×

Global dynamics in a tumor-immune model with an immune checkpoint inhibitor. (English) Zbl 1469.34067

Summary: In this paper, we fill several key gaps in the study of the global dynamics of a highly nonlinear tumor-immune model with an immune checkpoint inhibitor proposed by Nikolopoulou et al. (Letters in Biomathematics, 5 (2018), S137-S159). For this tumour-immune interaction model, it is known that the model has a unique tumour-free equilibrium and at most two tumorous equilibria. We present sufficient and necessary conditions for the global stability of the tumour-free equilibrium or the unique tumorous equilibrium. The global dynamics is obtained by employing a new Dulac function to establish the nonexistence of nontrivial positive periodic orbits. Our analysis shows that we can almost completely classify the global dynamics of the model with two critical values \(C_{K0},C_{K1} (C_{K0} > C_{K1})\) for the carrying capacity \(C_K\) of tumour cells and one critical value \(d_{T0}\) for the death rate \(d_T\) of T cells. Specifically, the following are true. (i) When no tumorous equilibrium exists, the tumour-free equilibrium is globally asymptotically stable. (ii) When \(C_K \leq C_{K1}\) and \(d_T > d_{T0}\), the unique tumorous equilibrium is globally asymptotically stable. (iii) When \(C_K >C_{K1}\), the model exhibits saddle-node bifurcation of tumorous equilibria. In this case, we show that when a unique tumorous equilibrium exists, tumor cells can persist for all positive initial densities, or can be eliminated for some initial densities and persist for other initial densities. When two distinct tumorous equilibria exist, we show that the model exhibits bistable phenomenon, and tumor cells have alternative fates depending on the positive initial densities. (iv) When \(C_K > C_{K0}\) and \(d_T=d_{T0}\), or \(d_T> d_{T0}\), tumor cells will persist for all positive initial densities.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92C37 Cell biology
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. E. Alexander; S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189, 75-96 (2004) · Zbl 1073.92040 · doi:10.1016/j.mbs.2004.01.003
[2] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17, 257-278 (1955) · doi:10.1007/BF02477753
[3] J. Hamanishi; M. Mandai; N. Matsumura; K. Abiko; T. Baba; I. Konishi, PD-1/PD-L1 blockade in cancer treatment: Perspectives and issues, Int. J. Clin. Oncol., 21, 462-473 (2016) · doi:10.1007/s10147-016-0959-z
[4] S.-B. Hsu; T. W. Hwang; Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42, 489-506 (2001) · Zbl 0984.92035 · doi:10.1007/s002850100079
[5] J. Huang; Y. Gong; S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18, 2101-2121 (2013) · Zbl 1417.34092 · doi:10.3934/dcdsb.2013.18.2101
[6] J. Huang; S. Ruan; J. Song, Bifurcaitons in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Diff. Equat., 257, 1721-1752 (2014) · Zbl 1326.34082 · doi:10.1016/j.jde.2014.04.024
[7] W. O. Kermack; A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 115, 700-721 (1927) · JFM 53.0517.01
[8] Y. Kuang; E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36, 389-406 (1998) · Zbl 0895.92032 · doi:10.1007/s002850050105
[9] Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, Chapman and Hall/CRC, 2016. · Zbl 1341.92002
[10] X. Lai; A. Friedman, Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: A mathematical model, PLoS ONE, 12, 1-24 (2017)
[11] M. Lu; J. Huang; S. Ruan; P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Diff. Equat., 267, 1859-1898 (2019) · Zbl 1421.92034 · doi:10.1016/j.jde.2019.03.005
[12] K. M. Mahoney; G. J. Freeman; D. F. McDermott, The next immune-checkpoint inhibitors: PD-1/PD-L1 blockade in melanoma, Clin. Ther., 37, 764-782 (2015) · doi:10.1016/j.clinthera.2015.02.018
[13] M. Mazel; W. Jacot; K. Pantel; K. Bartkowiak; D. Topart; L. Cayrefourcq; C. Alix-Panabières, Frequent expression of PD-L1 on circulating breast cancer cells, Mol. Oncol., 9, 1773-1782 (2015) · doi:10.1016/j.molonc.2015.05.009
[14] E. Nikolopoulou, L. R. Johnson, D. Harris, J. D. Nagy, E. C. Stites and Y. Kuang, Tumour-immune dynamics with an immune checkpoint inhibitor, Lett. Biomath., 5 (2018), S137-S159.
[15] E. Nikolopoulou, S. E. Eikenberry, J. L. Gevertz and Y. Kuang, Mathematical modeling of an immune checkpoint inhibitor and its synergy with an immunostimulant, Discrete Contin. Dyn. Syst. Ser. B, (2020), in press.
[16] P. A. Ott; F. S. Hodi; H. L. Kaufman; J. M. Wigginton; J. D. Wolchok, Combination immunotherapy: A road map, J. Immunother. Cancer., 5, 16-30 (2017) · doi:10.1186/s40425-017-0218-5
[17] L. Perko, Differential Equations and Dynamical Systems, \(3^{rd}\) edition, Springer, New York, 2001. · Zbl 0973.34001
[18] S. A. Patel; A. J. Minn, Combination Cancer Therapy with Immune Checkpoint Blockade: Mechanisms and Strategies, Immunity, 48, 417-433 (2018) · doi:10.1016/j.immuni.2018.03.007
[19] M. L. Rosenzweig; R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97, 209-223 (1963) · doi:10.1086/282272
[20] M. Swart; I. Verbrugge; J. B. Beltman, Combination approaches with immune-checkpoint blockade in cancer therapy, Front. Oncol., 6, 233-248 (2016) · doi:10.3389/fonc.2016.00233
[21] M. Swart; I. Verbrugge; J. B. Beltman, Combination approaches with immune-checkpoint blockade in cancer therapy, Front. Oncol., 6, 233-248 (2016) · doi:10.3389/fonc.2016.00233
[22] Suzanne L. Topalian, F. S. Hodi, J. R. Brahmer, S. N. Gettinger, D. C. Smith, D. F. McDermott, ... and P. D. Leming, Safety, activity, and immune correlates of anti-PD-1 antibody in cancer, N. Engl. J. Med., 366 (2012), 2443-2454. · doi:10.1073/pnas.0813367106
[23] O. Talay; C. H. Shen; L. Chen; J. Chen, B7-H1 (PD-L1) on T cells is required for T-cell-mediated conditioning of dendritic cell maturation, Proc. Natl. Acad. Sci. U. S. A., 106, 2741-2746 (2009) · doi:10.1073/pnas.0813367106
[24] H. Wimberly; J. R. Brown; K. Schalper; H. Haack; M. R. Silver; C. Nixon; D. L. Rimm, PD-L1 expression correlates with tumor-infiltrating lymphocytes and response to neoadjuvant chemotherapy in breast cancer, Cancer Immunol. Res., 3, 326-332 (2015) · Zbl 1119.92042 · doi:10.1158/2326-6066.CIR-14-0133
[25] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. · Zbl 1119.92042
[26] Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equation, Transl. Math. Monogr., 101, Amer. Math. Soc., Providence, RI, 1992. · Zbl 0779.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.