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A Third-order Two Stage Numerical Scheme and Neural Network Simulations for SEIR Epidemic Model: A Numerical Study

Muhammad Shoaib Arif, Kamaleldin Abodayeh, Yasir Nawaz

Abstract


This study focuses on the cutting-edge field of epidemic modeling, providing a comprehensive investigation of a third-order two-stage numerical approach combined with neural network simulations for the SEIR (Susceptible-Exposed-Infectious-Removed) epidemic model. An explicit numerical scheme is proposed in this work for dealing with both linear and nonlinear boundary value problems. The scheme is built on two grid points, or two time levels, and is third-order. The main advantage of the scheme is its order of accuracy in two stages. Third-order precision is not only not provided by most existing explicit numerical approaches in two phases, but it also necessitates the computation of an additional derivative of the dependent variable. The proposed scheme's consistency and stability are also examined and presented. Nonlinear SEIR (susceptible-exposed-infected-recovered) models are used to implement the scheme. The scheme is compared with the non-standard finite difference and forward Euler methods that are already in use. The graph shows that the plan is more accurate than non-standard finite difference and forward Euler methods that are already in use. The solution obtained is then looked at through the lens of the neural network. The neural network is trained using an optimization approach known as the Levenberg-Marquardt backpropagation (LMB) algorithm. The mean square error across the total number of iterations, error histograms, and regression plots are the various graphs that can be created from this process. This work conducts thorough evaluations to not only identify the strengths and weaknesses of the suggested approach but also to examine its implications for public health intervention. The results of this study make a valuable contribution to the continuously developing field of epidemic modeling. They emphasize the importance of employing modern numerical techniques and machine learning algorithms to enhance our capacity to predict and effectively control infectious diseases.

 

Doi: 10.28991/ESJ-2024-08-01-023

Full Text: PDF


Keywords


Explicit Scheme; Stability; Consistency; SEIR Model; Matlab ode45; Neural Network.

References


Thomas, R., Jose, S. A., Raja, R., Alzabut, J., Cao, J., & Balas, V. E. (2022). Modeling and analysis of SEIRS epidemic models using homotopy perturbation method: A special outlook to 2019-nCoV in India. International Journal of Biomathematics, 15(8), 2250059. doi:10.1142/S1793524522500590.

Alkhazzan, A., Wang, J., Nie, Y., Khan, H., & Alzabut, J. (2023). An effective transport-related SVIR stochastic epidemic model with media coverage and Lévy noise. Chaos, Solitons & Fractals, 175, 113953. doi:10.1016/j.chaos.2023.113953.

Hussain, S., Tunç, O., Rahman, G. Ur, Khan, H., & Nadia, E. (2023). Mathematical analysis of stochastic epidemic model of MERS-corona & application of ergodic theory. Mathematics and Computers in Simulation, 207, 130–150. doi:10.1016/j.matcom.2022.12.023.

Arif, M. S., Abodayeh, K., & Nawaz, Y. (2024). Construction of a Computational Scheme for the Fuzzy HIV/AIDS Epidemic Model with a Nonlinear Saturated Incidence Rate. Computer Modeling in Engineering & Sciences, 138(2), 1405–1425. doi:10.32604/cmes.2023.028946.

Baazeem, A. S., Arif, M. S., & Abodayeh, K. (2023). An Efficient and Accurate Approach to Electrical Boundary Layer Nanofluid Flow Simulation: A Use of Artificial Intelligence. Processes, 11(9), 2736. doi:10.3390/pr11092736.

Arif, M. S., Abodayeh, K., & Nawaz, Y. (2023). Design of Finite Difference Method and Neural Network Approach for Casson Nanofluid Flow: A Computational Study. Axioms, 12(6), 527. doi:10.3390/axioms12060527.

Ashyralyev, A., Agirseven, D., & Agarwal, R. P. (2020). Stability Estimates for Delay Parabolic Differential and Difference Equations. Applied and Computational Mathematics, 19(2), 175–204.

Ashyralyev, A., Erdogan, A. S., & Tekalan, S. N. (2019). an Investigation on Finite Difference Method for the First Order Partial Differential Equation with the Nonlocal Boundary Condition. Applied and Computational Mathematics, 18(3), 247–260.

Odibat, Z. (2020). Fractional Power Series Solutions of Fractional Differential Equations by Using Generalized Taylor Series. Applied and Computational Mathematics, 19(1), 47–58.

Khalsaraei, M. M., & Shokri, A. (2020). the New Classes of High Order Implicit Six-Step P-Stable Multiderivative Methods for the Numerical Solution of Schrödinger Equation. Applied and Computational Mathematics, 19(1), 59–86.

KHALSARAEI, M. M., & SHOKRI, A. (2020). A New Explicit Singularly P-Stable Four-Step Method for the Numerical Solution of Second-Order IVPs. Iranian Journal of Mathematical Chemistry, 11(1), 17–31. doi:10.22052/ijmc.2020.207671.1472.

Ramos, H., & Popescu, P. (2018). How many k-step linear block methods exist and which of them is the most efficient and simplest one? Applied Mathematics and Computation, 316, 296–309. doi:10.1016/j.amc.2017.08.036.

Ramos, H., & Rufai, M. A. (2020). Numerical solution of boundary value problems by using an optimized two-step block method. Numerical Algorithms, 84(1), 229–251. doi:10.1007/s11075-019-00753-3.

Lambert, J.D. (1973) Computational Methods in Ordinary Differential Equations. John Wiley and Sons, Hoboken, United States.

Mickens, R. E. (1993). Nonstandard Finite Difference Models of Differential Equations. World Scientific, Singapore. doi:10.1142/2081.

Mickens, R. E. (2002). Nonstandard finite difference schemes for differential equations. Journal of Difference Equations and Applications, 8(9), 823–847. doi:10.1080/1023619021000000807.

Piyawong, W., Twizell, E. H., & Gumel, A. B. (2003). An unconditionally convergent finite-difference scheme for the SIR model. Applied Mathematics and Computation, 146(2–3), 611–625. doi:10.1016/S0096-3003(02)00607-0.

Ramos, H. (2010). Contributions to the development of differential systems exactly solved by multistep finite-difference schemes. Applied Mathematics and Computation, 217(2), 639–649. doi:10.1016/j.amc.2010.05.101.

Shokri, A., Khalsaraei, M. M., & Molayi, M. (2022). Nonstandard Dynamically Consistent Numerical Methods for MSEIR Model. Journal of Applied and Computational Mechanics, 8(1), 196–205. doi:10.22055/jacm.2021.36545.2863.

Shokri, A., Khalsaraei, M. M., & Molayi, M. (2021). Dynamically Consistent NSFD Methods for Predator-prey System. Journal of Applied and Computational Mechanics, 7(3), 1565–1574. doi:10.22055/jacm.2021.36537.2862.

Roeger, L. I. W., & Barnard, R. W. (2007). Preservation of local dynamics when applying central difference methods: Application to SIR model. Journal of Difference Equations and Applications, 13(4), 333–340. doi:10.1080/10236190601079134.

Khalsaraei, M. M., Shokri, A., Ramos, H., & Heydari, S. (2021). A positive and elementary stable nonstandard explicit scheme for a mathematical model of the influenza disease. Mathematics and Computers in Simulation, 182, 397–410. doi:10.1016/j.matcom.2020.11.013.

Rihan, F. A., & Alsakaji, H. J. (2021). Analysis of a stochastic HBV infection model with delayed immune response. Mathematical Biosciences and Engineering, 18(5), 5194–5220. doi:10.3934/mbe.2021264.

Asif, M., Jan, S. U., Haider, N., Al-Mdallal, Q., & Abdeljawad, T. (2020). Numerical modeling of NPZ and SIR models with and without diffusion. Results in Physics, 19(103512). doi:10.1016/j.rinp.2020.103512.

Rihan, F. A., Al-Mdallal, Q. M., AlSakaji, H. J., & Hashish, A. (2019). A fractional-order epidemic model with time-delay and nonlinear incidence rate. Chaos, Solitons and Fractals, 126, 97–105. doi:10.1016/j.chaos.2019.05.039.

Khan, A., Alshehri, H. M., Abdeljawad, T., Al-Mdallal, Q. M., & Khan, H. (2021). Stability analysis of fractional Nabla difference COVID-19 model. Results in Physics, 22(103888). doi:10.1016/j.rinp.2021.103888.

Asif, M., Ali Khan, Z., Haider, N., & Al-Mdallal, Q. (2020). Numerical simulation for solution of SEIR models by meshless and finite difference methods. Chaos, Solitons & Fractals, 141, 110340. doi:10.1016/j.chaos.2020.110340.

Ahmad, S., Ullah, A., Al-Mdallal, Q. M., Khan, H., Shah, K., & Khan, A. (2020). Fractional order mathematical modeling of COVID-19 transmission. Chaos, Solitons & Fractals, 139, 110256. doi:10.1016/j.chaos.2020.110256.

Ahmad, I., Khan, M. N., Inc, M., Ahmad, H., & Nisar, K. S. (2020). Numerical simulation of simulate an anomalous solute transport model via local meshless method. Alexandria Engineering Journal, 59(4), 2827–2838. doi:10.1016/j.aej.2020.06.029.

Ahmad, H., Akgül, A., Khan, T. A., Stanimirović, P. S., & Chu, Y. M. (2020). New perspective on the conventional solutions of the nonlinear time-fractional partial differential equations. Complexity, 2020, 1–10. doi:10.1155/2020/8829017.

Ahmad, H., Khan, T. A., Stanimirovic, P. S., & Ahmad, I. (2020). Modified variational iteration technique for the numerical‎ solution of fifth order KdV-type equations. Journal of Applied and Computational Mechanics, 6(Special Issue), 1220-1227. doi:10.22055/JACM.2020.33305.2197.

Liu, X., Ahsan, M., Ahmad, M., Nisar, M., Liu, X., Ahmad, I., & Ahmad, H. (2021). Applications of HAAR wavelet-finite difference hybrid method and its convergence for hyperbolic nonlinear schrödinger equation with energy and mass conversion. Energies, 14(23), 7831. doi:10.3390/en14237831.

Ahsan, M., Lin, S., Ahmad, M., Nisar, M., Ahmad, I., Ahmed, H., & Liu, X. (2021). A Haar wavelet-based scheme for finding the control parameter in nonlinear inverse heat conduction equation. Open Physics, 19(1), 722–734. doi:10.1515/phys-2021-0080.

Pasha, S. A., Nawaz, Y., & Arif, M. S. (2023). On the nonstandard finite difference method for reaction–diffusion models. Chaos, Solitons & Fractals, 166, 112929. doi:10.1016/j.chaos.2022.112929.


Full Text: PDF

DOI: 10.28991/ESJ-2024-08-01-023

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