Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions

References

[1] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor, and A. S. Perelson, “Nonlinear dynamics of immungenic tumor: parameter estimation and global bifurcation analysis,” Bull. Math. Biol., vol. 56, pp. 295–321, 1994. https://doi.org/10.1016/s0092-8240(05)80260-5.Search in Google Scholar

[2] V. A. Kuznetsov and G. D. Knott, “Modeling tumor regrowth and immunotherapy,” Math. Comput. Model., vol. 33, pp. 1275–1287, 2001. https://doi.org/10.1016/s0895-7177(00)00314-9.Search in Google Scholar

[3] N. Bellomo and L. Preziosi, “Modelling and mathematical problems related to tumor evolution and its interaction with the immune system,” Math. Comput. Model., vol. 32, pp. 413–452, 2000. https://doi.org/10.1016/s0895-7177(00)00143-6.Search in Google Scholar

[4] L. Arlotti, A. Gamba, and M. Lachowicz, “A kinetic model of tumor/immune system cellular interactions,” J. Theor. Med., vol. 4, pp. 39–50, 2002. https://doi.org/10.1080/10273660290015170.Search in Google Scholar

[5] E. De Angelis, M. Delitala, A. Marasco, and A. Romano, “Bifurcation analysis for a mean field modeling of tumor and immune system competition,” Math. Comput. Model., vol. 37, pp. 1131–1142, 2003. https://doi.org/10.1016/s0895-7177(03)00125-0.Search in Google Scholar

[6] Y. Dong, R. Miyazaki, and Y. Takeuchi, “Mathematical modeling on helper T cells in a tumor immune system,” Discrete Continuous Dyn. Syst., vol. 19, pp. 55–72, 2014. https://doi.org/10.3934/dcdsb.2014.19.55.Search in Google Scholar

[7] J. Yang, S. Y. Tang, and R. A. Cheke, “Modelling pulsed immunotherapy of tumour-immune interaction,” Math. Comput. Simulat., vol. 109, pp. 92–112, 2015. https://doi.org/10.1016/j.matcom.2014.09.001.Search in Google Scholar

[8] Y. Dong, G. Huang, R. Miyazaki, and Y. Takeuchi, “Dynamics in a tumor immune system with time delays,” Appl. Math. Comput., vol. 252, pp. 99–113, 2015. https://doi.org/10.1016/j.amc.2014.11.096.Search in Google Scholar

[9] D. Kirschner and J. C. Panetta, “Modeling immunotherapy to the tumor-immune interaction,” J. Math. Biol., vol. 37, pp. 235–252, 1998. https://doi.org/10.1007/s002850050127.Search in Google Scholar PubMed

[10] L. G. de Pillis and A. Radunskaya, “The dynamics of an optimally controled tumor model: a case study,” Math. Comput. Model., vol. 37, pp. 1221–1244, 2003. https://doi.org/10.1016/s0895-7177(03)00133-x.Search in Google Scholar

[11] U. Forys, “Marchuk’s model of immune system dynamics with application to tumor growth,” J. Theor. Med., vol. 4, pp. 85–93, 2002. https://doi.org/10.1080/10273660290052151.Search in Google Scholar

[12] Z. Szymanska, “Analysis of immunotherapy models in the context of cancer dynamics,” Appl. Math. Comput. Sci., vol. 13, pp. 407–418, 2003.Search in Google Scholar

[13] L. G. De Pillis, K. R. Fister, W. Gu, T. Head, K. Maples, T. Neal, and K. Kozai, “Optimal control of mixed immunotherapy and chemotherapy of tumors,” J. Biol. Syst., vol. 16, pp. 51–80, 2008. https://doi.org/10.1142/s0218339008002435.Search in Google Scholar

[14] S. Sabir, N. Raissi, and M. Serhani, “Chemotherapy and immunotherapy for tumors: a study of quadratic optimal control,” Int. J. Appl. Comput. Math., vol. 6, pp. 1–13, 2020. https://doi.org/10.1007/s40819-020-00838-x.Search in Google Scholar

[15] J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise, and V. Cristini, “Nonlinear modeling of cancer: bridging the gap between cells and tumors,” Nonlinearity, vol. 23, pp. R1–R9, 2010. https://doi.org/10.1088/0951-7715/23/1/r01.Search in Google Scholar PubMed PubMed Central

[16] C. Letellier, S. K. Sasmal, C. Draghi, F. Denis, and D. Ghosh, “A chemotherapy combined with an anti-angiogenic drug applied to a cancer model including angiogenesis,” Chaos, Solit. Fractals, vol. 99, pp. 297–311, 2017. https://doi.org/10.1016/j.chaos.2017.04.013.Search in Google Scholar

[17] J. Yang, Y. Tan, and R. A. Cheke, “Thresholds for extinction and proliferation in a stochastic tumor-immune model with pulsed comprehensive therapy,” Commun. Nonlinear Sci. Numer. Simulat., vol. 73, pp. 363–378, 2019. https://doi.org/10.1016/j.cnsns.2019.02.025.Search in Google Scholar

[18] J. Yang, Y. Tan, and R. A. Cheke, “Modelling effects of a chemotherapeutic dose response on a stochastic tumor-immune model,” Chaos, Solit. Fractals, vol. 123, pp. 1–13, 2019. https://doi.org/10.1016/j.chaos.2019.03.029.Search in Google Scholar

[19] W.-L. Duan, H. Fang, and C. Zeng, “The stability analysis of tumor-immune response to chemotherapy system with Gaussian white noises,” Chaos, Solit. Fractals, vol. 127, pp. 96–102, 2019. https://doi.org/10.1016/j.chaos.2019.06.030.Search in Google Scholar

[20] W.-L. Duan, “The stability analysis of tumor-immune response to chemotherapy system driven by Gaussian colored noises,” Chaos, Solit. Fractals, vol. 141, p. 110303, 2020. https://doi.org/10.1016/j.chaos.2020.110303.Search in Google Scholar

[21] M. Fahimi, K. Nouri, and L. Torkzadeh, “Chaos in a stochastic cancer model,” Physica A, vol. 454, p. 123810, 2020. https://doi.org/10.1016/j.physa.2019.123810.Search in Google Scholar

[22] T. A. Phan and J. P. Tian, “Basic stochastic model for tumor virotherapy,” Math. Biosci. Eng., vol. 17, pp. 4271–4294, 2020. https://doi.org/10.3934/mbe.2020236.Search in Google Scholar PubMed PubMed Central

[23] W.-L. Duan and L. Lin, “Noise and delay enhanced stability in tumor-immune response to chemotherapy system,” Chaos, Solit. Fractals, vol. 148, p. 111019, 2021. https://doi.org/10.1016/j.chaos.2021.111019.Search in Google Scholar

[24] I. Bashkirtseva, L. Ryashko, J. Duarte, J. M. Seoane, and M. A. F. Sanjuan, “The role of noise in the tumor dynamics under chemotherapy treatment,” Eur. Phys. J. Plus, vol. 136, p. 1123, 2021. https://doi.org/10.1140/epjp/s13360-021-02061-z.Search in Google Scholar

[25] R. A. Gatenby and E. T. Gawlinski, “A reaction-diffusion model of cancer invasion,” Cancer Res., vol. 56, pp. 5745–5753, 1996.Search in Google Scholar

[26] M. R. Owen and J. A. Sherratt, “Pattern formation and spatiotemporal irregularity in a model for macrophage-tumor interactions,” J. Theor. Biol., vol. 189, pp. 63–80, 1997. https://doi.org/10.1006/jtbi.1997.0494.Search in Google Scholar PubMed

[27] A. Matzavinos, M. A. J. Chaplain, and V. A. Kuznetsov, “Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumor,” Math. Med. Biol., vol. 21, pp. 1–34, 2004. https://doi.org/10.1093/imammb/21.1.1.Search in Google Scholar PubMed

[28] A. Matzavinos and M. A. J. Chaplain, “Travelling-wave analysis of a model of the immune response to cancer,” Comptes Rendus Biol., vol. 327, pp. 995–1008, 2004. https://doi.org/10.1016/j.crvi.2004.07.016.Search in Google Scholar PubMed

[29] I. G. Pearce, M. A. J. Chaplain, P. G. Schofield, A. R. A. Anderson, and S. F. Hubbard, “A general reaction-diffusion model of acidity in cancer invasion,” J. Theor. Biol., vol. 241, pp. 876–886, 2006. https://doi.org/10.1016/j.jtbi.2006.01.026.Search in Google Scholar PubMed

[30] B. Mukhopadhyay and R. Bhattacharyya, “Temporal and spatiotemporal variations in a mathematical model of macrophage-tumor interaction,” Nonlinear Anal. Hybrid Syst., vol. 2, pp. 819–831, 2008. https://doi.org/10.1016/j.nahs.2007.11.011.Search in Google Scholar

[31] U. Stamper, M. R. Owen, P. K. Maini, and H. M. Byrne, “Oscillatory dynamics in a model of vascular tumor growth-implications for chemotherapy,” Biol. Direct, vol. 5, pp. 1–17, 2010.10.1186/1745-6150-5-27Search in Google Scholar PubMed PubMed Central

[32] J. Malinzi, P. Sibanda, and H. Mambili-Mamboundou, “Response of immunotherapy to tumor-TICLs interactions: a traveling wave analysis,” Abstr. Appl. Anal., vol. 2014, Art no. 137015, 2014.10.1155/2014/137015Search in Google Scholar

[33] V. Bitsouni, D. Trucu, M. A. J. Chaplain, and R. Eftimie, “Aggregation and traveling wave dynamics in a two-population model of cancer cell growth and invasion,” Math. Med. Biol., vol. 35, pp. 541–577, 2018.10.1093/imammb/dqx019Search in Google Scholar PubMed

[34] P. R. Nyarko and M. Anokye, “Mathematical modeling and numerical simulation of a multiscale cancer invasion of host,” AIMS Math, vol. 5, pp. 3111–3124, 2020.10.3934/math.2020200Search in Google Scholar

[35] A. Ali, M. Hussain, A. Ghaffar, Z. Ali, K. S. Nisar, M. R. Alharthi, and W. Jamshed, “Numerical simulations and mathematical model of avascular tumor growth using Gompertz growth rate function,” Alex. Eng. J., vol. 60, pp. 3731–3740, 2021. https://doi.org/10.1016/j.aej.2021.02.040.Search in Google Scholar

[36] R. V. Craster and R. Sassi, “Spectral algorithms for reaction-diffusion equations,” in Technical Report, 2006, p. 99.Search in Google Scholar

[37] H. W. Press, B. P. Flannery, A. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, London New York, Cambridge University Press, 1986.Search in Google Scholar

[38] G. J. Lord and V. Thummler, “Commutating stochastic traveling waves,” SIAM J. Sci. Compt., vol. 34, pp. B24–B43, 2012. https://doi.org/10.1137/100784734.Search in Google Scholar