Deterministic and Stochastic Model for the Role of the Immune Response Time Delay in Periodic Therapy of the Tumors

Abstract

We consider the deterministic model of interaction between the immune system and tumor cells including a memory function that reflects the influence of the past states, to simulate the time needed by the latter to develop a chemical and cell mediated response to the presence of the tumor. The memory function is called delay kernel. The results are compared with those from other papers, concluding that the memory function introduces new instabilities in the system leading to an uncontrollable growth of the tumor. If the coefficient of the memory function is used as a bifurcation parameter, it is found that Hopf bifurcation occurs for kernel. The direction and stability of the bifurcating periodic solutions are determined. The deterministic model with delay allows stochastic perturbation. Mean value and square mean value of the linearized model are analyzed for the variables of the stochastic model. Some numerical simulations for justifying the theoretical analysis are also given.

Mathematical Modeling of Cell Proliferation Using Stochastic Processes

“…[KUPIEC, 2008, GEVERTZ et al, 2012 The same authors argue that the process of differentiation is also a random process, because of the internal random diffusion at the molecular level. [ZHU et al, 2012, MOHAMMADI et al, 2008, SOTTORIVA et al, 2010 Using stochastic processes as an important instrument for modeling the "order in disorder" is significant, regarding the observation of the "in vitro" tissue, not only in the regenerative medicine, [ALBANO et al, 2013, CHIŞ et al, 2011 , but also in studying the tumoral tissue and comparing normal cells to tumoral cells in terms of proliferation time. [CARAVAGNA et al, 2013, CAPASSO et al, 2009 In this paper, the process of cell proliferation will be analyzed using stochastic processes.…”

Mathematical Modeling of Cell Proliferation Using Stochastic Processes

“…[KUPIEC, 2008, GEVERTZ et al, 2012 The same authors argue that the process of differentiation is also a random process, because of the internal random diffusion at the molecular level. [ZHU et al, 2012, MOHAMMADI et al, 2008, SOTTORIVA et al, 2010 Using stochastic processes as an important instrument for modeling the "order in disorder" is significant, regarding the observation of the "in vitro" tissue, not only in the regenerative medicine, [ALBANO et al, 2013, CHIŞ et al, 2011 , but also in studying the tumoral tissue and comparing normal cells to tumoral cells in terms of proliferation time. [CARAVAGNA et al, 2013, CAPASSO et al, 2009 In this paper, the process of cell proliferation will be analyzed using stochastic processes.…”