Effects of Lévy noise and immune delay on the extinction behavior in a tumor growth model

doi:10.1088/1674-1056/23/9/090501

Abstract The combined effects of Lévy noise and immune delay on the extinction behavior in a tumor growth model are explored. The extinction probability of tumor with certain density is measured by exit probability. The expression of the exit probability is obtained using the Taylor expansion and the infinitesimal generator theory. Based on numerical calculations, it is found that the immune delay facilitates tumor extinction when the stability index α<1, but inhibits tumor extinction when the stability index α>1. Moreover, larger stability index and smaller noise intensity are in favor of the extinction for tumor with low density. While for tumor with high density, the stability index and the noise intensity should be reduced to promote tumor extinction.

Keywords: exit probability Lé vy noise immune delay tumor growth model

Received: 17 January 2014
Revised: 09 March 2014
Accepted manuscript online:

PACS:	05.40.Fb	(Random walks and Levy flights)
	82.39.-k	(Chemical kinetics in biological systems)
	02.50.-r	(Probability theory, stochastic processes, and statistics)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11172233, 11272258, and 11302170).

Corresponding Authors: Xu Wei
E-mail: weixu@nwpu.edu.cn

Cite this article:

Hao Meng-Li (郝孟丽), Xu Wei (徐伟), Gu Xu-Dong (谷旭东), Qi Lu-Yuan (戚鲁媛) Effects of Lévy noise and immune delay on the extinction behavior in a tumor growth model 2014 Chin. Phys. B 23 090501

[1]	Burnet M 1957 Br. Med. J. 1 841
[2]	Kirschner D and Panett J C 1998 J. Math. Biol. 37 235
[3]	Dunn G P, Old L J and Schreiber R D 2004 Annu. Rev. Immunol. 22 329
[4]	Garay R P and Lefever R 1978 J. Theor. Biol. 73 417
[5]	Li F G and Ai B Q 2012 Eur. Phys. J. B 85 74
[6]	Fiasconaro A and Spagnolo B 2006 Phys. Rev. E 74 041904
[7]	Li D X, Xu W, Guo Y F and Xu Y 2011 Phys. Lett. A 375 886
[8]	Li D X, Xu W, Sun C Y and Wang L 2012 Phys. Lett. A 376 1771
[9]	Fiasconaro A, Ochab-Marcinek A, Spagnolo B and Gudowska-Nowak E 2008 Eur. Phys. J. B 65 435
[10]	Applebaum D 2009 Lévy Processes and Stochastic Calculus (UK: Cambridge University Press)
[11]	Lü Y and Bao J D 2011 Phys. Rev. E 84 051108
[12]	Lü Y and Bao J D 2011 Chin. Phys. Lett. 28 110201
[13]	Bao J D, Wang H Y, Jia Y and Zhuo Y Z 2005 Phys. Rev. E 72 051105
[14]	Liu J and Bao J D 2013 Chin. Phys. Lett. 30 020202
[15]	Humphries N E et al. 2010 Nature 465 1066
[16]	Bartumeus F, Catalan J, Fulco U, Lyra M L and Viswanathan G M 2002 Phys. Rev. Lett. 88 097901
[17]	Xu Y, Feng J, Li J J and Zhang H Q 2013 Chaos 23 013110
[18]	Ren J, Li C J, Gao T, Kan X Y and Duan J Q 2012 Int. J. Bifurcat. Chaos 22 1250090
[19]	Jiang L L, Luo X Q, Wu D and Zhu S Q 2012 Chin. Phys. B 21 090503
[20]	Jiang L L, Luo X Q, Wu D and Zhu S Q 2012 Modern Phys. Lett. B 26 1250149
[21]	Xu Y, Feng J, Li J J and Zhang H Q 2013 Physica A 392 4739
[22]	Tian J and Chen Y 2010 Chin. Phys. Lett. 27 030502
[23]	Jia Z L 2010 Chin. Phys. B 19 020504
[24]	Saleem M and Agrawal T 2012 J. Appl. Math. 16 891095
[25]	d'Onofrio A, Gatti F, Cerrai P and Freschi L 2010 Mathematical and Computer Modelling 51 572
[26]	Caravagna G, Graudenzi A, Antoniotti M, Mauri G and d'Onofrio A 2012 EPTCS 92 106
[27]	Li D X 2012 Stochastic Dynamics Research on Typical Tumor Immune System (MS Thesis) (Xi'an: Northwestern Polytechnical University) (in Chinese)
[28]	Qiao H J, Kan X Y and Duan J Q 2013 Springer Proceedings in Mathematics & Statistics 34 195
[29]	Chen H Q, Duan J Q, Li X F and Zhang C J 2011 Appl. Math. Comput. 218 1845
[30]	Buchmann F M and Petersen W P 2006 SIAM J. Sci. Comput. 28 1153
[31]	Guillouzic S, L'Heureux I and Longtin A 1999 Phys. Rev. E 59 3970
[32]	Milošević M and Jovanović M 2011 J. Comput. Appl. Math. 235 4439
[33]	Huang J M, Tao W M and Xu B H 2011 J. Phys. A: Math. Theor. 44 385101
[34]	Sun X and Duan J Q 2012 J. Math. Phys. 53 072701
[35]	Gao T, Duan J Q, Li X F and Song R M 2012 arXiv:1201.6015v1[math.NA]

[1]	Dynamics and near-optimal control in a stochastic rumor propagation model incorporating media coverage and Lévy noise Liang'an Huo(霍良安) and Yafang Dong(董雅芳). Chin. Phys. B, 2022, 31(3): 030202.
[2]	Dynamics of a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise Liang-An Huo(霍良安), Ya-Fang Dong(董雅芳), and Ting-Ting Lin(林婷婷). Chin. Phys. B, 2021, 30(8): 080201.
[3]	Analysis of weak signal detection based on tri-stable system under Levy noise Li-Fang He(贺利芳), Ying-Ying Cui(崔莹莹), Tian-Qi Zhang(张天骐), Gang Zhang(张刚), Ying Song(宋莹). Chin. Phys. B, 2016, 25(6): 060501.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Effects of Lévy noise and immune delay on the extinction behavior in a tumor growth model

Cite this article:

Online attention