Modeling the Transmission Routes of Hepatitis E Virus as a Zoonotic Disease Using Fractional-Order Derivative

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Introduction
Zoonotic diseases (zoonoses) are basically infectious diseases that are transmitted from animals to humans or from humans to animals.This is any infection that can naturally be transmitted from vertebrate animals to humans.Hepatitis E virus (HEV) is an emerging zoonotic disease for which domestic pigs are considered the main reservoir [1,2].A third of the world's population is thought to have been exposed to the virus as HEV antibodies have been discovered in serum, with workers in slaughterhouses and veterinarians having higher HEV prevalence [3,4].The zoonotic transmission of the HEV raises serious zoonotic and food safety concerns for the general public [5,6].The HEV is a kind of infection that can cause liver diseases [7].Twenty million HEV cases and approximately more than three million hepatitis E symptomatic cases are thought to occur annually globally, according to authors in [8].The World Health Organization (WHO) estimates that 44 000 people died in 2015 as a result of HEV.The fecal-oral route of hepatitis E transmission is primarily through tainted water.East and South Asia have the highest prevalence of this disease overall.The HEV vaccination was first created in China and is solely available there [7].These locally acquired illnesses are frequently linked to the human and animal-shared genotypes 3 and 4 [9], which are particularly common in domestic and wild pigs.Some cases of unique HEV have been linked to the eating of raw meat, liver, or offal from domestic pigs or wild animals (such as wild boars or deer) [9,10], even though the source of the majority of these cases is still unknown.Additionally, some research found that the sequences of local swine HEV strains and human HEV strains were nearly identical [11,12].Although the exact causes of liver damage are yet unknown, it is probable that the interaction of hormonal and immunologic changes during pregnancy with a high HEV virus load makes the pregnant woman more susceptible [13].Due to immunologic changes that occur during pregnancy that support the maintenance of the fetus in the maternal environment by decreasing T cellmediated immunity, pregnant women are more susceptible to viral infections like HEV infection.HEV infection is mostly a waterborne sickness that produces serious outbreaks in underdeveloped countries because of contaminated water and water supplies, poor sanitation standards, and other factors [14].In China, authors in [15] investigated the occurrence of hepatitis E in Shanghai, and a mathematical model using both a back-propagation neural network (BPNN) and an autoregressive integrated moving average model (ARIMA) was created.The model was used to anticipate the incidence of hepatitis E from January 2013 to December 2013 utilizing 144 months of morbidity data from 2000 to 2012, 12 months of data from 2012 to 2012, and validation using these data sets.
HEV is endemic in pig farms, according to a [16] study on HEV transmission in commercial pig farms in six European nations.Growers had an average prevalence of 20% to 44% compared to weaners' prevalence of 8% to 30%.The prevalence of fatteners ranged from 8% to 73% on average.The study confirmed the presence of HEV in pig farms, from weaners to fatteners, using real-time RT-PCR and a SIR model to assess data.The authors in [17] estimated the rate of infection in four European nations while simulating the foodborne transmission of the HEV from pork to people.They also provided a proof-of-concept for reducing HEV transmission through vaccination in both human and pig populations, offering essential information for creating efficient mitigation measures.A study conducted by authors in [18] looked at the dynamics of a hepatitis E outbreak that occurred in Sudan and Uganda between 2007 and 2009, and they found out that 16% and 17%, respectively, of latrines and boreholes are crucial levels of coverage.The benefits and usefulness of fractional operators have been demonstrated in several fields of science and engineering applications, such as the modeling of actual physical phenomena, plasma physics, diffusion process models, fractional-order controllers, viscoelastoplastic material modeling, and membrane vibration modeling in viscoelastic surroundings [19][20][21][22][23][24].Fractional operators are more effective in simulating dynamical systems with memory or hereditary properties due to their nonlocal nature [19,20,[22][23][24][25].The most well-known operators are the Caputo fractional derivatives and integrals, which have traditionally been used to model a wide range of real-world problems.The authors [23] introduced two new operational matrices of fractional Legendre function vectors, focusing on generalized Caputo-type fractional derivative and generalized Riemann-Liouville-type fractional integral operators.Furthermore, there exists a strong correlation between the Caputo fractional derivative and the Riemann-Liouville derivative operator.These operators may obstruct getting better findings when utilized to examine the structure of various models.The singularity quality of the Riemann-Liouville and Caputo operators' kernels is the fundamental problem [26].
As a result, scientists have felt that fractional operators with nonsingular kernels are necessary to comprehend model dynamics more fully.Several researchers have been able to offer fractional operators with nonsingular kernels to this degree; one such example is the Caputo-Fabrizio fractional operator [22].Recently, a novel fractional derivative operator (ABC) with a one-parameter Mittag-Leffler kernel was proposed by [20].This operator's primary benefit is its nonlocal and nonsingular kernel, and it gives those who work in numerical modeling of real-world applications an advantage.This operator also captures crossing behavior more effectively than other operators.The above motivations encouraged the researchers to employ the Atangana-Baleanu and Caputo fractional derivatives in this paper.For several studies using the Atangana-Baleanu-Caputo derivation, we refer the reader to see ( [19,21,27]) and the references therein.The previous works [17,28,29] on HEV have considered environment and human-to-human transmission only.In this work, we consider the transmission of HEV in three phases: the contaminated environment, the pig population, and the human population by employing fractional-order derivatives defined in the Atangana-Baleanu in the Caputo sense.
Section 2 of the work presents the HEV schematic model, the governing nonlinear order fractional differential equations, and the parameter descriptions.The preliminary information that will be used in this study is covered in Section 3. In Section 4, we determine the equilibrium points, the invariant region, and the fundamental reproductive number.The existence and uniqueness of the HEV model using Banach's fixed point theorems and the Ulam-Hyers stability are also presented in this section.Section 5 demonstrates the numerical findings that validate the analytical solution.This study concludes with a discussion and conclusion of the results in Section 6.

Model Formulation and Description
The model is made up of three components.Contaminated environment, pigs, and humans.The population of humans is denoted by N h , that of pigs is denoted by N p , and G represents a contaminated environment with pathogens.The human population and pig population are both subdivided into three classes, respectively, as shown in Table 1.Table 2 shows the parameters used in the model formulation.
2.1.Model Assumptions.The following assumptions were made when formulating the model flow diagram in Figure 1: • HEV is transmitted by environmental contamination.
• There are no vertical transmissions.
• Susceptible pigs can only become infected by ingesting the HEV pathogens.
• Some infected pigs can exhibit symptoms of the disease and can be treated.
• Newborns and other susceptible pigs are vaccinated.
2.1.1.Force of Infection.It is the rate at which each susceptible individual in the population contracts a HEV infection during a given period.The force of infection on the human population is found to be where β θ hh S h I h /N h represents force of infection of human to human population, β θ hp S h I p /N p represents pig to human population, and β θ hg GS h / G + K denotes environment to human population.
The force of infection of the pig population is

Parameters Description
where β θ pp S p I p /N p represents the force of infection of pig to pig population, and β θ pg GS p / G + K denotes environment to the pig population.
Here, we consider environmental HEV exposure, as in the case of the majority of models involving free-living pathogens in the environment, Michaelis-Menten or Holling type II functional responses are used to simulate the environmental-related forces of infection hg and pg [14,30].K is the amount of HEV present in the environment to ensure a 50% risk of developing the illness.The HEV dynamics in relation to the shedding of the virus from humans and pigs are covered by the system's final Equation (2).

Preliminaries
In this section, we provide the preliminary results that we will use in this current study.
Definition 1 (see [19,20]).The Liouville-Caputo (LC) notion states that the fractional derivative of order θ is given by Definition 2 (see [19][20][21]).The Atangana-Baleanu definition in the C sense is given by where B θ = 1 − θ + θ/Γ θ is the normalisation function, M θ represents the Mittag-Leffler function.The Mittag-Leffler function with complex variables is given by where θ, b ∈ ℂ. Equations ( 3) and ( 4) become zero when h v is constant.Proof 1.The Mittag-Leffler fractional derivative of the function x m equals 0 when m = 0, 1, ⋯, n − 1.This is in line with Definition 2. The following is derived from the inequality: Using the Mittag-Leffler function, we have The scheme and convergent of Definition 2 can be found in [31].
3.1.Example.Consider the ABC derivative of a basic fractional differential equation, as shown in [31,32] The exact solution is given by If θ = 1/2, and the boundary condition is given by

HEV Model Analysis
In this section, we provide a qualitative analysis of the model such as the computation of equilibrium points, the invariant region, and the reproductive number.The existence and uniqueness of the HEV model using Banach's fixed point theorems and the Ulam-Hyers stability are also covered in this section.
4.1.Invariant Region.We prove in the section that the feasible area, R 7  + , is a positive invariant region with S h t , I h t , R h t , G t , S p t , I p t , R h t all positive in the system of equation ( 1).
Theorem 1.The Atangana-Baleanu fractional model's (Equation ( 1)) feasible zone can be identified through Ω, where Ω is defined as We seek to prove that the set is positively invariant about the starting data The following lemma proves Theorem 1.
The following L transform exists: The Mittag-Leffler function [20,33] is bounded and due to its asymptotic character which results in the conclusion that N t ≤ Λ θ h /μ θ h as t ⟶ ∞.Hence, all the state variables in the model ( 1) are bounded since N t is bounded in a region Ω.
As a result, the appropriate solution of model ( 1) in R 3 + approaches asymptotically as t ⟶ ∞ and remains in for any set of nonnegative starting data.This indicates that all R 3 + solutions are drawn to the area.The proposed model ( 1) is therefore biologically accurate because the region is positively invariant [34][35][36] for the model.Similarly, the approach can be used to prove pig populations and contaminant environments.
Equation ( 13) can be rewritten as where The solution I h t is still valid for all t ≥ 0 because the right-hand side of Equation ( 15) has two favorable terms.A similar approach can be used to demonstrate that other states are positive for all t ≥ 0 against any beginning data in θ.As a result, the solutions in R 3  + are always affirmative.

4.3.
Existence and Uniqueness of the Solutions.The existence and uniqueness of the solutions to the system (1) are established in this section using Banach's fixed point theorem.
Using Z Y as a Banach space, where Y = 0, a and where S h = sup t∈Y S h t , I h = sup t∈Y I h t , R h = sup t∈Y R h t , G = sup t∈Y G t , S p = sup t∈Y S p t , I p = sup t∈Y I p t , R p = sup t∈Y R p t , and applying the ABC integral operation to the system (1), we have Applying Equation ( 5) on Equation ( 16) gives where Furthermore, the Atangana-Baleanu in Caputo derivatives only satisfies the Lipschitz criterion if and only if S h t , I h t , R h t , G t , S p t , I p t , and R p t have an upper bound.
Assuming pair functions for each such as S h t and S * h t where where Considering Equations ( 18) and ( 19), as well as  18) and ( 19) hold.The and ZS h,n t ⟶ 0, ZI h,n t ⟶ 0, ZR h,n t ⟶ 0, Z G n t ⟶ 0, ZS p,n t ⟶ 0, ZI p,n t ⟶ 0, ZR p,n t ⟶ 0 as n ⟶ ∞.
The triangle inequality is taken into consideration and for every τ, in system (23).

S h,n+τ t
where It follows that system (23) has a unique solution.
• For 0 < θ < 1, and B θ = 1, we have • If h is a Lipschitz function with a Lipschitz constant of L ≤ 1/2 , then the fractional initial value problem (2) has a unique solution in H 1 0, 1 for all 0 ≤ θ ≤ 1.
4.4.Hyers-Ulam (HU) Stability.HU-type stability for the propagation of HEV illness has been employed in this section since it has the advantage of providing an approximate solution to a complex problem [21,37].First, we demonstrate that model ( 1) is (HU) stable.

Definition 4.
The ABC fractional HEV model is HU stable if for σ 1 < 0, i ∈ N 7 , i = 1, 2, ⋯, 7 matching the following conditions exist for In a population that is vulnerable to infection, the basic reproduction number is the total number of secondary cases that one infected person may produce over the course of the infection (1).If the disease spreads throughout a population, this is a key determining factor.The I h , I P , and G are the contaminated compartments in this model.Using the nextgeneration operator approach [21,39,40], designate F and V as the right-hand side of the system (1) that corresponds to the infected compartments, respectively.
We have The characteristic polynomial is Therefore, the R 0 is computed as are the negative eigenvalues.When R 0 < 1, the HEV-free equilibrium is consequently locally asymptotically stable.
The model system's (2) Jacobian matrix is presented below. where

Numerical Simulation
This section validates the fractional-order transmission of the HEV model using published parameter values in [3,28,29,[45][46][47][48][49] and others assumed.Table 3 shows the parameters, values, and sources.Our model includes the entire human population under study with the following values:

Parameters
Value day −1 Source Λ θ h 100 [28] when the susceptible pig population increases, the number of infected pig populations rises with a change in θ.However, the dynamics of both recovered humans and recovered pig populations show a rise in population with changes in θ.  humans, infectious pigs, susceptible humans, and susceptible pigs.The population dynamics confirmed the theoretical analysis.A slight change in value corresponds to an increase or decrease in the population dynamics.

Susceptible and Infected Population Dynamics of Both
Human and Pig. Figure 5 shows the population dynamics of susceptible human and infected pig populations for the period of the epidemics.Humans' susceptibility to HEV 18 Journal of Applied Mathematics infection decreases with time as infected humans increase with time.There has been an exponential increase in the human population infected with the HEV infection in the first 5 days of epidemics as shown in Figure 5.This is because as more leave the susceptible compartment, more get infected with the diseases.This can be controlled by ensuring that the susceptible individuals do not contract the HEV infection.Also, there should be effective treatment measures for those infected with the virus.As the human population decreases, more humans are infected with the disease.This explains why the infectious population increases with time.The human population decreases exponentially as infectious humans increase exponentially with time.

Susceptible and Recovered Population Dynamics of Both
Human and Pig. Figure 6 shows the population dynamics of susceptible human and recovered pig populations for the period of epidemics.From Figure 6(a), it can be observed that, as the susceptible human population decreases with time, the recovered population remains constant.This can be attributed to continuous infection without recovery.However, as the susceptible pig population increases exponentially in the first 10 days, the infected pig population remains steady in the first 10 days and increases exponentially in the last 10 days of the epidemics as shown in Figure 6(b).The number of recovered human populations can be achieved further by using some control measures.However, the exponential increase in the infected pig 19 Journal of Applied Mathematics population can be controlled by the treatment of the infected pig population and the prevention of susceptible pigs from getting infected.The susceptible human population decreases steadily as the recovered population shows no changes.This is attributed to the increase in the number of infected humans.More people are infected without recovering.Moreover, as the susceptible pig population decreases, more pigs are infected.This explains the increase in the population of infected pigs with a steady recovery population.

Conclusion
In this paper, an ABC fractional-order derivative model for the HEV disease transmission was considered.The system of Equation ( 1) was used to establish the model's reproduction number and equilibrium points.The conditions necessary for the system to be stable at those points have been established.Using Banach's fixed point theorem, the existence and uniqueness of solutions to the suggested model (1) were achieved.The Ulam-Hyers criteria was used to determine the stability of solutions.The predictor-corrector numerical methodology of Adam Bashforth was employed, and approximations for different fractional orders θ were considered.Considering θ approaching 1, we investigated the HEV model system's dynamic behaviors.It was observed that a few variations in the fractional derivative order did not alter the function's overall behavior with the results of numerical simulations.Again, when the susceptible pig

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Journal of Applied Mathematics population increases, the number of infected pig populations rises with a change in θ.Moreover, as the number of recovered human populations increases, there is a corresponding increase in the population of recovered pigs with a change in θ.In this current study, the authors did not consider the optimal control strategy and cost-effectiveness analysis.Further studies on optimal control and cost-effectiveness analysis for HEV will help identify the best control strategy that comes with less cost in combating the disease.An optimal control strategy will help minimize the HEV model in three phases, and we recommend it for future studies.

4. 2 .Theorem 2 .
Positivity of Solutions.The positivity of the model solution is established via the following theorem.Any positive initial data will result in the model's ((1)) solution.ϕ = S h t , I h t , R h t , G t S p t , I p t , R p t always has a positive value.Proof 3. Taking the second equation from (1) into consideration.Rewriting it as ABC 0 2000, S h 0 = 500, I p 0 = 150, R h = 90, and G 0 = 200.The initial conditions are as follows: N p 0 = 200, S p 0 = 20, I p 0 = 20, and R p 0 = 10.5.1.Compartment Population Dynamics.The simulation results are shown in Figures 2-4 for a period of 20 days of dynamic behavior of human and pig populations.The solutions of system (1) for five different values of θ in 0, 1 at step-size 0.2 during a 20-day period are shown in Figures 2-4.A drop in the value of the fractional operator is exactly related to a decrease in the population of infected pigs and humans.Figures 2(a) and 3(a) show that when susceptible human population declines, the number of infected human populations rises with change in θ.Figures 2(b) and 3(b) show that

Figures 4 (Figure 2 :
Figure 2: (a) Dynamics of infected human population and (b) dynamics of infected pig population.

Figure 3 :
Figure 3: (a) Dynamics of susceptible human population and (b) dynamics of susceptible pig population.

Figure 4 :
Figure 4: (a) Dynamics of recovered human population and (b) dynamics of recovered pig population.

Figure 5 :
Figure 5: (a) Dynamics of susceptible and infected human population and (b) dynamics of susceptible and infected pig population.

Figure 6 :
Figure 6: (a) Dynamics of susceptible and recovered human population and (b) dynamics of susceptible and recovered pig population.

Table 2 :
Model parameter description.