Projective Synchronization and H ∞ Control of a T-S Fuzzy-Based COVID-19 SEIR Model

Abstract

This paper addresses the projective synchronization and $H_{\infty}$ control of the Susceptible-Exposed-Infectious-Recovered (SEIR) COVID-19 model using Takagi-Sugeno (T-S) fuzzy control techniques. A fuzzy controller design is proposed to synchronize the SEIR nonlinear model by using the Linear Matrix Inequality (LMI) technique and the Lyapunov function. Based on the LMI technique, Lyapunov function, and $H_{\infty}$ control theory, the system's stability and $H_{\infty}$ performance is ensured by a set of necessary criteria. The advantage of this research lies in its ability to enhance system stability, defined via Lyapunov-based LMI conditions, and robustness, measured through the $H_{\infty}$ disturbance attenuation level, both of which are crucial for effective pandemic management. This paper presents simulations and numerical data to highlight the benefits of the proposed method, demonstrating its effectiveness in mitigating the negative consequences of the COVID-19 pandemic by comparing controlled and uncontrolled scenarios.

Keywords

SEIR COVID-19 model T-S fuzzy systems projective synchronization Linear matrix inequalities

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References

Ahmad

Atta

Farman

Nisar

K. S.

Ahmad

Hincal

(2025). Investigation of lassa fever with relapse and saturated incidence rate: Mathematical modeling and control. Modeling Earth Systems and Environment, 11(3), 1–28. https://doi.org/10.1007/s40808-025-02370-7

Alonso-Quesada

De la Sen

Ibeas

(2012). A vaccination strategy based on a state feedback control law for linearizing SEIR epidemic models. In Proc int joint conf biomed eng syst technol (pp. 195–209). Springer.

Al-Talib

Z. S.

A.-A.

(2020). Projective and hybrid projective synchronization of 4-D hyperchaotic system via nonlinear controller strategy. TELKOMNIKA Telecommun Comput Electron Control, 18(2), 1012–1020. https://doi.org/10.12928/telkomnika.v18i2.14664

Annas

Pratama

M. I.

Rifandi

Sanusi

Side

(2020). Stability analysis and numerical simulation of SEIR model for pandemic COVID-19 spread in Indonesia. Chaos Solitons & Fractals, 139(1), 110072. https://doi.org/10.1016/j.chaos.2020.110072

Arifi

Bouallegue

(2024). Takagi-Sugeno fuzzy-based approach for modeling and control of an activated sludge process. International Journal of Dynamics and Control, 12(8), 3123–3138. https://doi.org/10.1007/s40435-024-01398-4

Asamoah

J. K. K.

Owusu

M. A.

Jin

Oduro

F. T.

Abidemi

Opoku Gyasi

(2020). Global stability and cost-effectiveness analysis of COVID-19 considering the impact of the environment: Using data from Ghana. Chaos, Solitons & Fractals, 140, 110103. https://doi.org/10.1016/j.chaos.2020.110103

Assawinchaichote

Junhom

(2011).

H_{\infty}

fuzzy controller design for HIV/AIDS infection system with dual drug dosages via an LMI approach. International Energy Journal, 5(2), 27–33.

Bououden

Chadli

Filali

El Hajjaji

(2012). Fuzzy model based multivariable predictive control of a variable speed wind turbine: LMI approach. Renew Energy, 37(1), 434–439. https://doi.org/10.1016/j.renene.2011.06.025

Boyd

El Ghaoui

Feron

Balakrishnan

(1994). Linear matrix inequalities in systems and control theory. SIAM.

10.

Brauer

(2017). Mathematical epidemiology: Past, present, and future. Infectious Disease Modelling, 2(2), 113–127.

11.

Buesa

(2021). Grand challenge in viral disease investigation: An endless endeavor. Frontiers in Virology, 1(1), 692105. https://doi.org/10.3389/fviro.2021.692105

12.

Chang

X. H.

(2014). Robust output feedback H∞ control and filtering for uncertain linear systems. Springer.

13.

Chang

X. H.

Yang

Xiong

(2018). Quantized fuzzy output feedback H∞ control for nonlinear systems with adjustment of dynamic parameters. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49(10), 2005–2015. https://doi.org/10.1109/TSMC.2018.2867213

14.

Chen

Zhao

Sprott

J. C.

(2013). Application of takagi-sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization. Nonlinear Dynamics, 73(3), 1495–1505. https://doi.org/10.1007/s11071-013-0880-1

15.

Demirci

Unal

Özalp

(2011). A fractional order SEIR model with density dependent death rate. Hacettepe Journal of Mathematics and Statistics, 40(2), 287–295.

16.

Diekmann

Heesterbeek

Britton

(2013). Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton University Press.

17.

Din

Khan

Zaman

(2020). Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China. Chaos Solitons & Fractals, 141(1), 110286. https://doi.org/10.1016/j.chaos.2020.110286

18.

Esfahani

S. H.

(2016). Improvement on the problem of output feedback fuzzy H∞-tracking control design for non-linear discrete-time systems with state and input delay. IET Control Theory & Applications, 10(1), 24–34. https://doi.org/10.1049/iet-cta.2014.1378

19.

Flores

(2018). Pale rider: The spanish flu of 1918 and how it changed the world, by laura spinney. Baylor Univ Med Cent Proc, 31(2), 254. https://doi.org/10.1080/08998280.2018.1435117

20.

Giap

V. N.

(2023). Text message secure communication based on fractional-order chaotic systems with takagi-sugeno fuzzy disturbance observer and sliding mode control. International Journal of Dynamics and Control, 11(1), 3109–3123. https://doi.org/10.1007/s40435-023-01170-0

21.

Kharitonov

V. L.

Chen

(2003). Stability of time-delay systems. Birkhäuser.

22.

Harko

Lobo

F. S. N.

Mak

M. K.

(2014). Exact analytical solutions of the susceptible-infected-recovered (SIR) epidemic model and of the SIR model with equal death and birth rates. Applied Mathematics and Computation, 236(1), 184–194. https://doi.org/10.1016/j.amc.2014.03.030

23.

Hethcote

H. W.

(2000). The mathematics of infectious diseases. SIAM Review, 42(4), 599–653. https://doi.org/10.1137/S0036144500371907

24.

Huang

Cao

(2017). Active control strategy for synchronization and anti-synchronization of a fractional chaotic financial system. Physica A, 473(1), 262–275. https://doi.org/10.1016/j.physa.2017.01.009

25.

Jamil

Naik

P. A.

Farman

Saleem

M. U.

Ganie

A. H.

(2024). Stability and complex dynamical analysis of COVID-19 epidemic model with non-singular kernel of mittag-leffler law. Journal of Applied Mathematics and Computing, 70(4), 3441–3476. https://doi.org/10.1007/s12190-024-02105-4

26.

Jamshidi

Safa

Mirzajani

(2025). Continuous Nonsingular Fast Terminal Sliding Mode Control for a Free Gyro Stabilized Mirror System. IEEE Transactions on Automation Science and Engineering, 22(1), 10937–10947. https://doi.org/10.1109/TASE.2025.3530679

27.

Kamrujjaman

Saha

Islam

M. S.

Ghosh

(2022). Dynamics of SEIR model: A case study of COVID-19 in Italy. Results in Control and Optimization, 7(1), 100119. https://doi.org/10.1016/j.rico.2022.100119

28.

Keeling

M. J.

Rohani

(2011). Modeling Infectious Diseases in Humans and Animals. Princeton University Press.

29.

Khalili Amirabadi

Heydari

Zarrabi

(2016). Analysis and control of SEIR epidemic model via sliding mode control. Advanced Modeling and Optimization, 18(1), 153–162.

30.

Khan

Kumar

(2019). T-S fuzzy modeling and predictive control and synchronization of chaotic satellite systems. International Journal of Modelling and Simulation, 39(3), 203–213. https://doi.org/10.1080/02286203.2018.1563393

31.

Khan

M. A.

Atangana

(2020). Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alexandria Engineering Journal, 59(4), 2379–2389. https://doi.org/10.1016/j.aej.2020.02.033

32.

Khatun

M. S.

Das

(2023). Dynamics and control of an SITR COVID-19 model with awareness and hospital bed dependency. Chaos Solitons & Fractals, 175(1), 114010. https://doi.org/10.1016/j.chaos.2023.114010

33.

Kim

Y. M.

(2016). Robust data driven H∞ control for wind turbine. Journal of the Franklin Institute, 353(13), 3104–3117. https://doi.org/10.1016/j.jfranklin.2016.06.009

34.

Kocarev

Parlitz

(1996). Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Physical Review Letters, 76(11), 1816. https://doi.org/10.1103/PhysRevLett.76.1816

35.

Q. Q.

Wang

Z. P.

Huang

H. N.

H. X.

Qiao

(2023). Fault-tolerant stochastic sampled-data fuzzy control for nonlinear delayed parabolic PDE systems. IEEE Transactions on Fuzzy Systems, 31(8), 2679–2693. https://doi.org/10.1109/TFUZZ.2023.3235400

36.

Liu

(2015). Adaptive hybrid function projective synchronization of general chaotic complex systems with different orders. Journal of Computational and Nonlinear Dynamics, 10(2), 021018. https://doi.org/10.1115/1.4027975

37.

Liu

Gao

Kao

(2015). Robust H∞ control for a class of neutral-type systems via sliding mode observer. Applied Mathematics and Computation, 271(1), 669–681. https://doi.org/10.1016/j.amc.2015.09.007

38.

Stratton

C. W.

Tang

Y. W.

(2020). The Wuhan SARS-CoV-2 - what’s next for China. Journal of Medical Virology, 92(6), 546. https://doi.org/10.1002/jmv.25738

39.

Wang

Zhang

(2021). A takagi-sugeno fuzzy-model-based finite-time H∞ control for a hydraulic turbine governing system with time delay. International Journal of Electrical Power & Energy Systems, 132(1), 107152. https://doi.org/10.1016/j.ijepes.2021.107152

40.

Mahmoud

G. M.

Mahmoud

E. E.

(2010a). Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dynamics, 62(1), 875–882. https://doi.org/10.1007/s11071-010-9770-y

41.

Mahmoud

G. M.

Mahmoud

E. E.

(2010b). Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear systems. Nonlinear Dynamics, 61(1), 141–152. https://doi.org/10.1007/s11071-009-9637-2

42.

Muli

F. M.

Okelo

Magwanga

Ongati

(2024). Mathematical analysis of COVID-19 model incorporating vaccination of susceptible and isolation of symptomatic individuals. Journal of Applied Mathematics and Computing, 70(1), 461–488. https://doi.org/10.1007/s12190-023-01966-5

43.

Pecora

L. M.

Carroll

T. L.

(1990). Synchronization in chaotic systems. Physical Review Letters, 64(8), 821. https://doi.org/10.1103/PhysRevLett.64.821

44.

Precup

R. E.

Hellendoorn

(2011). A survey on industrial applications of fuzzy control. Comput Ind, 62(3), 213–226. https://doi.org/10.1016/j.compind.2010.10.001

45.

Rigatos

Siano

Abbaszadeh

Ademi

Melkikh

(2018). Nonlinear H∞ control for underactuated systems: The furuta pendulum example. International Journal of Dynamics and Control, 6(1), 835–847. https://doi.org/10.1007/s40435-017-0348-0

46.

Roohi

Mirzajani

Haghighi

A. R.

Basse-O’Connor

(2024). Robust stabilization of fractional-order hybrid optical system using a single-input TS-fuzzy sliding mode control strategy with input nonlinearities. AIMS Mathematics, 9(9), 25879–25907. https://doi.org/10.3934/math.20241264

47.

Shi

Zhang

Chadli

Agarwal

R. K.

(2015). Mixed H∞ and passive filtering for discrete fuzzy neural networks with stochastic jumps and time delays. IEEE Transactions on Neural Networks and Learning Systems, 27(4), 903–909. https://doi.org/10.1109/TNNLS.2015.2425962

48.

Siettos

C. I.

Russo

(2013). Mathematical modeling of infectious disease dynamics. Virulence, 4(4), 295–306. https://doi.org/10.4161/viru.24041

49.

Syafruddin

Noorani

M. S. M.

(2013). Lyapunov function of SIR and SEIR model for transmission of dengue fever disease. International Journal of Simulation and Process Modelling, 8(2-3), 177–184. https://doi.org/10.1504/IJSPM.2013.057544

50.

Takagi

Sugeno

(1985). Fuzzy identification of systems and its applications to modelling and control. IEEE Transactions on Systems, Man, and Cybernetics, 15(1), 116–132. https://doi.org/10.1109/TSMC.1985.6313399

51.

Tanaka

Wang

H. O.

(2004). Fuzzy control systems design and analysis: a linear matrix inequality approach. John Wiley & Sons.

52.

Tuan

H. D.

Apkarian

Narikiyo

Yamamoto

(2001). Parameterized linear matrix inequality techniques in fuzzy control system design. IEEE Transactions on Fuzzy Systems, 9(2), 324–332. https://doi.org/10.1109/91.919253

53.

Vafamand

Asemani

M. H.

Khayatiyan

Khooban

M. H.

Dragicevic

(2017). TS Fuzzy model-based controller design for a class of nonlinear systems including nonsmooth functions. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 50(1), 233–244. https://doi.org/10.1109/TSMC.2017.2773664

54.

Vembarasan

(2022). Passivity based on synchronization of T-S fuzzy energy resources system. J Math Comput Sci, 25(3), 269–283. https://doi.org/10.22436/jmcs.025.03.06

55.

Vembarasan

Balasubramaniam

(2013). Chaotic synchronization of Rikitake system based on T-S fuzzy control techniques. Nonlinear Dynamics, 74(1), 31–44. https://doi.org/10.1007/s11071-013-0946-0

56.

Vembarasan

Balasubramaniam

Joo

E. M

2014.

H_{\infty}

state-feedback control of time-delay systems using reciprocally convex approach. Journal of Process Control, 24(6), 892–904. https://doi.org/10.1016/j.jprocont.2014.04.005

57.

M. T.

Kim

S. H.

Pham

D. H.

Thanh

H. L.

Pham

V. H.

Roohi

(2025). Adaptive dynamic programming-based intelligent finite-time flexible SMC for stabilizing fractional-order four-wing chaotic systems. Mathematics, 13(13), 2078. https://doi.org/10.3390/math13132078

58.

Wang

Fan

Wang

Zhang

(2012). Adaptive fuzzy synchronization for a class of chaotic systems with unknown nonlinearities and disturbances. Nonlinear Dynamics, 69(1), 1167–1176. https://doi.org/10.1007/s11071-012-0338-x

59.

H. N.

Wang

Z. P.

(2017). Observer-based H∞ sampled-data fuzzy control for a class of nonlinear parabolic PDE systems. IEEE Transactions on Fuzzy Systems, 26(2), 454–473. https://doi.org/10.1109/TFUZZ.2017.2686337

60.

S. J.

(2007). Affine TS-model-based fuzzy regulating/servo control design. Fuzzy Sets and Systems, 158(20), 2288–2305. https://doi.org/10.1016/j.fss.2007.05.012

61.

Yadav

Pallav

Handa (2023). Projective synchronization for a new class of chaotic/hyperchaotic systems with and without parametric uncertainty. Transactions of the Institute of Measurement and Control, 45(10), 1975–1985. https://doi.org/10.1177/01423312221150294

62.

Zhu

Z. Y.

Zhao

Z. S.

Zhang

Wang

R. K.

(2020). Adaptive fuzzy control design for synchronization of chaotic time-delay system. Information Sciences, 535, 225–241. https://doi.org/10.1016/j.ins.2020.05.056