xt,t−ℓ2X1t,t=rασWtX−X1+ασγ0σWtYt−μ0σ+βσ+γ1σX−X1,ℓ3Yt,t−ℓ3Y1t,t=1−rασWtXt+ασγ0σWtY−Y1+sβσXt−μ0σ+μ1σ+γ2σY−Y1,ℓ4Zt,t−ℓ4Z1t,t=1−sβσXt+νσWt+γ2σYt−μ0σZ−Z1.
Recursively, we obtain(13)Wt=21−σℓ1Wn−1t,t2−σKσ+2σ2−σKσ∫0tℓ1Wn−1y,ydy,Xt=21−σℓ2Xn−1t,t2−σKσ+2σ2−σKσ∫0tℓ2Xn−1y,ydy,Yt=21−σℓ3Yn−1t,t2−σKσ+2σ2−σKσ∫0tℓ3Yn−1y,ydy,Zt=21−σℓ4Zn−1t,t2−σKσ+2σ2−σKσ∫0tℓ4Zn−1y,ydy.
The difference between consecutive terms with majorizing the norm application implies(14)Ent=Wnt−W1,n−1t≤21−σKσ2−σℓ1Wn−1t,t−ℓ1W1,n−2t,t+2σKσ2−σ∫0tℓ1Wn−1y,y−ℓ1W1,n−2y,ydy,Fnt=Xnt−X1,n−1t≤21−σKσ2−σℓ2Xn−1t,t−ℓ2X1,n−2t,t+2σKσ2−σ∫0tℓ2Xn−1y,y−ℓ2X1,n−2y,ydy,Gnt=Ynt−Y1,n−1t≤21−σKσ3−σℓ3Yn−1t,t−ℓ3Y1,n−2t,t+2σKσ2−σ∫0tℓ3Yn−1y,y−ℓ3Y1,n−2y,ydy,Hnt=Znt−Z1,n−1t≤21−σKσ2−σℓ4Zn−1t,t−ℓ4Z1,n−2t,t+2σKσ2−σ∫0tℓ4Zn−1y,y−ℓ4Z1,n−2y,ydy,where(15)∑i=0∞Eit=Wnt,∑i=0∞Fit=Xnt,∑i=0∞Git=Ynt,∑i=0∞Hit=Znt.
Following the work of Qureshi et al., [41], the kernels ℓ1,…,ℓ4 satisfy the Lipschitz conditions, so(16)Ent=Wnt−W1,n−1t≤21−σKσ2−στ1Wn−1t−W1,n−2t+2σKσ2−στ2∫0tWn−1y−W1,n−2ydy,Fnt=Xnt−X1,n−1t≤21−σKσ2−στ3Xn−1t−X1,n−2t+2σKσ2−στ4∫0tXn−1y−X1,n−2ydy,Gnt=Ynt−Y1,n−1t≤21−σKσ2−στ5Yn−1t−Y1,n−2t+2σKσ2−στ6∫0tYn−1y−Y1,n−2ydy,Hnt=Znt−Z1,n−1t≤21−σKσ2−στ7Zn−1t−Z1,n−2t+2σKσ2−στ8∫0tZn−1y−Z1,n−2ydy.Theorem 2.
The solution of the proposed model (7) exists.Proof.
By applying the equation (11) with iterative scheme, we obtain(17)Ent≤W0+2τ11−σKσ2−σn+2τ2σtKσ2−σn,Fnt≤X0+21−στ32−σKσn+2στ4t2−σKσn,Gnt≤Y0+2τ51−σKσ2−σn+2τ6σtKσ2−σn,Hnt≤Z0+21−στ72−σKσn+2στ8t2−σKσn.
We investigate that the (17) are the solutions of the model (7) consequently by making the substitutions(18)Wt=Wnt−Υ1,nt,Xt=Xnt−Υ2,nt,Yt=Ynt−Υ3,nt,Zt=Znt−Υ4,nt,where Υ1,nt, Υ2,nt, Υ3,nt, and Υ4,nt show the rest of the terms of the series; then,(19)Wt−Wn−1t=21−σℓ1Wt−Υ1,ntKσ2−σ+2σKσ2−σ∫0tℓ1Wy−Υ1,nydy,Xt−Xn−1t=2ℓ2Xt−Υ2,nt1−σKσ2−σ+2σ2−σKσ∫0tℓ2Xy−Υ2,nydy,Yt−Yn−1t=2ℓ3Yt−Υ3,nt1−σKσ2−σ+2σ2−σKσ∫0tℓ3Yy−Υ3,nydy,Zt−Zn−1t=21−σℓ4Zt−Υ4,nt2−σKσ+2σ2−σKσ∫0tℓ4Zy−Υ4,nydy.
Applying Lipschitz axiom and taking norm on both sides, then(20)Wt−21−σℓ1Wt,t2−σKσ−W0−2σ2−σKσ∫0tℓ1Wy,ydy≤Υ1,nt1+21−στ12−σKσ+2στ2t2−σKσ,Xt−21−σℓ2Xt,t2−σKσ−X0−2σ2−σKσ∫0tℓ2Xy,ydy≤Υ2,nt1+21−στ32−σKσ+2στ4t2−σKσ,Yt−21−σℓ3Yt,t2−σKσ−Y0−2σ2−σKσ∫0tℓ3Yy,ydy≤Υ3,nt1+21−στ52−σKσ+2στ6t2−σKσ,Zt−21−σℓ4Zt,tKσ2−σ−Z0−2σ2−σKσ∫0tℓ4Zy,ydy≤Υ4,nt1+21−στ72−σKσ+2στ8t2−σKσ.
Applyinglim as t tends ∞, we get(21)Wt=21−σℓ1Wt,tKσ2−σ+2σKσ2−σ∫0tℓ1Wy,ydy+W0,Xt=21−σℓ2Xt,t2−σKσ+2σ2−σKσ∫0tℓ2Xy,ydy+X0,Yt=21−σℓ3Yt,t2−σKσ+2σ2−σKσ∫0tℓ3Yy,ydy+Y0,Zt=21−σℓ4Zt,tKσ2−σ+2σKσ2−σ∫0tℓ4Zy,ydy+Z0,which 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 W+t,X+t,Y+t,Z+t; then,(22)Wt−W+t=21−σℓ1Wt,t−ℓ1W+t,tKσ2−σ+2σKσ2−σ∫0tℓ1Wy,y−ℓ1W+y,ydy,Xt−X+t=21−σℓ2Xt,t−ℓ2X+t,t2−σKσ+2σKσ2−σ∫0tℓ2Xy,y−ℓ2X+y,ydy,Yt−Y+t=21−σℓ3Yt,t−ℓ3Y+t,tKσ2−σ+2σ2−σKσ∫0tℓ3Yy,y−ℓ3Y+y,ydy,Zt−Z+t=21−σℓ4Zt,t−ℓ4Z+t,tKσ2−σ+2σ2−σKσ∫0tℓ4Zy,y−ℓ4Z+y,ydy.
By majorizing, one may derive(23)Wt−W+t=21−σℓ1Wt,t−ℓ1W+t,tKσ2−σ+2σKσ2−σ∫0tℓ1Wy,y−ℓ1W+y,ydy,Xt−X+t=21−σℓ2Xt,t−ℓ2X+t,tKσ2−σ+2σKσ2−σ∫0tℓ2Xy,y−ℓ2X+y,ydy,Yt−Y+t=21−σℓ3Yt,t−ℓ3Y+t,tKσ2−σ+2σKσ2−σ∫0tℓ3Yy,y−ℓ3Y+y,ydy,Zt−Z+t=21−σℓ4Zt,t−ℓ4Z+t,tKσ2−σ+2σKσ2−σ∫0tℓ4Zy,y−ℓ4Z+y,ydy.
By applying Theorems1 and 2, one may get(24)Wt−W+t≤2τ1ψ11−σKσ2−σ+2τ2σϕ2tKσ2−σn,Xt−X+t≤2τ31−σψ3Kσ2−σ+2τ4σϕ4t2−σKσn,Yt−Y+t≤21−στ5ψ52−σKσ+2στ6ϕ6t2−σKσn,Zt−Z+t≤2τ7ψ71−σKσ2−σ+2στ8ϕ8tKσ2−σn.
The inequalities as reported by the above (24) hold for each value of n; thus, we get(25)Wt=W+t,Xt=X+t,Yt=Y+t,Zt=Z+t.
Here, 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.
AsWt,Xt,Yt,Zt are the solutions of the proposed model (7) and suppose that the model having non-negative initial conditions, then Wt,Xt,Yt,Zt are non-negative solutions for every t≥0.Proof.
For the proposed system given by (7), we take a general order ω fractional model as(26)GD0,tωWt=Πσ−ασWtXt−ασγ0σWtYt−μ0σ+νσWt,GD0,tωXt=rασWtXt+ασγ0σWtYt−μ0σ+βσ+γ1σXt,GD0,tωYt=1−rασWtXt+ασγ0σWtYt+sβσXt−μ0σ+μ1σ+γ2σYt,GD0,tωZt=1−sβσXt+νσWt+γ2σYt−μ0σZt.Here, ω denotes the order and G represents the operator under consideration. Therefore, (26) takes the form(27)GD0,tωWt|κS=Πσ>0,GD0,tωXt|κA=rασWtXt+ασγ0σWtYt≥0GD0,tωYt|κC=1−rασWtXt+ασγ0σWtYt+sβσXt≥0,GD0,tωZt|κR=1−sβσXt+νσWt+γ2σYt>0,
Asκξ=ξ=0andW,X,Y,Z contained in CR+×R+ and ξ∈W,X,Y,Z. By following [42], we conclude that for every non-negative t, the solutions Wt,Xt,Yt,Zt are positive.Lemma 2.
LetΩ denote the region, in which the dynamics of the proposed model (7) is positively invariant; then,(28)Ω=Wt,Xt,Yt,Zt∈R+4:W+X+Y+Z≤Πϑω.Proof.
AsN denotes the total consider population, therefore, we have(29)GD0,tωTt+ϑωTt≤Πω.
The solution for the (29) is given by(30)Tt≤T0Uω−ϑωtω+Πϑω1−Uω−ϑωtω.
Here, the Mittag–Leffler function is denoted byU. with UωM=∑n=0∞Mn/Γωi+1. The above (30) whenever time increases without bound, then Tt⟶Π/ϑω. Thus, if T0≤Π/ϑω, then Tt≤Π/ϑω for every t>0, while if T0>Π/ϑω, then T contained in Ω. So, we concluded that the dynamics of the reported model can be analyzed in the feasible portion Ω.


## 5. Steady State

The model as presented by (7) studied for equilibria such as endemic and disease free. Let D2 be the state of disease free of the reported model; for studding D2 point, the population is taken to be noninfectious. Therefore, the reported system given by (7) has a disease-free point D2=W0,X0,Y0,Z0, where W0=πσ/μ0σ+νσ, W0=X0=0 and Z0=πσνσ/μ0σμ0σ+νσ. Now, to find the basic reproductive number, suppose J=X,YT, then system (7) gives(31)dXdt|D2=F1−V1,where(32)F1=rασW0rασγ0σW01−rασW01−rασγ0σW0,V1=μ0σ+βσ+γ1σ0−sβσμ0σ+μ1σ+γ2σ.Thus, the spectral radius ofρFV−1, i.e., R0=R1+R2+R3, is the basic reproductive number.(33)R1=1−rW0ασγ0σq3,R2=W0ασrσq2,R3=W0ασβγ0σrsq2q3.SupposeD3 is the endemic equilibrium and let W=W∗, X=X∗, Y=Y∗. and Z=Z∗, then the endemic equilibrium becomes(34)W∗=q2q2ασrq3+βσsγ0σr+q2γ0σ1−r,X∗=rq1q3R0−1ασrq3+βσsγ0σr+q2γ0σ1−r,Y∗=q1q21−r+rβσsR0−1ασrq3+βσsγ0σr+q2γ0σ1−r,Z∗=1−sβσX∗+νσW∗+γ2σY∗μ0σ.Thus, concerning to the local and global dynamics of the reported system, we find the dynamics results given below:Theorem 4.
The disease-free pointD2 of the reported model described by (7) is stable locally asymptotically, if R0<1 where endemic equilibrium D3 is stable locally asymptotically if R0>1.Proof.
Following [41], the linearizable description of the reported hepatitis B-virus model is given by (7) around D1 guide to a matrix that is under(35)J0|D2=−q1−ασW0−ασγ0σW00rασW0−q2rασγ0σW001−rασW0+sβσ1−rασγ0σW0−q3.
One eigenvalue is−q1<0 which is negative. For the remaining, we have(36)J1|D2=rασW0−q2rασγ0σW01−rασW0+sβσ1−rασγ0σW0−q3.
Here, we need to show thatH0:=traceJ1<0 and detJ1>0, so(37)traceJ1=−q21−R2−q31−R1,and(38)detJ1=q2q31−1−rW0ασγ0σq3+W0ασrq2+W0ασβσγ0σrsq2q3,which implies that(39)detJ1=q2q31−R0.
Clearly, we examine thatH0 holds if R0<1; thus, we obtain the result that the equilibrium point D2 of the reported model is stable locally asymptotically with condition R0<1.
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 pointD2 of the reported model given by (7) is stable globally asymptotically if R0≤1, where then the endemic point D3 is stable globally asymptotically if R0>1.Proof.
To study the global analysis of the epidemic model, we use the same methodology reported in [43]. Let us assume that Nt=Wt,Xt,Yt,Zt and let it has a finite limit if t varies without bound, so equation first of the reported model (7) becomes(40)CFD0,tσWt≤Πσ−q1Wt.
Since, fort≥0 and for any x, x≤xet, so by following Theorem 3.1 in [44], it then leads to(41)Wt≤aCexp−q11/σ+1t,where a=S0e−T+KTσe−T/σΓσ+Πσ/q1, t≥T and C>0. Consequently, we get(42)limt⟶∞Wt≤CΠσ/q1.
In a same way, thelim of Xt, Yt and Zt can be proved. Furthermore, let us assume(43)limt⟶∞Nt=W∞,X∞,Y∞,Z∞,and(44)fN=f1Nf2Nf3Nf4N=Πσ−ασWtXt−ασγ0σWtYt−μ0σ+νσWtrασWtXt+ασγ0σWtYt−μ0σ+βσ+γ1σXt1−rασWtXt+ασγ0σWtYt+sβσXt−μ0σ+μ1σ+γ2σYt1−sβXt+νWt+γ2Yt−μ0Zt.
So, by using the mean value theorem, there existsC1 and C2 such that(45)fN≤C1+C2N.
Thus, by following the work proposed in [45], we can write that CFD0,tσWt,Xt,Yt,Zt is continuous uniformly. So, Barbalat’s Lemma [46]) gives that(46)limt⟶∞CFD0,tσNt=0,0,0,0,0,and(47)Πσ−ασWtXt−ασγ0σWtYt−μ0σ+νσWt=0,rασWtXt+ασγ0σWtYt−μ0σ+βσ+γ1σXt=0,1−rασWtXt+ασγ0σWtYt+sβσXt−μ0σ+μ1σ+γ2σYt=0,1−sβXt+νWt+γ2Yt−μ0Zt=0.
Thus,W∞,X∞,Y∞,Z∞ is the equilibrium of the given fractional model given by (7) and with a same argument of Theorem 3. In the work presented in [47], we get(48)limt⟶∞Nt=D2orlimt⟶∞Nt=D3.
Thus, the disease endemic stateD3 does not exists if R0<1, and so limYt=D2 whenever t tends to ∞ and if R0=1, then D2=D3, and limNt=D2, whenever t varies unboundedly. On the other hand, for R0>1, the equilibrium point D3 exists and limNt=D2 if t⟶∞.


## 6. Numerical Simulation

We solve the epidemic problem of hepatitis B as stated by (7) to properly understand the dynamical behavior of the disease and to differentiate between fractional and integer order. We will also show the validity of our analytical work with the help of large-scale simulation.Following the work reported in [48–50] for the numerical solution of non-integer problem, let us assume the interval 0,t with time step h=10−3 for integration. We also assume that u=0,1,2,…,n and n=T/h, n∈ℕ; then, the algorithm for the epidemic problem may take the form(49)CFWu+1=W0+1−σΠσ−ασWtXt−ασγ0σWtYt−μ0σ+νσWt+σh∑k=0uΠσ−ασWtXt−ασγ0σWtYt−μ0σ+νσWt,CFXu+1=X0+1−σrασWtXt+ασγ0σWtYt−μ0σ+βσ+γ1σXt,+σh∑k=0urασWtXt+ασγ0σWtYt−μ0σ+βσ+γ1σXt,CFYu+1=Y0+1−σ1−rασWtXt+ασγ0σWtYt+sβσXt−μ0σ+μ1σ+γ2σYttt,+σh∑k=0u1−rσσWtXt+σσγ0σWtYt+sβσXt−μ0σ+μ1σ+γ2σYtCFZu+1=Z0+1−σ1−sβXt+νWt+γ2Yt−μ0Zt+σh∑k=0u1−sβXt+νWt+γ2Yt−μ0Zt.We chose the value of epidemic parameters in a feasible way to perform the visualization of the analytical results. More specifically, the values are as follows:Π=0.4, α=0.0975, γ0=0.00000456, μ0=0.3454, ν=0.022, r=0.48, β=0.5, γ1=0.5, μ1=0.45, γ2=0.8613, and s=0.1428, while 100,90,80,70,60 are the initial compartmental sizes for various populations categorized in our proposed model.The application of the above scheme as reported above with the numerical values and various orders of fractional order the following results is obtained as presented in Figures1–4, which represent the temporal behaviors of the compartmental populations. We demonstrate the dynamics of the susceptible group of individuals against the various value of σ in Figure 1. It could be observed that the dynamics of susceptible and fractional parameter σ are proportional inversely. If we increase the value of σ, then the susceptible population decreases, as shown in Figure 1. Moreover, in real world situation, there will be always susceptible population; however, this is not in case of classical order. This verifies that the fractional order is more appropriate for representing the temporal dynamics of hepatitis B virus transmission. Likewise, the temporal dynamics of acute as well as chronic individuals also described that Xt, Yt are also proportional inversely to the σ as respectively described by Figures 2 and 3. We noted that whenever the values of fractional order decreases, the number of infected population goes to zero sharply which suggest that the fractional order parameter has a significant impact on the infected population of hepatitis B. Moreover, Zt and fractional parameter are directly proportional, as shown in Figure 4, and when the value of fractional parameter σ increases, then the number of Zt also increases, see Figure 4. It could be quite clear from the discussion that CF-fractional order epidemic problem describes valuable outputs to present the temporal behavior of the various compartments of the considered model. These usually are not derived in the classical case.Figure 1
The graph visualize the time dynamics ofWt for various value of the fractional parameter σ.Figure 2
This visualizes the time dynamics ofXt for distinct value of the fractional parameter σ.Figure 3
The graph demonstrate the dynamicsYt verses distinct value of the fractional parameter σ.Figure 4
The graph demonstrate the time dynamicsZt against different value of the fractional parameter σ.


## 7. Conclusion

We formulated a mathematical model to study the temporal dynamics of the hepatitis B virus. The model has been derived from the characteristic of the disease. Particularly, we developed the model with two infected compartments acute and chronic, and took the probabilistic transmission of the disease accordingly because this was not considered, while having a major influence on the transmission of hepatitis B. We analyzed the considered model and then fractionalized due to the increased application of fractional calculus. We applied the well-known CF operator to perform the temporal dynamics of the considered model. For the uniqueness as well as existence rigorously, we used the concept of fixed point theory. The detailed biological feasibility with mathematical validness of the model that is under consideration is discussed. We proved the boundedness as well as positivity of the model to make the model meaningful as we are dealing with population dynamics. The threshold parameter reproductive number via the next-generation method is calculated and performed in the local and global analysis of the fractional epidemiological problem. We supported our analytical work with a certain level of verification via using numerical visualization. We also showed the effect of fractional order parameters and differentiated between fractional and nonfractional order graphically. We found that the fractional operator has a significant impact on the transmission of hepatitis B virus. We proved finally that the CF fractional-order operator provides more valuable results and is one of the best choices of all other classical orders.



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*Source: 6202049-2022-06-15.xml*