APPLICATION OF NON-STANDARD FINITE DIFFERENCE METHOD ON COVID-19 MATHEMATICAL MODEL WITH FEAR OF INFECTION

This study presents a novel application of Non -Standard Finite Difference (NSFD) Method to solve a COVID-19 epidemic mathematical model with the impact of fear due to infection. The mathematical model is governed by a system of first-order non-linear ordinary differential equations and is shown to possess a unique positive solution that is bounded. The proposed numerical scheme is used to obtain an approximate solution for the COVID-19 model. Graphical results were displayed to show that the solution obtained by NSFD agrees well with those obtained by the Runge-Kutta-Fehlberg method built-in Maple 18


INTRODUCTION
The use of differential equations to model the transmission dynamics of infectious disease can be traced back to 1970 when Daniel Bernoulli justified the use of inoculation to curb the spread of smallpox (Dietz and Heesterbeek, 2002;Foppa, 2017).These models are usually nonlinear (Peter et al., 2020;Gu et al., 2023;Akinyemi et al., 2023;Kambali et al., 2023;Ochi et al., 2023) and are difficult to obtain their exact solution (Onwubuoya et al., 2018b;Riyapan et al., 2021;ur Rehman et al., 2023).Thus, numerical methods are used to obtain approximate solutions.Some of the numerical techniques are Euler (Ashigi et al., 2021;Mohammed et al., 2021;Reza et al., 2022), Euler Predictor Corrector (Onwubuoya et al., 2018a), Non-Standard Finite Difference (Raza et al., 2022;Butt et al., 2023;ur Rehman et al., 2023).The Non-Standard Finite Difference (NSFD) method developed by Ronald E. Mickens is a discrete representation of a continuous model (Mickens andWashington, 2012, Qui et al., 2014).Apart from predicting the behaviour of the dynamical system correctly, the NSFD method is known to preserve the dynamical properties of an epidemic model and is less difficult to implement when compared with the aforementioned numerical methods (Qui et al., 2014).Applications of NSFD method are found in financial theory (Mehdizadeh et al., 2022;Mehdizadeh et al., 2023), epidemiology (ur Rehman et al., 2023, Butt et al., 2023), enzymology (Miller & O'Riordan, 2020;Zafar et al., 2023), pharmacology (Egbelowo, 2018;Ebgelowo & Hoang, 2021), immunology (Costa et al., 2023;Elaiw et al., 2023).The purpose of this study is to apply the NSFD scheme to solve a mathematical model presented in Ibrahim (2023).The mathematical model proposed by Ibrahim (2023), describes the spread of COVID-19 in the presence of fear of infection and is governed by the following system of nonlinear differential equations.

MATERIALS AND METHODS
This section deals with the introduction to NSFD, the dynamical properties of Model (1) and the application of NSFD on Model (1).

Basic Concept of NSFD
First, we consider an autonomous ordinary differential equation of the form   = (()) (2) Definition 1: A discretized form of ( 2) is called an NSFD scheme provided at least one of these conditions is satisfied.1.The discretized representation of ( 2 the numerator function (ℎ) = 1 + (ℎ), and the denominator function 2. The nonlinear term () in ( 2) should be approximated using the nonlocal discretized form.For instance,  2 ≈    +1 (4) Here,  is the final time, ℎ the time step size and  the number of iterations.Again, we consider a system of firstorder nonlinear differential equations subject to  1 (0) =  1 and  2 (0) =  2 .Discretized (5) using the semi-implicit finite scheme while ensuring that the above condition conditions are met to have Ahmed (2011) and Sweilam et al. (2017), to have  1 =  2 = 1 and an exponential denominator function , are used.Hence, (6) becomes Remark [Sweilam et al. (2017)] : Whenever the denominator function (ℎ) = ℎ, the scheme is called NSFD-I, otherwise it is called NSFD-II.Thus, this study utilizes the NSFD-II scheme.Next, the dynamical properties such as the existence and uniqueness, positivity and boundedness solution of Model (1) are examined.

Existence and Uniqueness Solution of the Covid-19 Model
Theorem 2.1: The system (1) has a unique solution in the region (, , , , , , , , , ) ∈ ℝ + 10 Proof: We write the right-hand side of Model (1) as Then the following are obtained  Since, all the partial derivatives are continuous and bounded, then by Derrick and Grossman's theorem in Derrick and Grossman (1987) and Rabiu and Akinyemi (2016), the unique solution of Model ( 1) is established.

𝜇𝜀
. Therefore, is positively invariant since ,  and  are bounded.

RESULTS AND DISCUSSION
We simulated the COVID-19 model ( 29)-( 38) for  = 150 days while using the initial conditions mentioned above by setting the stepsize ℎ = 0.01 for NSFD-II.To validate the reliability of NSFD-II, the result obtained by NSFD-II was compared with the Runge-Kutta-Fehlberg (RKF45) method built-in Maple 18 software.
The results generated by NSFD-II and RKF45 methods for the population of susceptible individuals are displayed in Figure 1.Both methods show a gradual decrease in the population of susceptible humans for about 10 days and become steady for the remaining simulation period.The comparison shows that both methods are in excellent agreement even though the RKF45 is more cumbersome and not easy to implement when compared with NSFD-II.Thus, the use NSFD-II method is reliable and efficient and should be applied to solve other nonlinear real phenomena.
0.1086Adewole et al., 2021 The rest of this paper is arranged as follows: Section 2 presents the material and methods.Results and discussion are addressed in Section 3. Section 4 gives the conclusion of the study.

Figure 2 :
Figure 2: Graphical Comparison for ( ) Vt Figures 3-4 depict the population profile for the exposed and quarantined humans respectively.The figures show that both methods describe that the population of the exposed and quarantined decreases to zero.

Figure 8 :
Figure 8: Graphical Comparison for ( ) Rt The population and concentration profiles for COVID-19 deceased individuals and COVID-19 viruses in the environment are depicted in Figures 9 -10 respectively.Figures 9 -10 show that () and () decreases to zero.