Stability analysis of an incommensurate fractional-order SIR model


  • Bahatdin Daşbaşı Kayseri University, Faculty of Engineering, Architecture and Design, Department of Engineering Basic Sciences, 38039, Kayseri, Turkey



SIR mathematical model, incommensurate order differential equation, fractional-derivative, stability analysis


In this paper, a fractional-order generalization of the susceptible-infected-recovered (SIR) epidemic model for predicting the spread of an infectious disease is presented. Also, an incommensurate fractional-order differential equations system involving the Caputo meaning fractional derivative is used. The equilibria are calculated and their stability conditions are investigated. Finally, numerical simulations are presented to illustrate the obtained theoretical results.


Download data is not yet available.


Yavuz, M., Ozdemir, N. & Baskonus, H. Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel. The European Physical Journal Plus, 133(6), 1-11, (2018).

Kashkynbayev, A. & Rihan, F. Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay. Mathematics, 9(15), 1829, (2021).

Razminia, A., Majd, V. & Baleanu, D. Chaotic incommensurate fractional order Rössler system: active control and synchronization. Advances In Difference Equations, 2011(1), 1-12, (2011).

Ji, Y., Lu, J. & Qiu, J. Stability of equilibrium points for incommensurate fractional-order nonlinear systems. 2016 35th Chinese Control Conference (CCC), pp. 10453-10458, (2016, July).

Daşbaşi, B. Stability analysis of the hiv model through incommensurate fractional-order nonlinear system. Chaos, Solitons & Fractals, 137, 109870, (2020).

Deng, W., Li, C. & Guo, Q. Analysis of fractional differential equations with multi-orders. Fractals, 15(02), 173-182, (2007).

Hamou, A.A., Azroul, E. & Hammouch, Z. On dynamics of fractional incommensurate model of Covid-19 with nonlinear saturated incidence rate, medRxiv, (2021).

Wang, Y. & Li, T. Stability analysis of fractional-order nonlinear systems with delay. Mathematical Problems In Engineering, (2014).

Petras, I. Stability of fractional-order systems with rational orders. ArXiv Preprint ArXiv:0811.4102, (2008).

Rivero, M., Rogosin, S.V., Tenreiro Machado, J.A. & Trujillo, J.J. Stability of fractional order systems. Mathematical Problems In Engineering, (2013).

Brandibur, O., Garrappa, R. & Kaslik, E. Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives. Mathematics, 9(8), 914, (2021).

Lekdee, N., Sirisubtawee, S. & Koonprasert, S. Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders. Advances In Difference Equations, 2019(1), 1-22, (2019).

Debbouche, N., Ouannas, A., Batiha, I.M. & Grassi, G. Chaotic Dynamics in a Novel COVID-19 Pandemic Model Described by Commensurate and Incommensurate Fractional-Order Derivatives. (2021).

Wang, X., Wang, Z. & Xia, J. Stability and bifurcation control of a delayed fractional-order eco-epidemiological model with incommensurate orders. Journal Of The Franklin Institute, 356(15), 8278-8295, (2019).

Veeresha, P. A Numerical Approach to the Coupled Atmospheric Ocean Model Using a Fractional Operator. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 1-10, (2021).

Yavuz, M., & Yaşkıran, B. Conformable Derivative Operator in Modelling Neuronal Dynamics. Applications & Applied Mathematics, 13(2), 803-817, (2018).

Yokuş, A. Construction of Different Types of Traveling Wave Solutions of the Relativistic Wave Equation Associated with the Schrödinger Equation. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 24-31, (2021).

Yavuz, M., & Sene, N. Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model. Journal of Ocean Engineering and Science, 6(2), 196-205, (2021).

Hammouch, Z., Yavuz, M., & Özdemir, N. Numerical Solutions and Synchronization of a Variable-Order Fractional Chaotic System. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 11-23, (2021).

Sene, N. Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Journal of Fractional Calculus and Nonlinear Systems, 2(1), 60-75, (2021).

Al-Mdallal, Q.M., Hajji, M.A., & Abdeljawad, T. On the iterative methods for solving fractional initial value problems: new perspective. Journal of Fractional Calculus and Nonlinear Systems, 2(1), 76-81, (2021).

Yavuz, M., Coşar, F. Ö., Günay, F., & Özdemir, F. N. A new mathematical modeling of the COVID-19 pandemic including the vaccination campaign. Open Journal of Modelling and Simulation, 9(3), 299-321, (2021).

Yavuz, M., Sulaiman, T.A., Yusuf, A. & Abdeljawad, T. The Schrödinger-KdV equation of fractional order with Mittag-Leffler nonsingular kernel. Alexandria Engineering Journal, 60(2), 2715-2724, (2021).

Nazir, G., Zeb, A., Shah, K., Saeed, T., Khan, R.A. & Khan, S.I.U. Study of COVID-19 mathematical model of fractional order via modified Euler method. Alexandria Engineering Journal, 60(6), 5287-5296, (2021).

Zarin, R., Ahmed, I., Kumam, P., Zeb, A. & Din, A. Fractional modeling and optimal control analysis of rabies virus under the convex incidence rate. Results in Physics, 28, 104665, (2021).

Alqudah, M.A., Abdeljawad, T., Zeb, A., Khan, I.U. & Bozkurt, F. Effect of Weather on the Spread of COVID-19 Using Eigenspace Decomposition. Cmc-Computers Materials & Continua, 3047-3063, (2021).

Angstmann, C.N., Henry, B.I. & McGann, A.V. A fractional-order infectivity and recovery SIR model. Fractal And Fractional, 1(1), 11, (2017).

Liu, N., Fang, J., Deng, W. & Sun, J.W. Stability analysis of a fractional-order SIS model on complex networks with linear treatment function. Advances In Difference Equations, 2019(1), 1-10, (2019).

Alqahtani, R.T. Mathematical model of SIR epidemic system (COVID-19) with fractional derivative: stability and numerical analysis. Advances In Difference Equations, 2021(1), 1-16 (2021).

Naik, P.A., Yavuz, M., Qureshi, S., Zu, J. & Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42, (2020).

Tavares, D., Almeida, R. & Torres, D.F. Caputo derivatives of fractional variable order: numerical approximations. Communications In Nonlinear Science And Numerical Simulation, 35, 69-87, (2016).

Tavazoei, M.S. & Haeri, M. Chaotic attractors in incommensurate fractional order systems. Physica D: Nonlinear Phenomena, 237(20), 2628-2637, (2008).

Odibat, Z.M. Analytic study on linear systems of fractional differential equations. Computers & Mathematics With Applications, 59(3), 1171-1183, (2010).

Owolabi, K.M. Riemann-Liouville fractional derivative and application to model chaotic differential equations. Progress in Fractional Differentiation and Applications, 4, 99-110, (2018).

Alshomrani, A.S., Ullah, M.Z. & Baleanu, D. Caputo SIR model for COVID-19 under optimized fractional order. Advances In Difference Equations, 2021(1), 1-17, (2021).

Daşbaşı, B. & Öztürk, İ. Mathematical modelling of bacterial resistance to multiple antibiotics and immune system response. SpringerPlus, 5(1), 1-17, (2016).

Daşbaşı, B. Stability analysis of mathematical model including pathogen-specific immune system response with fractional-order differential equations. Computational And Mathematical Methods In Medicine, (2018).

Daşbaşı, B. Çoklu Kesirli Mertebeden Diferansiyel Denklem Sistemlerinin Kalitatif Analizi, Analizdeki Bazi Özel Durumlar ve Uygulamasi: Av-Avci Modeli. Fen Bilimleri ve Matematik’te Akademik Araştırmalar (1. B., S. 127-157). Içinde Ankara: Gece Kitaplığı, (2018).




How to Cite

Daşbaşı, B. (2021). Stability analysis of an incommensurate fractional-order SIR model. Mathematical Modelling and Numerical Simulation With Applications, 1(1), 44–55.



Research Articles