Nonstandard Finite Difference Scheme Research Articles

A Non-Standard Finite Difference Discretization Scheme Applied to a Malaria Model

In this research work, a dynamically consistent non-standard finite difference (NSFD) scheme is developed to solve a continuous-time model of malaria transmission with herbal medicine as control strategy. We compared results from NSFD scheme with the standard finite difference methods (4th order Runge-kutta and forward Euler methods). The numerical investigation showed that the proposed NSFD method remains consistent, preserves the positivity of solutions and converges to true equilibrium points of the continuous model independent of the step size h.

Dynamics of a delayed discrete-time predator prey model proposed from a nonstandard finite difference scheme

Existing literature established stability in a delayed logistic Lotka–Volterra predator- prey model in terms of equilibrium analysis. However, several researchers did not construct the time-series analysis in such models. It has also been observed that both the RK4 methods and the inbuilt ‘dde23’ MATLAB solver were unable to generate stable solutions. This motivated us to develop a nonstandard scheme to capture the numerical solutions which are well consistent with the analytical equilibrium analysis of continuous delayed predator–prey model. In this paper, we will propose a nonstandard finite difference (NSFD) scheme for a delayed predator–prey model. We shall prove that the developed scheme preserves the qualitative behavior of the system, including the local stability of the equilibrium, and stability switching for any step size h=1m,m∈Z+. It is observed that the discretized system shows the occurrence of a Neimark-Sacker bifurcation. Moreover, the convergence analysis of the numerical scheme establishes first-order convergence. The bifurcation diagram and comparison of delay τ−sequence generated by NSFD with the ones obtained by analytical means have been discussed graphically.

A metapopulation model with exit screening measure for the 2014–2016 West Africa Ebola virus outbreak

We construct a new metapopulation model for the transmission dynamics and control of the Ebola Virus Disease (EVD) in an environment characterized by considerable migrations and travels of people. It is an extended SEIR model modified by the addition of Quarantine and Isolated compartments to account for travelers who undergo the exit screening. The model is well-fitted by using the reported cases from the neighboring countries Guinea, Liberia and Sierra Leone where the 2014–2016 Ebola outbreak simultaneously arose. We show that the unique disease-free equilibrium (DFE) of the model is unstable or locally asymptotically stable (LAS) depending on whether the control reproduction number is larger or less than unity. In the latter case, we prove that the DFE is globally asymptotically stable (GAS) provided that the exit screening is 100% negative. We also prove the GAS of the DFE by introducing more explicit thresholds, thanks to which the existence of at least one boundary equilibrium is established. We design two new nonstandard finite difference (NSFD) schemes, which preserve the dynamics of the continuous model. Numerical simulations that support the theory highlight that exit screening is useful to mitigate the infection. They also suggest that the disease is controlled or the explicit threshold is less than unity provided that the migration and the exit screening parameters are above a critical value.

Stability and computational analysis of Influenza-A epidemic model through double time delay

Delay factors demonstration has a significant role in controlling a strain of infectious disease instead of a pharmaceutical strategy. According to the World Health Organization (WHO), 3–5 million cases are reported annually and approximately 290,000 to 650,000 respiratory deaths annually. So, in the present study, we develop a delayed mathematical model based on delay differential equations (DDEs) for the influenza epidemic using a deterministic approach by introducing double delay parameters. The four distinct sub-populations are considered susceptible, exposed, infected, and recovered. For the rigorous analysis, the fundamental properties of the model like positivity, boundedness, existence, and uniqueness, were studied. The influenza-free equilibrium (IFE) and influenza-existing equilibrium (IEE), are the two nonnegative equilibrium points that the model demonstrates. Both locally and globally, the asymptotic stability of the equilibrium points of the model is established and shown under specific situations of reproduction number. Additionally, investigated the model's parameter sensitivity and determined the relative sensitivity of each parameter. Both standard and nonstandard methods—such as Euler, Runge-Kutta, and nonstandard finite difference with a delayed sense—are presented to make computational analysis support a dynamical analysis and the best visualization of results. The stability of the non-standard finite difference scheme is thoroughly analyzed around the steady states of the model. Additionally, the results show that the nonstandard finite difference approximation is an efficient, cost-effective method, independent of time step size, to solve such highly nonlinear and complex real-world problems.

Exact solution for a discrete-time SIR model

We propose a nonstandard finite difference scheme for the Susceptible–Infected–Removed (SIR) continuous model. We prove that our discretized system is dynamically consistent with its continuous counterpart and we derive its exact solution. We end with the analysis of the long-term behavior of susceptible, infected and removed individuals, illustrating our results with examples. In contrast with the SIR discrete-time model available in the literature, our new model is simultaneously mathematically and biologically sound.

A Non-Standard Finite Difference Schemes for the Solution of Stiff Initial Value Problems

In this study, we introduce a novel non-standard finite difference (NSFD) scheme designed to address the challenges posed by stiff initial value problems. Stiffness in differential equations often leads to numerical instability and requires specialized methods for stable and accurate solutions. A novel set of numerical schemes for solving stiff ordinary differential equations caused by the decay of radioactive substances developed. This paper demonstrates the power of normalization in the discretization function. We employed non-local approximation and renormalization of the denominator function to create qualitatively stable schemes for a stiff ordinary differential equation. The schemes' stability properties were verified using numerical experiments. The schemes' performance is evaluated in comparison to other typical finite difference schemes

Global dynamics and numerical simulation of a modified epidemiological model for viral marketing on social networks

The aim of this work is to conduct a rigorous mathematical analysis for global dynamics and numerical simulation of a recognized viral marketing (VM) model, which is described by a system of ordinary differential equations (ODEs). We first establish positivity and boundedness of solutions and then investigate local and global asymptotic stability properties of possible equilibrium points. As an important consequence, complex dynamics of the VM model is determined fully.Secondly, we develop the Mickens’ methodology to design a nonstandard finite difference (NSFD) scheme, which is useful in numerical simulation of the VM model. The main advantage of the constructed NSFD scheme is that it has the ability to preserve important mathematical features of the continuous-time model for all finite values of the step size. These features include the positivity and boundedness of solutions, positively invariant sets, equilibrium points and their asymptotic stability properties. Consequently, the NSFD scheme is not only effective to simulate dynamics of the VM model, but also easy to be implemented.Thirdly, to emphasize implications of the constructed mathematical analysis, an extended version combining the integer-order ODE model under consideration with the Caputo fractional derivative is considered and analyzed. From the mathematical analysis performed for the integer-order VM model, global dynamics of the fractional-order VM model is also investigated rigorously.Finally, the theoretical insights are supported by a set of illustrative numerical experiments.The findings of this research not only improve some existing results in the literature, but may also provide several useful real-life applications.

Nonstandard finite difference schemes for some epidemic optimal control problems

We construct and analyse nonstandard finite difference (NSFD) schemes for two epidemic optimal control problems. Firstly, we consider the well-known MSEIR system that can be used to model childhood diseases such as the measles, with the vaccination as a control intervention. The second optimal control problem is related to the 2014–2016 West Africa Ebola Virus Disease (EVD) outbreak, that came with the unprecedented challenge of the disease spreading simultaneously in three different countries, namely Guinea, Liberia and Sierra Leone, where it was difficult to control the considerable migrations and travels of people inbound and outbound. We develop an extended SEIRD metapopulation model modified by the addition of compartments of quarantined and isolated individuals. The control parameters are the exit screening of travelers and the vaccination of the susceptible individuals. For the two optimal control problems, we provide the results on: (i) the (global) stability of the disease-free and/or endemic equilibria of the state variable systems; (ii) the positivity and boundedness of solutions of the state variables systems; (iii) the existence, uniqueness and characterization of the optimal control solutions that minimizes the cost functional. On the other hand: (iv) we design Euler-based nonstandard finite difference versions of the Forward-Backward Sweep Method (NSFD-FBSM) that are dynamically consistent with the state variable systems; (v) we provide numerical simulations that support the theory and show the superiority of the nonstandard approach over the classical FBSM. The numerical simulations suggest that significantly increasing the coverage of the vaccine with its implementation for adults as well is essential if the recurrence of measles outbreaks is to be stopped in South Africa. They also show that the optimal control vaccination for the 2014-2016 EVD is more efficient than the exit screening intervention.

Dynamical analysis of methicillin-resistant Staphylococcus aureus infection in North Cyprus with optimal control: prevalence and awareness

The number of Methicillin-resistant Staphylococcus aureus (MRSA) cases in communities and hospitals is on the rise worldwide. In this work, a nonlinear deterministic model for the dynamics of MRSA infection in society was developed to visualize the significance of awareness in interventions that could be applied in the prevention of transmission with and without optimal control. Positivity and uniqueness were verified for the proposed corruption model to identify the level of resolution of infection factors in society. Furthermore, how various parameters affect the reproductive number R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}$$\\end{document} and sensitivity analysis of the proposed model was explored through mathematical techniques and figures. The global stability of model equilibria analysis was established by using Lyapunov functions with the first derivative test. A total of seven years of data gathered from a private hospital consisting of inpatients and outpatients of MRSA were used in this model for numerical simulations and for observing the dynamics of infection by using a non-standard finite difference (NSFD) scheme. When optimal control was applied as a second model, it was determined that increasing awareness of hand hygiene and wearing a mask were the key controlling measures to prevent the spread of community-acquired MRSA (CA-MRSA) and hospital-acquired MRSA (HA-MRSA). Lastly, it was concluded that both CA-MRSA and HA-MRSA cases are on the rise in the community, and increasing awareness concerning transmission is extremely significant in preventing further spread.

Stability analysis of a nonlinear malaria transmission epidemic model using an effective numerical scheme

Malaria is a fever condition that results from Plasmodium parasites, which are transferred to humans by the attacks of infected female Anopheles mosquitos. The deterministic compartmental model was examined using stability theory of differential equations. The reproduction number was obtained to be asymptotically stable conditions for the disease-free, and the endemic equilibria were determined. More so, the qualitatively evaluated model incorporates time-dependent variable controls which was aimed at reducing the proliferation of malaria disease. The reproduction number R o\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left(o\\right)$$\\end{document} was determined to be an asymptotically stable condition for disease free and endemic equilibria. In this paper, we used various schemes such as Runge–Kutta order 4 (RK-4) and non-standard finite difference (NSFD). All of the schemes produce different results, but the most appropriate scheme is NSFD. This is true for all step sizes. Various criteria are used in the NSFD scheme to assess the local and global stability of disease-free and endemic equilibrium points. The Routh–Hurwitz condition is used to validate the local stability and Lyapunov stability theorem is used to prove the global asymptotic stability. Global asymptotic stability is proven for the disease-free equilibrium when R0≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}\\le 1$$\\end{document}. The endemic equilibrium is investigated for stability when R0≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}\\ge 1$$\\end{document}. All of the aforementioned schemes and their effects are also numerically demonstrated. The comparative analysis demonstrates that NSFD is superior in every way for the analysis of deterministic epidemic models. The theoretical effects and numerical simulations provided in this text may be used to predict the spread of infectious diseases.

MODELING THE EFFECTS OF PESTS AND PESTICIDE ON CROP YIELDS IN A MULTIPLE CROPPING SYSTEM

Pest infestation poses a significant threat to agricultural crop yields, and to control it, farmers spray chemical pesticides. The persistent use of these chemical agents not only leads to pesticide residues within crops but also exerts collateral damage on the beneficial pest population. In this research work, we formulate a nonlinear mathematical model to assess the impacts of pesticide on crop yields within a multiple cropping system. Model analysis illustrates that crop consumption rates destabilize, and the spraying rate of pesticide stabilizes the system. Furthermore, we determine conditions for the global stability of the coexisting equilibrium and conduct a global sensitivity analysis to identify model parameters that significantly influence pest population density. Our findings emphasize that, for effective pest population control and enhanced crop yields, farmers should choose either pesticides with a high pest abatement rate or those with a higher pesticide uptake rate. Considering the spraying rate of pesticide as time-dependent, we also suggest an optimal control strategy to minimize the pest population and associated costs. We provide analytical results backed by numerical simulations implemented through the non-standard finite difference scheme to support our findings.

Analysis of a diffusive chemical reaction model in three space dimensions

This article proposes an implicit operator splitting nonstandard finite difference (OS-NSFD) scheme for numerical treatment of two species in three space dimensions reaction-diffusion glycolysis model. Since, the unknown state variables exhibiting the concentrations of species in glycolysis models and they cannot be negative and obtaining their positive solutions is a challenging task. The established theoretical result ensures that our proposed OS-NSFD scheme is unconditionally convergent at equilibrium point and fulfills the condition of positivity of solutions on contrary to other methods. Further, we analyze the existence and uniqueness of the solution obtained for the underlying system. To highlight the effectiveness of OS-NSFD scheme we compare the simulation results of OS-NSFD scheme with three well-known existing operators splitting finite difference (FD) schemes, namely, forward Euler explicit, backward Euler implicit and Crank Nicolson splitting schemes. Many existing techniques provide with the restricted positive solutions which do not work always. These techniques are only applicable if certain conditions on the discretized parameters are considered otherwise; they produce negative solutions, which is not the physical feature of the real system. The current work bridges this gap by catering the unconditional positive solutions to the reaction diffusion models.

A fitted parameter convergent finite difference scheme for two-parameter singularly perturbed parabolic differential equation

The objective of this paper is to develop a numerical scheme that is uniform in its parameters for a specific type of time-dependent parabolic problem with two perturbation parameters. The existence of these two parameters in the terms with the highest-order derivatives results in the formation of boundary layer(s) in the solution of such problems. Solving these model problems using classical methods does not yield satisfactory results due to the layer behavior. Therefore, nonstandard finite difference schemes have been developed as a means to obtain numerical solutions for these problems. To develop the scheme, we employ the Crank-Nicolsondiscretization on a uniform time mesh and apply a fitted operator method with a uniform spatial mesh. We have established the stability and convergence of the proposed scheme. The proposed scheme exhibits uniform convergence of second order in the temporal direction and first order in the spatial direction. However, temporal mesh refinements is employed to enhance the order to two in both directions. Model examples are provided to validate the practicality of the proposed numerical scheme.

Some novel exact and non-standard finite difference schemes for a class of diffusion–advection–reaction equation

In this paper, a generalized diffusion–advection–reaction equation (GDARE) is considered and by using the solitary wave solution two new class of exact finite difference (EFD) schemes are proposed under certain functional relationship between the temporal and space step sizes. Suitable denominator functions are identified from these EFD schemes and a new non-standard finite difference (NSFD) scheme is developed. This NSFD scheme preserves positivity and boundedness properties of the solution of GDARE. The L 2 , L ∞ errors and CPU times are obtained for the proposed NSFD and EFD schemes to show the proficiency and accuracy of the schemes.

A multi-population approach to epidemiological modeling of Listeriosis transmission dynamics incorporating food and environmental contamination

Listeriosis is a food-borne disease that mainly affects pregnant women and newborns. We propose and analyze a deterministic model of Listeriosis by considering three groups of individuals: newborns, pregnant women, and others. Mathematical analysis of the model is performed, and equilibrium points are determined. The model has three equilibria, namely, the disease-free equilibrium, the bacteria-free equilibrium, and the endemic equilibrium. We use Castillo-Chavez theorem to establish the global stability of the disease-free equilibrium when the basic reproduction number is less than 1. The local asymptotic stability of the bacteria-free, and endemic equilibria are also established using the sign of the eigenvalues of the Jacobian matrix. We use the non-standard finite difference scheme and carried numerical simulations to confirm the theoretical results. We further show the impact of specific parameters on the dynamics of infectious individuals and observe that intervention is required in all the sub-populations by reducing the contact rate and vertical transmission to mininmize the number of infectious.

Dynamical behavior of a hepatitis B epidemic model and its NSFD scheme

Hepatitis is inflammation of the liver, and one of its types, hepatitis B, is a contagious infection. Using mathematical models, the nature of the spread of the Hepatitis B virus can be predicted. In the present paper, a hepatitis B epidemic model with a Beddington–DeAngelis type incidence rate and a constant vaccination rate is considered. Some dynamical properties of this model, such as non-negativity, boundedness character, the basic reproduction number R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0$$\\end{document}, stability nature, and the bifurcation phenomenon, are investigated. By the Bendixson theorem, it is demonstrated that the disease-free equilibrium is globally asymptotically stable. It is shown that a transcritical bifurcation phenomenon occurs when R0=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0=1$$\\end{document}. It is concluded that the endemic equilibrium is globally asymptotically stable when R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0>1$$\\end{document}, by utilizing Dulac’s criteria. Also, a discrete system of difference equations is obtained by constructing a non-standard finite difference (NSFD) scheme for the continuous model. It is shown that the solutions of this discrete system are dynamically consistent for all finite step sizes. The theoretical results obtained are also supported and visualized by numerical simulations. These simulations also demonstrate that the NSFD scheme produces much more efficient results than the Euler or RK4 schemes, as shown in the theoretical results obtained.

Comparative analysis of a fractional co-infection model using nonstandard finite difference and two-step Lagrange polynomial methods

In this work, Caputo fractional model for co-dynamics of two-strains (Alpha and Delta variants) of SARS-CoV-2 and tuberculosis (TB) is formulated. We investigates the basic mathematical analysis, existence of unique solution, stability analysis, sensitivity analysis and simulations of the proposed model. In the basic mathematical analysis, it is proved that the model’s solutions are non-negative, bounded and locally asymptotically stable at disease free equilibrium (DFE) point. The existence of unique solution is proved by using Banach fixed point theorem (BFPT). Stability analysis is carried out by employing Hyers–Ulam stability criteria. Sensitivity analysis is performed to assess the impact of the model parameters on the general dynamics of the model by implementing partial rank correlation coefficient (PRCC) technique. Simulations of the proposed model obtained from non-standard finite difference scheme (NSFDs) are compared with the simulations obtained from two-step Lagrange polynomial method (TLPM). It is concluded that NSFDs gives more accurate and realistic results than TLPM. To assess some control measures necessary for reducing the co-spread of both diseases, comparative simulations are carried out for the infected classes. Also, numerical assessments showed that preventive efforts against SARS-CoV-2 variants could results in the reduction of TB prevalence and the co-infection of both diseases.

Analyzing the dynamic patterns of COVID-19 through nonstandard finite difference scheme

This paper presents a novel approach to analyzing the dynamics of COVID-19 using nonstandard finite difference (NSFD) schemes. Our model incorporates both asymptomatic and symptomatic infected individuals, allowing for a more comprehensive understanding of the epidemic's spread. We introduce an unconditionally stable NSFD system that eliminates the need for traditional Runge–Kutta methods, ensuring dynamical consistency and numerical accuracy. Through rigorous numerical analysis, we evaluate the performance of different NSFD strategies and validate our analytical findings. Our work demonstrates the benefits of using NSFD schemes for modeling infectious diseases, offering advantages in terms of stability and efficiency. We further illustrate the dynamic behavior of COVID-19 under various conditions using numerical simulations. The results from these simulations demonstrate the effectiveness of the proposed approach in capturing the epidemic's complex dynamics.

Ebola virus disease model with a nonlinear incidence rate and density-dependent treatment

This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change. The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment. Firstly, we provide a theoretical study of the nonlinear differential equations model obtained. More precisely, we derive the effective reproduction number and, under suitable conditions, prove the stability of equilibria. Afterwards, we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one. We find that the bi-stability and backward-bifurcation are not automatically connected in epidemic models. In fact, when a backward-bifurcation occurs, the disease-free equilibrium may be globally stable. Numerically, we use well-known standard tools to fit the model to the data reported for the 2018–2020 Kivu Ebola outbreak, and perform the sensitivity analysis. To control Ebola epidemics, our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy. Secondly, we propose and analyze a fractional-order Ebola epidemic model, which is an extension of the first model studied. We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme, and show its advantages.

Fractional epidemic model of coronavirus disease with vaccination and crowding effects

Most of the countries in the world are affected by the coronavirus epidemic that put people in danger, with many infected cases and deaths. The crowding factor plays a significant role in the transmission of coronavirus disease. On the other hand, the vaccines of the covid-19 played a decisive role in the control of coronavirus infection. In this paper, a fractional order epidemic model (SIVR) of coronavirus disease is proposed by considering the effects of crowding and vaccination because the transmission of this infection is highly influenced by these two factors. The nonlinear incidence rate with the inclusion of these effects is a better approach to understand and analyse the dynamics of the model. The positivity and boundedness of the fractional order model is ensured by applying some standard results of Mittag Leffler function and Laplace transformation. The equilibrium points are described analytically. The existence and uniqueness of the non-integer order model is also confirmed by using results of the fixed-point theory. Stability analysis is carried out for the system at both the steady states by using Jacobian matrix theory, Routh–Hurwitz criterion and Volterra-type Lyapunov functions. Basic reproductive number is calculated by using next generation matrix. It is verified that disease-free equilibrium is locally asymptotically stable if R0<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}<1$$\\end{document} and endemic equilibrium is locally asymptotically stable if R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}>1$$\\end{document}. Moreover, the disease-free equilibrium is globally asymptotically stable if R0<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}<1$$\\end{document} and endemic equilibrium is globally asymptotically stable if R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}>1$$\\end{document}. The non-standard finite difference (NSFD) scheme is developed to approximate the solutions of the system. The simulated graphs are presented to show the key features of the NSFD approach. It is proved that non-standard finite difference approach preserves the positivity and boundedness properties of model. The simulated graphs show that the implementation of control strategies reduced the infected population and increase the recovered population. The impact of fractional order parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} is described by the graphical templates. The future trends of the virus transmission are predicted under some control measures. The current work will be a value addition in the literature. The article is closed by some useful concluding remarks.