A dynamical approach for a fractional-order SEIL epidemic model embedding time delay

Suganya Dhandapani; Bhuvaneswari Venkatasubramaniam; Ibraheem M. Alsulami; Amer Alsulami; Hariharan Soundararajan; Shangerganesh Lingeshwaran; Suganya Dhandapani; Bhuvaneswari Venkatasubramaniam; Ibraheem M. Alsulami; Amer Alsulami; Hariharan Soundararajan; Shangerganesh Lingeshwaran

doi:10.3934/math.2026647

AIMS Mathematics

2026, Volume 11, Issue 6: 15725-15744. doi: 10.3934/math.2026647

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Research article

A dynamical approach for a fractional-order SEIL epidemic model embedding time delay

Suganya Dhandapani ¹,
Bhuvaneswari Venkatasubramaniam ^{2
,
,},
Ibraheem M. Alsulami ^{3
,
,},
Amer Alsulami ⁴,
Hariharan Soundararajan ⁵,
Shangerganesh Lingeshwaran ⁶

1.
Department of Mathematics, PSG College of Arts & Science, Coimbatore 641 014, India
2.
Department of Mathematics with Computer Applications, PSG College of Arts & Science, Coimbatore 641 014, India
3.
Mathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia
4.
Department of Mathematics, Turabah University College, Taif University, Taif 21944, Saudi Arabia
5.
Department of Mathematics, School of Engineering Dayananda Sagar University, Bangalore 562 112, India
6.
Department of Applied Sciences, National Institute of Technology Goa, Cuncolim, Goa 403 703, India

Received: 16 February 2026 Revised: 10 May 2026 Accepted: 25 May 2026 Published: 04 June 2026
MSC : 34A08, 34A12, 34D20, 49J15

In this paper, we suggest a fractional-order susceptible, exposed, infectious, and latent (SEIL) epidemic model with discrete time delays that provide for biological factors and memory effects in the spread of disease. Caputo derivatives and fixed-point theory are implemented, and we identify the existence and uniqueness of solutions, as well as the basic reproduction number for both endemic and disease-free equilibria. Local stability is analyzed through characteristic equations and linearization, while numerical simulations confirm theoretical results and illustrate the influence of fractional order, delays, and parameters. The findings show that fractional-delay models provide a more flexible and effective framework for studying disease dynamics and control.
- time-delay system,
- Caputo fractional derivatives,
- stability analysis,
- basic reproduction number,
- optimal control,
- Pontryagin maximum principle
Citation: Suganya Dhandapani, Bhuvaneswari Venkatasubramaniam, Ibraheem M. Alsulami, Amer Alsulami, Hariharan Soundararajan, Shangerganesh Lingeshwaran. A dynamical approach for a fractional-order SEIL epidemic model embedding time delay[J]. AIMS Mathematics, 2026, 11(6): 15725-15744. doi: 10.3934/math.2026647

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Abstract

In this paper, we suggest a fractional-order susceptible, exposed, infectious, and latent (SEIL) epidemic model with discrete time delays that provide for biological factors and memory effects in the spread of disease. Caputo derivatives and fixed-point theory are implemented, and we identify the existence and uniqueness of solutions, as well as the basic reproduction number for both endemic and disease-free equilibria. Local stability is analyzed through characteristic equations and linearization, while numerical simulations confirm theoretical results and illustrate the influence of fractional order, delays, and parameters. The findings show that fractional-delay models provide a more flexible and effective framework for studying disease dynamics and control.

References

[1]	S. Abbas, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electronic Journal of Diﬀerential Equations, 2011 (2011), 1–11.
[2]	O. P. Agrawal, General formulation for the numerical solution of optimal control problems, Int. J. Control, 50 (1989), 627–638. https://doi.org/10.1080/00207178908953385 doi: 10.1080/00207178908953385
[3]	P. Agarwal, U. Baltaeva, Y. Alikulov, Solvability of the boundary-value problem for a linear loaded integro-differential equation in an infinite three-dimensional domain, Chaos Soliton. Fract., 140 (2020), 110108. https://doi.org/10.1016/j.chaos.2020.110108 doi: 10.1016/j.chaos.2020.110108
[4]	M. Al-Refai, Y. Luchko, Maximum principle for the fractional diffusion equations with the Riemann–Liouville fractional derivative and its applications, Fract. Calc. Appl. Anal., 17 (2014), 483–498. https://doi.org/10.2478/s13540-014-0181-5 doi: 10.2478/s13540-014-0181-5
[5]	D. Aksim, D. Pavlov, On the extension of Adams–Bashforth–Moulton methods for numerical integration of delay differential equations and application to the moon's orbit, Math. Comput. Sci., 14 (2020), 103–109. https://doi.org/10.1007/s11786-019-00447-y doi: 10.1007/s11786-019-00447-y
[6]	F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numer. Algor., 54 (2010), 395–409. https://doi.org/10.1007/s11075-009-9342-8 doi: 10.1007/s11075-009-9342-8
[7]	E. B. M. Bashier, K. C. Patidar, Optimal control of an epidemiological model with multiple time delays, Appl. Math. Comput., 292 (2017), 47–56. https://doi.org/10.1016/j.amc.2016.07.009 doi: 10.1016/j.amc.2016.07.009
[8]	E. Bazhlekova, I. Bazhlekov, Viscoelastic flows with fractional derivative models: Computational approach by convolutional calculus of Dimovski, Fract. Calc. Appl. Anal., 17 (2014), 954–976. https://doi.org/10.2478/s13540-014-0209-x doi: 10.2478/s13540-014-0209-x
[9]	C. Castillo-Chavez, Z. Feng, Treat or not to treat: the case of tuberculosis, J. Math. Biol., 35 (1997), 629–656. https://doi.org/10.1007/s002850050069 doi: 10.1007/s002850050069
[10]	N. D. Cong, H. T. Tuan, Existence, uniqueness and exponential boundedness of global solutions to delay fractional differential equations, Mediterr. J. Math., 14 (2017), 193. https://doi.org/10.1007/s00009-017-0997-4 doi: 10.1007/s00009-017-0997-4
[11]	W. Deng, C. Li, J. Lü, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dyn., 48 (2007), 409–416. https://doi.org/10.1007/s11071-006-9094-0 doi: 10.1007/s11071-006-9094-0
[12]	S. Dong, L. Xu, Y. A, Z.-Z. Lan, D. Xiao, B. Gao, Application of a time-delay SIR model with vaccination in COVID-19 prediction and its optimal control strategy, Nonlinear Dyn., 111 (2023), 10677–10692. https://doi.org/10.1007/s11071-023-08308-x doi: 10.1007/s11071-023-08308-x
[13]	C. Giannantoni, The problem of the initial conditions and their physical meaning in linear differential equations of fractional order, Appl. Math. Comput., 141 (2003), 87–102. https://doi.org/10.1016/s0096-3003(02)00323-5 doi: 10.1016/s0096-3003(02)00323-5
[14]	S. Hariharan, L. Shangerganesh, Optimal control problem on cancer–obesity dynamics, Int. J. Biomath., 18 (2025), 2450032. https://doi.org/10.1142/S1793524524500323 doi: 10.1142/S1793524524500323
[15]	S. Hariharan, K. P. Sreesiva, L. Shangerganesh, N. Barani Balan, Prey-predator model with an infection in both population: stability analysis and an optimal control study, Discontinuity, Nonlinearity, and Complexity, 13 (2024), 257–268. https://doi.org/10.5890/DNC.2024.06.004 doi: 10.5890/DNC.2024.06.004
[16]	T. T. Hartley, C. F. Lorenzo, Dynamics and control of initialized fractional-order systems, Nonlinear Dyn., 29 (2002), 201–233. https://doi.org/10.1023/a:1016534921583 doi: 10.1023/a:1016534921583
[17]	H. Ismail, S. Hariharan, M. Jeyaraj, L. Shangerganesh, L. Rong, Hopf bifurcation and optimal control studies for avian-influenza virus with multi-delay model, Adv. Cont. Discr. Mod., 2025 (2025), 77. https://doi.org/10.1186/s13662-025-03936-6 doi: 10.1186/s13662-025-03936-6
[18]	D. Kereyu, S. Demie, Transmission dynamics model of tuberculosis with optimal control strategies in haramaya district, Ethiopia, Adv. Differ. Equ., 2021 (2021), 289. https://doi.org/10.1186/s13662-021-03448-z doi: 10.1186/s13662-021-03448-z
[19]	S. Kim, A. A. De Los Reyes V, E. Jung, Mathematical model and intervention strategies for mitigating tuberculosis in the Philippines, J. Theor. Biol., 443 (2018), 100–112. https://doi.org/10.1016/j.jtbi.2018.01.026 doi: 10.1016/j.jtbi.2018.01.026
[20]	D. Kirschner, S. Lenhart, S. Serbin, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997), 775–792. https://doi.org/10.1007/s002850050076 doi: 10.1007/s002850050076
[21]	S. Lenhart, J. T. Workman, Optimal control applied to biological models, New York: Chapman and Hall/CRC, 2007. https://doi.org/10.1201/9781420011418
[22]	Z. Liu, X. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM J. Control Optim., 53 (2015), 1920–1933. https://doi.org/10.1137/120903853 doi: 10.1137/120903853
[23]	P. K. Mondal, T. K. Kar, Optimal treatment control and bifurcation analysis of a tuberculosis model with effect of multiple re-infections, Int. J. Dyn. Control, 5 (2017), 367–380. https://doi.org/10.1007/s40435-015-0176-z doi: 10.1007/s40435-015-0176-z
[24]	C. Milici, G. Drăgănescu, J. Tenreiro Machado, Numerical methods, In: Introduction to fractional differential equations, Cham: Springer, 2019,121–185 https://doi.org/10.1007/978-3-030-00895-6_6
[25]	B. Meltzer, L. S. Pontryagin, V. G. Boltanskii, R. S. Gamkrelidze and E. F. Mischenko, The mathematical theory of optimal processes, Translated by D. E. Brown (Pergamon Press), 338 pp., 80s, Proc. Edinburgh Math. Soc., 15 (1966), 157–158. https://doi.org/10.1017/s0013091500011524 doi: 10.1017/s0013091500011524
[26]	K. O. Okosun, O. D. Makinde, I. Takaidza, Impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives, Appl. Math. Model., 37 (2013), 3802–3820. https://doi.org/10.1016/j.apm.2012.08.004 doi: 10.1016/j.apm.2012.08.004
[27]	S. Paul, A. Mahata, S. Mukherjee, P. C. Mali, B. Roy, Dynamical behavior of fractional order SEIR epidemic model with multiple time delays and its stability analysis, Examples and Counterexamples, 4 (2023), 100128. https://doi.org/10.1016/j.exco.2023.100128 doi: 10.1016/j.exco.2023.100128
[28]	C. Phang, Y. T. Toh, F. S. M. Nasrudin, An operational matrix method based on Poly-Bernoulli polynomials for solving fractional delay differential equations, Computation, 8 (2020), 82. https://doi.org/10.3390/computation8030082 doi: 10.3390/computation8030082
[29]	S. A. Rakhshan, S. Effati, A generalized Legendre–Gauss collocation method for solving nonlinear fractional differential equations with time varying delays, Appl. Numer. Math., 146 (2019), 342–360. https://doi.org/10.1016/j.apnum.2019.07.016 doi: 10.1016/j.apnum.2019.07.016
[30]	S. Rekhviashvili, A. Pskhu, P. Agarwal, S. Jain, Application of the fractional oscillator model to describe damped vibrations, Turk. J. Phys., 43 (2019), 236–242. https://doi.org/10.3906/fiz-1811-16 doi: 10.3906/fiz-1811-16
[31]	S. Ruan, On nonlinear dynamics of predator-prey models with discrete delay, Math. Model. Nat. Phenom., 4 (2009), 140–188. https://doi.org/10.1051/mmnp/20094207 doi: 10.1051/mmnp/20094207
[32]	N. Senu, K. C. Lee, A. Ahmadian, S. N. I. Ibrahim, Numerical solution of delay differential equation using two-derivative Runge-Kutta type method with Newton interpolation, Alex. Eng. J., 61 (2022), 5819–5835. https://doi.org/10.1016/j.aej.2021.11.009 doi: 10.1016/j.aej.2021.11.009
[33]	P. T. Sowndarrajan, L. Shangerganesh, A. Debbouche, D. F. M. Torres, Optimal control of a heroin epidemic mathematical model, Optimization, 71 (2022), 3107–3131. https://doi.org/10.1080/02331934.2021.2009823 doi: 10.1080/02331934.2021.2009823
[34]	S. Suganya, V. Parthiban, L. Shangerganesh, S. Hariharan, Transmission dynamics of fractional order SVEIR model for African swine fever virus with optimal control analysis, Sci. Rep., 14 (2024), 27185. https://doi.org/10.1038/s41598-024-78140-9 doi: 10.1038/s41598-024-78140-9
[35]	P. Sunthrayuth, R. Ullah, A. Khan, R. Shah, J. Kafle, I. Mahariq, et al., Numerical analysis of the Fractional-order nonlinear system of Volterra integro-differential equations, J. Funct. Space., 2021 (2021), 1537598. https://doi.org/10.1155/2021/1537958 doi: 10.1155/2021/1537958
[36]	P. Van Den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/s0025-5564(02)00108-6 doi: 10.1016/s0025-5564(02)00108-6
[37]	J. L. Willems, M. Pandit, Stability theory of dynamical systems, IEEE Trans. Syst. Man. Cybern. Syst., SMC-1 (1971), 408–408. https://doi.org/10.1109/tsmc.1971.4308335 doi: 10.1109/tsmc.1971.4308335
[38]	Y. Yang, J. Wu, J. Li, X. Xu, Tuberculosis with relapse: a model, Math. Popul. Stud., 24 (2017), 3–20. https://doi.org/10.1080/08898480.2014.998550 doi: 10.1080/08898480.2014.998550
[39]	D. Young, J. Stark, D. Kirschner, Systems biology of persistent infection: tuberculosis as a case study, Nat. Rev. Microbiol., 6 (2008), 520–528. https://doi.org/10.1038/nrmicro1919 doi: 10.1038/nrmicro1919

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