CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT

doi:10.1007/s10473-022-0319-1

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (3): 1141-1159.doi: 10.1007/s10473-022-0319-1

CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT

黄明湛¹, 刘守宗¹, 宋新宇¹, 邹秀芬²

1. College of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China;
2. School of Mathematics and Statistics, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China

收稿日期:2020-12-18 修回日期:2021-04-11 发布日期:2022-06-24
通讯作者: Xinyu SONG,E-mail:xysong88@163.com E-mail:xysong88@163.com
基金资助:
This work was supported by the National Natural Science Foundation of China (12071407, 11901502), Training plan for young backbone teachers in Henan Province (2019GGJS157), Foundation of Henan Educational Committee under Contract (21A110022), Program for Science & Technology Innovation Talents in Universities of Henan Province (21HASTIT026), Scientific and Technological Key Projects of Henan Province (212102110025) and Nanhu Scholars Program for Young Scholars of XYNU.

CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT

Mingzhan HUANG¹, Shouzong LIU¹, Xinyu SONG¹, Xiufen ZOU²

1. College of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China;
2. School of Mathematics and Statistics, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China

Received:2020-12-18 Revised:2021-04-11 Published:2022-06-24
Contact: Xinyu SONG,E-mail:xysong88@163.com E-mail:xysong88@163.com
Supported by:
This work was supported by the National Natural Science Foundation of China (12071407, 11901502), Training plan for young backbone teachers in Henan Province (2019GGJS157), Foundation of Henan Educational Committee under Contract (21A110022), Program for Science & Technology Innovation Talents in Universities of Henan Province (21HASTIT026), Scientific and Technological Key Projects of Henan Province (212102110025) and Nanhu Scholars Program for Young Scholars of XYNU.

摘要/Abstract

摘要： This paper mainly studies the stochastic character of tumor growth in the presence of immune response and periodically pulsed chemotherapy. First, a stochastic impulsive model describing the interaction and competition among normal cells, tumor cells and immune cells under periodically pulsed chemotherapy is established. Then, sufficient conditions for the extinction, non-persistence in the mean, weak and strong persistence in the mean of tumor cells are obtained. Finally, numerical simulations are performed which not only verify the theoretical results derived but also reveal some specific features. The results show that the growth trend of tumor cells is significantly affected by the intensity of noise and the frequency and dose of drug deliveries. In clinical practice, doctors can reduce the randomness of the environment and increase the intensity of drug input to inhibit the proliferation and growth of tumor cells.

关键词: Tumor-immune system, chemotherapy, stochastic impulsive model, extinction, persistence

Abstract: This paper mainly studies the stochastic character of tumor growth in the presence of immune response and periodically pulsed chemotherapy. First, a stochastic impulsive model describing the interaction and competition among normal cells, tumor cells and immune cells under periodically pulsed chemotherapy is established. Then, sufficient conditions for the extinction, non-persistence in the mean, weak and strong persistence in the mean of tumor cells are obtained. Finally, numerical simulations are performed which not only verify the theoretical results derived but also reveal some specific features. The results show that the growth trend of tumor cells is significantly affected by the intensity of noise and the frequency and dose of drug deliveries. In clinical practice, doctors can reduce the randomness of the environment and increase the intensity of drug input to inhibit the proliferation and growth of tumor cells.

Key words: Tumor-immune system, chemotherapy, stochastic impulsive model, extinction, persistence

中图分类号:

92C50

黄明湛, 刘守宗, 宋新宇, 邹秀芬. CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT[J]. 数学物理学报(英文版), 2022, 42(3): 1141-1159.

Mingzhan HUANG, Shouzong LIU, Xinyu SONG, Xiufen ZOU. CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT[J]. Acta mathematica scientia,Series B, 2022, 42(3): 1141-1159.

参考文献

[1] Ahmedin J, Freddie B, Melissa M C, et al. Global cancer statistics. CA:A Cancer Journal for Clinicians, 2011, 61(2):69-90
[2] Freddie B, Jacques F, Isabelle S, et al. Global cancer statistics 2018:globocan estimates of incidence and mortality worldwide for 36 cancers in 185 countries. CA:A Cancer Journal for Clinicians, 2018, 68(6):394-424
[3] World Health Organization. World health statistics 2016:mMonitoring hHealth for the SDGs sustainable development goals. Geneva, Switzerland:World Health Organization, 2016
[4] Lindsey A T, Freddie B, Rebecca L S, et al. Global cancer statistics, 2012. CA:A Cancer Journal for Clinicians, 2015, 65(2):87-108
[5] Marek B, Monika J P. Stability analysis of the family of tumour angiogenesis models with distributed time delays. Commun Nonlinear Sci Numer Simul, 2016, 31(1/8):124-142
[6] Liu X D, Li Q Z, Pan J X. A deterministic and stochastic model for the system dynamics of tumor-immune responses to chemotherapy. Physica A:Statistical Mechanics and its Applications, 2018, 500:162-176
[7] Bellomo N, Preziosi L. Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math Comput Modell, 2000, 32(3/8):413-452
[8] Pillis L G D, Radunskaya A E, Wiseman C L. A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res, 2005, 65:235-252
[9] Radouane Y. Hopf bifurcation in differential equations with delay for tumor-immune system competition model. SIAM J Appl Math, 2007, 67(6):1693-1703
[10] Saleem M, Tanuja A. Chaos in a tumor growth model with delayed responses of the immune system. J Appl Math, 2012, 2012:1-16
[11] Wang S L, Wang S L, Song X Y. Hopf bifurcation analysis in a delayed oncolytic virus dynamics with continuous control. Nonlinear Dynam, 2012, 67(1):629-640
[12] Dong Y P, Rinko M, Yasuhiro T. Mathematical modeling on helper t cells in a tumor immune system. Discrete Contin Dyn Syst -Ser B, 2014, 19(1):55-72
[13] Fuat G, Senol K, Ilhan O, Fatma B. Stability and bifurcation analysis of a mathematical model for tumor cimmune interaction with piecewise constant arguments of delay. Chaos Solitons Fractals, 2014, 68:169-179
[14] Subhas K, Sandip B. Stability and bifurcation analysis of delay induced tumor immune interaction model. Appl Math Comput, 2014, 248:652-671
[15] Rihan F A, Rahman D H A, Lakshmanan S, Alkhajeh A S. A time delay model of tumour-immune system interactions:global dynamics, parameter estimation, sensitivity analysis. Appl Math Comput, 2014, 232(1):606-623
[16] Subhas K. Bifurcation analysis of a delayed mathematical model for tumor growth. Chaos Solitons Fractals, 2015, 77:264-276
[17] Dong Y P, Huang G, Rinko M, Yasuhiro T. Dynamics in a tumor immune system with time delays. Appl Math Comput, 2015, 252:99-113
[18] Pang L Y, Shen L, Zhao Z. Mathematical modelling and analysis of the tumor treatment regimens with pulsed immunotherapy and chemotherapy. Comput Math Methods Med, 2016 (2016)
[19] López A G, Seoane J M, Sanjuán M A F. Bifurcation analysis and nonlinear decay of a tumor in the presence of an immune response. Int J Bifurcat Chaos, 2017, 27(14):1750223
[20] Ansarizadeh F, Singh M, Richards D. Modelling of tumor cells regression in response to chemotherapeutic treatment. Applied Mathematical Modelling, 2017, 48:96-112
[21] López A G, Iarosz K C, Batista A M, et al. Nonlinear cancer chemotherapy:modelling the NortonSimon hypothesis. Commun Nonlinear Sci Numer Simulat, 2019, 70:307-317
[22] Lisette G D P, Radunskaya A. The dynamics of an optimally controlled tumor model:a case study. Math Comput Modelling, 2003, 37(11):1221-1244
[23] Lisette G D P, Gu W, Fister K R, et al. Chemotherapy for tumors:an analysis of the dynamics and a study of quadratic and linear optimal controls. Math Biosci, 2007, 209(1):292-315
[24] Alberto D O, Urszula L, Helmut M, Heinz S. On optimal delivery of combination therapy for tumors. Math Biosci, 2009, 222(1):13-26
[25] Mehmet I, Metin U S, Stephen P B. Optimal control of drug therapy in cancer treatment. Nonlinear Analysis:Theory. Methods & Applications, 2009, 71(12):e1473-e1486
[26] Rihan F A, Abdelrahman D H, Al-Maskari F, et al. Delay differential model for tumour-immune response with chemoimmunotherapy and optimal control. Comput Math Methods Med, 2014 (2014):1-15
[27] Pang L Y, Zhao Z, Song X Y. Cost-effectiveness analysis of optimal strategy for tumor treatment. Chaos Solitons Fractals, 2016, 87:293-301
[28] Sarkar R R, Sandip B. Cancer self remission and tumor stability-a stochastic approach. Math Biosci, 2005, 196(1):65-81
[29] Albano G, Giorno V. A stochastic model in tumor growth. J Theoret Biol, 2006, 242(2):329-336
[30] Thomas B, Steffen T. Stochastic model for tumor growth with immunization. Phys Rev E, 2009, 79(5):051903
[31] Xu Y, Feng J, Li J J, Zhang H Q. Stochastic bifurcation for a tumor-immune system with symmetric lvy noise. Physica A:Statal Mechanics and its Applications, 2013, 392(20):4739-4748
[32] Kim K S, Kim S, Jung I H. Dynamics of tumor virotherapy:a deterministic and stochastic model approach. Stoch Anal Appl, 2016, 34(3):483-495
[33] Deng Y, Liu M. Analysis of a stochastic tumor-immune model with regime switching and impulsive perturbations. Applied Mathematical Modelling, 2020, 78:482-504
[34] Bashkirtseva I, Ryashko L, Dawson K A, et al. Analysis of noise-induced phenomena in the nonlinear tumor-mmune system. Physica A:Statal Mechanics and its Applications, 2019, 549:123923
[35] Samanta G P, Ricardo G A, Sharma S. Analysis of a mathematical model of periodically pulsed chemotherapy treatment. International Journal of Dynamics and Control, 2015, 5(3):842-857
[36] Samanta G P, Sen P, Maiti A. A delayed epidemic model of diseases through droplet infection and direct contact with saturation incidence and pulse vaccination. Systems Science and Control Engineering, 2016, 4(1):320-333
[37] Samanta G P, Ricardo G A. Analysis of a delayed epidemic model of diseases through droplet infection and direct contact with pulse vaccination. International Journal of Dynamics and Control, 2015, 3(3):275-287
[38] Samanta G P. Mathematical Analysis of a Chlamydia Epidemic Model with Pulse Vaccination Strategy. Acta Biotheoretica, 2015, 63(1):1-21
[39] Samanta G P, Sharma S. Analysis of a delayed Chlamydia epidemic model with pulse vaccination. Applied Mathematics and Computation, 2014, 230:555-569
[40] Samanta G P. Analysis of a delayed epidemic model with pulse vaccination. Chaos, Solitons and Fractals, 2014, 66:74-85
[41] Samanta G P, Bera S P. Analysis of a Chlamydia epidemic model with pulse vaccination strategy in a random environment. Nonlinear Analysis:Modelling and Control, 2018, 23(4):457-474
[42] Jing Y, Mei L Q, Song X Y, Tian W J, Ding X M. Analysis of an impulsive epidemic model with time delays and nonlinear incidence rate. Acta Mathematica Scientia, 2012, 32A(4):670-684
[43] Ling L, Liu S Y, Jiang G R. Bifurcation analysis of a SIRS epidemic model with saturating contact rate and vertical transmission. Acta Mathematica Scientia, 2014, 34A(6):1415-1425
[44] Ma Z E, Cui G R, Wang W D. Persistence and extinction of a population in a polluted environment. Math Biosci, 1990, 101:75-97
[45] Ma Z E, Hallam T G. Effects of parameter fluctuations on community survival. Math Biosci, 1987, 86(1):35-49
[46] Liu M, Wang K. Persistence and extinction in stochastic non-autonomous logistic systems. J Math Anal Appl, 2011, 375(2):443-57
[47] Li D X, Cheng F J. Threshold for extinction and survival in stochastic tumor immune system. Communications in Nonlinear ence & Numerical Simulations, 2017, 51(OCT):1-12
[48] Lan G J, Ye C, Zhang S W, Wei C J. Dynamics of a stochastic glucose-insulin model with impulsive injection of insulin. Commun Math Biol Neurosci, 2020, 2020:6
[49] Kloeden P E, Platen E. Numerical solution of stochastic differential equations. New York:Springer-Verlag, 1992
[50] Shochat E, Hart D, Agur Z. Using computer simulations for evaluating the efficacy of breast cancer chemotherapy protocols. Mathematical Models and Methods in Applied Sciences, 1999, 9(4):599-615

CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT

CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT

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