Stochastic tumor-immune interaction model with external treatments and time delays: An optimal control problem

H. J. Alsakaji; F. A. Rihan; K. Udhayakumar; F. El Ktaibi; H. J. Alsakaji; F. A. Rihan; K. Udhayakumar; F. El Ktaibi

doi:10.3934/mbe.2023852

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 11: 19270-19299. doi: 10.3934/mbe.2023852

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Stochastic tumor-immune interaction model with external treatments and time delays: An optimal control problem

1.
Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al-Ain 15551, UAE
2.
College of Natural and Health Sciences, Zayed University, Abu Dhabi 144534, UAE

Academic Editor: Yang Kuang

Received: 20 June 2023 Revised: 02 October 2023 Accepted: 08 October 2023 Published: 17 October 2023

Herein, we discuss an optimal control problem (OC-P) of a stochastic delay differential model to describe the dynamics of tumor-immune interactions under stochastic white noises and external treatments. The required criteria for the existence of an ergodic stationary distribution and possible extinction of tumors are obtained through Lyapunov functional theory. A stochastic optimality system is developed to reduce tumor cells using some control variables. The study found that combining white noises and time delays greatly affected the dynamics of the tumor-immune interaction model. Based on numerical results, it can be shown which variables are optimal for controlling tumor growth and which controls are effective for reducing tumor growth. With some conditions, white noise reduces tumor cell growth in the optimality problem. Some numerical simulations are conducted to validate the main results.
- tumor-immure interactions,
- optimal control,
- stochastic noise,
- stationary distribution,
- time-delays
Citation: H. J. Alsakaji, F. A. Rihan, K. Udhayakumar, F. El Ktaibi. Stochastic tumor-immune interaction model with external treatments and time delays: An optimal control problem[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 19270-19299. doi: 10.3934/mbe.2023852

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Abstract

Herein, we discuss an optimal control problem (OC-P) of a stochastic delay differential model to describe the dynamics of tumor-immune interactions under stochastic white noises and external treatments. The required criteria for the existence of an ergodic stationary distribution and possible extinction of tumors are obtained through Lyapunov functional theory. A stochastic optimality system is developed to reduce tumor cells using some control variables. The study found that combining white noises and time delays greatly affected the dynamics of the tumor-immune interaction model. Based on numerical results, it can be shown which variables are optimal for controlling tumor growth and which controls are effective for reducing tumor growth. With some conditions, white noise reduces tumor cell growth in the optimality problem. Some numerical simulations are conducted to validate the main results.

References

[1]	World Health Organization, Cancer, 2022. Available from: https://www.who.int/news-room/fact-sheets/detail/cancer
[2]	A. Desai, T. Mohammed, N. Duma, M. Garassino, L. Hicks, N. Kuderer, et al., COVID-19 and cancer: A review of the registry-based pandemic response, JAMA Oncol., 7 (2021), 1882–1890. https://doi.org/10.1001/jamaoncol.2021.4083 doi: 10.1001/jamaoncol.2021.4083
[3]	K. Dehingia, H. Sarmah, Y. Alharbi, K. Hosseini, Mathematical analysis of a cancer model with time-delay in tumor-immune interaction and stimulation processes, Adv. Differ. Equation, 2021 (2021), 1–27. https://doi.org/10.1186/s13662-021-03621-4 doi: 10.1186/s13662-021-03621-4
[4]	F. A. Rihan, K. Udhayakumar, Fractional order delay differential model of a tumor-immune system with vaccine efficacy: Stability, bifurcation and control, Chaos Solitons Fractals, 173 (2023) 113670. https://doi.org/10.1016/j.chaos.2023.113670 doi: 10.1016/j.chaos.2023.113670
[5]	F. A. Rihan, G. Velmurugan, Dynamics of fractional-order delay differential model for tumor-immune system, Chaos Solitons Fractals, 132 (2020), 109592. https://doi.org/10.1016/j.chaos.2019.109592 doi: 10.1016/j.chaos.2019.109592
[6]	V. Bitsouni, V. Tsilidis, Mathematical modeling of tumor-immune system interactions: The effect of rituximab on breast cancer immune response, J. Theor. Biol., 539 (2022), 111001. https://doi.org/10.1016/j.jtbi.2021.111001 doi: 10.1016/j.jtbi.2021.111001
[7]	M. Itik, S. Banks, Chaos in a three-dimensional cancer model, Int. J. Bifurcat. Chaos, 20 (2010), 71–79. https://doi.org/10.1142/S0218127410025417 doi: 10.1142/S0218127410025417
[8]	R. Yafia, A study of differential equation modeling malignant tumor cells in competition with immune system, Int. J. Biomath., 4 (2011), 185–206. https://doi.org/10.1142/S1793524511001404 doi: 10.1142/S1793524511001404
[9]	Y. Radouane, Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response, Discrete Dyn. Nat. Soc., 2006 (2006), 095296. https://doi.org/10.1155/DDNS/2006/95296 doi: 10.1155/DDNS/2006/95296
[10]	F. Najm, R. Yafia, M. A. Aziz-Alaoui, Hopf bifurcation in oncolytic therapeutic modeling: Viruses as anti-tumor means with viral lytic cycle, Int. J. Bifurcat. Chaos, 32 (2022), 2250171. https://doi.org/10.1142/S0218127422501711 doi: 10.1142/S0218127422501711
[11]	R. Brady, H. Enderling, Mathematical models of cancer: When to predict novel therapies, and when not to, Bull. Math. Biol., 81 (2019), 3722–3731. https://doi.org/10.1007/s11538-019-00640-x doi: 10.1007/s11538-019-00640-x
[12]	T. Phan, S. Crook, A. Bryce, C. Maley, E. Kostelich, Y. Kuang, Mathematical modeling of prostate cancer and clinical application, Appl. Sci., 10 (2020), 2721. https://www.mdpi.com/2076-3417/10/8/2721
[13]	O. Nave, Adding features from the mathematical model of breast cancer to predict the tumour size, Int. J. Comput. Math. Comput. Syst. Theory, 5 (2020), 159–174. https://doi.org/10.1080/23799927.2020.1792552 doi: 10.1080/23799927.2020.1792552
[14]	D. Kirschner, J. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235–252. https://doi.org/10.1007/s002850050127 doi: 10.1007/s002850050127
[15]	V. Kuznetsov, L. Makalkin, M. Taylor, A. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295–321. https://doi.org/10.1016/S0092-8240(05)80260-5 doi: 10.1016/S0092-8240(05)80260-5
[16]	A. Omame, C. Nnanna, S. Inyama, Optimal control and cost-effectiveness analysis of an HPV-chlamydia trachomatis co-infection model, Acta Biotheor., 69 (2021), 185–223. 10.1007/s10441-020-09401-z doi: 10.1007/s10441-020-09401-z
[17]	U. Ijeoma, S. Inyama, A. Omame, Mathematical model and optimal control of new-castle disease (ND), Appl. Math. Comput., 9 (2020), 70–84. doi: 10.11648/j.acm.20200903.14 doi: 10.11648/j.acm.20200903.14
[18]	F. A. Rihan, S. Lakshmanan, H. Maurer, Optimal control of tumor-immune model with time-delay and immuno-chemotherapy, Appl. Math. Comput., 353 (2019), 147–165. https://doi.org/10.1016/j.amc.2019.02.002 doi: 10.1016/j.amc.2019.02.002
[19]	F. A. Rihan, H. J. Alsakaji, S. Kundu, O. Mohamed, Dynamics of a time-delay differential model for tumor-immune interactions with random noise, Alex. Eng. J., 61 (2022), 11913–11923. https://doi.org/10.1016/j.aej.2022.05.027 doi: 10.1016/j.aej.2022.05.027
[20]	M. Yu, Y. Dong, Y. Takeuchi, Dual role of delay effects in a tumour–immune system, J. Biol. Dyn., 11 (2017), 334–347. https://doi.org/10.1080/17513758.2016.1231347 doi: 10.1080/17513758.2016.1231347
[21]	X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. https://doi.org/10.1016/B978-1-904275-34-3.50014-1
[22]	M. Baar, L. Coquille, H. Mayer, M. Hölzel, M. Rogava, T. Tüting, et al., A stochastic model for immunotherapy of cancer, Sci. Rep., 6 (2016), 1–10. https://doi.org/10.1038/srep24169 doi: 10.1038/srep24169
[23]	L. Han, C. He, Y. Kuang, Dynamics of a model of tumor-immune interaction with time delay and noise, DCDS-S, 13(2020). http://dx.doi.org/10.3934/dcdss.2020140 doi: 10.3934/dcdss.2020140
[24]	H. J. Alsakaji, F. A. Rihan, A. Hashish, Dynamics of a stochastic epidemic model with vaccination and multiple time-delays for COVID-19 in the UAE, Complexity, 2022 (2022), 1–15. https://doi.org/10.1155/2022/4247800 doi: 10.1155/2022/4247800
[25]	C. Odoux, H. Fohrer, T. Hoppo, L. Guzik, D. Stolz, D. Lewis, et al., A stochastic model for cancer stem cell origin in metastatic colon cancer, Cancer Res., 68 (2008), 6932–6941. https://doi.org/10.1158/0008-5472.CAN-07-5779 doi: 10.1158/0008-5472.CAN-07-5779
[26]	Y. Deng, M. Liu, Analysis of a stochastic tumor-immune model with regime switching and impulsive perturbations, Appl. Math. Model., 78 (2020), 482–504. https://doi.org/10.1016/j.apm.2019.10.010 doi: 10.1016/j.apm.2019.10.010
[27]	A. Raza, J. Awrejcewicz, M. Rafiq, N. Ahmed, M. Mohsin, Stochastic analysis of nonlinear cancer disease model through virotherapy and computational methods, Mathematics, 10 (2022), 368. https://doi.org/10.3390/math10030368 doi: 10.3390/math10030368
[28]	K. Dehingia, H. Sarmah, K. Hosseini, K. Sadri, S. Salahshour, C. Park, An optimal control problem of immuno-chemotherapy in presence of gene therapy, AIMS Math., 6 (2021), 11530–11549. https://doi.org/10.3934/math.2021669 doi: 10.3934/math.2021669
[29]	F. A. Rihan, Delay Differential Equations and Applications to Biology, Springer, 2021. https://doi.org/10.1007/978-981-16-0626-7
[30]	C. Orrieri, E. Rocca, L. Scarpa, Optimal control of stochastic phase-field models related to tumor growth, ESAIM Control Optim. Calc. Var., 26 (2020), 104. https://doi.org/10.1051/cocv/2020022 doi: 10.1051/cocv/2020022
[31]	M. Huang, S. Liu, X. Song, X. Zou, Control strategies for a tumor-immune system with impulsive drug delivery under a random environment, Acta Math. Sci., 42 (2022), 1141–1159. https://doi.org/10.1007/s10473-022-0319-1 doi: 10.1007/s10473-022-0319-1
[32]	L. J. Allen, An Introduction to Stochastic Processes with Applications to Biology, CRC press, 2010. https://doi.org/10.1201/b12537
[33]	F. Rihan, H. Alsakaji, Persistence and extinction for stochastic delay differential model of prey predator system with hunting cooperation in predators, Adv. Differ. Equations, 2020 (2020), 1–22 https://doi.org/10.1186/s13662-020-02579-z doi: 10.1186/s13662-020-02579-z
[34]	X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, World Scientific, 2006. https://doi.org/10.1142/p473
[35]	R. Hasminskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. https://doi.org/10.1007/978-3-642-23280-0
[36]	S. Rajasekar, M. Pitchaimani, Qualitative analysis of stochastically perturbed SIRS epidemic model with two viruses, Chaos Solitons Fractals, 118 (2019), 207–221. https://doi.org/10.1016/j.chaos.2018.11.023 doi: 10.1016/j.chaos.2018.11.023
[37]	E. Beretta, V. Kolmanovskii, L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simul., 45 (1998), 269–277. https://doi.org/10.1016/S0378-4754(97)00106-7 doi: 10.1016/S0378-4754(97)00106-7
[38]	M. Kinnally, Stationary Distributions for Stochastic Delay Differential Equations with Non-negativity Constraints, University of California, San Diego, 2009.
[39]	G. Milstein, Numerical Integration of Stochastic Differential Equations, Springer Science & Business Media, 1994. https://doi.org/10.1007/978-94-015-8455-5
[40]	Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69–84. https://doi.org/10.1016/j.jmaa.2006.12.032 doi: 10.1016/j.jmaa.2006.12.032
[41]	Q. An, E. Beretta, Y. Kuang, C. Wang, H. Wang, Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters, J. Differ. Equation, 266 (2019), 7073–7100. https://doi.org/10.1016/j.jde.2018.11.025 doi: 10.1016/j.jde.2018.11.025
[42]	Q. Sun, M. Xiao, M. B. Tao, Local bifurcation analysis of a fractional-order dynamic model of genetic regulatory networks with delays, Neural Process. Lett., 47 (2018), 1285–1296. https://doi.org/10.1007/s11063-017-9690-7 doi: 10.1007/s11063-017-9690-7
[43]	L. Li, Z. Wang, Y. Li, H. Shen, J. Lu, Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays, Appl. Math. Comput., 330 (2018), 152–169. https://doi.org/10.1016/j.amc.2018.02.029 doi: 10.1016/j.amc.2018.02.029
[44]	C. Xu, M. Liao, P. Li, Y. Guo, Q. Xiao, S. Yuan, Influence of multiple time delays on bifurcation of fractional-order neural networks, Appl. Math. Comput., 361 (2019), 565–582. https://doi.org/10.1016/j.amc.2019.05.057 doi: 10.1016/j.amc.2019.05.057

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