
-
Previous Article
Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium
- DCDS-B Home
- This Issue
-
Next Article
Global analysis of a model of competition in the chemostat with internal inhibitor
Global dynamics in a tumor-immune model with an immune checkpoint inhibitor
1. | School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China |
2. | School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, USA |
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.
References:
[1] |
M. E. Alexander and S. M. Moghadas,
Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96.
doi: 10.1016/j.mbs.2004.01.003. |
[2] |
R. FitzHugh,
Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278.
doi: 10.1007/BF02477753. |
[3] |
J. Hamanishi, M. Mandai, N. Matsumura, K. Abiko, T. Baba and I. Konishi,
PD-1/PD-L1 blockade in cancer treatment: Perspectives and issues, Int. J. Clin. Oncol., 21 (2016), 462-473.
doi: 10.1007/s10147-016-0959-z. |
[4] |
S.-B. Hsu, T. W. Hwang and Y. Kuang,
Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489-506.
doi: 10.1007/s002850100079. |
[5] |
J. Huang, Y. Gong and S. Ruan,
Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121.
doi: 10.3934/dcdsb.2013.18.2101. |
[6] |
J. Huang, S. Ruan and J. Song,
Bifurcaitons in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Diff. Equat., 257 (2014), 1721-1752.
doi: 10.1016/j.jde.2014.04.024. |
[7] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 115 (1927), 700-721. Google Scholar |
[8] |
Y. Kuang and E. Beretta,
Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.
doi: 10.1007/s002850050105. |
[9] |
Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, Chapman and Hall/CRC, 2016. |
[10] |
X. Lai and A. Friedman, Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: A mathematical model, PLoS ONE, 12 (2017), 1-24. Google Scholar |
[11] |
M. Lu, J. Huang, S. Ruan and P. Yu,
Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Diff. Equat., 267 (2019), 1859-1898.
doi: 10.1016/j.jde.2019.03.005. |
[12] |
K. M. Mahoney, G. J. Freeman and D. F. McDermott,
The next immune-checkpoint inhibitors: PD-1/PD-L1 blockade in melanoma, Clin. Ther., 37 (2015), 764-782.
doi: 10.1016/j.clinthera.2015.02.018. |
[13] |
M. Mazel, W. Jacot, K. Pantel, K. Bartkowiak, D. Topart, L. Cayrefourcq and C. Alix-Panabières,
Frequent expression of PD-L1 on circulating breast cancer cells, Mol. Oncol., 9 (2015), 1773-1782.
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.
doi: 10.1080/23737867.2018.1440978. |
[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. Google Scholar |
[16] |
P. A. Ott, F. S. Hodi, H. L. Kaufman, J. M. Wigginton and J. D. Wolchok,
Combination immunotherapy: A road map, J. Immunother. Cancer., 5 (2017), 16-30.
doi: 10.1186/s40425-017-0218-5. |
[17] |
L. Perko, Differential Equations and Dynamical Systems, 3$^{rd}$ edition, Springer, New York, 2001.
doi: 10.1007/978-1-4613-0003-8. |
[18] |
S. A. Patel and A. J. Minn,
Combination Cancer Therapy with Immune Checkpoint Blockade: Mechanisms and Strategies, Immunity, 48 (2018), 417-433.
doi: 10.1016/j.immuni.2018.03.007. |
[19] |
M. L. Rosenzweig and R. H. MacArthur,
Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.
doi: 10.1086/282272. |
[20] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge university press, 1995.
doi: 10.1017/CBO9780511530043.![]() ![]() |
[21] |
M. Swart, I. Verbrugge and J. B. Beltman,
Combination approaches with immune-checkpoint blockade in cancer therapy, Front. Oncol., 6 (2016), 233-248.
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: 10.1056/NEJMoa1200690. |
[23] |
O. Talay, C. H. Shen, L. Chen and 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 (2009), 2741-2746.
doi: 10.1073/pnas.0813367106. |
[24] |
H. Wimberly, J. R. Brown, K. Schalper, H. Haack, M. R. Silver, C. Nixon and D. L. Rimm,
PD-L1 expression correlates with tumor-infiltrating lymphocytes and response to neoadjuvant chemotherapy in breast cancer, Cancer Immunol. Res., 3 (2015), 326-332.
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.
doi: 10.1016/j.mbs.2006.09.025. |
[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. |
show all references
References:
[1] |
M. E. Alexander and S. M. Moghadas,
Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96.
doi: 10.1016/j.mbs.2004.01.003. |
[2] |
R. FitzHugh,
Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278.
doi: 10.1007/BF02477753. |
[3] |
J. Hamanishi, M. Mandai, N. Matsumura, K. Abiko, T. Baba and I. Konishi,
PD-1/PD-L1 blockade in cancer treatment: Perspectives and issues, Int. J. Clin. Oncol., 21 (2016), 462-473.
doi: 10.1007/s10147-016-0959-z. |
[4] |
S.-B. Hsu, T. W. Hwang and Y. Kuang,
Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489-506.
doi: 10.1007/s002850100079. |
[5] |
J. Huang, Y. Gong and S. Ruan,
Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121.
doi: 10.3934/dcdsb.2013.18.2101. |
[6] |
J. Huang, S. Ruan and J. Song,
Bifurcaitons in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Diff. Equat., 257 (2014), 1721-1752.
doi: 10.1016/j.jde.2014.04.024. |
[7] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 115 (1927), 700-721. Google Scholar |
[8] |
Y. Kuang and E. Beretta,
Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.
doi: 10.1007/s002850050105. |
[9] |
Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, Chapman and Hall/CRC, 2016. |
[10] |
X. Lai and A. Friedman, Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: A mathematical model, PLoS ONE, 12 (2017), 1-24. Google Scholar |
[11] |
M. Lu, J. Huang, S. Ruan and P. Yu,
Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Diff. Equat., 267 (2019), 1859-1898.
doi: 10.1016/j.jde.2019.03.005. |
[12] |
K. M. Mahoney, G. J. Freeman and D. F. McDermott,
The next immune-checkpoint inhibitors: PD-1/PD-L1 blockade in melanoma, Clin. Ther., 37 (2015), 764-782.
doi: 10.1016/j.clinthera.2015.02.018. |
[13] |
M. Mazel, W. Jacot, K. Pantel, K. Bartkowiak, D. Topart, L. Cayrefourcq and C. Alix-Panabières,
Frequent expression of PD-L1 on circulating breast cancer cells, Mol. Oncol., 9 (2015), 1773-1782.
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.
doi: 10.1080/23737867.2018.1440978. |
[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. Google Scholar |
[16] |
P. A. Ott, F. S. Hodi, H. L. Kaufman, J. M. Wigginton and J. D. Wolchok,
Combination immunotherapy: A road map, J. Immunother. Cancer., 5 (2017), 16-30.
doi: 10.1186/s40425-017-0218-5. |
[17] |
L. Perko, Differential Equations and Dynamical Systems, 3$^{rd}$ edition, Springer, New York, 2001.
doi: 10.1007/978-1-4613-0003-8. |
[18] |
S. A. Patel and A. J. Minn,
Combination Cancer Therapy with Immune Checkpoint Blockade: Mechanisms and Strategies, Immunity, 48 (2018), 417-433.
doi: 10.1016/j.immuni.2018.03.007. |
[19] |
M. L. Rosenzweig and R. H. MacArthur,
Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.
doi: 10.1086/282272. |
[20] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge university press, 1995.
doi: 10.1017/CBO9780511530043.![]() ![]() |
[21] |
M. Swart, I. Verbrugge and J. B. Beltman,
Combination approaches with immune-checkpoint blockade in cancer therapy, Front. Oncol., 6 (2016), 233-248.
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: 10.1056/NEJMoa1200690. |
[23] |
O. Talay, C. H. Shen, L. Chen and 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 (2009), 2741-2746.
doi: 10.1073/pnas.0813367106. |
[24] |
H. Wimberly, J. R. Brown, K. Schalper, H. Haack, M. R. Silver, C. Nixon and D. L. Rimm,
PD-L1 expression correlates with tumor-infiltrating lymphocytes and response to neoadjuvant chemotherapy in breast cancer, Cancer Immunol. Res., 3 (2015), 326-332.
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.
doi: 10.1016/j.mbs.2006.09.025. |
[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. |







[1] |
Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419 |
[2] |
Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923 |
[3] |
Gladis Torres-Espino, Claudio Vidal. Periodic solutions of a tumor-immune system interaction under a periodic immunotherapy. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020301 |
[4] |
Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203 |
[5] |
Denise E. Kirschner, Alexei Tsygvintsev. On the global dynamics of a model for tumor immunotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 573-583. doi: 10.3934/mbe.2009.6.573 |
[6] |
Giulio Caravagna, Alex Graudenzi, Alberto d’Onofrio. Distributed delays in a hybrid model of tumor-Immune system interplay. Mathematical Biosciences & Engineering, 2013, 10 (1) : 37-57. doi: 10.3934/mbe.2013.10.37 |
[7] |
J.C. Arciero, T.L. Jackson, D.E. Kirschner. A mathematical model of tumor-immune evasion and siRNA treatment. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 39-58. doi: 10.3934/dcdsb.2004.4.39 |
[8] |
Mohammad A. Tabatabai, Wayne M. Eby, Karan P. Singh, Sejong Bae. T model of growth and its application in systems of tumor-immune dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 925-938. doi: 10.3934/mbe.2013.10.925 |
[9] |
Lifeng Han, Changhan He, Yang Kuang. Dynamics of a model of tumor-immune interaction with time delay and noise. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2347-2363. doi: 10.3934/dcdss.2020140 |
[10] |
Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233 |
[11] |
Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031 |
[12] |
Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215 |
[13] |
Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21 |
[14] |
W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351 |
[15] |
Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069 |
[16] |
Min Yu, Gang Huang, Yueping Dong, Yasuhiro Takeuchi. Complicated dynamics of tumor-immune system interaction model with distributed time delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2391-2406. doi: 10.3934/dcdsb.2020015 |
[17] |
Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 787-802. doi: 10.3934/mbe.2013.10.787 |
[18] |
Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler. Dynamics of tumor-immune interaction under treatment as an optimal control problem. Conference Publications, 2011, 2011 (Special) : 971-980. doi: 10.3934/proc.2011.2011.971 |
[19] |
Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282 |
[20] |
Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]