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

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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.

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

Document Type: Research Article

Publication date: December 1, 2011

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  • Current Computer-Aided Drug Design aims to publish all the latest developments in drug design based on computational techniques. The field of computer-aided drug design has had extensive impact in the area of drug design. Current Computer-Aided Drug Design is an essential journal for all medicinal chemists who wish to be kept informed and up-to-date with all the latest and important developments in computer-aided methodologies and their applications in drug discovery. Each issue contains a series of timely, in-depth reviews written by leaders in the field, covering a range of computational techniques for drug design, screening, ADME studies, etc., providing excellent rationales for drug development.
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