The effect of a psychological scare on the dynamics of the tumor-immune interaction with optimal control strategy

Rafel Ibrahim  Salih; Shireen Jawad; Kaushik  Dehingia; Anusmita Das

doi:10.11121/ijocta.1520

Authors

Rafel Ibrahim Salih Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq https://orcid.org/0009-0004-4706-6979
Shireen Jawad Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq https://orcid.org/0000-0002-3090-8357
Kaushik Dehingia $Department of Mathematics, Sonari College, Sonari 785690, Assam, India https://orcid.org/0000-0002-8042-4166
Anusmita Das Department of Mathematics, Udalguri College, Udalguri 784509, Assam, India https://orcid.org/0000-0003-4022-8053

DOI:

https://doi.org/10.11121/ijocta.1520

Keywords:

Immune system, Local stability, Global stability, Bifurcation analysis, Numerical simulation

Abstract

Contracting cancer typically induces a state of terror among the individuals who are affected. Exploring how chemotherapy and anxiety work together to affect the speed at which cancer cells multiply and the immune system’s response model is necessary to come up with ways to stop the spread of cancer. This paper proposes a mathematical model to investigate the impact of psychological scare and chemotherapy on the interaction of cancer and immunity. The proposed model is accurately described. The focus of the model’s dynamic analysis is to identify the potential equilibrium locations. According to the analysis, it is possible to establish three equilibrium positions. The stability analysis reveals that all equilibrium points consistently exhibit stability under the defined conditions. The bifurcations occurring at the equilibrium sites are derived. Specifically, we obtained transcritical, pitchfork, and saddle-node bifurcation. Numerical simulations are employed to validate the theoretical study and ascertain the minimum therapy dosage necessary for eradicating cancer in the presence of psychological distress, thereby mitigating harm to patients. Fear could be a significant contributor to the spread of tumors and weakness of immune functionality.

Downloads

Download data is not yet available.

Author Biographies

Rafel Ibrahim Salih, Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq

Rafel Ibrahim Salih Bachelor in applied science in department of mathematics, university of technology, Teacher at ministry of education - Second Karkh Education Directorate - Al-Suwaib Girls Secondary School, Teacher at ministry of education - Second Karkh Education Directorate - Al-Nafi Secondary School, Student M.Sc in applied mathematics science at university of Baghdad from 2022/9/4 and present.

Shireen Jawad, Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq

Shireen Jawad is a faculty member of the University of Baghdad, College of Science. She graduated from the University of Baghdad, College of Science, Department of Mathematics, in 2005. Then she received her PhD in applied mathematics-dynamical systems from Brunel University, London. Her research interest is mathematical modeling and analysis of dynamical systems.

Kaushik Dehingia, $Department of Mathematics, Sonari College, Sonari 785690, Assam, India

Kaushik Dehingia obtained his PhD in Mathematical Biology and Dynamical Systems from Gauhati University, Guwahati, India, and MSc from Tezpur University, Tezpur, India. Currently, he is working as an Assistant Professor at Sonari College, Sonari, India. His research interests are in the areas of mathematical modeling, dynamic systems, mathematical biology, and nonlinear dynamics.

Anusmita Das, Department of Mathematics, Udalguri College, Udalguri 784509, Assam, India

Anusmita Das received her PhD in Mathematical Biology and Dynamical Systems from Gauhati University, Guwahati, India, and MSc from Gauhati University, Guwahati, India. Currently, she is working as an Assistant Professor at Udalguri College, Udalguri, India. Her research interests are in the areas of mathematical modeling, dynamic systems, and mathematical biology.

References

Gershenfeld, N. A. (1999). The nature of mathematical modeling. Cambridge university press, Cambridge, United Kingdom.

Thirthar, A. A. (2023). A mathematical modelling of a plant-herbivore community with additional effects of food on the environment. Iraqi Journal of Science, 64(7), 3551-3566.

Murray, J. D. (2002). Models for Interacting Populations. In: J. D. Murray, ed., Mathematical Biology: I. An Introduction. Springer, New York, 79-118. https://doi.org/10.1007/978-0-387-22437-4_3

Murray, J. D. (2002). Continuous Population Models for Single Species. In: J. D. Murray, eds., Mathematical Biology: I. An Introduction. Springer, New York, 1-43. https://doi.org/10.1007/978-0-387-22437-4_1

Shalan, R. N., Shireen, R., & Lafta, A. H. (2021). Discrete an SIS model with immigrants and treatment. Journal of Interdisciplinary Mathematics, 24(5), 1201-1206. https://doi.org/10.1080/09720502.2020.1814496

Sk, N., Mondal, B., Thirthar, A. A., Alqudah, M. A., & Abdeljawad, T. (2023). Bistability and tristability in a deterministic prey-predator model: Transitions and emergent patterns in its stochastic counterpart. Chaos, Solitons & Fractals, 176, 114073. https://doi.org/10.1016/j.chaos.2023.114073

Chatterjee, A., & Pal, S. (2023). A predator-prey model for the optimal control of fish harvesting through the imposition of a tax. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 13(1), 68-80. https://doi.org/10.11121/ijocta.2023.1218

Sene, N. (2022). Theory and applications of new fractional-order chaotic system under Caputo operator. An International Journal of Optimization and Control, 12(1), 20-38. https://doi.org/10.11121/ijocta.2022.1108

Hoang, M. T., Ngo, T. K. Q., & Truong, H. H. (2023). A simple method for studying asymptotic stability of discrete dynamical systems and its applications. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 13(1), 10-25. https://doi.org/10.11121/ijocta.2023.1243

Courchamp, F., Berec, L., & Gascoigne, J. (2008). Allee effects in ecology and conservation. OUP Oxford, Oxford, England. https://doi.org/10.1093/acprof:oso/9780198570301.001.0001

Allee, W. C., & Bowen, E. S. (1932). Studies in animal aggregations: mass protection against colloidal silver among goldfishes. Journal of Experimental Zoology, 61(2), 185-207. https://doi.org/10.1002/jez.1400610202

Gomez-Llano, M., Germain, R. M., Kyogoku, D., McPeek, M. A., & Siepielski, A. M. (2021). When ecology fails: how reproductive interactions promote species coexistence. Trends in Ecology & Evolution, 36(7), 610-622. https://doi.org/10.1016/j.tree.2021.03.003

Jawad, S., Sultan, D., & Winter, M. (2021). The dynamics of a modified Holling-Tanner prey-predator model with wind effect. International Journal of Nonlinear Analysis and Applications, 12(Special Issue), 2203-2210.

Al Nuaimi, M., & Jawad, S. (2022). Modelling and stability analysis of the competitional ecological model with harvesting. Communications in Mathematical Biology and Neuroscience, 2022, 1-29.

Hassan, S. K., & Jawad, S. R. (2022). The Effect of Mutual Interaction and Harvesting on Food Chain Model. Iraqi Journal of Science, 63(6), 2641-2649. https://doi.org/10.24996/ijs.2022.63.6.29

Dawud, S., & Jawad, S. (2022). Stability analysis of a competitive ecological system in a polluted environment. Communications in Mathematical Biology and Neuroscience, 2022, 1-34.

Hollingsworth, T. D. (2009). Controlling infectious disease outbreaks: Lessons from mathematical modelling. Journal of public health policy, 30, 328-341. https://doi.org/10.1057/jphp.2009.13

White, P. J., & Enright, M. C. (2010). Mathematical models in infectious disease epidemiology. Infectious Diseases, 70-75. https://doi.org/10.1016/B978-0-323-04579-7.00005-8

Huppert, A., & Katriel, G. (2013). Mathematical modelling and prediction in infectious disease epidemiology. Clinical microbiology and infection, 19(11), 999-1005. https://doi.org/10.1111/1469-0691.12308

Kareem, A. M., & Al-Azzawi, S. N. (2021). A stochastic differential equations model for the spread of coronavirus COVID-19): the case of Iraq. Iraqi Journal of Science, 63(3), 1025-1035. https://doi.org/10.24996/ijs.2021.62.3.31

Hameed, H. H., & Al-Saedi, H. M. (2021). Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function. Baghdad Science Journal, 18(2), 0296-0296. https://doi.org/10.21123/bsj.2021.18.2.0296

Kareem, A. M., & Al-Azzawi, S. N. (2022). Comparison Between Deterministic and Stochastic Model for Interaction (COVID-19) With Host Cells in Humans. Baghdad Science Journal, 19(5), 1140-1140. https://doi.org/10.21123/bsj.2022.6111

Kirschner, D., & Panetta, J. C. (1998). Modeling immunotherapy of the tumor-immune interaction. Journal of mathematical biology, 37, 235-252. https://doi.org/10.1007/s002850050127

Frascoli, F., Kim, P. S., Hughes, B. D., & Landman, K. A. (2014). A dynamical model of tumour immunotherapy. Mathematical biosciences, 253, 50-62. https://doi.org/10.1016/j.mbs.2014.04.003

Villasana, M., & Radunskaya, A. (2003). A delay differential equation model for tumor growth. Journal of mathematical biology, 47, 270-294. https://doi.org/10.1007/s00285-003-0211-0

Huang, M., Liu, S., Song, X., & Zou, X. (2022). Control strategies for a tumor-immune system with impulsive drug delivery under a random environment. Acta Mathematica Scientia, 42(3), 1141-1159. https://doi.org/10.1007/s10473-022-0319-1

Saeed, T., Djeddi, K., Guirao, J. L., Alsulami, H. H., & Alhodaly, M. S. (2022). A discrete dynamics approach to a tumor system. Mathematics, 10(10), 1774. https://doi.org/10.3390/math10101774

Iarosz, K. C., Borges, F. S., Batista, A. M., Baptista, M. S., Siqueira, R. A., Viana, R. L., & Lopes, S. R. (2015). Mathematical model of brain tumour with glia-neuron interactions and chemotherapy treatment. Journal of theoretical biology, 368, 113-121. https://doi.org/10.1016/j.jtbi.2015.01.006

Colli, P., Gilardi, G., & Sprekels, J. (2019). A distributed control problem for a fractional tumor growth model. Mathematics, 7(9), 792. https://doi.org/10.3390/math7090792

Alharbi, S. A., & Rambely, A. S. (2020). A new ODE-based model for tumor cells and immune system competition. Mathematics, 8(8), 1285. https://doi.org/10.3390/math8081285

Ghanbari, B. (2020). On the modeling of the interaction between tumor growth and the immune system using some new fractional and fractional-fractal operators. Advances in Difference Equations, 2020(1), 1-32. https://doi.org/10.1186/s13662-020-03040-x

Arshad, S., Yildiz, T. A., Baleanu, D., & Tang, Y. (2020). The role of obesity in fractional order tumor-immune model. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys, 82(2), 181-196.

Akman Yildiz, T., Arshad, S., & Baleanu, D. (2018). Optimal chemotherapy and immunotherapy schedules for a cancer-obesity model with Caputo time fractional derivative. Mathematical Methods in the Applied Sciences, 41(18), 9390-9407. https://doi.org/10.1002/mma.5298

Alqudah, M. A. (2020). Cancer treatment by stem cells and chemotherapy as a mathematical model with numerical simulations. Alexandria Engineering Journal, 59(4), 1953-1957. https://doi.org/10.1016/j.aej.2019.12.025

Jawad, S., Winter, M., Rahman, Z. A. S., Al-Yasir, Y. I., & Zeb, A. (2023). Dynamical behavior of a cancer growth model with chemotherapy and boosting of the immune system. Mathematics, 11(2), 406. https://doi.org/10.3390/math11020406

Letellier, C., Sasmal, S. K., Draghi, C., Denis, F., & Ghosh, D. (2017). A chemotherapy combined with an anti-angiogenic drug applied to a cancer model including angiogenesis. Chaos, Solitons & Fractals, 99, 297-311. https://doi.org/10.1016/j.chaos.2017.04.013

De Pillis, L. G., & Radunskaya, A. (2001). A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. Computational and Mathematical Methods in Medicine, 3(2), 79-100. https://doi.org/10.1080/10273660108833067

De Pillis, L. G., Gu, W., & Radunskaya, A. E. (2006). Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations. Journal of theoretical biology, 238(4), 841-862. https://doi.org/10.1016/j.jtbi.2005.06.037

Suraci, J. P., Clinchy, M., Dill, L. M., Roberts, D., & Zanette, L. Y. (2016). Fear of large carnivores causes a trophic cascade. Nature communications, 7(1), 10698. https://doi.org/10.1038/ncomms10698

Pal, S., Pal, N., Samanta, S., & Chattopadhyay, J. (2019). Effect of hunting cooperation and fear in a predator-prey model. Ecological Complexity, 39, 100770. https://doi.org/10.1016/j.ecocom.2019.100770

Sarkar, K., & Khajanchi, S. (2020). Impact of fear effect on the growth of prey in a predator-prey interaction model. Ecological Complexity, 42, 100826. https://doi.org/10.1016/j.ecocom.2020.100826

He, M., & Li, Z. (2022). Stability of a fear effect predator-prey model with mutual interference or group defense. Journal of Biological Dynamics, 16(1), 480-498. https://doi.org/10.1080/17513758.2022.2091800

Yousef, A., Thirthar, A. A., Alaoui, A. L., Panja, P., & Abdeljawad, T. (2022). The hunting cooperation of a predator under two prey’s competition and fear-effect in the prey-predator fractional-order model. AIMS Math, 7(4), 5463-5479. https://doi.org/10.3934/math.2022303

Thirthar, A. A., Abboubakar, H., Khan, A., & Abdeljawad, T. (2023). Mathematical modeling of the COVID-19 epidemic with fear impact. AIMS Math, 8(3), 6447-6465. https://doi.org/10.3934/math.2023326

Doshi, D., Karunakar, P., Sukhabogi, J. R., Prasanna, J. S., & Mahajan, S. V. (2021). Assessing coronavirus fear in Indian population using the fear of COVID-19 scale. International journal of mental health and addiction, 19, 2383-2391. https://doi.org/10.1007/s11469-020-00332-x

Gormley, M., Knobf, M. T., Vorderstrasse, A., Aouizerat, B., Hammer, M., Fletcher, J., & D’Eramo Melkus, G. (2021). Exploring the effects of genomic testing on fear of cancer recurrence among breast cancer survivors. Psycho-Oncology, 30(8), 1322-1331. https://doi.org/10.1002/pon.5679

Niknamian, S. (2019). The impact of stress, anxiety, fear and depression in the cause of cancer in humans. American Journal of Biomedical Science and Research, 3(4), 363-370. https://doi.org/10.34297/AJBSR.2019.03.000696

Epstein, J. M., Parker, J., Cummings, D., & Hammond, R. A. (2008). Coupled contagion dynamics of fear and disease: mathematical and computational explorations. PloS one, 3(12), e3955. https://doi.org/10.1371/journal.pone.0003955

Vrinten, C., McGregor, L. M., Heinrich, M., von Wagner, C., Waller, J., Wardle, J., & Black, G. B. (2017). What do people fear about cancer? A systematic review and meta-synthesis of cancer fears in the general population. Psycho-oncology, 26(8), 1070-1079. https://doi.org/10.1002/pon.4287

Lebel, S., Tomei, C., Feldstain, A., Beattie, S., & McCallum, M. (2013). Does fear of cancer recurrence predict cancer survivors’ health care use?. Supportive Care in Cancer, 21, 901-906. https://doi.org/10.1007/s00520-012-1685-3

De Pillis, L. G., & Radunskaya, A. (2003). - A mathematical model of immune response to tumor invasion. In: K. J. Bathe, ed., Computational Fluid and Solid Mechanics 2003. Elsevier Science Ltd, 1661-1668. https://doi.org/10.1016/B978-008044046-0.50404-8

Wang, X., Zanette, L., & Zou, X. (2016). Modelling the fear effect in predator-prey interactions. Journal of mathematical biology, 73(5), 1179-1204. https://doi.org/10.1007/s00285-016-0989-1

Das, A., Dehingia, K., Ray, N., & Sarmah, H. K. (2023). Stability analysis of a targeted chemotherapy-cancer model. Mathematical Modelling and Control, 3(2), 116-126. https://doi.org/10.3934/mmc.2023011

Hubbard, J. H., & West, B. H. (2012). Differential equations: a dynamical systems approach: higher-dimensional systems. Vol. 18. Springer Science & Business Media, New York.

Perko, L. (2013). Differential equations and dynamical systems. Vol. 7. Springer Science & Business Media, New York.

Hirsch, M. W., Smale, S., & Devaney, R. L. (2012). Differential equations, dynamical systems, and an introduction to chaos. Academic press, New York. https://doi.org/10.1016/B978-0-12-382010-5.00015-4

Place, C. M. (2017). Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour. Routledge, London.

Jawad, S. R., & Al Nuaimi, M. (2023). Persistence and bifurcation analysis among four species interactions with the influence of competition, predation and harvesting. Iraqi Journal of Science, 64(3) 1369-1390. https://doi.org/10.24996/ijs.2023.64.3.30

Jawad, S., & Hassan, S. K. (2023). Bifurcation analysis of commensalism intraction and harvisting on food chain model. Brazilian Journal of Biometrics, 41(3), 218-233. https://doi.org/10.28951/bjb.v41i3.609

Lukes, D. L. (1969). Optimal regulation of nonlinear dynamical systems. SIAM Journal on Control, 7(1), 75-100. https://doi.org/10.1137/0307007

Kopp, R. E. (1962). Pontryagin Maximum Principle. In: G. Leitmann, ed., Mathematics in Science and Engineering. Elsevier, 255-279. https://doi.org/10.1016/S0076-5392(08)62095-0