Transmission Analysis of Hepatitis B Epidemic Model using Standard and Non-standard Schemes

Mathematical modeling is a vast field that has interdisciplinary implications for research. These models help to investigate the basic dynamics and quantitative behavior of infectious diseases that affect human beings, such as COVID-19, hepatitis B virus (HBV), and human immunodeficiency virus (HIV). The current study investigates the spread of HBV by using the basic virus model. In order to determine the stability of disease-free and endemic equilibria, the basic reproduction number is determined. The convergence and divergence of disease-free and endemic equilibria are demonstrated by using standard finite difference (SFD) and non-standard finite difference (NSFD) schemes. Arguably, SFD schemes, namely Euler and Runge-Kutta order four (RK-4) schemes, converge for lower step sizes, while the NSFD scheme converges for all step sizes. The latter is a strong, efficient, and reliable method that shows a clear picture of the continuous model. All the results are validated using numerical simulations in order to better comprehend the dynamics of the disease. The theoretical and numerical findings in this work can be applied as a useful tool for tracking the prevalence of HBV infectious disease.


INTRODUCTION
HBV is a devastating liver infection and an epidemic that poses a considerable health hazard, specifically in developing nations [1,2]. HBV damages the liver and causes numerous shocking disorders. Almost 360 million people worldwide are affected by this infectious disease and among them are more than 150 million HBV transporters (who transmit the disease to healthy people) [3,4]. Also, more than 150 million of them have chronic School of Science Volume 7 Issue 1, 2023 hepatitis C virus (HCV) infection and another 150-200 million are HBV chronic liver infection carriers [5]. HBV is transmitted by exposed sex, blood transfusion, and spreads into the recently born infant during pregnancy from the infected mother.
Mathematical models [6][7][8][9][10][11][12][13][14][15][16][17][18][19] are very useful for analyzing the dynamics of real-world problems. Researchers utilize ordinary differential equations [6][7][8][9][10][11] as well as partial differential equations [12][13][14][15] to investigate different characteristics of physical objects. Fractional order [16][17][18][19] mathematical models are also among the most investigated topics in the world in this regard. Mathematical modeling also helps in the analysis of experimental results, as well as understanding the fundamental biological mechanisms of disease transmission. Many mathematical models have been suggested to improve the understanding and knowledge of HBV disease. Zeuzem et al. [6] and Nowak et al. [7] recommended simple virus infection models in which clear immune responses are not required. These models are mostly applied in the analysis of virus infection dynamics. In [8], the author proposed and examined the stochastic SACR model for the transmission of HBV disease. In [9,10], the authors focused on population characteristics, infectious diseases, infection characteristics, and related social factors of HBV. Aja et al. [11] constructed a model based on statistical data which is not only useful for forecasting disease transmission but also helps to identify the key factors that affect the disease spread.
In early 1980s, mathematical modelling was used to investigate the dynamics of HBV and the efficiency of disease control. To date, several systematic methods have been applied for the prevention of HBV transmission [20][21][22]. The most attractive and effective way of reducing the prevalence of HBV in newborns is vaccination. In case of vaccination, the emphasis shifts towards the use of a mathematical model to forecast the long-term consequences of immunization on hepatitis B regulation. Many scientists have analyzed the transmission mechanism of HBV and evaluated the effect of various vaccine policies [23,24]. Recently, in [25], the author proposed and analyzed a non-linear basic adaptation system for HBV spread. The stability of disease-free and endemic equilibria was discussed for the continuous model. The author used a mathematical framework for the continuous model to study the stability of equilibria which strictly depend on the basic reproduction number. The aim of the current study is to prevent the spread of HBV and explore its threats to the health of general people. Multiple numerical schemes, such as Euler, RK-4, and the NSFD scheme have been deployed to assess the various properties of the model and demonstrate their ecological feasibility and accuracy. The findings demonstrate that the aforementioned schemes give a thorough description of the continuous model.
The structure of the current paper is organized as follows. The HBV disease model and corresponding parameters are described in Section 2. In Section 3, the expression for 0 , the equilibria of the system, and their stability are explored for the continuous model. SFD schemes, including Euler, RK-4, and the NSFD scheme are produced for the continuous model in Section 4. The fundamental dynamic properties of the continuous model containing the positivity of solutions cannot be preserved by Euler and RK-4 schemes for higher step sizes, leading to numerical solutions that are different from the solutions of the original system. However, the NSFD scheme gives positive solutions for all finite step sizes and basically overcomes the shortcomings of the SFD schemes. The analysis also demonstrates that the NSFD scheme is unconditionally convergent and provides better results in all aspects than Euler and RK-4 schemes. The theoretical findings are strengthened by numerical simulations at each stage. In Section 5, brief conclusions are offered to sum up the whole work. Figure 1 depicts the non-linear basic HBV epidemic model [25].

Parameters
The parameters used in system (2.1) are described as follows: shows

STABILITY ANALYSIS OF CONTINUOUS MODEL (1) EQUILIBRIA
The following subsection gives the equilibria of model (1).

Equilibria of Model
If all other classes are equal to zero aside from the susceptible class, the above described system (2.1) has the disease-free equilibrium (DFE) point 0 = � , 0,0�. If the system (2.1) is solved simultaneously, then the endemic equilibrium (EE) point occurs which is * = ( * , * , * ) = ( , ( 0 −1) , ( 0 −1) ). In the following, 0 is provided which is crucial for the stability of the equilibria of model (1).

Basic Reproductive Number ( )
The most essential threshold quantity for any infectious disease is 0 . It helps to assess whether a contagious disease will spread among the people or not. 0 is a significant biological quantity that may be determined using the modified technique known as the "next generation matrix" [26]. The transmission matrix ( ) and translation matrix ( ) for system (2.1) can be described as follows: Scientific Inquiry and Review Volume 7 Issue 1, 2023 From the above, we can easily get As 0 = ( −1 ), so using the above information, 0 becomes 0 = .
Theorem 3.2 [25]: The EE point of system (1) is LAS whenever 0 > 1, as displayed in Figure 2(b). According to the numerical simulations depicted in Figure 2(a), the DFE point of the continuous system (2.1) is stable if 0 < 1. On the other hand, as illustrated in Figure 2(b), the EE point exists and becomes stable if 0 >1. It is evident from Figure 2 and the preceding discussion that the EE point emerges and remains stable when the DFE point becomes unstable, that is, when 0 passes the critical value 0 = 1, the system exhibits transcritical bifurcation and stability exchange from one equilibrium point to another. It shows that 0 =1 is the saddle point for system (2.1). More details can be found in [25].

NUMERICAL ANALYSIS OF SYSTEM (1)
In the current section, different discrete schemes for the continuous model (1) are constructed. The main aim is to better understand the dynamics of HBV disease transmission. Three schemes, namely Euler, RK-4, and NSFD schemes [27,28] are constructed. It is demonstrated that the NSFD scheme is not only independent of step size but also more preferable as compared to the other defined schemes in every aspect.

Euler Scheme
Subsequent steps can be employed to construct the Euler scheme for system (2.1).
Therefore, the Euler scheme becomes Scientific Inquiry and Review Once the step size is small, then the numerical solutions offered by the Euler scheme are positive, as illustrated in Figure 3(a-c). According to Figure 3(d), stability is disrupted as the step size increases. Therefore, the stability and positivity of Euler scheme does not maintain for all finite step sizes.

Fourth-order Runge-Kutta Scheme (RK-4)
To develop the RK-4 scheme, it is assumed that = 1 , = 1 and = School of Science

��.
Stage 4 Finally, we get   Figure. 4(a-c). Figure 4(d) illustrates that the stability of model (1) is destroyed as the step size increases, that is, the RK-4 scheme diverges for large values of ℎ.

Non-standard Finite Difference (NSFD) Scheme
Mickens [29] propagated the NSFD scheme which approximates the solutions of ordinary and partial differential equations. According to Shokri et al. [30], there are two factors that affect the study of the NSFD scheme. The first is how to discretize the derivative and the second constitutes the best way to approximate the nonlinear terms. The forward finite difference approximation is one of the general methods of first order derivative discretization. The first order derivative is written in standard form as , where (ℎ) is referred to as the denominator function. There are various formulas for (ℎ) that can be viewed in [30]. In the current paper (ℎ) = ℎ. The following steps can be taken to construct the NSFD scheme for system (2.1).
The above equations can be rewritten as

Investigating the Stability of Equilibria for the NSFD Scheme.
In order to discuss the local stability, suppose To verify the eigenvalues, we consider System (4.6) has the following characteristic equation (4.7) Using one of the eigenvalues from (4.7), λ 1 = According to Routh Hurwitz criterion [31,32], the roots of the above equation are less than one if 0 < 1. As a result, the DFE point for the NSFD scheme is LAS.

Proof:
In a manner analogous to Theorem 4.1, the Jacobian matrix is obtained as (4.8) By putting all the derivatives from Theorem (4.1), the Jacobian matrix (4.8) becomes Furthermore, by replacing EE point * in matrix (4.9), we obtain To check the eigenvalues, we take (4.10) From (4.10), λ 1 = 1 (1+ℎ +ℎ * ) is obtained. The other two eigenvalues can be obtained from (4.11) The Routh Hurwitz criterion [31,32] demonstrates that the roots of equation (4.11) are smaller than one if 0 > 1. This demonstrates that * for the discrete NSFD scheme (4.3) is LAS whenever 0 > 1.  Figure 5(a-d) depicts that the NSFD scheme (4.3) remains stable for each step size. The SFD scheme, however, shows convergence for lower step sizes. This demonstrates that for each step size the NSFD scheme is positive forever and unconditionally convergent.

CONCLUSION
The basic disease model for HBV has been proposed and examined in the current paper. For the proposed epidemic system, the equilibria and threshold quantity 0 have been determined. In order to verify the convergence and divergence of the DFE and EE points, various numerical approaches have been employed. It has been demonstrated that RK-4 and Euler schemes depend on step sizes and are conditionally convergent which makes them ineffective. On the other hand, the NSFD approach developed for the current model is not step size dependent. This shows that the NSFD scheme is incredibly effective in comparison to the notable Euler and RK-4 numerical schemes. This scheme is a simple technique that demonstrates how discrete and continuous models behave appropriately and provide mathematically accurate results. By employing it, the spread of HBV epidemic diseases can be monitored effectively. The findings presented in this paper are beneficial to humanity in the field of medicine and can be applied also as a useful tool to predict the appearance of HBV epidemic disease. Each section includes numerical simulations to validate the theoretical findings.