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From the guest editors: "Delay Differential Equations: Theory, Applications and New Trends"
September  2020, 13(9): 2347-2363. doi: 10.3934/dcdss.2020140

## Dynamics of a model of tumor-immune interaction with time delay and noise

 School of Mathematical and Statistical Sciences, Arizona state University, Tempe, AZ 85287-1804, USA

* Corresponding author: Yang Kuang

Received  January 2019 Published  September 2020 Early access  November 2019

Fund Project: The authors are supported by NSF grant 161587 and NIH grant 1R01GM131405-01

We propose a model of tumor-immune interaction with time delay in immune reaction and noise in tumor cell reproduction. Immune response is modeled as a non-monotonic function of tumor burden, for which the tumor is immunogenic at nascent stage but starts inhibiting immune system as it grows large. Without time delay and noise, this system demonstrates bistability. The effects of response time of the immune system and uncertainty in the tumor innate proliferation rate are studied by including delay and noise in the appropriate model terms. Stability, persistence and extinction of the tumor are analyzed. We find that delay and noise can both induce the transition from low tumor burden equilibrium to high tumor equilibrium. Moreover, our result suggests that the elimination of cancer depends on the basal level of the immune system rather than on its response speed to tumor growth.

Citation: Lifeng Han, Changhan He, Yang Kuang. Dynamics of a model of tumor-immune interaction with time delay and noise. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2347-2363. doi: 10.3934/dcdss.2020140
##### References:
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Chen, C. Wei and S. Zhang, Stationary distribution of a stochastic SIQR epidemic model with saturated incidence and degenerate diffusion, Phys. A, 511 (2018), 61–77, Available from: https://www.sciencedirect.com/science/article/pii/S0378437118309130. doi: 10.1016/j.physa.2018.07.041. [20] R. Lefever and W. Horsthemke, Bistability in fluctuating environments. Implications in tumor immunology, Bulletin of Mathematical Biology, 41 (1979), 469–490, Available from: https://www.sciencedirect.com/science/article/pii/S0092824079800038. [21] D. Li and F. Cheng, Threshold for extinction and survival in stochastic tumor immune system, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 1–12, Available from: https://www.sciencedirect.com/science/article/pii/S1007570417300850. doi: 10.1016/j.cnsns.2017.03.007. [22] K. M. Mahoney, G. J. Freeman and D. F. McDermott, The next immune-checkpoint inhibitors: PD-1/PD-L1 blockade in melanoma, Clinical Therapeutics, 37 (2015), 764-782.  doi: 10.1016/j.clinthera.2015.02.018. [23] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. [24] J. D. Murray, Mathematical Biology: I. An Introduction, Third edition. Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. [25] 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. [26] B. Øksendal, Stochastic Differential Equations, An introduction with applications. Sixth edition. Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [27] C. R. Parish, Cancer immunotherapy: The past, the present and the future, Immunology and Cell Biology, 81 (2003), 106-113.  doi: 10.1046/j.0818-9641.2003.01151.x. [28] F. Rihan, D. Abdel Rahman, S. Lakshmanan and A. Alkhajeh, A time delay model of tumour-immune system interactions: Global dynamics, parameter estimation, sensitivity analysis, Appl. Math. Comput., 232 (2014), 606–623, Available from: https://www.sciencedirect.com/science/article/pii/S0096300314001568. doi: 10.1016/j.amc.2014.01.111. [29] L. Wang, D. Jiang, G. S. K. Wolkowicz and D. O'Regan, Dynamics of the stochastic chemostat with Monod-Haldane response function, Scientific Reports, 7 (2017), 13641, Available from: http://www.nature.com/articles/s41598-017-13294-3. [30] T. L. Whiteside, Immune suppression in cancer: Effects on immune cells, mechanisms and future therapeutic intervention, Seminars in Cancer Biology, 16 (2006), 3–15, Available from: https://www.sciencedirect.com/science/article/pii/S1044579X0500060X?via{%}3Dihub. doi: 10.1016/j.semcancer.2005.07.008. [31] C. Zeng and H. Wang, Noise and large time delay: Accelerated catastrophic regime shifts in ecosystems, Ecological Modelling, 233 (2012), 52–58, Available from: https://www.sciencedirect.com/science/article/pii/S030438001200141X. doi: 10.1016/j.ecolmodel.2012.03.025.

show all references

##### References:
 [1] S. Banerjee and R. R. Sarkar, Delay-induced model for tumor-immune interaction and control of malignant tumor growth, Biosystems, 91 (2008), 268–288, Available from: https://www.sciencedirect.com/science/article/pii/S0303264707001499. doi: 10.1016/j.biosystems.2007.10.002. [2] T. Bose and S. Trimper, Stochastic model for tumor growth with immunization, Phys. Rev. E (3), 79 (2009), 051903, 10 pp. doi: 10.1103/PhysRevE.79.051903. [3] A. D'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Modelling, 47 (2008), 614–637, Available from: https://www.sciencedirect.com/science/article/pii/S0895717707001951. doi: 10.1016/j.mcm.2007.02.032. [4] A. D'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction, Math. Comput. Modelling, 51 (2010), 572–591, Available from: https://www.sciencedirect.com/science/article/pii/S089571770900404X. doi: 10.1016/j.mcm.2009.11.005. [5] 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. [6] R. Fisher, L. Pusztai and C. Swanton, Cancer heterogeneity: Implications for targeted therapeutics, British journal of cancer, 108 (2013), 479-485.  doi: 10.1038/bjc.2012.581. [7] T. D. Frank, Delay Fokker-Planck equations, perturbation theory, and data analysis for nonlinear stochastic systems with time delays, Physical Review E, 71 (2005), 031106. doi: 10.1103/PhysRevE.71.031106. [8] M. Gałach, Dynamics of the tumor–immune system competition–the effect of time delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395–406, Available from: https://pdfs.semanticscholar.org/88fb/d5af40f9ebdba3e4d262b0d5bf80963199b4.pdf. [9] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X. [10] S. Guillouzic, I. L'Heureux and A. Longtin, Small delay approximation of stochastic delay differential equations, Physical Review E, 59 (1999), 3970-3982.  doi: 10.1103/PhysRevE.59.3970. [11] W. Guo and D.-C. Mei, Stochastic resonance in a tumor-immune system subject to bounded noises and time delay, Phys. A, 416 (2014), 90–98, Available from: https://www.sciencedirect.com/science/article/pii/S0378437114006748. doi: 10.1016/j.physa.2014.08.003. [12] D. J. Higham., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [13] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor - immune interaction, Journal of Mathematical Biology, 37 (1998), 235-252.  doi: 10.1007/s002850050127. [14] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. [15] Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, CRC Press, Boca Raton, FL, 2016. [16] U. Küchler and B. Mensch, Langevins stochastic differential equation extended by a time-delayed term, Stochastics Stochastics Rep., 40 (1992), 23-42.  doi: 10.1080/17442509208833780. [17] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology, 56 (1994), 295–321, Available from: https://www.sciencedirect.com/science/article/pii/S0092824005802605. [18] X. Lai and A. Friedman, Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: A mathematical model, PLOS ONE, 12 (2017), e0178479. doi: 10.1371/journal.pone.0178479. [19] G. Lan, Z. Chen, C. Wei and S. Zhang, Stationary distribution of a stochastic SIQR epidemic model with saturated incidence and degenerate diffusion, Phys. A, 511 (2018), 61–77, Available from: https://www.sciencedirect.com/science/article/pii/S0378437118309130. doi: 10.1016/j.physa.2018.07.041. [20] R. Lefever and W. Horsthemke, Bistability in fluctuating environments. Implications in tumor immunology, Bulletin of Mathematical Biology, 41 (1979), 469–490, Available from: https://www.sciencedirect.com/science/article/pii/S0092824079800038. [21] D. Li and F. Cheng, Threshold for extinction and survival in stochastic tumor immune system, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 1–12, Available from: https://www.sciencedirect.com/science/article/pii/S1007570417300850. doi: 10.1016/j.cnsns.2017.03.007. [22] K. M. Mahoney, G. J. Freeman and D. F. McDermott, The next immune-checkpoint inhibitors: PD-1/PD-L1 blockade in melanoma, Clinical Therapeutics, 37 (2015), 764-782.  doi: 10.1016/j.clinthera.2015.02.018. [23] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. [24] J. D. Murray, Mathematical Biology: I. An Introduction, Third edition. Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. [25] 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. [26] B. Øksendal, Stochastic Differential Equations, An introduction with applications. Sixth edition. Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [27] C. R. Parish, Cancer immunotherapy: The past, the present and the future, Immunology and Cell Biology, 81 (2003), 106-113.  doi: 10.1046/j.0818-9641.2003.01151.x. [28] F. Rihan, D. Abdel Rahman, S. Lakshmanan and A. Alkhajeh, A time delay model of tumour-immune system interactions: Global dynamics, parameter estimation, sensitivity analysis, Appl. Math. Comput., 232 (2014), 606–623, Available from: https://www.sciencedirect.com/science/article/pii/S0096300314001568. doi: 10.1016/j.amc.2014.01.111. [29] L. Wang, D. Jiang, G. S. K. Wolkowicz and D. O'Regan, Dynamics of the stochastic chemostat with Monod-Haldane response function, Scientific Reports, 7 (2017), 13641, Available from: http://www.nature.com/articles/s41598-017-13294-3. [30] T. L. Whiteside, Immune suppression in cancer: Effects on immune cells, mechanisms and future therapeutic intervention, Seminars in Cancer Biology, 16 (2006), 3–15, Available from: https://www.sciencedirect.com/science/article/pii/S1044579X0500060X?via{%}3Dihub. doi: 10.1016/j.semcancer.2005.07.008. [31] C. Zeng and H. Wang, Noise and large time delay: Accelerated catastrophic regime shifts in ecosystems, Ecological Modelling, 233 (2012), 52–58, Available from: https://www.sciencedirect.com/science/article/pii/S030438001200141X. doi: 10.1016/j.ecolmodel.2012.03.025.
Nullclines of (4). Open circles denote unstable fixed points and filled circle denotes stable fixed points. Parameter values used: $\rho = 2.5, {\beta = 0.02, \gamma = 5, K = 10}$
Time course of $u$ and $v$ with and without delay. There is a stability switch as $\tau$ increases and eventually the solution settles down to high tumor equilibrium. The observation indicates that responsiveness of the immune system is important to contain the tumor in its nascent size. If there is long time delay in immune response, the tumor can grow in oscillatory fashion and eventually escape the control of the immune system. Parameter values used to generate the plots: $\rho = 2.5, {\beta = 0.02, \gamma = 5, K = 10}$
Computer simulated sample paths of the stochastic system (12) in comparison with its deterministic version (4). (a) tumor extinction in small noise regime; (b) tumor extinction in big noise regime; (c) monostable fluctuation; (d) bistable switching. Parameter values used here are summarized in Table 1
stationary distribution of (15) with $\tau = 0, 0.5, 1, 2$. The histogram is formed by 5000 samples of $u(1000)$
Parameter values used in Figure 3
 small noise induced extinction (a) big noise induced extinction (b) monostable (c) bistable (d) $\rho$ 0.12 2.5 1.5 2.5 $\beta$ 0.02 0.02 0.02 0.02 $\gamma$ 5 5 5 5 $K$ 10 10 10 10 $\sigma$ 0.3 5.66 0.65 0.65
 small noise induced extinction (a) big noise induced extinction (b) monostable (c) bistable (d) $\rho$ 0.12 2.5 1.5 2.5 $\beta$ 0.02 0.02 0.02 0.02 $\gamma$ 5 5 5 5 $K$ 10 10 10 10 $\sigma$ 0.3 5.66 0.65 0.65
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