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

Oana      Chis; Mihaela      Neamtu; Dumitru      Opris

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.

Keywords: Tumor dynamics, growth rate, delay kernels, stochastic differential equation with delay, eigenvalue, eigenvector, Hopf bifurcation, local stability, local asymptotic stability, bifurcation parameter, bifurcation direction