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Stability of a mathematical model with piecewise constant arguments for tumor-immune interaction under drug therapy. (English) Zbl 1415.34128

Summary: This paper studies a mathematical model for the interaction between tumor cells and Cytotoxic T lymphocytes (CTLs) under drug therapy. We obtain some sufficient conditions for the local and global asymptotical stabilities of the system by using Schur-Cohn criterion and the theory of Lyapunov function. In addition, it is known that the system without any treatment may undergo Neimark-Sacker bifurcation, and there may exist a chaotic region of values of tumor growth rate where the system exhibits chaotic behavior. So it is important to narrow the chaotic region. This may be done by increasing the intensity of the treatment to some extent. Moreover, for a fixed value of tumor growth rate in the chaotic region, a threshold value \(\gamma_0\) is predicted of the treatment parameter \(\gamma\). We can see Neimark-Sacker bifurcation of the system when \(\gamma=\gamma_0\), and the chaotic behavior for tumor cells ends and the system becomes locally asymptotically stable when \(\gamma>\gamma_0\).

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
92C37 Cell biology
34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
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[1] Banerjee, S. & Sarkar, R. [2008] “ Delay-induced model for tumor-immune interaction and control of malignant tumor growth,” Biosystems91, 268-288.
[2] Bozkurt, F. [2013] “ Modeling a tumor growth with piecewise constant arguments,” Discr. Dyn. Nat. Soc.2013, 841764-1-8. · Zbl 1264.34094
[3] Cooke, K. & Huang, W. [1991] “ A theorem of George seifert and an equation with state-dependent delay,” Delay and Differential Equations, eds. Fink, A. M., Miller, R. K. & Kliemann, W. (World Scientific, Singapore). · Zbl 0820.34042
[4] Costa, O., Molina, L., Perez, D., Antoranz, J. & Reyes, M. [2003] “ Behavior of tumors under nonstationary therapy,” Physica D178, 242-253. · Zbl 1011.92028
[5] Cull, P. [1981] “ Global stability of population models,” Bull. Math. Biol.43, 47-58. · Zbl 0451.92011
[6] Feng, Z., Li, F. & Liu, J. [2018] “ Notes on a boundary value problem with a periodic nonlinearity,” Optik-Int. J. Light Electron Opt.156, 439-446.
[7] Firmani, B., Guerri, L. & Preziosi, L. [1999] “ Tumor/immune system competition with medically induced activation/deactivation,” Math. Model. Meth. Appl. Sci.9, 491-512. · Zbl 0934.92016
[8] Galach, M. [2003] “ Dynamics of the tumor-immune system competition — The effect of time delay,” Int. J. Appl. Math. Comput. Sci.13, 395-406. · Zbl 1035.92019
[9] Gatenby, R. [1995] “ Models of tumor-host interaction as competing populations: Implications for tumor biology and treatment,” J. Theoret. Biol.176, 447-455.
[10] Gopalsamy, K. [1992] Stability and Oscillation in Delay Differential Equations of Population Dynamics (Kluwer Academic Publishers, Dodrecht, The Netherlands). · Zbl 0752.34039
[11] Gopalsamy, K. & Liu, P. [1998] “ Persistence and global stability in a population model,” J. Math. Anal. Appl.224, 59-80. · Zbl 0912.34040
[12] Gurcan, F. & Bozkurt, F. [2009] “ Global stability in a population model with piecewise constant arguments,” J. Math. Anal. Appl.360, 334-342. · Zbl 1177.34097
[13] Gurcan, F., Kartal, S., Ozturk, I. & Bozkurt, F. [2014] “ Stability and bifurcation analysis of a mathematical model for tumor-immune interaction with piecewise constant arguments of delay,” Chaos Solit. Fract.68, 169-179. · Zbl 1354.92037
[14] Gyori, I. & Ladas, G. [1991] Oscillation Theory of Delay Differential Equations with Applications (Clarendon Press, Oxford). · Zbl 0780.34048
[15] Han, M., Hou, X., Sheng, L. & Wang, C. [2018a] “ Theory of rotated equations and applications to a population model,” Discr. Contin. Dyn. Syst.38, 2171-2185. · Zbl 1396.37026
[16] Han, M., Sheng, L. & Zhang, X. [2018b] “ Bifurcation theory for finitely smooth planar autonomous differential systems,” J. Diff. Eqs.264, 3596-3618. · Zbl 1410.34116
[17] Han, M., Tian, H., Xu, B. & Bai, Y. [2018c] “ On the number of periodic solutions of delay differential equations,” Int. J. Bifurcation and Chaos28, 1850051-1-10. · Zbl 1391.34115
[18] Kirschner, D. & Panetta, J. [1998] “ Modeling immunotherapy of the tumor-immune interaction,” J. Math. Biol.37, 235-252. · Zbl 0902.92012
[19] Kocic, V. & Ladas, G. [1993] Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (Kluwer Academic Publishers, Dordrecht, Boston, London). · Zbl 0787.39001
[20] Kuznetsov, V., Makalkin, I., Taylor, M. & Perelson, A. [1994] “ Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,” Bull. Math. Biol.56, 295-321. · Zbl 0789.92019
[21] Li, X., Mou, C., Niu, W. & Wang, D. [2011] “ Stability analysis for discrete biological models using algebraic methods,” Math. Comput. Sci.5, 247-262. · Zbl 1270.92023
[22] Li, J., Oprocha, P. & Wu, X. [2017] “ Furstenberg families, sensitivity and the space of probability measures,” Nonlinearity30, 987-1005. · Zbl 1420.54059
[23] Li, M. & Wang, J. [2018] “ Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations,” Appl. Math. Comput.324, 254-265. · Zbl 1426.34110
[24] Liu, P. & Gopalsamy, K. [1999] “ Global stability and chaos in a population model with piecewise constant arguments,” Appl. Math. Comput.101, 63-88. · Zbl 0954.92020
[25] Liu, L., Sun, F., Zhang, X. & Wu, Y. [2017] “ Bifurcation analysis for a singular differential system with two parameters via to topological degree theory,” Nonlin. Anal.22, 31-50. · Zbl 1420.34048
[26] Liu, W., Cui, J. & Xin, J. [2018] “ A block-centered finite difference method for an unsteady asymptotic coupled model in fractured media aquifer system,” J. Comput. Appl. Math.337, 319-340. · Zbl 1524.76460
[27] May, R. [1975] “ Biological populations obeying difference equations: Stable points, stable cycles and chaos,” J. Theoret. Biol.51, 511-524.
[28] May, R. & Oster, G. [1976] “ Bifurcations and dynamics complexity in simple ecological models,” Amer. Nat.110, 573-599.
[29] Muroya, Y. [2002] “ Persistence, contractivity and global stability in logistic equations with piecewise constant delays,” J. Math. Anal. Appl.270, 602-635. · Zbl 1012.34076
[30] Onofrio, A. [2005] “ A general framework for modeling tumor-immune system competition and immunotherapy, mathematical analysis and biomedical inferences,” Physica D208, 220-235. · Zbl 1087.34028
[31] Onofrio, A. [2008] “ Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy,” Math. Comput. Model.47, 614-637. · Zbl 1148.92026
[32] Oprocha, P. & Wu, X. [2017] “ On average tracing of periodic average pseudo orbits,” Discr. Contin. Dyn. Syst.37, 4943-4957. · Zbl 1369.37031
[33] Ozturk, I. & Bozkurt, F. [2011] “ Stability analysis of a population model with piecewise constant arguments,” Nonlin. Anal.12, 1532-1545. · Zbl 1402.34076
[34] Ozturk, I., Bozkurt, F. & Gurcan, F. [2012] “ Stability analysis of a mathematical model in a microcosm with piecewise constant arguments,” Math. Biosci.240, 85-91. · Zbl 1316.92071
[35] Pillis, L. & Radunskaya, A. [2001] “ A mathematical tumor model with immune resistance and drug therapy: An optimal control approach,” J. Theor. Med.3, 79-100. · Zbl 0985.92023
[36] Sarkar, R. & Banerjee, S. [2006] “ A time delay model for control of malignant tumor growth,” Third Natl. Conf. Nonlinear Systems and Dynamic.
[37] Tian, H. & Han, M. [2017] “ Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems,” J. Diff. Eqs.263, 7448-7474. · Zbl 1419.37052
[38] Uesugi, K., Muroya, Y. & Ishiwata, E. [2004] “ On the global attractivity for a logistic equation with piecewise constant arguments,” J. Math. Anal. Appl.294, 560-580. · Zbl 1050.34116
[39] Villasana, M. & Radunskaya, A. [2003] “ A delay differential equation model for tumor growth,” J. Math. Biol.47, 270-294. · Zbl 1023.92014
[40] Wu, X. [2016] “ Chaos of transformations induced onto the space of probability measures,” Int. J. Bifurcation and Chaos26, 1650227-1-12. · Zbl 1354.37016
[41] Yafia, R. [2006] “ Stability of limit cycle in a delayed model for tumor immune system competition with negative immune response,” Discr. Dyn. Nat. Soc.2006, 58463-1-13. · Zbl 1106.92042
[42] Yu, P., Han, M. & Li, J. [2018a] “ An improvement on the number of limit cycles bifurcating from a nondegenerate center of homogeneous polynomial systems,” Int. J. Bifurcation and Chaos28, 1850078-1-31. · Zbl 1394.34075
[43] Yu, X., Wang, Q. & Bai, Y. [2018b] “ Permanence and almost periodic solutions for N-species non-autonomous Lotka-Volterra competitive systems with delays and impulsive perturbations on time scales,” Complexity2018, 2658745. · Zbl 1407.34098
[44] Zhang, J. & Wang, J. [2018] “ Numerical analysis for Navier-Stokes equations with time fractional derivatives,” Appl. Math. Comput.336, 481-489. · Zbl 1427.76177
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