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## Complicated dynamics of tumor-immune system interaction model with distributed time delay

 1 College of Science and Engineering, Aoyama Gakuin University, Sagamihara 252-5258, Japan 2 School of Mathematics and Physics, China University of Geoscience, Wuhan 430074, China 3 School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China 4 School of Mathematics and Physics, Aoyama Gakuin University, Sagamihara 252-5258, Japan

*Corresponding author: ypdong@mail.ccnu.edu.cn

Received  April 2019 Revised  August 2019 Published  December 2019

In this paper, we propose a distributed delay model to investigate the dynamics of the interactions between tumor and immune system. And we choose a special form of delay kernel which combines two delay kernels: a monotonic delay kernel representing a fading memory and a nonmonotonic delay kernel describing a peaking memory. Then, we discuss the effect of such delay kernel on system dynamics. The results show that the introduction of nonmonotonic delay kernel does not change the stability of tumor-free equilibrium, but it can induce stability switches of tumor-presence equilibrium and cause a rich pattern of dynamical behaviors including stabilization. Moreover, our numerical simulation results reveal that the nonmonotonic delay kernel has more complicated effects on the stability compared with the monotonic delay kernel.

Citation: 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, doi: 10.3934/dcdsb.2020015
##### References:
 [1] J. A. Adam and N. Bellomo, A Survey of Models for Tumor-Immune System Dynamics, Birkhauser, Boston, 1997. doi: 10.1007/978-0-8176-8119-7.  Google Scholar [2] A. Arabameri, D. Asemani and J. Hadjati, A structural methodology for modeling immune-tumor interactions including pro- and anti-tumor factors for clinical applications, Math. Biosci., 304 (2018), 48-61.  doi: 10.1016/j.mbs.2018.07.006.  Google Scholar [3] S. Banerjee and R. R. Sarkar, Delay-induced model for tumor-immune interaction and control of malignant tumor growth, BioSystems, 91 (2008), 268-288.  doi: 10.1016/j.biosystems.2007.10.002.  Google Scholar [4] P. Bi and S. G. Ruan, Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Appl. Dyn. Syst., 12 (2013), 1847-1888.  doi: 10.1137/120887898.  Google Scholar [5] P. Bi, S. G. Ruan and X. N. Zhang, Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays, Chaos, 24 (2014), 023101, 16 pp. doi: 10.1063/1.4870363.  Google Scholar [6] M. Bodnar and U. Foryś, Delays do not cause oscillations in a corrected model of humoral me diate d immune response, Appl. Math. Comput., 289 (2016), 7-21.  doi: 10.1016/j.amc.2016.05.006.  Google Scholar [7] L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958.  doi: 10.1158/0008-5472.CAN-05-0564.  Google Scholar [8] L. G. de Pillis and A. E. Radunskaya, Modeling tumor-immune dynamics, in Mathematical Modeling of Tumor-Immune System Dynamics, Springer Proc. Math. Stat., Springer, New York, 107 (2014), 59-108. doi: 10.1007/978-1-4939-1793-8.  Google Scholar [9] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235.  doi: 10.1016/j.physd.2005.06.032.  Google Scholar [10] A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model., 47 (2008), 614-637.  doi: 10.1016/j.mcm.2007.02.032.  Google Scholar [11] A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction, Math. Comput. Model., 51 (2010), 572-591.  doi: 10.1016/j.mcm.2009.11.005.  Google Scholar [12] Y. P. Dong, R. Miyazaki and Y. Takeuchi, Mathematical modeling on helper T cells in a tumor immune system, Discrete Continuous Dynam. Systems-B, 19 (2014), 55-72.  doi: 10.3934/dcdsb.2014.19.55.  Google Scholar [13] Y. P. Dong, G. Huang, R. Miyazaki and Y. Takeuchi, Dynamics in a tumor immune system with time delays, Appl. Math. Comput., 252 (2015), 99-113.  doi: 10.1016/j.amc.2014.11.096.  Google Scholar [14] 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.  Google Scholar [15] S. Eikenberry, C. Thalhauser and Y. Kuang, Tumor-immune interaction, surgical treatment, and cancer recurrence in a mathematical model of melanoma, PLoS Comput. Biol., 5 (2009), e1000362, 18 pp. doi: 10.1371/journal.pcbi.1000362.  Google Scholar [16] B. Ermentrout, Simulating, Analyzing, and Animating DynamicalSystems: A Guide to XPPAUT for Researchers and Students, Software, Environments, and Tools, 14. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898718195.  Google Scholar [17] U. Foryś, J. Waniewski and P. Zhivkov, Anti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapy, J. Biol. Syst., 14 (2006), 13-30.  doi: 10.1142/S0218339006001702.  Google Scholar [18] M. Galach, Dynamics of the tumor-immune system competition-the effect of time delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395-406.   Google Scholar [19] M. Iannelli and A. Pugliese, An Introduction to Mathematical Population Dynamics, Along the Trail of Volterra and Lotka. Unitext, 79. La Matematica per il 3+2. Springer, Cham, 2014. doi: 10.1007/978-3-319-03026-5.  Google Scholar [20] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252.  doi: 10.1007/s002850050127.  Google Scholar [21] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993.   Google Scholar [22] 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.   Google Scholar [23] N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge Studies in Mathematical Biology, 8. Cambridge University Press, Cambridge, 1989.   Google Scholar [24] K. J. Mahasa, R. Ouifki, A. Eladdadi and L. de Pillis, Mathematical model of tumor-immune surveillance, J. Theor. Biol., 404 (2016), 312-330.  doi: 10.1016/j.jtbi.2016.06.012.  Google Scholar [25] D. G. Mallet and L. G. de Pillis, A cellular automata model of tumor-immune system interactions, J. Theor. Biol., 239 (2006), 334-350.  doi: 10.1016/j.jtbi.2005.08.002.  Google Scholar [26] M. J. Piotrowska, M. Bodnar, J. Poleszczuk and U. Foryś, Mathematical modelling of immune reaction against gliomas: sensitivity analysis and influence of delays, Nonlinear Anal. Real World Appl., 14 (2013), 1601-1620.  doi: 10.1016/j.nonrwa.2012.10.020.  Google Scholar [27] M. J. Piotrowska, An immune system-tumour interactions model with discrete time delay: Model analysis and validation, Commun. Nonlinear Sci. Numer. Simulat., 34 (2016), 185-198.  doi: 10.1016/j.cnsns.2015.10.022.  Google Scholar [28] M. J. Piotrowska and M. Bodnar, Influence of distributed delays on the dynamics of a generalized immune system cancerous cells interactions model, Commun. Nonlinear Sci. Numer. Simulat., 54 (2018), 389-415.  doi: 10.1016/j.cnsns.2017.06.003.  Google Scholar [29] F. A. Rihan, D. H. Abdel Rahman, S. Lakshmanan and A. S. Alkhajeh, A time delay model of tumour-immune system interactions: global dynamics, parameter estimation, sensitivity analysis, Appl. Math. Comput., 232 (2014), 606-623.  doi: 10.1016/j.amc.2014.01.111.  Google Scholar [30] E. D. Sontag, A dynamic model of immune responses to antigen presentation predicts different regions of tumor or pathogen elimination, Cell Syst., 4 (2017), 231-241.  doi: 10.1016/j.cels.2016.12.003.  Google Scholar [31] J. Stark, C. Chan and A. J. T. George, Oscillations in the immune system, Immunol. Rev., 216 (2007), 213-231.  doi: 10.1111/j.1600-065X.2007.00501.x.  Google Scholar [32] K. Thurley, L. F. Wu and S. J. Altschuler, Modeling cell-to-cell communication networks using response-time distributions, Cell Syst., 6 (2018), 355-367.  doi: 10.1016/j.cels.2018.01.016.  Google Scholar [33] A. Tsygvintsev, S. Marino and D. E. Kirschner, A mathematical model of gene therapy for the treatment of cancer, Mathematical Methods and Models in Biomedicine, Lect. Notes Math. Model. Life Sci., Springer, New York, (2013), 367–385. doi: 10.1007/978-1-4614-4178-6.  Google Scholar [34] 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.  Google Scholar [35] K. P. Wilkie and P. Hahnfeldt, Modeling the dichotomy of the immune response to cancer: Cytotoxic effects and tumor-promoting inflammation, Bull. Math. Biol., 79 (2017), 1426-1448.  doi: 10.1007/s11538-017-0291-4.  Google Scholar [36] M. Yu, Y. P. Dong and Y. Takeuchi, Dual role of delay effects in a tumor-immune system, J. Biol. Dyn., 11 (2017), 334-347.  doi: 10.1080/17513758.2016.1231347.  Google Scholar

show all references

##### References:
 [1] J. A. Adam and N. Bellomo, A Survey of Models for Tumor-Immune System Dynamics, Birkhauser, Boston, 1997. doi: 10.1007/978-0-8176-8119-7.  Google Scholar [2] A. Arabameri, D. Asemani and J. Hadjati, A structural methodology for modeling immune-tumor interactions including pro- and anti-tumor factors for clinical applications, Math. Biosci., 304 (2018), 48-61.  doi: 10.1016/j.mbs.2018.07.006.  Google Scholar [3] S. Banerjee and R. R. Sarkar, Delay-induced model for tumor-immune interaction and control of malignant tumor growth, BioSystems, 91 (2008), 268-288.  doi: 10.1016/j.biosystems.2007.10.002.  Google Scholar [4] P. Bi and S. G. Ruan, Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Appl. Dyn. Syst., 12 (2013), 1847-1888.  doi: 10.1137/120887898.  Google Scholar [5] P. Bi, S. G. Ruan and X. N. Zhang, Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays, Chaos, 24 (2014), 023101, 16 pp. doi: 10.1063/1.4870363.  Google Scholar [6] M. Bodnar and U. Foryś, Delays do not cause oscillations in a corrected model of humoral me diate d immune response, Appl. Math. Comput., 289 (2016), 7-21.  doi: 10.1016/j.amc.2016.05.006.  Google Scholar [7] L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958.  doi: 10.1158/0008-5472.CAN-05-0564.  Google Scholar [8] L. G. de Pillis and A. E. Radunskaya, Modeling tumor-immune dynamics, in Mathematical Modeling of Tumor-Immune System Dynamics, Springer Proc. Math. Stat., Springer, New York, 107 (2014), 59-108. doi: 10.1007/978-1-4939-1793-8.  Google Scholar [9] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235.  doi: 10.1016/j.physd.2005.06.032.  Google Scholar [10] A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model., 47 (2008), 614-637.  doi: 10.1016/j.mcm.2007.02.032.  Google Scholar [11] A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction, Math. Comput. Model., 51 (2010), 572-591.  doi: 10.1016/j.mcm.2009.11.005.  Google Scholar [12] Y. P. Dong, R. Miyazaki and Y. Takeuchi, Mathematical modeling on helper T cells in a tumor immune system, Discrete Continuous Dynam. Systems-B, 19 (2014), 55-72.  doi: 10.3934/dcdsb.2014.19.55.  Google Scholar [13] Y. P. Dong, G. Huang, R. Miyazaki and Y. Takeuchi, Dynamics in a tumor immune system with time delays, Appl. Math. Comput., 252 (2015), 99-113.  doi: 10.1016/j.amc.2014.11.096.  Google Scholar [14] 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.  Google Scholar [15] S. Eikenberry, C. Thalhauser and Y. Kuang, Tumor-immune interaction, surgical treatment, and cancer recurrence in a mathematical model of melanoma, PLoS Comput. Biol., 5 (2009), e1000362, 18 pp. doi: 10.1371/journal.pcbi.1000362.  Google Scholar [16] B. Ermentrout, Simulating, Analyzing, and Animating DynamicalSystems: A Guide to XPPAUT for Researchers and Students, Software, Environments, and Tools, 14. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898718195.  Google Scholar [17] U. Foryś, J. Waniewski and P. Zhivkov, Anti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapy, J. Biol. Syst., 14 (2006), 13-30.  doi: 10.1142/S0218339006001702.  Google Scholar [18] M. Galach, Dynamics of the tumor-immune system competition-the effect of time delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395-406.   Google Scholar [19] M. Iannelli and A. Pugliese, An Introduction to Mathematical Population Dynamics, Along the Trail of Volterra and Lotka. Unitext, 79. La Matematica per il 3+2. Springer, Cham, 2014. doi: 10.1007/978-3-319-03026-5.  Google Scholar [20] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252.  doi: 10.1007/s002850050127.  Google Scholar [21] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993.   Google Scholar [22] 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.   Google Scholar [23] N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge Studies in Mathematical Biology, 8. Cambridge University Press, Cambridge, 1989.   Google Scholar [24] K. J. Mahasa, R. Ouifki, A. Eladdadi and L. de Pillis, Mathematical model of tumor-immune surveillance, J. Theor. Biol., 404 (2016), 312-330.  doi: 10.1016/j.jtbi.2016.06.012.  Google Scholar [25] D. G. Mallet and L. G. de Pillis, A cellular automata model of tumor-immune system interactions, J. Theor. Biol., 239 (2006), 334-350.  doi: 10.1016/j.jtbi.2005.08.002.  Google Scholar [26] M. J. Piotrowska, M. Bodnar, J. Poleszczuk and U. Foryś, Mathematical modelling of immune reaction against gliomas: sensitivity analysis and influence of delays, Nonlinear Anal. Real World Appl., 14 (2013), 1601-1620.  doi: 10.1016/j.nonrwa.2012.10.020.  Google Scholar [27] M. J. Piotrowska, An immune system-tumour interactions model with discrete time delay: Model analysis and validation, Commun. Nonlinear Sci. Numer. Simulat., 34 (2016), 185-198.  doi: 10.1016/j.cnsns.2015.10.022.  Google Scholar [28] M. J. Piotrowska and M. Bodnar, Influence of distributed delays on the dynamics of a generalized immune system cancerous cells interactions model, Commun. Nonlinear Sci. Numer. Simulat., 54 (2018), 389-415.  doi: 10.1016/j.cnsns.2017.06.003.  Google Scholar [29] F. A. Rihan, D. H. Abdel Rahman, S. Lakshmanan and A. S. Alkhajeh, A time delay model of tumour-immune system interactions: global dynamics, parameter estimation, sensitivity analysis, Appl. Math. Comput., 232 (2014), 606-623.  doi: 10.1016/j.amc.2014.01.111.  Google Scholar [30] E. D. Sontag, A dynamic model of immune responses to antigen presentation predicts different regions of tumor or pathogen elimination, Cell Syst., 4 (2017), 231-241.  doi: 10.1016/j.cels.2016.12.003.  Google Scholar [31] J. Stark, C. Chan and A. J. T. George, Oscillations in the immune system, Immunol. Rev., 216 (2007), 213-231.  doi: 10.1111/j.1600-065X.2007.00501.x.  Google Scholar [32] K. Thurley, L. F. Wu and S. J. Altschuler, Modeling cell-to-cell communication networks using response-time distributions, Cell Syst., 6 (2018), 355-367.  doi: 10.1016/j.cels.2018.01.016.  Google Scholar [33] A. Tsygvintsev, S. Marino and D. E. Kirschner, A mathematical model of gene therapy for the treatment of cancer, Mathematical Methods and Models in Biomedicine, Lect. Notes Math. Model. Life Sci., Springer, New York, (2013), 367–385. doi: 10.1007/978-1-4614-4178-6.  Google Scholar [34] 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.  Google Scholar [35] K. P. Wilkie and P. Hahnfeldt, Modeling the dichotomy of the immune response to cancer: Cytotoxic effects and tumor-promoting inflammation, Bull. Math. Biol., 79 (2017), 1426-1448.  doi: 10.1007/s11538-017-0291-4.  Google Scholar [36] M. Yu, Y. P. Dong and Y. Takeuchi, Dual role of delay effects in a tumor-immune system, J. Biol. Dyn., 11 (2017), 334-347.  doi: 10.1080/17513758.2016.1231347.  Google Scholar
The black curves represent relatively small $b$ and imply strong delay. The red curves represent relatively large $b$ and imply weak delay. (a) The dashed curves correspond to the monotonic function $F_1(t)$ with respect to $t$. (b) The solid curves correspond to the nonmonotonic function $F_2(t)$ with respect to $t$
(a) The monotonic decreasing function $D_4$ with respect to $c_2$. (b) The quadratic function $H(D_4)$ with respect to $D_4$
Log plots for the stability regions of tumor-presence equilibrium $E^+$ and stability boundary curves representing by black curves in $c_2-b$ parameter plane. (a) $E^+$ is stable in the area outside these two stability boundary curves, while $E^+$ is unstable in the area between these two stability boundary curves. (b) $E^+$ is stable in the area below the stability boundary curve, while $E^+$ is unstable in the area above the stability boundary curve
Stability boundary curves of $E^+$ are shown in $\rho-\omega$ parameter plane for different $b$. $E^+$ is stable below each stability boundary curve, while $E^+$ is unstable above each stability boundary curve. (a) We fix $c_2 = 0.2(<0.5)$. (b) We fix $c_2 = 0.8(>0.5)$
Stability boundary curves of $E^+$ are shown in $\rho-\omega$ parameter plane for different $c_2$. $E^+$ is stable below each stability boundary curve, while $E^+$ is unstable above each stability boundary curve. (a) We fix $b = 0.001$. (b) We fix $b = 0.1$
Schematic diagram of $F(t)$ with respect to $t$ for different $c_2$ and fixed $b$
The function $F(t)$ with respect to $t$. The black curves are given for relatively small $b$ and imply strong delay. The red curves are given for relatively large $b$ and imply weak delay. (a) We fix $c_2 = 0.2(<0.5)$. The dashed curves correspond to relatively small $c_2$. (b) We fix $c_2 = 0.8(>0.5)$. The solid curves correspond to relatively large $c_2$
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