Advanced Search
Article Contents
Article Contents

Gompertz model with delays and treatment: Mathematical analysis

Abstract Related Papers Cited by
  • In this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy. We mainly consider two types of delayed models, the one with the delay introduced in the per capita growth rate (we call it the single delayed model) and the other with the delay introduced in the net growth rate (the double delayed model). We focus on stability and possible stability switches with increasing delay for the positive steady state. Moreover, we study a Hopf bifurcation, including stability of arising periodic solutions for a constant treatment. The analytical results are extended by numerical simulations for a pharmacokinetic treatment function.
    Mathematics Subject Classification: Primary: 34K11, 34K13, 34K18, 34K20, 34K28, 37N25; Secondary: 92B05, 92B25, 92C50.


    \begin{equation} \\ \end{equation}
  • [1]

    M. Bodnar, The nonnegativity of the solutions of delay differential equations, Appl. Math. Lett., 13 (2000), 91-95.doi: 10.1016/S0893-9659(00)00061-6.


    M. Bodnar and U. Foryś, Three types of simple DDE's describing tumor growth, J. Biol. Sys., 15 (2007), 1-19.doi: 10.1142/S0218339007002313.


    O. Diekmann, S. van Giles and S.M.V. Lunel, "Delay Equations," Springer-Verlag, New York, 1995.doi: 10.1007/978-1-4612-4206-2.


    A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999), Math. Biosci., 191 (2004), 159-184.doi: 10.1016/j.mbs.2004.06.003.


    A. d'Onofrio and A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Math. Med. Biol., 26 (2009), 63-95.doi: 10.1093/imammb/dqn024.


    A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosci., 222 (2009), 13-26.doi: 10.1016/j.mbs.2009.08.004.


    U. Foryś, J. Poleszczuk and T. LiuLogistic tumor growth with delay and impulsive treatment, Accepted for Math. Pop. Studies.


    G. Gompertz, On the nature of the function expressive of the law of human mortality, and on the new mode of determining the value of life contingencies, Philos. Trans. R. Soc. London, 115 (1825), 513-585.


    P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Res., 59 (1999), 4770-4775.


    J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations," Springer, New York, 1993.


    G. E. Hutchinson, Circular casual systems in ecology, Ann. N. Y. Acad. Sci., 50 (1948), 221-246.


    U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control Optim., 46 (2007), 1052-1079.doi: 10.1137/060665294.


    U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. Theor. Biol., 252 (2008), 295-312.doi: 10.1016/j.jtbi.2008.02.014.


    J. D. Murray, "Mathematical Biology: {I.} An Introduction," Springer, Berlin-Heidelberg, 2007.


    M. J. Piotrowska, M. Bodnar and U. ForyśLogistic equation with treatment function and discrete delays, (submitted).


    M. J. Piotrowska and U. Foryś, Analysis of the Hopf bifurcation for the family of angiogenesis models, J. Math. Anal. Appl., 382 (2011), 180-203.doi: 10.1016/j.jmaa.2011.04.046.


    M. J. Piotrowska and U. Foryś, The nature of Hopf bifurcation for the Gompertz model with delays, Math. and Comp. Modelling, 54 (2011), 2183-2198.doi: 10.1016/j.mcm.2011.05.027.


    J. Poleszczuk, M. Bodnar and U. Foryś, New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model, Math. Biosci. Eng., 8 (2011), 591-603.doi: 10.3934/mbe.2011.8.591.


    R. Schuster and H. Schuster, Reconstruction models for the Ehrlich Ascites tumor for the mouse, in "Mathematical Population Dynamics" (eds. O. Arino, D. Axelrod and M. Kimmel), Wuertz, Winnipeg, Canada, (1995), 335-348.

  • 加载中

Article Metrics

HTML views() PDF downloads(74) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint