Khajanchi Subhas, Perc Matjaž, Ghosh Dibakar
Department of Mathematics, Presidency University, Kolkata 700073, India.
Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia.
Chaos. 2018 Oct;28(10):103101. doi: 10.1063/1.5052496.
The tumor-immune interactive dynamics is an evergreen subject that continues to draw attention from applied mathematicians and oncologists, especially so due to the unpredictable growth of tumor cells. In this respect, mathematical modeling promises insights that might help us to better understand this harmful aspect of our biology. With this goal, we here present and study a mathematical model that describes how tumor cells evolve and survive the brief encounter with the immune system, mediated by effector cells and host cells. We focus on the distribution of eigenvalues of the resulting ordinary differential equations, the local stability of the biologically feasible singular points, and the existence of Hopf bifurcations, whereby the time lag is used as the bifurcation parameter. We estimate analytically the length of the time delay to preserve the stability of the period-1 limit cycle, which arises at the Hopf bifurcation point. We also perform numerical simulations, which reveal the rich dynamics of the studied system. We show that the delayed model exhibits periodic oscillations as well as chaotic behavior, which are often indicators of long-term tumor relapse.
肿瘤 - 免疫相互作用动力学是一个常青的课题，一直吸引着应用数学家和肿瘤学家的关注，尤其是由于肿瘤细胞的不可预测生长。在这方面，数学建模有望提供一些见解，帮助我们更好地理解生物学中这一有害的方面。出于这个目标，我们在此提出并研究一个数学模型，该模型描述了肿瘤细胞如何在效应细胞和宿主细胞介导的情况下，在与免疫系统的短暂接触中进化和存活。我们关注所得常微分方程的特征值分布、生物学可行奇点的局部稳定性以及霍普夫分岔的存在性，其中时间延迟被用作分岔参数。我们通过解析估计时间延迟的长度，以保持在霍普夫分岔点出现的1周期极限环的稳定性。我们还进行了数值模拟，揭示了所研究系统丰富的动力学特性。我们表明，延迟模型表现出周期性振荡以及混沌行为，这些通常是长期肿瘤复发的指标。
