Finite-Time Synchronization in a Novel Discrete Fractional SIR Model for COVID-19

Iqbal Batiha; Issam  Bendib; Adel  Ouannas; Praveen  Agarwal; Nidal Anakira; Iqbal H. Jebril; Shaher Momani

doi:10.5269/bspm.75295

Iqbal Batiha Al Zaytoonah University of Jordan
Issam Bendib
Adel Ouannas
Praveen Agarwal
Nidal Anakira
Iqbal H. Jebril
Shaher Momani

Résumé

This paper presents a novel discrete fractional Susceptible-Infected-Recovered (SIR) model tailored for analyzing the dynamics of COVID-19 transmission. The model categorizes the population into four groups: susceptible (S), infected (I), recovered (R), and deceased (D), employing fractional calculus to encapsulate the memory effects and non-local interactions inherent in disease spread. A comprehensive mathematical framework is developed, incorporating the fractional sum and Caputo fractional difference operator to characterize the dynamics of the system. The study explores the conditions for finite-time synchronization among interconnected SIR models, leveraging Lyapunov stability theory to derive sufficient conditions for convergence within finite time. Numerical simulations illustrate the model's effectiveness and its sensitivity to key parameters, such as infection and recovery rates. The findings have significant implications for understanding disease dynamics and informing public health strategies, including vaccination optimization and outbreak control. Future research directions are suggested, emphasizing the enhancement of synchronization techniques and the robustness of the model in more complex epidemic scenarios.

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Références

N. Djenina, A. Ouannas, I. M. Batiha, G. Grassi, T.-E. Oussaeif, and S. Momani, A novel fractional-order discrete SIR model for predicting COVID-19 behavior. Mathematics 10, 2224, (2022). doi: 10.3390/math10132224.

M. A. Acu˜na-Zegarra, S. D´ıaz-Infante, D. Baca-Carrasco, and D. Olmos-Liceaga, COVID-19 optimal vaccination policies: A modeling study on efficacy, natural and vaccine-induced immunity responses. Math. Biosci. 337, 108614, (2021).

F. Farooq, J. Khan, and M. U. G. Khan, Effect of Lockdown on the spread of COVID-19 in Pakistan. arXiv:2005.09422v1, (2020).

O. Iyiola, B. Oduro, T. Zabilowicz, B. Iyiola, and D. Kenes, System of Time Fractional Models for COVID-19: Modeling, Analysis and Solutions. Symmetry 13, 787, (2021).

N. H. Shah, A. H. Suthar, E. N. Jayswal, A. Sikarwar, Fractional SIR-Model for Estimating Transmission Dynamics of COVID-19 in India. J. 4, 86–100, (2021).

Z.-Y. He, A. Abbes, H. Jahanshahi, N. D. Alotaibi, and Y. Wang, Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity. Mathematics 10, 165, (2022).

J. M. Blackledge, On the Evolution Equation for Modelling the Covid-19 Pandemic. In Analysis of Infectious Disease Problems (COVID-19) and Their Global Impact, P. Agarwal, J. J. Nieto, M. Ruzhansky, D. F. M. Torres, Eds., Infosys Science Foundation Series, Springer: Singapore, 2021.

P. Agarwal, J. J. Nieto, M. Ruzhansky, and D. F. M. Torres, Analysis of Infectious Disease Problems (COVID-19) and Their Global Impact. Infosys Science Foundation Series in Mathematical Sciences, Springer: Singapore, 2021.

N. R. Anakira, A. Almalki, D. Katatbeh, G. B. Hani, A. F. Jameel, K. S. Al Kalbani, and M. Abu-Dawas, An algorithm for solving linear and non-linear Volterra integro-differential equations, International Journal of Advances in Soft Computing and Its Applications, 15 (3) (2023), 77–83.

G. Farraj, B. Maayah, R. Khalil, and W. Beghami, An algorithm for solving fractional differential equations using conformable optimized decomposition method, International Journal of Advances in Soft Computing and Its Applications, 15 (1) (2023).

M. Berir, Analysis of the effect of white noise on the Halvorsen system of variable-order fractional derivatives using a novel numerical method, International Journal of Advances in Soft Computing and Its Applications, 16 (3) (2024), 294–306.

S. Ahmad, S. Javeed, H. Ahmad, J. Khushi, S. K. Elagan, and A. Khames, Analysis and numerical solution of novel fractional model for dengue. Results Phys. 28, 104669, (2021).

N. Djenina, G. Grassi, A. Ouannas, and Z. Dibi, A New COVID-19 Model Using Fractional Calculus: Stability, Mitigate Pandemic And Simulations. IFAC-PapersOnLine 58, 12, (2024). doi: 10.3390/computation12070144.

T. Hamadneh, A. Hioual, R. Saadeh, M. A. Abdoon, D. K. Almutairi, T. A. Khalid, and A. Ouannas, General Methods to Synchronize Fractional Discrete Reaction-Diffusion Systems Applied to the Glycolysis Model. Fractal Fract. 7, 11, (2023). doi: 10.3390/computation12070144.

A. O. Almatroud, N. Djenina, A. Ouannas, and G. Grassi, The SEIR COVID-19 Model Described by Fractional-Order Difference Equations: Analysis and Application with Real Data in Brazil. J. Differ. Equ. Appl. 29, 9–12, (2023). doi: 10.3390/computation12070144.

I. Al-Shbeil, N. Djenina, A. Jaradat, A. Al-Husban, A. Ouannas, and G. Grassi, A New COVID-19 Pandemic Model Including the Compartment of Vaccinated Individuals: Global Stability of the Disease-Free Fixed Point. Mathematics 11, 3, (2023). doi: 10.3390/computation12070144.

A. Abbes, A. Ouannas, N. Shawagfeh, and H. Jahanshahi, The Fractional-Order Discrete COVID-19 Pandemic Model: Stability and Chaos. Nonlinear Dyn. 111, 1, (2023). doi: 10.3390/computation12070144.

A. Dababneh, N. Djenina, A. Ouannas, G. Grassi, I. M. Batiha, and I. H. Jebril, A New Incommensurate Fractional-Order Discrete COVID-19 Model with Vaccinated Individuals Compartment. Fractal Fract. 6, 8, (2022). doi: 10.3390/computation12070144.

A. O. Almatroud, N. Djenina, A. Ouannas, G. Grassi, and M. M. Al-Sawalha, A Novel Discrete-Time COVID-19 Epidemic Model Including the Compartment of Vaccinated Individuals. Math. Biosci. Eng. 19, 1, (2022). doi: 10.3390/computation12070144.

N. Debbouche, A. Ouannas, I. M. Batiha, and G. Grassi, Chaotic Dynamics in a Novel COVID-19 Pandemic Model Described by Commensurate and Incommensurate Fractional-Order Derivatives. Nonlinear Dyn. 1, 1–13, (2021). doi: 10.3390/computation12070144.

K. Kozio l, R. Stanis lawski, and G. Bialic, Fractional-Order SIR Epidemic Model for Transmission Prediction of COVID-19 Disease. Appl. Sci. 10, 8316, (2020).

I. M. Batiha, O. Ogilat, I. Bendib, A. Ouannas, I. H. Jebril, and N. Anakira, Finite-Time Dynamics of the Fractional-Order Epidemic Model: Stability, Synchronization, and Simulations. Chaos Solitons Fractals X 13, 100118, (2024). doi: 10.3390/computation12070144.

A. Al-Husban, N. Djenina, R. Saadeh, A. Ouannas, and G. Grassi, A New Incommensurate Fractional-Order COVID-19 Model: Modelling and Dynamical Analysis. Mathematics 11, 3, (2023). doi: 10.3390/computation12070144.

L. Xiang, Y. Zhang, and J. Huang, Stability analysis of a discrete SIRS epidemic model with vaccination. J. Differ. Equ. Appl. 26, 309–327, (2020).

M. Parsamanesh, M. Erfanian, and S. Mehrshad, Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics. BMC Bioinform. 21, 1–5, (2020).

M. Parsamanesh, M. Erfanian, Stability and bifurcations in a discrete-time SIVS model with saturated incidence rate. Chaos Solitons Fractals 150, 111178, (2021).

I. Abu Falahah, A. Hioual, M. O. Al-Qadri, Y. A. Al-Khassawneh, A. Al-Husban, T. Hamadneh, and A. Ouannas, Synchronization of Fractional Partial Difference Equations via Linear Methods. Axioms 12, 8, (2023). doi: 10.3390/computation12070144.

R. Hatamleh, A. Hioual, A. Qazza, R. Saadeh, and A. Ouannas, Synchronization in Fractional Discrete Neural Networks: Constant and Variable Orders Cases. Arab J. Basic Appl. Sci. 31, 1, (2024). doi: 10.3390/computation12070144.

A. Ouannas, G. Grassi, A. T. Azar, and A. A. Khennaoui, Synchronization Control in Fractional Discrete-Time Systems with Chaotic Hidden Attractors. Adv. Mach. Learn. Technol. Appl.: Proc. AMLTA 2020, 661–669, (2021). doi: 10.3390/computation12070144.

A. Ouannas, G. Grassi, A. T. Azar, A.-A. Khennaoui, and V.-T. Pham, Synchronization of Fractional-Order Discrete-Time Chaotic Systems. Int. Conf. Adv. Intell. Syst. Inform., 218–228, (2019). doi: 10.3390/computation12070144.

A. Ouannas, Z. Odibat, and N. Shawagfeh, A New Q-S Synchronization Results for Discrete Chaotic Systems. Differ. Equ. Dyn. Syst. 27, 4, (2019). doi: 10.3390/computation12070144.

A. Ouannas, L. Jouini, O. Zehrour, On New Generalized Hybrid Synchronization in Chaotic and Hyperchaotic Discrete-Time Dynamical Systems. J. Appl. Nonlinear Dyn. 8, 3, (2019). doi: 10.3390/computation12070144.

A. Ouannas, V.-T. Pham, New Trends in Synchronization of Fractional-Order Chaotic Systems. Handb. Fract. Calc. Appl., 397, (2019). doi: 10.3390/computation12070144.

I. M. Batiha, I. Bendib, A. Ouannas, I. H. Jebril, S. Alkhazaleh, S. Momani, On New Results of Stability and Synchronization in Finite-Time for FitzHugh-Nagumo Model Using Gronwall Inequality and Lyapunov Function. J. Robot. Control 5, 6, (2024). doi: 10.3390/computation12070144.

L. Jouini, A. Ouannas, Increased and Reduced Synchronization between Discrete-Time Chaotic and Hyperchaotic Systems. Nonlinear Dyn. Syst. Theory 19, 2, (2019). doi: 10.3390/computation12070144.

M. A. Hammad, I. Bendib, W. G. Alshanti, A. Alshanty, A. Ouannas, A. Hioual, S. Momani, Fractional-Order Degn–Harrison Reaction–Diffusion Model: Finite Time Dynamics of Stability and Synchronization. Computation 12, 144, (2023). doi: 10.3390/computation12070144.

W. Pan, T. Li, Finite-Time Synchronization of Fractional-Order Chaotic Systems with Different Structures under Stochastic Disturbances. J. Comput. Commun. 9, 120–137, (2021). doi: 10.4236/jcc.2021.96007.

H. M. Almimi, M. Abu Hammad, G. Farraj, I. Bendib, A. Ouannas, A New Investigation on Dynamics of the Fractional Lengyel-Epstein Model: Finite Time Stability and Finite Time Synchronization. Computation 12, 197, (2024). doi: 10.3390/computation12100197.

S. Momani, I. M. Batiha, I. Bendib, A. Ouannas, A. Hioual, D. Mohamed, Examining finite-time behaviors in the fractional Gray-Scott model: Stability, synchronization, and simulation analysis. Int. J. Cogn. Comput. Eng.

F. Liu, S. Huang, S. Zheng, H. O. Wang, Stability Analysis and Bifurcation Control for a Fractional Order SIR Epidemic Model with Delay. In Proceedings of the 2020 39th Chinese Control Conference (CCC), Shenyang, China, 27–29 July 2020, pp. 724–729.

I. M. Batiha, S. Momani, A. Ouannas, Z. Momani, S. B. Hadid, Fractional-order COVID-19 pandemic outbreak: Modeling and stability analysis. Int. J. Biomath. 15, 2150090, (2021). Finite-Time Synchronization in a Novel Discrete Fractional SIR Model for COVID-19 15

O. A. Almatroud, I. Bendib, A. Hioual, A. Ouannas, On stability of a reaction-diffusion system described by difference equations. J. Differ. Equ. Appl. 1-15, (2024).

I. Bendib, I. M. Batiha, A. Hioual, N. Anakira, M. Dalah, A. Ouannas, On a new version of the Gierer-Meinhardt model using fractional discrete calculus. Results Nonlinear Anal. 7(2), 1-15, (2024).

T. Abdeljawad, On Riemann and Caputo fractional differences. Comput. Math. Appl. 62, 1602–1611, (2011).

G.-C. Wu, D. Baleanu, S.-D. Zeng, Finite–time stability of discrete fractional delay systems: Gronwall inequality and stability criterion. Commun. Nonlinear Sci. Numer. Simul., (2017). doi: 10.1016/j.cnsns.2017.09.001.

A. Abdurahman, H. Jiang, Z. Teng, Finite-time synchronization for fuzzy cellular neural networks with time-varying delays. Fuzzy Sets Syst. 297, 96–111, (2016).

S. Zhao, K. Li, W. Hu, Y. Wang, Finite-time synchronization of discontinuous fuzzy neural networks with mixed timevarying delays and impulsive disturbances. Results Control Optim. 12, 100253, (2023). doi: 10.1016/j.rico.2023.100253.