Stability and bifurcation of fractional tumor-immune model with time delay

Document Type : Research Paper


Department of Mathematical Sciences, Shahrekord University, Shahrekord, Iran.


‎The present study aims are to analyze a delay tumor-immune fractional-order system to describe the rivalry among the immune system and tumor cells. Given that the dynamics of this system depend on the time delay parameter, we examine the impact of time delay on this system to attain better compatibility with actuality. For this purpose, we analytically evaluated the stability of the system’s equilibrium points. It is shown that Hopf bifurcation occurs in the fractional system when the delay parameter passes a certain value. Finally, by using numerical simulations, the analytical results were compared to the numerical results to acquire several dynamical behaviors of this system.


[1] J. Alidousti and M. M. Ghahfarokhi, Stability and bifurcation for time delay fractional predator prey system by incorporating the dispersal of prey, Appl. Math. Modelling., 72 (2019), 385–402.
[2] J. Alidousti and R. K. Ghaziani, Spiking and bursting of a fractional order of the modified FitzHugh-Nagumo neuron model, Math. Mod. Comput. Simul., 9(3) (2017), 390–403.
[3] E. Ahmed, A. EI-Sayed, and H. A. EI-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl., 325(1) (2003), 542–553.
[4] M. Bodnar and U. Fory, Periodic dynamics in a model of immune system, Appl. Math., 27 (2000), 113–126.
[5] H. M. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144(2) (1997), 83–117.
[6] R. Caponettoi, Fractional order systems: modeling and control applications, World. Sci., (2010).
[7] A. Carvalho and C. M. Pinto, A delay fractional order model for the co-infection of malaria and HIV/AIDS, Int. J. Dynam. Control., 5(1) (2017), 168–186.
[8] K. Diethelm, The analysis of fractional differential equations, Springer, (2010).
[9] K. Diethelm, N. J. Ford, and A. D. Freed, Detailed error analysis for a fractional Adams method Analysis, Numer. algorithms., 36(1) (2004), 31–52.
[10] U. Fory, Marchuks model of immune system dynamics with application to tumor growth, Comput. Math. Method. M., 4(1) (2002), 85–93.
[11] M. Galach, Dynamics of the Tumor-Immune System Competition-the Effect of Time Delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395–406.
[12] R. Hilfer, Applications of Fractional Calculus in Physics, World. Sci. Publ. Co., Singapore, (2000).
[13] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37(3) (1998), 235–252.
[14] P. Kumar and O. P. Agrawal, An approximate method for numerical solution of fractional differential equations, Signal. Process., 86(10) (2006), 2602–2610.
[15] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor, and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56(2) (1994), 295–321.
[16] C. Li and Y. Ma, Fractional dynamical system and its linearization theorem, Nonlinear. Dynam., 71 (2013), 621–633.
[17] H. Mayer, K. S. Zaenker, and U. An Der Heiden, A basic mathematical model of the immune response, Int. J. Nonlinear. Sci., 5(1) (1995), 155–161.
[18] E. Reyes-Melo, J. Martinez-Vega, C. Guerrero-Salazar, and U. Ortiz-Mendez, Application of fractional calculus to modeling of dielectric relaxation phenomena in polymeric materials, J. Appl. Poly. Sci., 98(2) (2005), 923–935.
[19] K. M. Saad, H. M. Srivastava, and J. F. Gmez-Aguilar, A Fractional Quadratic autocatalysis associated with chemical clock reactions involving linear inhibition, Chaos. Solutions. Fractals., 132 (2020), 109–557.
[20] J. Sabatier, O.P. Agrawal, and J.A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, (2007).
[21] A. Sapora, P. Cornetti, and A. Carpinteri, Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach, Commun. Nonlinear. Sci., 18(1) (2013), 63–74.
[22] R. Yafia, Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response, Nonlinear. Anal. Real. World. Appl., 8(5) (2007), 1359–1369.
[23] R. Yafia, Hopf bifurcation in differential equations with delay for tumor-immune system competition model, SIAM. J. Appl. Math., 67(6) (2007), 1693–1703.
[24] R. Yafia, stability of limit cycle in a delayed model for tumor immune system competition with negative immune response, Discrete. Dyn. Nat. Soc., (2006).