Non-Gaussian dynamics of a tumor growth system with immunization

Mengli Hao; Ting Gao; Jinqiao Duan; Wei Xu

doi:10.3934/ipi.2013.7.697

Article Contents

2013, Volume 7, Issue 3: 697-716. Doi: 10.3934/ipi.2013.7.697

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Non-Gaussian dynamics of a tumor growth system with immunization

1.
Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710129
2.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, Los Angeles, CA 90095
3.
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616

Received: June 2012

Revised: March 2013

Published: August 2013

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Abstract

This paper is devoted to exploring the effects of non-Gaussian fluctuations on dynamical evolution of a tumor growth model with immunization, subject to non-Gaussian $\alpha$-stable type Lévy noise. The corresponding deterministic model has two meaningful states which represent the state of tumor extinction and the state of stable tumor, respectively. To characterize the time for different initial densities of tumor cells staying in the domain between these two states and the likelihood of crossing this domain, the mean exit time and the escape probability are quantified by numerically solving differential-integral equations with appropriate exterior boundary conditions. The relationships between the dynamical properties and the noise parameters are examined. It is found that in the different stages of tumor, the noise parameters have different influences on the time and the likelihood inducing tumor extinction. These results are relevant for determining efficient therapeutic regimes to induce the extinction of tumor cells.

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Mathematics Subject Classification: 60H15, 62P10, 65C50, 92C45.

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