Asymmetric non-Gaussian effects in a tumor growth model with immunization

Abstract The dynamical evolution of a tumor growth model, under immune surveillance and subject to asymmetric non-Gaussian α -stable Levy noise, is explored. The lifetime of a tumor staying in the range between the tumor-free state and the stable tumor state, and the likelihood of noise-induced tumor extinction, are characterized by the mean residence time and the escape probability, respectively. For various initial densities of tumor cells, the mean residence time and the escape probability are computed with different noise parameters. It is observed that unlike the Gaussian noise or symmetric non-Gaussian noise, the asymmetric non-Gaussian noise plays a constructive role in the tumor evolution in this simple model. By adjusting the noise parameters, the mean residence time can be shortened and the escape probability can be increased, simultaneously. This suggests that a tumor may be mitigated with higher probability in a shorter time, under certain external environmental stimuli.

[1]  A. Ochab-Marcinek,et al.  Population growth and control in stochastic models of cancer development , 2004, q-bio/0403041.

[2]  Peter Tankov,et al.  Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes , 2007 .

[3]  Sergio Albeverio,et al.  Invariant Measures and Symmetry Property of Lévy Type Operators , 2000 .

[4]  Xiaofan Li,et al.  A computational analysis for mean exit time under non-Gaussian Lévy noises , 2011, Appl. Math. Comput..

[5]  O. Barndorff-Nielsen,et al.  Lévy processes : theory and applications , 2001 .

[6]  A. Weron,et al.  Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes , 1993 .

[7]  Zhenhui He,et al.  Pure multiplicative stochastic resonance of a theoretical anti-tumor model with seasonal modulability. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Werner Horsthemke,et al.  Noise-induced transitions , 1984 .

[9]  C. Zeng,et al.  Cross-correlation enhanced stability in a tumor cell growth model with immune surveillance driven by cross-correlated noises , 2009 .

[10]  D. Kirschner,et al.  Modeling immunotherapy of the tumor – immune interaction , 1998, Journal of mathematical biology.

[11]  Stationary states in Langevin dynamics under asymmetric Lévy noises. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  A. Kyprianou.  Introductory Lectures on Fluctuations of Lévy Processes with Applications , 2006 .

[13]  P. Lee,et al.  14. Simulation and Chaotic Behaviour of α‐Stable Stochastic Processes , 1995 .

[14]  Chunhua Zeng,et al.  Colored Noise Enhanced Stability in a Tumor Cell Growth System Under Immune Response , 2010 .

[15]  B. Ai,et al.  Fractional Gaussian noise-induced evolution and transition in anti-tumor model , 2012 .

[16]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[17]  Steffen Trimper,et al.  Stochastic model for tumor growth with immunization. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  John P. Nolan,et al.  Parameterizations and modes of stable distributions , 1998 .

[19]  Liao Ming,et al.  The Dirichlet problem of a discontinuous Markov process , 1989 .

[20]  W Horsthemke,et al.  Bistability in fluctuating environments. Implications in tumor immunology. , 1979, Bulletin of mathematical biology.

[21]  Simone Cifani,et al.  On numerical methods and error estimates for degenerate fractional convection–diffusion equations , 2012, Numerische Mathematik.

[22]  Bernardo Spagnolo,et al.  Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Ting Gao,et al.  Mean Exit Time and Escape Probability for a Tumor Growth System under Non-Gaussian noise , 2011, Int. J. Bifurc. Chaos.

[24]  Jinqiao Duan,et al.  Escape Probability for Stochastic Dynamical Systems with Jumps , 2011, 1111.6419.

[25]  Huiqing Zhang,et al.  Stochastic bifurcation for a tumor–immune system with symmetric Lévy noise , 2013 .

[26]  R. Lefever,et al.  A kinetic approach to the immunology of cancer: stationary states properties of effector-target cell reactions. , 1978, Journal of theoretical biology.

[27]  B. Spagnolo,et al.  Monitoring noise-resonant effects in cancer growth influenced by external fluctuations and periodic treatment , 2007, 0710.1317.

[28]  Ting Gao,et al.  Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noises , 2012, SIAM J. Sci. Comput..

[29]  Meng Chen,et al.  External noise control in inherently stochastic biological systems. , 2012, Journal of mathematical physics.

[30]  Xiaoqin Luo,et al.  STOCHASTIC PROPERTIES OF TUMOR GROWTH DRIVEN BY WHITE LÉVY NOISE , 2012 .

[31]  H. Breusers,et al.  Applied Mathematical Modelling , 1976 .

[32]  Peter Imkeller,et al.  Limit Theorems for p-Variations of Solutions of SDEs Driven by Additive Stable Lévy Noise and Model Selection for Paleo-Climatic Data , 2010 .

[33]  R. Evans.  Environmental control and immunotherapy for allergic disease. , 1992, The Journal of allergy and clinical immunology.

[34]  Yong Xu,et al.  Fluctuations induced extinction and stochastic resonance effect in a model of tumor growth with periodic treatment , 2011 .

[35]  René Lefever,et al.  Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology , 2007 .

[36]  J. Huston McCulloch,et al.  Measuring Tail Thickness to Estimate the Stable Index α: A Critique , 1997 .