Threshold behavior of an age-structured tumor immune model

Zhonghu Luo¹ ; Zijian Liu¹ ; Yuanshun Tan¹ ; Jin Yang¹ ; Huanhuan Qiu¹

¹ College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, PR China

Mathematical modelling of natural phenomena, Tome 18 (2023), article no. 6.

Résumé

In this paper, we present and analyze an age-structured tumor immune model. Based on the fact that tumor cells of different ages tend to exhibit different physiological behaviors, we consider the age structure of tumor cells, age-based proliferation function and age-dependent death function in the model. The threshold R0 for the existence of tumor-free steady state is derived. It is found that if R0 > 1, the tumor-free steady state is not only locally stable but also globally stable. Moreover, numerical simulation shows that the threshold R0 may be regarded as an index to reflect the ability of “tumor immune surveillance”, that is, the smaller the R0, the better the ability of tumor immune surveillance. If R0 > 1, it is proved that the tumor steady state is existent and uniformly persistent. The local stability of the tumor steady state is investigated under some further conditions besides R0 > 1. In the end, we estimate the system parameters, verify the theoretical results and analyze some system parameters’ sensitivities.

Détail

DOI : 10.1051/mmnp/2023001

Bibliographie

[1] O. Arino, E. Sonchez, G.F. Webb Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence 1997 499 513

[2] B. Basse, B.C. Baguley, E.S. Marshall, W.R. Joseph, B.V. Brunt, G. Wake, D. Wall A mathematical model for analysis of the cell cycle in cell lines derived from human tumors 2003 295 312

[3] F. Billy, J. Clairambaultt, O. Fercoq, S. Gaubertt, T. Lepoutre, T. Ouillon, S. Saito Synchronisation and control of proliferation in cycling cell population models with age structure 2014 66 94

[4] F.B. Brikci, J. Clairambault, B. Ribba, B. Perthame An age-and-cyclin-structured cell population model for healthy and tumoral tissues 2008 91 110

[5] Y. Chen, J. Yang, F. Zhang The global stability of an SIRS model with infection age 2014 449 469

[6] K.J. Denise, P. Carl Modeling immunotherapy of the tumor C immune interaction 1998 235 252

[7] L. Depillis, A. Eladdadi, A. Radunskaya Modeling cancer-immune responses to therapy 2014 461 478

[8] J. Dyson, R. Villella-Bressan, G.F. Webb Asynchronous exponential growth in an age structured population of proliferating and quiescent cells 2002 73 83

[9] M. Gaach Dynamics of the tumor-immune system competition: the effect of time delay 2003 395 406

[10] P. Gabriel, S.P. Garbett, V. Quaranta, D.R. Tyson, G.F. Webb The contribution of age structure to cell population responses to targeted therapeutics 2012 19 27

[11] A. Golubev Exponentially modified Gaussian (EMG) relevance to distributions related to cell proliferation and differentiation 2010 257 266

[12] M. Gyllenberg, G.F. Webb Age-size structure in populations with quiescence 1987 67 95

[13] J.K. Hale, P. Waltman Persistence in infinite-dimensional systems 1989 388 395

[14] H. Inaba Threshold and stability for an age-structured epidemic model 1989 411 434

[15] R.J. Jang, H.C. Wei On a mathematical model of tumor-immune system interactions with an oncolytic virus therapy 2021 1559 1587

[16] V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor, A.S. Perelson Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis 1994 295 321

[17] Z. Liu, C. Guo, H. Li, L. Zhao Analysis of a nonlinear age-structured tumor cell population model 2019 283 300

[18] Z. Liu, C. Guo, J. Yang, H. Li Steady states analysis of a nonlinear age-structured tumor cell population model with quiescence and bidirectional transition 2020 455 474

[19] Z. Liu, C. Jing, J. Pang, B. Ping, S. Ruan Modeling and analysis of a nonlinear age-structured model for tumor cell populations with quiescence 2018 1763 1791

[20] P. Magal, X.Q. Zhao Global attractors and steady states for uniformly persistent dynamical systems 2005 251 275

[21] K.J. Mahasa, R. Ouifki, A. Eladdadi, L.D. Pillis Mathematical model of tumor-immune surveillance 2016 312 330

[22] Norhayati, G.C. Wake The solution and the stability of a nonlinear age-structured population model 2003 153 165

[23] S. Ostrand-Rosenberg Immune surveillance: a balance between protumor and antitumor immunity 2008 11 18

[24] J. Pang, J. Chen, Z. Liu, P. Bi, S. Ruan Local and global stabilities of a viral dynamics model with infection-age and immune response 2019 793 813

[25] L.G.D. Pillis, A.E. Radunskaya, C.L. Wiseman A validated mathematical model of cell-mediated immune response to tumor growth 2005 7950 7958

[26] S. Ruan Nonlinear dynamics in tumor-immune system interaction models with delays 2021 541 602

[27] M. Villasana, A. Radunskaya A delay differential equation model for tumor growth 2003 270 294

[28] D. Xiao, S. Ruan, Y. Yang Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function 2015 859 877

[29] H. Yang, Y. Tan, J. Yang, Z. Liu Extinction and persistence of a tumor-immune model with white noise and pulsed comprehensive therapy 2021 456 470

[30] J. Yang, Y. Tan, R.A. Cheke Modelling effects of a chemotherapeutic dose response on a stochastic tumour-immune model 2019 1 13