February  2021, 26(2): 1149-1170. doi: 10.3934/dcdsb.2020157

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

* Corresponding author: Jicai Huang

Received  August 2019 Revised  February 2020 Published  February 2021 Early access  May 2020

Fund Project: Research of JH and SS was partially supported by NSFC (No. 11871235) and the Fundamental Research Funds for the Central Universities (CCNU19TS030). Research of YK is partially supported by NSF grants DMS-1615879, DEB-1930728 and an NIH grant 5R01GM131405-02

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.

Citation: Shujing Shi, Jicai Huang, Yang Kuang. Global dynamics in a tumor-immune model with an immune checkpoint inhibitor. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1149-1170. doi: 10.3934/dcdsb.2020157
References:
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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.

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J. HamanishiM. MandaiN. MatsumuraK. AbikoT. 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.

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S.-B. HsuT. 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.

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J. HuangY. 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.

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J. HuangS. 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.

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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. 

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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.

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Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, Chapman and Hall/CRC, 2016.

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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. 

[11]

M. LuJ. HuangS. 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. MahoneyG. 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. MazelW. JacotK. PantelK. BartkowiakD. TopartL. 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.

[16]

P. A. OttF. S. HodiH. L. KaufmanJ. 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. SwartI. 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. TalayC. H. ShenL. 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. WimberlyJ. R. BrownK. SchalperH. HaackM. R. SilverC. 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. HamanishiM. MandaiN. MatsumuraK. AbikoT. 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. HsuT. 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. HuangY. 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. HuangS. 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. 

[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. 

[11]

M. LuJ. HuangS. 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. MahoneyG. 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. MazelW. JacotK. PantelK. BartkowiakD. TopartL. 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.

[16]

P. A. OttF. S. HodiH. L. KaufmanJ. 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. SwartI. 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. TalayC. H. ShenL. 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. WimberlyJ. R. BrownK. SchalperH. HaackM. R. SilverC. 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.

Figure 1.  A unique positive root $ y_0 $ of $ G_1(y) = 0 $
Figure 2.  A unique boundary equilibrium $ E_0 $ which is (a) hyperbolic saddle if $ b<(c+1)e-a $; (b) stable hyperbolic node if $ b>(c+1)e-a $; (c) saddle-node with a stable parabolic sector in the right half plane if $ b = (c+1)e-a $ and $ d< 2c+\frac{a}{e} $; (d) saddle-node with a stable parabolic sector in the left half plane if $ b = (c+1)e-a $ and $ d>2c+\frac{a}{e} $; (e) stable degenerate node if $ b = (c+1)e-a $ and $ d = 2c+\frac{a}{e} $
Figure 3.  The positive roots of $ G_2(y) = 0 $ when $ d>c $: (a) no positive root; (b) one double positive root $ y_1 $; (c) two simple positive roots $ y_2 $ and $ y_3 $
Figure 4.  The phase portraits of system $ (3) $ with no positive equilibrium when $ d>c $
Figure 5.  The phase portraits of system $ (3) $ with a unique positive equilibrium $ E_1 $ which is a saddle-node when $ d>c $
Figure 6.  The phase portraits of system $ (3) $ with two distinct positive equilibria: a hyperbolic saddle $ E_3 $ and a stable hyperbolic node $ E_2 $ when $ d>c $
Figure 7.  The positive root of $ G_2(y) = 0 $. (a) $ d<c $; (b) $ d = c $
Figure 8.  The tumour-free equilibrium is globally asymptotically stable when $ C_K>C_{K1} $
Figure 9.  The unique tumorous equilibrium is globally asymptotically stable when $ C_K>C_{K1} $
Figure 10.  Two tumorous equilibria and bistable phenomenon when $ C_K>C_{K1} $
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