In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article


1 Department of Mathematics‎, ‎Payame Noor University (PNU), Tehran‎, ‎Iran

2 Department of Statistics, Payame Noor University (PNU), Tehran, Iran,

3 Department of Mathematics‎, ‎Maragheh University‎, ‎Maragheh‎, ‎Iran‎.



This paper presents an extension of the SEIR mathematical model for infectious disease‎ ‎transmission to a fractional-order model‎. ‎The model is formulated using the Caputo derivative of order α ∈ (0, 1]‎. ‎We study the stability of equilibrium points‎, ‎including the disease-free equilibrium $(E_{f})$‎, ‎and the‎ ‎infected steady-state equilibrium $(E_{e})$ using the‎ ‎stability theorem of Fractional Differential Equations‎. ‎The model is also analyzed under certain conditions‎, ‎and‎ ‎it is shown that the disease-free equilibrium is locally asymptotically‎ ‎stable‎. ‎Additionally‎, ‎the extended Barbalat’s lemma is applied to the‎ ‎fractional-order system‎, ‎and a suitable Lyapunov functional is constructed‎ ‎to demonstrate the global asymptotic stability of the infected‎ ‎steady-state equilibrium‎. ‎To validate the theoretical results‎, ‎a numerical simulation of the problem is conducted‎.


Main Subjects

[1] Ahmad, A.S., Owyed, S., Aty, A.H.A., Mahmoud, E.E., Shah, K., Alrabaiahg, H. (2021). “Mathematical analysis of COVID-19 via new mathematical model”, Nonlinear Analysis, Chaos, Solitonsand Fractals, 143, 110585.
[2] Almeida, R. (2018). “Analysis of a fractional SEIR model with treatment”, Applied Mathematics Letters, 84, 56-62.
[3] Almeida, R., Cruz, A.B.D., Martins, N., Monteiro, M.T.T. (2019). “An epidemiological MSEIR model described by the Caputo fractional derivative”, International Journal of Dynamics and Control, 7(2), 776-784.
[4] Boukhouima, A., Zine, H., Lotfi, E.M., Mahrouf Zhu, Q.J. (1999). “A survey of subdifferential calculus with applications”, Nonlinear Analysis, 38, 687-773.
[5] Brauer, F. (2017). “Mathematical epidemiology: Past, present, and future”, Infectious Disease Modelling, 2(2), 113-127.
[6] Cardoso,L.C.,Camargo,R.F.,DosSantos,F.L.P.,DosSantos,J.P.C.(2021).“Globalstabilityanalysis of a fractional differential system in hepatitis B”, Chaos, Solitons and Fractals, 143, 110619.
[7] Chen, L., Chai, Y., Wu, R. (2014). “New results on stability and stabilization of a class of nonlinear fractional-order systems”, Nonlinear Dynamics, 75(4), 633-641.
[8] Cruz, V.D.L. (2015). “Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun”, Nonlinear Sci., 24, 75-85.
[9] Daley, D.J., Gani, J. (2001). “Epidemic modelling: an introduction”, Cambridge University Press.
[10] Edrisi, Y. (2022). “Using the integral operational matrix of B-spline functions to solve fractional optimal control problems”, Control and Optimization in Applied Mathematics (COAM), 7(2), 77-98.
[11] Haldrup, N., Valdés, J.E.V. (2017). “Long memory, fractional integration, and cross-sectional aggregation”, Journal of Econometrics, 199 (1), 1-11.
[12] Huang, L.L., Baleanu, D., Mo, Z.W., Wu, G.C. (2018). “Fractional discrete-time diffusion equation with uncertainty: Applications of fuzzy discrete fractional calculus”, Physica A: Statistical Mechanics and its Applications, 508, 166-175.
[13] Huo, J., Zhao, H., Zhu, L. (2015). “The effect of vaccines on backward bifurcation in a fractional order HIV model”, Nonlinear Analysis: Real World Applications, 26, 289-305.
[14] Kamyad, A.V., Akbari, R., Heydari, A.A., Heydari, A. (2014). “Mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis B virus”, Computational and Mathematical Methods in Medicine.
[15] Li, C., Ma, Y. (2013). “Fractional dynamical system and its linearization theorem”, Nonlinear Dynamics, 71(4), 621-633.
[16] Li, M.Y., Muldowney, J.S. (1995). “Global stability for the SEIR model in epidemiology”, Mathematical Biosciences, 125(2), 155-164.
[17]Li,Y.,Chen,Y.,Podlubny,I.(2009).“Mittag-Lefflerstabilityoffractionalordernonlineardynamic Systems”, Automatica, 45(8), 1965-1969.
[18] Li, Y., Chen, Y., Podlubny, I. (2010). “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability”, Computers and Mathematics with Applications, 59(5), 1810-1821.
[19] Lin, W. (2007). “Global existence theory and chaos control of fractional differential equations”, Journal of Mathematical Analysis and Applications, 332(1), 709-726.
[20] McCluskey, C.C., Driessche, P.V.D. (2004). “Global analysis of two tuberculosis models”, Journal of Dynamics and Differential Equations, 16(1), 139-166.
 [21] Nemati, A., Alizadeh, A., Soltanian, F. (2019). “Solution of fractional optimal control problems with noise function using the bernstein functions”, Control and Optimization in Applied Mathematics (COAM), 4(1), 37-51.
[22] Nigmatullin, R.R., Omay, T., Baleanu, D. (2010). “On fractional filtering versus conventional filtering in economics”, Communications in Nonlinear Science and Numerical Simulation, 15(4), 979-986.
[23] Obembe, A.D., Al-Yousef, H.Y., Hossain, M.E., Abu-Khamsin, S.A. (2017). “Fractional derivatives and their applications in reservoir engineering problems: A review”, Journal of Petroleum Science and Engineering, 157, 312-327.
[24] Rostamy, D., Mottaghi, E. (2016). “Stability analysis of a fractional-order epidemics model with multiple equilibriums”, Advances in Difference Equations, 2016(1), 1-11.
[25] Salehi, Y., Darvishi, M.T. (2016). “An investigation of fractional Riccati differential equation”, Optik, 127(23), 11505-11521.
[26] Shafiof, S.M., Askari, J., Shams Solary, M. (2018). “A numerical solution of fractional optimal control problems using spectral method and hybrid functions”, Control and Optimization in Applied Mathematics (COAM), 3(1), 1-25.
[27] Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.Q. (2018). “A new collection of real world applications of fractional calculus in science and engineering”, Communications in Nonlinear Science and Numerical Simulation, 64, 213-231.
[28] Yang, Y., Xu, L. (2020). “Stability of a fractional order SEIR model with general incidence”, Applied Mathematics Letters, 105.
[29] Zhang, F., Li, C., Chen, Y. (2011). “Asymptotical stability of nonlinear fractional differential system with Caputo derivative”, International Journal of Differential Equations, 011:12. Article ID 635165.