Abstract
This research explores a delay differential model to describe the dynamics of tumour-immune interactions in presence of immuno-chemotherapy. The model includes a constant delay in the recruitment term of the immune cells to illustrate the time lag between the stimulated accumulations of immune cells in the vicinity of cancer cells. The efficiency of solutions and the boudness have been supported by the findings. Also, the conditions for local stability and the existence of Hopf bifurcation are investigated. In particular, sufficient conditions dependent on the delay parameter under which the interior equilibrium is asymptotically stable are constructed. Afterwards, we established the length of delay to preserve stability. The numerical simulations show that the combination of immuno-chemotherapy protocol reduces the tumor load in a few months of therapy. The numerical simulations are presented to illustrate our theoretical results and show that the blend of immuno-chemotherapy convention decreases the tumour load in a couple of long periods of treatment.
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Banerjee, S., Sarkar, R.R.: Delay-induced model for tumor-immune interaction and control of malignant tumor growth. Biosystems 91(1), 268–288 (2008)
Bi, P., Ruan, S.: Bifurcations in delay differential equations and applications to tumor and immune system interaction models. SIAM Journal on Applied Dynamical Systems 12(4), 1847–1888 (2013)
Burić, N., Todorović, D.: Dynamics of delay-differential equations modelling immunology of tumor growth. Chaos, Solitons & Fractals 13(4), 645–655 (2002)
Byrne, H.: The effect of time delays on the dynamics of avascular tumor growth. Mathematical biosciences 144(2), 83–117 (1997)
Chen, Y., Cheng, J., Jiang, Y., Liu, K.: A time delay dynamic system with external source for the local outbreak of 2019-ncov. Applicable Analysis pp. 1–12 (2020)
Cooke, K.L., Van Den Driessche, P.: On zeroes of some transcendental equations. Funkcialaj Ekvacioj 29(1), 77–90 (1986)
DeJesus, E.X., Kaufman, C.: Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations. Physical Review A 35(12), 5288 (1987)
Freedman, H., Rao, V.S.H.: The trade-off between mutual interference and time lags in predator-prey systems. Bulletin of Mathematical Biology 45(6), 991–1004 (1983)
Gałach, M.: Dynamics of the tumor–immune system competition–the effect of time delay. International Journal of Applied Mathematics and Computer Science 13, 395–406 (2003)
Geng, C., Paganetti, H., Grassberger, C.: Prediction of treatment response for combined chemo-and radiation therapy for non-small cell lung cancer patients using a bio-mathematical model. Scientific reports 7(1), 1–12 (2017)
Ghosh, D., Khajanchi, S., Mangiarotti, S., Denis, F., Dana, S.K., Letellier, C.: How tumor growth can be influenced by delayed interactions between cancer cells and the microenvironment? BioSystems 158, 17–30 (2017)
of Health, U.N.I., et al.: Cancer trends progress report–2011/2012 update. Bethesda: National Cancer Institute 19 (2012)
Kamke, E.: A new proof of sturm’s comparison theorms. The American Mathematical Monthly 46(7), 417–421 (1939)
Khajanchi, S., Banerjee, S.: Stability and bifurcation analysis of delay induced tumor immune interaction model. Applied Mathematics and Computation 248, 652–671 (2014)
Khajanchi, S., Banerjee, S.: Influence of multiple delays in brain tumor and immune system interaction with t11 target structure as a potent stimulator. Mathematical biosciences 302, 116–130 (2018)
Khajanchi, S., Nieto, J.J.: Mathematical modeling of tumor-immune competitive system, considering the role of time delay. Applied Mathematics and Computation 340, 180–205 (2019)
Kirschner, D., Panetta, J.C.: Modeling immunotherapy of the tumor-immune interaction. Journal of mathematical biology 37(3), 235–252 (1998)
Kuang, Y.: Delay differential equations: with applications in population dynamics, vol. 191. Academic press (1993)
Kuznetsov, V.A., Makalkin, I.A., Taylor, M.A., Perelson, A.S.: Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bulletin of mathematical biology 56(2), 295–321 (1994)
Lai, X., Friedman, A.: Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: A mathematical model. PLoS One 12(5), e0178479 (2017)
Mahaffy, J.: A test for stability of linear differential delay equations. Quarterly of Applied Mathematics 40(2), 193–202 (1982)
Mahlbacher, G.E., Reihmer, K.C., Frieboes, H.B.: Mathematical modeling of tumor-immune cell interactions. Journal of Theoretical Biology 469, 47–60 (2019)
Moghtadaei, M., Golpayegani, M.R.H., Malekzadeh, R.: Periodic and chaotic dynamics in a map-based model of tumor-immune interaction. Journal of theoretical biology 334, 130–140 (2013)
Moiola, J.L., Chen, G.: Hopf bifurcation analysis: a frequency domain approach, vol. 15. World Scientific (1996)
Neves, H., Kwok, H.F.: Recent advances in the field of anti-cancer immunotherapy. BBA clinical 3, 280–288 (2015)
Nyquist, H.: Regeneration theory. Bell system technical journal 11(1), 126–147 (1932)
Piotrowska, M.: A remark on the ode with two discrete delays. Journal of Mathematical Analysis and Applications 329(1), 664–676 (2007)
Rihan, F.A., Abdelrahman, D., Al-Maskari, F., Ibrahim, F., Abdeen, M.A.: Delay differential model for tumour-immune response with chemoimmunotherapy and optimal control. Computational and mathematical methods in medicine 2014, (2014)
Rodrigues, D.S., Mancera, P.F., Carvalho, T., Gonçalves, L.F.: A mathematical model for chemoimmunotherapy of chronic lymphocytic leukemia. Applied Mathematics and Computation 349, 118–133 (2019)
SWAN, G.W.: Optimal control applications in the chemotherapy of multiple myeloma. Mathematical Medicine and Biology: A Journal of the IMA 2(3), 139–160 (1985)
Villasana, M., Radunskaya, A.: A delay differential equation model for tumor growth. Journal of Mathematical Biology 47(3), 270–294 (2003)
Xu, C., Li, P., Liao, M., Yuan, S.: Bifurcation analysis for a fractional-order chemotherapy model with two different delays. Mathematical Methods in the Applied Sciences 43(3), 1053–1083 (2020)
Yafia, R.: Hopf bifurcation in differential equations with delay for tumor-immune system competition model. SIAM Journal on Applied Mathematics 67(6), 1693–1703 (2007)
Yang, X., Chen, L., Chen, J.: Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models. Computers & Mathematics with Applications 32(4), 109–116 (1996)
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Cherraf, A., Li, M. & Moulai-Khatir, A. Interaction tumor-immune model with time-delay and immuno-chemotherapy protocol. Rend. Circ. Mat. Palermo, II. Ser 72, 869–887 (2023). https://doi.org/10.1007/s12215-021-00615-9
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DOI: https://doi.org/10.1007/s12215-021-00615-9