An Accurate Predictor-Corrector-Type Nonstandard Finite Difference Scheme for an SEIR Epidemic Model

Faculty of Science, Yibin University, Yibin 644000, China School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China Wah Medical College POF Hospital, Wah Cantt 47040, Pakistan Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia Department of Mathematics, Cankaya University, Ankara 06790, Turkey Institute of Space Sciences, Magurele 077125, Romania Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan Department of Mathematics, Faculty of Sciences University of Central Punjab, Lahore, Pakistan Department of Mathematics and Statistics, University of Lahore, Lahore 54590, Pakistan


Introduction
Mathematicians and biologists have been working for a long time on the biological process of life science. ey succeeded in evaluating some remarkable results from their work. An important mark in mathematical biology is the mathematical modeling of infectious diseases [1]. Measles is one of such a highly infectious childhood disease, caused by respiratory infection by a Morbilli virus, measles virus. Arthur Ransom first observed the irregular cyclic behavior of measles that is considered as a mainly conspicuous facet of measles. e age structure of the population, contact, immigration rate, and the school seasons were known as a crucial phase for the swell of measles [2][3][4]. William Hammer in 1906 published a discrete numerical model for the transmission of measles epidemic. Later assumption of "Mass Action" is applied to that model which is the basic rule to the current theory of deterministic modeling of infectious diseases [5].
Society has a keen concern in knowing the major evolution for the spread of diseases. Analytical results give a solution to these problems but for limited cases and causes many problems. e homotopy perturbation method and variational iteration method can be used for the solution of the epidemic models [6]. However, the first choice to solve these laws of nature is the numerical method based on a difference scheme for good approximations [7][8][9]. In general, already developed numerical schemes such as Euler, Runge-Kutta, and others at times stop working by generating nonphysical results. ese unnecessary oscillations contrived chaos and false fixed points [10]. Moreover, some methods are unsuccessful if we check them on larger step sizes [11]. To avoid such discrepancies, the numerical schemes based on the "nonstandard finite difference method (NSFD)" are established. ese techniques were first developed by R. E. Mickens [7,12,13]. e created numerical schemes preserve the essential properties such as dynamical consistency, stability, and equilibrium points [14][15][16][17][18][19][20][21]. Researchers have developed competitive NSFD schemes for epidemic diseases. Many of these NSFD schemes are consistent for small step sizes with the continuous model, but for large step sizes, the unwanted oscillations have been observed. Piyanwong, Jansen, and Twizel have constructed a positive and unconditionally stable scheme for SIR and SEIR models, respectively [22,23]. Nevertheless, the lack of application of conservation law in their developed schemes explicitly caused impracticable and unrealistic solutions, while Abraham and Gilberto have developed NSFD schemes of the SIR epidemic model to obtain the physically realistic solutions for all step sizes, where they apply the conservation law in addition to nonlocal approximation [24].
In this paper, we have developed a normalized NSFDM of predictor-corrector-(PC-) type inspired by the previous work discussed to double refine the numerical solution of a nonlinear dynamics regarding the transmission of measles. To keep the method explicit, we will use the forward difference approximations for the first derivative terms. e nonlocal approximations are used to tackle the nonlinear terms with φ(h) as a nonclassical denominator function. By using this idea, the measles model will converge to equilibrium points, even for the larger step sizes.

The Mathematical Frame of Work
In this work, the dynamics of the measles epidemic described by the SEIR mathematical model suggested by Jansen and E. H. Twizel is considered [23]. In the SEIR model, the total human population is categorized as susceptible, exposed, infectious, and recovered subpopulations denoted by S, E, I, and R, respectively. Consider the flow of the SEIR model for the measles epidemic, as shown in Figure 1.Here S � susceptible individuals E �exposed individuals I � infected individual R �recovered individual μ � birth rate and death rate β � the rate at which susceptible individuals are infected by those who are infectious σ � the rate at which exposed individuals become infected c � the rate at which infected individuals recover.
Here μ, β, σ, and c are considered as positive parameters. Furthermore, we assume that (E 0 , I 0 ≠ 0). Consider N is the constant size of the population so that the number of recovered individuals R � R(t) at time t is defined by A susceptible is a move to the exposed model, where the individual is infected but yet not infectious. After sometimes the individual becomes infectious and enters into the infected compartment, and in this way, the disease spreads into the population. e mathematical model is written as where Equation (2) shows that the total size of the population remains constant, which is connected with the continuous system (1). e following two points give the equilibrium points of (1): e disease-free equilibrium (DFE) point (N, 0, 0) e endemic equilibrium (EE) point (N/R 0 , μN/μ+ is the basic reproductive number associated with the measles model.
For the sake of brevity, we are not mentioning RK-4 and Euler-PC scheme.

Numerical Modeling
To the construction of NSFD scheme, the continuous system (1) is discretized using the forward difference approximation for the first-order time derivatives. us, if f(t) is differentiable, the f ′ (t) can be approximated by where φ(h) is a real-valued function satisfying the condition φ(h) ⟶ 0 as h ⟶ 0. We have  Journal of Mathematics us, the NSFD scheme for system (1) takes the form From equation (5), we have us, if S n + E n + I n + R n � N for all n ≥ 0, then S n+1 + E n+1 + I n+1 + R n+1 � N for all n ≥ 0.

NSFD Predictor-Corrector Scheme
is section is improved by the approach of a predictorcorrector-type NSFD scheme (7) to obtain the benefits of both methods. For the development of this scheme, firstly, system (7) is taken as a predictor scheme, i.e., , Now, we evaluate system (1) at time t + φ(h) and introduce the term ϵ − 1 S n+1 /φ(h) (where ϵ is just like the accelerating factor and its range is (0 < ϵ < 1)). us, the expression will be NSFDCL scheme that preserves conservation law represented as us, the corrector scheme is obtained as

Convergence Analysis
In this section, the unconditional convergence of the numerical solution is presented by the proposed method.
Taking the values as where F 1 , F 2 , and F 3 are given as in equation (8).
Journal of Mathematics e convergence and stability analysis of scheme (10) is carried out by calculating the eigenvalues of the Jacobian of the linearized scheme and studying its behavior and evaluated at the fixed points. If (S * , E * , I * ) is the fixed point of system (1), then the Jacobian matrix JG is given by where Journal of Mathematics . e values of functions and its derivatives at DFE point (N, 0, 0) are as , For ease in the calculation, we use the following conventions: Since the partial derivatives using the conventions are 6 Journal of Mathematics So, To calculate the eigenvalues, put det (J − λI) � 0, where To calculate the eigenvalues of J * , the following lemma is proved.

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Now, for As all the conditions of the lemma hold for R 0 < 1, therefore the absolute value of both eigenvalues of J * is less than one, whenever R 0 < 1. us, the numerical scheme in equations (8) and (10) will converge unconditionally to disease-free equilibrium (N, 0, 0) for any value of time step h whenever R 0 < 1 which is also computationally verified in Figure 2(a).

Endemic Equilibrium
e value of functions and its derivatives at an endemic point is as , , , Step size , It is tedious to calculate the eigenvalues of the Jacobian of endemic equilibrium analytically, so we have plotted the largest eigenvalue against each step size, and Figure 2(b) shows that, for all step sizes, the spectral radius of the Jacobian of EE remains less than one, if R 0 > 1, which implies that the numerical scheme in equations (8) and (10) is unconditionally convergent if R 0 > 1, for all step sizes.

Numerical Results and Discussion
All the methods showed the convergence for small step sizes in Table 1. However, for large values of step sizes, only the normalized NSFDPC converge to the correct disease-free point for β � 0.1 × 10 − 5 as well as the correct endemic point for β � 0.3 × 10 − 5 . is means that, for the value of the basic reproductive number (threshold parameter), R 0 < 1; then, DFE is stable, i.e., solution to the system (S, E, I, R) are converging to it. is can be seen from Figures 3(a)-3(c) for h � 0.01. Also, for the value of the basic reproductive number (threshold parameter) R 0 > 1, then EE is stable, i.e., solution to the system (S, E, I, R) is converging to it. is can be seen from For comparison with the well-known RK-4 method to system (1), using the parameter values given in Table 2, it is found that, in Figures 5(a)-5(f ), the numerical solution converges for h � 0.01 for both equilibrium points, and      Another popular numerical method is the Euler predictor-corrector (E-PC) technique, which employs the explicit Euler method as a predictor and the trapezoidal rule as the corrector. is combination is second-order accurate but has similar stability properties in PECE mode to the explicit Euler method alone. Figures 7(a)-7(f ) show convergence results for h � 0.01 in both cases that are DFE and EE. e value of h � 0.1 which produces overflow when solving the system with the parameter values of Table 2 for the PECE combination can be seen in Figure 8.
us, the presented numerical results demonstrated that NSFDPC has better convergence property following the Euler PC and RK-4, as shown in Figure 9. It is proved that the approximations made by other standard numerical methods experience difficulties in preserving either the stability or the positivity of the solutions or both but NSFDPC is unconditionally convergent.

Conclusions
In this paper, a normalized NSFDPC for the SEIR model concerning the transmission dynamics of measles is constructed and analysed. is proposed numerical scheme is very competitive. It is qualitatively stable, that is, it double refines the solution and gives realistic results even for large step sizes. It is dynamically consistent with a continuous system and unconditionally convergent and satisfies the positivity of the state variables involved in the system. Simulations are carried out and its usefulness is compared with a well-known numerical method of standard difference schemes such as RK4 and Euler predictor-corrector method. e standard finite difference schemes are highly dependent on step sizes and initial value problems, but NSFDPC is independent of these two features which make it more practical. Also, this method saves computation time and memory.

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Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.