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Global dynamics in a tumor-immune model with an immune checkpoint inhibitor

  • * Corresponding author: Jicai Huang

    * Corresponding author: Jicai Huang 

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

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  • 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.

    Mathematics Subject Classification: Primary: 34D23, 34C23; Secondary: 92B05.


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  • 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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