Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland.
Math Biosci Eng. 2013 Jun;10(3):551-63. doi: 10.3934/mbe.2013.10.551.
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.
本文研究了延迟的 Gompertz 模型，作为肿瘤生长的典型模型，其中包含一个描述外部干扰的项，可以反映治疗，例如化疗。我们主要考虑了两种延迟模型，一种是在人均增长率中引入延迟（我们称之为单延迟模型），另一种是在净增长率中引入延迟（双延迟模型）。我们关注正平衡点随延迟增加的稳定性和可能的稳定性转变。此外，我们研究了 Hopf 分支，包括对恒定治疗的出现的周期解的稳定性。通过数值模拟扩展了药物代谢动力学治疗函数的分析结果。
