# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022033
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## Complex dynamic behaviors of a tumor-immune system with two delays in tumor actions

 1 Department of Mathematics, Shaanxi University of Science and Technology, Xi'an, 710021, China 2 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada 3 Department of Immunology, Xi'an Medical University, Xi'an 710021, China

** Equal contributor: Dian Zhang

* Corresponding author: Jianquan Li

Received  August 2021 Revised  October 2021 Early access March 2022

Fund Project: This work is supported partially by the National Natural Science Foundation of PR China (Nos. 11971281, 12071268, 12071418), NSERC of Canada (No. RGPIN-2019-05892), and the Project of Xi'an Medical University (No. 2018GJFY05)

The action of a tumor on the immune system includes stimulation and neutralization, which usually have different time delays. In this work we propose a tumor-immune system to incorporate these two kinds of delays due to tumor actions. We explore effects of time delays on the model and find some different phenomena induced by them. When there is only the neutralization delay, the model has a uniform upper bound while when there is only the stimulation delay, the bound varies with the delay. The theoretic analysis suggests that, for the model only with the stimulation delay, the stability of its tumor-present equilibrium may change at most once as the delay increases, but the increase of the neutralization delay may lead to multiple stability switches for the model only with the neutralization delay. Numerical simulations indicate that, in the presence of the neutralization delay, the stimulation delay may induce multiple stability switches. Further, when the model has two tumor-present equilibria, numerical simulations also demonstrate that the model may present some interesting outcomes as each of the two delays increases. These phenomena include the onset of the cytokine storm, the almost global attractivity of the tumor-free equilibrium for sufficiently large time delays, and so on. These results show the complexity of the dynamic behaviors of the model and different effects of the two time delays.

Citation: Jianquan Li, Xiangxiang Ma, Yuming Chen, Dian Zhang. Complex dynamic behaviors of a tumor-immune system with two delays in tumor actions. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022033
##### References:
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Ma, Stability switches in a class of characteristic equations with delay-dependent parameters, Nonlinear Anal. Real World Appl., 5 (2004), 389-408.  doi: 10.1016/j.nonrwa.2003.06.001. [23] J. Li, X. Xie, Y. Chen and D. Zhang, Complex dynamics of a tumor-immune system with antigenicity, Appl. Math. Comput., 400 (2021), 126052.  doi: 10.1016/j.amc.2021.126052. [24] J. Li, X. Xie, D. Zhang, J. Li and X. Lin, Qualitative analysis of a simple tumor-immune system with time delay of tumor action, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5227-5249.  doi: 10.3934/dcdsb.2020341. [25] N. Nishida and M. Kudo, Immunological microenvironment of hepatocellular carcinoma and its clinical implication, Oncology, 92 (2017), 40-49.  doi: 10.1159/000451015. [26] L. Olien and J. Bélair., Bifurcations, stability, and monotonicity properties of a delayed neural network model, Phys. D, 102 (1997), 349-363.  doi: 10.1016/S0167-2789(96)00215-1. [27] J. Prieto, I. Melero and B. Sangro, Immunological landscape and immunotherapy of hepatocellular carcinoma, Nat. Rev. Gastroenterol. Hepatol., 12 (2015), 681-700.  doi: 10.1038/nrgastro.2015.173. [28] M. Robertson-Tessi, A. El-Kareh and A. Goriely, A mathematical model of tumor-immune interactions, J. Theoret. Biol., 294 (2012), 56-73.  doi: 10.1016/j.jtbi.2011.10.027. [29] S. Ruan, Nonlinear dynamics in tumor-immune system interaction models with delays, Discrete Contin. Dyn. Syst. Ser. B., 26 (2021), 541-602.  doi: 10.3934/dcdsb.2020282. [30] R. D. Schreiber, L. J. Old and M. J. Smyth, Cancer immunoediting: Integrating immunity's roles in cancer suppression and promotion, Science, 331 (2011), 1565-1570.  doi: 10.1126/science.1203486. [31] H. Shalabi, V. Sachdev and A. Kulshreshtha, Impact of cytokine release syndrome on cardiac function following CD19 CAR-T cell therapy in children and young adults with hematological malignancies, J. Immunother. Cancer, 8 (2020), e001159.  doi: 10.1136/jitc-2020-001159. [32] S. Stöcker and M. G. Curci, Modelling and simulating the effect of cytokines on the immune response to tumor cells, Math. Comput. Model., 28 (1998), 1-13.  doi: 10.1016/S0895-7177(98)00093-4. [33] S. Tang, S. Li, S. Zheng, Y. Ding, D. Zhu, C. Sun, Y. Hu, J. Qiao and H. Fang, Understanding of cytokines andgeted therapy in macrophage activation syndrome, Semin. Arthritis Rheum., 51 (2021), 198-210. [34] M. Waito, S. R. Walsh, A. Rasiuk, B. W. Bridle and A. R. Willms, A mathematical model of cytokine dynamics during a cytokine storm,, In Mathematical and Computational Approaches in Advancing Modern Science and Engineering, (eds. J. Bélair, I.A. Frigaard, H. Kunze, R. Makarov, R. Melnik, R.J. Spiteri), (2016), 331–339. doi: 10.1007/978-3-319-30379-6_31. [35] J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Phys. D, 130 (1999), 255-272.  doi: 10.1016/S0167-2789(99)00009-3. [36] L. Wenbo and J. Wang, Uncovering the underlying mechanism of cancer tumorigenesis and development under an immune microenvironment from global quantification of the landscape, J. R. Soc. Interface, 14 (2017), 20170105.  doi: 10.1098/rsif.2017.0105. [37] R. Yafia, A study of differential equation modeling malignant tumor cells in competition with immune system, Int. J. Biomath., 4 (2011), 185-206.  doi: 10.1142/S1793524511001404. [38] R. Yafia, Hopf bifurcation in differential equation with delay for tumor immune system competition model, SIAM J. Appl. Math., 67 (2007), 1693-1703.  doi: 10.1137/060657947. [39] M. Yu, Y. Dong and Y. Takeuchi, Dual role of delay effects in a tumour-immune system, J. Biol. Dynam., 11 (2017), 334-347.  doi: 10.1080/17513758.2016.1231347.

show all references

##### References:
 [1] S. Banerjee, S. Khajanchi and S. Chaudhuri, A mathematical model to elucidate brain tumor abrogration by immunotherapy with T11 target structure, PLoS ONE, 10 (2015), e0123611.  doi: 10.1371/journal.pone.0123611. [2] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086. [3] H. M. Byrne, The effect of time delay on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117.  doi: 10.1016/S0025-5564(97)00023-0. [4] I. A. Clark and B. Vissel, The meteorology of cytokine storms, and the clinical usefulness of this knowledge, Semin. Immunopathol., 39 (2017), 505-516.  doi: 10.1007/s00281-017-0628-y. [5] K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627.  doi: 10.1016/0022-247X(82)90243-8. [6] K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcial. Ekvac., 29 (1986), 77-90. [7] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Phys. D, 208 (2005), 220-235.  doi: 10.1016/j.physd.2005.06.032. [8] R. Eftimie, J. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models, Bull. Math. Biol., 73 (2011), 2-32.  doi: 10.1007/s11538-010-9526-3. [9] D. Ghosh, S. Khajanchi, S. Mangiarotti, F. Denis, S. K. Dana and C. Letellier, How tumor growth can be influenced by delayed interactions between cancer cells and the microenvironment?, BioSystems., 158 (2017), 17-30.  doi: 10.1016/j.biosystems.2017.05.001. [10] Y. Guo, B. Niu and J. P. Tian, Backward Hopf bifurcation in a mathematical model for oncolytic virotherapy with the infection delay and innate immune effects, J. Biol. Dyn., 13 (2019), 733-748.  doi: 10.1080/17513758.2019.1667443. [11] H. Y. Hao, A. L. Graham and R. F. Stengel, Dynamics of a cytokine storm, PLoS ONE, 7 (2012), e45027. [12] Q. Huang and Z. E. Ma, On stability of some transcendental equations, Ann. Differential Equations, 6 (1990), 21-31. [13] P. Johansen, T. Storni and L. Rettig, Antigen kinetics determines immune reactivity, Proc. Natl. Acad. Sci., 105 (2008), 5189-5194.  doi: 10.1073/pnas.0706296105. [14] S. Khajanchi and S. Banerjee, Influence of multiple delays in brain tumor and immune system interaction with T11 target structure as a potent stimulator, Math. Biosci., 302 (2018), 116-130.  doi: 10.1016/j.mbs.2018.06.001. [15] S. Khajanchi and D. Ghosh, The combined effects of optimal control in cancer remission, Appl. Math. Comput., 271 (2015), 375-388.  doi: 10.1016/j.amc.2015.09.012. [16] S. Khajanchi and J. J. Nieto, Mathematical modeling of tumor-immune competitive system, considering the role of time delay, Appl. Math. Comput., 340 (2019), 180-205.  doi: 10.1016/j.amc.2018.08.018. [17] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252.  doi: 10.1007/s002850050127. [18] N. Kronik, Y. Kogan, M. Elishmereni, K. Halevi-Tobias, S. Vuk-Pavolić and Z. Agur, Predicting outcomes of prostate cancer immunotherapy by personalized mathematical models, PLoS One, 5 (2010), e15482.  doi: 10.1371/journal.pone.0015482. [19] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Inc., Boston, MA, 1993. [20] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321. [21] C. Letellier, F. Denis and L. A. Aguirre, What can be learned from chaotic cancer model?, J. Theoret. Biol., 322 (2013), 7-16.  doi: 10.1016/j.jtbi.2013.01.003. [22] J. Li and Z. Ma, Stability switches in a class of characteristic equations with delay-dependent parameters, Nonlinear Anal. Real World Appl., 5 (2004), 389-408.  doi: 10.1016/j.nonrwa.2003.06.001. [23] J. Li, X. Xie, Y. Chen and D. Zhang, Complex dynamics of a tumor-immune system with antigenicity, Appl. Math. Comput., 400 (2021), 126052.  doi: 10.1016/j.amc.2021.126052. [24] J. Li, X. Xie, D. Zhang, J. Li and X. Lin, Qualitative analysis of a simple tumor-immune system with time delay of tumor action, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5227-5249.  doi: 10.3934/dcdsb.2020341. [25] N. Nishida and M. Kudo, Immunological microenvironment of hepatocellular carcinoma and its clinical implication, Oncology, 92 (2017), 40-49.  doi: 10.1159/000451015. [26] L. Olien and J. Bélair., Bifurcations, stability, and monotonicity properties of a delayed neural network model, Phys. D, 102 (1997), 349-363.  doi: 10.1016/S0167-2789(96)00215-1. [27] J. Prieto, I. Melero and B. Sangro, Immunological landscape and immunotherapy of hepatocellular carcinoma, Nat. Rev. Gastroenterol. Hepatol., 12 (2015), 681-700.  doi: 10.1038/nrgastro.2015.173. [28] M. Robertson-Tessi, A. El-Kareh and A. Goriely, A mathematical model of tumor-immune interactions, J. Theoret. Biol., 294 (2012), 56-73.  doi: 10.1016/j.jtbi.2011.10.027. [29] S. Ruan, Nonlinear dynamics in tumor-immune system interaction models with delays, Discrete Contin. Dyn. Syst. Ser. B., 26 (2021), 541-602.  doi: 10.3934/dcdsb.2020282. [30] R. D. Schreiber, L. J. Old and M. J. Smyth, Cancer immunoediting: Integrating immunity's roles in cancer suppression and promotion, Science, 331 (2011), 1565-1570.  doi: 10.1126/science.1203486. [31] H. Shalabi, V. Sachdev and A. Kulshreshtha, Impact of cytokine release syndrome on cardiac function following CD19 CAR-T cell therapy in children and young adults with hematological malignancies, J. Immunother. Cancer, 8 (2020), e001159.  doi: 10.1136/jitc-2020-001159. [32] S. Stöcker and M. G. Curci, Modelling and simulating the effect of cytokines on the immune response to tumor cells, Math. Comput. Model., 28 (1998), 1-13.  doi: 10.1016/S0895-7177(98)00093-4. [33] S. Tang, S. Li, S. Zheng, Y. Ding, D. Zhu, C. Sun, Y. Hu, J. Qiao and H. Fang, Understanding of cytokines andgeted therapy in macrophage activation syndrome, Semin. Arthritis Rheum., 51 (2021), 198-210. [34] M. Waito, S. R. Walsh, A. Rasiuk, B. W. Bridle and A. R. Willms, A mathematical model of cytokine dynamics during a cytokine storm,, In Mathematical and Computational Approaches in Advancing Modern Science and Engineering, (eds. J. Bélair, I.A. Frigaard, H. Kunze, R. Makarov, R. Melnik, R.J. Spiteri), (2016), 331–339. doi: 10.1007/978-3-319-30379-6_31. [35] J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Phys. D, 130 (1999), 255-272.  doi: 10.1016/S0167-2789(99)00009-3. [36] L. Wenbo and J. Wang, Uncovering the underlying mechanism of cancer tumorigenesis and development under an immune microenvironment from global quantification of the landscape, J. R. Soc. Interface, 14 (2017), 20170105.  doi: 10.1098/rsif.2017.0105. [37] R. Yafia, A study of differential equation modeling malignant tumor cells in competition with immune system, Int. J. Biomath., 4 (2011), 185-206.  doi: 10.1142/S1793524511001404. [38] R. Yafia, Hopf bifurcation in differential equation with delay for tumor immune system competition model, SIAM J. Appl. Math., 67 (2007), 1693-1703.  doi: 10.1137/060657947. [39] M. Yu, Y. Dong and Y. Takeuchi, Dual role of delay effects in a tumour-immune system, J. Biol. Dynam., 11 (2017), 334-347.  doi: 10.1080/17513758.2016.1231347.
For $\tau_2 = 0$, phase portraits of system (4) with the initial condition $(0.3, 2.7)$ for ${\tau}_1 = 0.16$, $0.2$, $0.3$, $0.4$ (the left column) and the corresponding $y$-components (right column)
For $\tau_1 = 0$, phase portraits of system (4) with the initial condition $(0.3, 2.7)$ for ${\tau}_2 = 1.2$, $1.5$, $2.5$, $3.5$ (left column) and the corresponding $y$-components (right column)
Bifurcation diagrams of system (4) with Set (PV1), where $\tau_1$ is bifurcation parameter with $\tau_2 = 0$
Bifurcation diagrams of system (4) with Set (PV1), where $\tau_2$ as bifurcation parameter with $\tau_1 = 0$
The bifurcation diagrams of (4) with Set (PV1), where, for (a) fixing ${\tau}_1 = 0.15$, ${\tau}_2$ is the bifurcation parameter; for (b) fixing ${\tau}_2 = 1.0$, ${\tau}_1$ is the bifurcation parameter
The bifurcation diagram of system (4)$_{{\tau}_1 = 0}$ with Set (PV2) and $r = 0.6$ for ${\tau}_2$, where the tumor-present equilibrium $P^*$ of (4) can recover stability once with the increase of ${\tau}_2$
The bifurcation diagram of system (4)$_{{\tau}_1 = 0}$ with Set (PV2) and $r = 2.8$ for ${\tau}_2$, where the tumor-present equilibrium $P^*$ of (4) can recover stability twice with the increase of ${\tau}_2$
Phase portraits of (4) with the same initial conditions $(0.05, 0.7)$, $(0.1, 0.7)$, $(0.175, 0.85)$, and $(0.175, 0.9)$ for ${\tau}_1 = 0.33$, $0.336$, $0.338$, $0.3385$ when $\tau_2 = 0$ (left column) and the corresponding graphs of $y = y(t)$ (right column)
Phase portraits (left column) of (4) and the corresponding graph of $y = y(t)$ (right column) with the initial conditions $(0.35, 1.1)$ (blue) and $(0.08, 0.55)$ (black) for different delays, ${\tau}_2 = 1.3$, $1.4$, $1.5$, $1.68$ with $\tau_1 = 0$
Phase portraits of system (4)$_{{\tau}_2 = 0}$ with ${\tau}_1 = 7.98$, where the black curve (C1), the red one (C2), the blue one (C3), and the manganese one (C4) correspond to the initial values $(0.04, 0.6)$, $(0.04, 0.9)$, $(0.24, 0.9)$, and $(0.14, 0.2)$, respectively
Phase portraits of system (4)$_{{\tau}_2 = 0}$ with ${\tau}_1 = 7.995$. The cytokine storm appears only for the solution with the initial value $(0.04, 0.6)$ (C1-black), which is depicted at the top row
Phase portraits of system (4)$_{{\tau}_2 = 0}$ with ${\tau}_1 = 8.05$. Here cytokine storm occurs for solutions with the initial values $(0.04, 0.6)$ (C1-black) and $(0.04, 0.9)$ (C2-red) shown at the top row
Periods of periodic solutions and the corresponding amplitudes of the component $y = y(t)$ for different values of ${\tau}_1$ and ${\tau}_2$
 ${\tau}_2=0$ ${\tau}_1=0$ $\tau_1$ Cycle Period Amplitude $\tau_2$ Cycle Period Amplitude 0.2 8.05 6.04 1.5 2.46 5.94 0.3 18.78 25.89 2.5 3.83 9.65 0.4 42.31 92.66 3.5 4.91 10.53
 ${\tau}_2=0$ ${\tau}_1=0$ $\tau_1$ Cycle Period Amplitude $\tau_2$ Cycle Period Amplitude 0.2 8.05 6.04 1.5 2.46 5.94 0.3 18.78 25.89 2.5 3.83 9.65 0.4 42.31 92.66 3.5 4.91 10.53
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