# American Institute of Mathematical Sciences

October  2021, 26(10): 5707-5722. doi: 10.3934/dcdsb.2021041

## Survival analysis for tumor growth model with stochastic perturbation

 College of Data Science, Taiyuan University of Technology, Taiyuan, 030024, China

* Corresponding author: Dongxi Li

Received  October 2019 Revised  December 2020 Published  October 2021 Early access  February 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant No.11571009), and Applied Basic Research Programs of Shanxi Province (Grant No. 201901D111086)

In this paper, we investigate the dynamical behavior of extinction and survival in tumor growth model with immunization under stochastic perturbation. Firstly, the model describing the growth of cancer cells monitored by immune cells is established. Then, the steady probability distribution of tumor cells for different noise intensities and immune parameter intensities, and necessary conditions for extinction and different survival of cancer cells are obtained by numerical and theoretical method. Besides, it is found that the extinction and survival of cancer cells rely on the state of immunization and noise. Finally, stochastic simulations are taken to test the theoretical analytical results. The results of our work are beneficial to discover the evolution mechanism and design effective immunotherapy of tumor.

Citation: Dongxi Li, Ni Zhang, Ming Yan, Yanya Xing. Survival analysis for tumor growth model with stochastic perturbation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5707-5722. doi: 10.3934/dcdsb.2021041
##### References:
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Garay and P. Lefever, A kinetic approach to the immunology of cancer: Stationary states properties of effector-target cell reactions, Journal of Theoretical Biology, 73 (1978), 417-438.  doi: 10.1016/0022-5193(78)90150-9. [13] Q. Han, T. Yang, C. Zeng and H. Wang, Impact of time delays on stochastic resonance in an ecological system describing vegetation, Physica A, 408 (2014), 96-105. [14] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [15] R. Lefever and W. Horsthemke, Bistability in fluctuating environments. Implications in tumor immunology, Bulletin of Mathematical Biology, 41 (1979), 469-490. [16] R. Lefever and W. Horsthemk, Multiple transitions induced by light intensity fluctuations in illuminated chemical systems, Proceedings of the National Academy of Sciences, 76 (1979), 2490-2494.  doi: 10.1073/pnas.76.6.2490. [17] D. Li, W. Xu, C. Sun and L. Wang, Stochastic fluctuation induced the competition between extinction and recurrence in a model of tumor growth, Physics Letters A, 376 (2012), 1771-1776.  doi: 10.1016/j.physleta.2012.04.006. [18] D. Li, W. Xu, Y. Guo and Y. Xu, Fluctuations induced extinction and stochastic resonance effect in a model of tumor growth with periodic treatment, Physics Letters A, 375 (2011), 886-890.  doi: 10.1016/j.physleta.2010.12.066. [19] M. Liu and K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, Journal of Theoretical Biology, 264 (2010), 934-944.  doi: 10.1016/j.jtbi.2010.03.008. [20] M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, Journal of Mathematical Analysis and Applications, 375 (2011), 443-457.  doi: 10.1016/j.jmaa.2010.09.058. [21] M. Liu and K. 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Spagnolo, Coexistence of resonant activation and noise enhanced stability in a model of tumor-host interaction: Statistics of extinction times, Acta Phys. Pol. B, 37 (2006), 1651-1666. [27] C. Parish, Cancer immunotherapy: The past, the present and the future, Immunol. Cell Biol., 81 (2003), 106-113.  doi: 10.1046/j.0818-9641.2003.01151.x. [28] A. L. Pankratov and S. Bernardo, Suppression of timing errors in short overdamped josephson junctions, Physical Review Letters, 93 (2004), 177001. doi: 10.1103/PhysRevLett.93.177001. [29] N. Pizzolato, D. P. Adorno, D. Valenti and B. Spagnolo, Stochastic dynamics of leukemic cells under an intermittent targeted therapy, Theory in Biosciences, 130 (2011), 203-210. [30] T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208.  doi: 10.1137/S0036144504446291. [31] S. A. Rosenberg, P. Spiess and R. Lafreniere, A new approach to the adoptive immunotherapy of cancer with tumor-infiltrating lymphocytes, Science, 233 (1986), 1318-1321.  doi: 10.1126/science.3489291. [32] M. Smyth, D. Godfrey and J. Trapani, A fresh look at tumor immunosurveillance and immunotherapy, Nat. Immunol., 2 (2001), 293-299.  doi: 10.1038/86297. [33] D. Valenti, L. Tranchina, M. Brai, A. Caruso, C. Cosentino and B. Spagnolo, Environmental metal pollution considered as noise: Effects on the spatial distribution of benthic foraminifera in two coastal marine areas of Sicily (Southern Italy), Ecological Modelling, 213 (2008), 449-462. [34] D. Valenti, L. Schimansky-Geier, X. Sailer and B. Spagnolo, Moment equations for a spatially extended system of two competing species, Eur. Phys. J. B, 50 (2006), 199-203. [35] Q. Xie, T. Wang, C. Zeng, X. Dong and L. Guan, Predicting fluctuations-caused regime shifts in a time delayed dynamics of an invading species, Physica A, 493 (2018), 69-83. [36] Y. Xu, J. Feng, J. J. Li and H. Zhang, Stochastic bifurcation for a tumor-immune system with symmetric Levy noise, Physical A, 392 (2013), 4739-4748.  doi: 10.1016/j.physa.2013.06.010. [37] P. Zhivkov and J. Waniewski, Modelling tumour-immunity interactions with different stimulation functions, International Journal of Applied Mathematics and Computer Science, 13 (2003), 307-315. [38] P. Zhivkov and J. Waniewski, Modelling tumour-immunity interactions with different stimulation functions, Guangdong Journal of Animal and Veterinary Science, 13 (2003), 307-315. [39] C. Zeng and H. Wang, Noise and large time delay: Accelerated catastrophic regime shifts in ecosystems, Ecological Modelling, 233 (2012), 52-58. [40] C. Zeng, Q. Han, T. Yang, H. Wang and Z. Jia, Noise- and delay-induced regime shifts in an ecological system of vegetation, Journal of Statistical Mechanics: Theory and Experiment, 2013 (10)(2013), P10017. [41] C. Zeng, C. Zhang, J. Zeng and H. Luo, Noises-induced regime shifts and -enhanced stability under a model of lake approaching eutrophication, Ecological complexity, 22 (2015), 102-108. [42] J. Zeng, C. Zeng, Q. Xie, L. Guan, X. Dong and F. Yang, Different delays-induced regime shifts in a stochastic insect outbreak dynamics, Physica A, 462 (2016), 1273-1285.  doi: 10.1016/j.physa.2016.06.115. [43] C. Zeng, Q. Xie, T. Wang and C. Zhang, Stochastic ecological kinetics of regime shifts in a time-delayed lake eutrophication ecosystem, Ecosphere, 8 (2017), e01805. [44] W. R. Zhong, Y. Z. Shao and Z. H. He, Pure multiplicative stochastic resonance of a theoretical anti-tumor model with seasonal modulability, Physical Review E, 73 (2006), 06090. doi: 10.1103/PhysRevE.73.060902.

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##### References:
 [1] B. Q. Ai, X. J. Wang and L. G. Liu, Reply to "comment on correlated noise in a logistic growth model", Phys. Rev. E, 77 (2008), 013902. doi: 10.1103/PhysRevE.77.013902. [2] S. Banerjee and R. R. Sarkar, Delay-induced model for tumor–immune interaction and control of malignant tumor growth, Bio. Systems, 91 (2008), 268-288.  doi: 10.1016/j.biosystems.2007.10.002. [3] A. L. Barbera and B. Spagnolo, Spatio-temporal patterns in population dynamics, Physica A, 314 (2002), 120-124. [4] G. Berke, D. Gabison and M. Feldman, The frequency of effector cells in populations containing cytotoxic lymphocytes, European Journal of Immunology, 5 (2005), 813-818.  doi: 10.1002/eji.1830051204. [5] O. A. Chichigina, A. A. Dubkov, D. Valenti and B. Spagnolo, Stability in a system subject to noise with regulated periodicity, Phys. Rev. E, 84 (2011), 021134. [6] A. A. Dubkov and B. Spagnolo, Verhulst model with lévy white noise excitation, Eur. Phys. J. B, 65 (2008), 361-367. [7] L. C. Evans, An Introduction to Stochastic Differential Equations, Amer Mathematical Society, Providence, RI, 2013. doi: 10.1090/mbk/082. [8] A. Fiasconaro, B. Spagnolo, A. Ochab-Marcinek and E. Gudowska-Nowak, Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response, Physical Review E, 74 (2006), 159-163. [9] A. Fiasconaro, A. Ochab-Marcinek, B. Spagnolo and E. Gudowska-Nowak, Monitoring noise-resonant effects in cancer growth influenced by external fluctuations and periodic treatment, Eur. Phys. J. B, 65 (2008), 435-442. [10] A. Fiasconaro, B. Spagnolo and S. Boccaletti, Signatures of noise-enhanced stability in metastable states, Physical Review E Statistical Nonlinear and Soft Matter Physics, 72 (2006), 061110. [11] A. Fiasconaro, D. Valenti and B. Spagnolo, Nonmonotonic behavior of spatiotemporal pattern formation in a noisy Lotka-Volterra system, Acta Physica Polonica B, 35 (2004), 1491-1500. [12] R. P. Garay and P. Lefever, A kinetic approach to the immunology of cancer: Stationary states properties of effector-target cell reactions, Journal of Theoretical Biology, 73 (1978), 417-438.  doi: 10.1016/0022-5193(78)90150-9. [13] Q. Han, T. Yang, C. Zeng and H. Wang, Impact of time delays on stochastic resonance in an ecological system describing vegetation, Physica A, 408 (2014), 96-105. [14] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [15] R. Lefever and W. Horsthemke, Bistability in fluctuating environments. Implications in tumor immunology, Bulletin of Mathematical Biology, 41 (1979), 469-490. [16] R. Lefever and W. Horsthemk, Multiple transitions induced by light intensity fluctuations in illuminated chemical systems, Proceedings of the National Academy of Sciences, 76 (1979), 2490-2494.  doi: 10.1073/pnas.76.6.2490. [17] D. Li, W. Xu, C. Sun and L. Wang, Stochastic fluctuation induced the competition between extinction and recurrence in a model of tumor growth, Physics Letters A, 376 (2012), 1771-1776.  doi: 10.1016/j.physleta.2012.04.006. [18] D. Li, W. Xu, Y. Guo and Y. Xu, Fluctuations induced extinction and stochastic resonance effect in a model of tumor growth with periodic treatment, Physics Letters A, 375 (2011), 886-890.  doi: 10.1016/j.physleta.2010.12.066. [19] M. Liu and K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, Journal of Theoretical Biology, 264 (2010), 934-944.  doi: 10.1016/j.jtbi.2010.03.008. [20] M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, Journal of Mathematical Analysis and Applications, 375 (2011), 443-457.  doi: 10.1016/j.jmaa.2010.09.058. [21] M. Liu and K. Wang, A note on stability of stochastic logistic equation, Applied Mathematics Letters, 26 (2013), 601-606.  doi: 10.1016/j.aml.2012.12.015. [22] Z. Ma and T. G. Hallam, Effects of parameter fluctuations on community survival, Math. Biosci., 86 (1987), 35-49.  doi: 10.1016/0025-5564(87)90062-9. [23] X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in populations dynamics, Stochastic Process. Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0. [24] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997. [25] A. Ochab-Marcinek and E. Gudowska-Nowak, Population growth and control in stochastic models of cancer development, Physica A: Statistical Mechanics and Its Applications, 343 (2004), 557-572.  doi: 10.1016/j.physa.2004.06.071. [26] A. Ochab-Marcinek, E. Gudowska-Nowak, A. Fiasconaro and B. Spagnolo, Coexistence of resonant activation and noise enhanced stability in a model of tumor-host interaction: Statistics of extinction times, Acta Phys. Pol. B, 37 (2006), 1651-1666. [27] C. Parish, Cancer immunotherapy: The past, the present and the future, Immunol. Cell Biol., 81 (2003), 106-113.  doi: 10.1046/j.0818-9641.2003.01151.x. [28] A. L. Pankratov and S. Bernardo, Suppression of timing errors in short overdamped josephson junctions, Physical Review Letters, 93 (2004), 177001. doi: 10.1103/PhysRevLett.93.177001. [29] N. Pizzolato, D. P. Adorno, D. Valenti and B. Spagnolo, Stochastic dynamics of leukemic cells under an intermittent targeted therapy, Theory in Biosciences, 130 (2011), 203-210. [30] T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208.  doi: 10.1137/S0036144504446291. [31] S. A. Rosenberg, P. Spiess and R. Lafreniere, A new approach to the adoptive immunotherapy of cancer with tumor-infiltrating lymphocytes, Science, 233 (1986), 1318-1321.  doi: 10.1126/science.3489291. [32] M. Smyth, D. Godfrey and J. Trapani, A fresh look at tumor immunosurveillance and immunotherapy, Nat. Immunol., 2 (2001), 293-299.  doi: 10.1038/86297. [33] D. Valenti, L. Tranchina, M. Brai, A. Caruso, C. Cosentino and B. Spagnolo, Environmental metal pollution considered as noise: Effects on the spatial distribution of benthic foraminifera in two coastal marine areas of Sicily (Southern Italy), Ecological Modelling, 213 (2008), 449-462. [34] D. Valenti, L. Schimansky-Geier, X. Sailer and B. Spagnolo, Moment equations for a spatially extended system of two competing species, Eur. Phys. J. B, 50 (2006), 199-203. [35] Q. Xie, T. Wang, C. Zeng, X. Dong and L. Guan, Predicting fluctuations-caused regime shifts in a time delayed dynamics of an invading species, Physica A, 493 (2018), 69-83. [36] Y. Xu, J. Feng, J. J. Li and H. Zhang, Stochastic bifurcation for a tumor-immune system with symmetric Levy noise, Physical A, 392 (2013), 4739-4748.  doi: 10.1016/j.physa.2013.06.010. [37] P. Zhivkov and J. Waniewski, Modelling tumour-immunity interactions with different stimulation functions, International Journal of Applied Mathematics and Computer Science, 13 (2003), 307-315. [38] P. Zhivkov and J. Waniewski, Modelling tumour-immunity interactions with different stimulation functions, Guangdong Journal of Animal and Veterinary Science, 13 (2003), 307-315. [39] C. Zeng and H. Wang, Noise and large time delay: Accelerated catastrophic regime shifts in ecosystems, Ecological Modelling, 233 (2012), 52-58. [40] C. Zeng, Q. Han, T. Yang, H. Wang and Z. Jia, Noise- and delay-induced regime shifts in an ecological system of vegetation, Journal of Statistical Mechanics: Theory and Experiment, 2013 (10)(2013), P10017. [41] C. Zeng, C. Zhang, J. Zeng and H. Luo, Noises-induced regime shifts and -enhanced stability under a model of lake approaching eutrophication, Ecological complexity, 22 (2015), 102-108. [42] J. Zeng, C. Zeng, Q. Xie, L. Guan, X. Dong and F. Yang, Different delays-induced regime shifts in a stochastic insect outbreak dynamics, Physica A, 462 (2016), 1273-1285.  doi: 10.1016/j.physa.2016.06.115. [43] C. Zeng, Q. Xie, T. Wang and C. Zhang, Stochastic ecological kinetics of regime shifts in a time-delayed lake eutrophication ecosystem, Ecosphere, 8 (2017), e01805. [44] W. R. Zhong, Y. Z. Shao and Z. H. He, Pure multiplicative stochastic resonance of a theoretical anti-tumor model with seasonal modulability, Physical Review E, 73 (2006), 06090. doi: 10.1103/PhysRevE.73.060902.
The potential $U(x)$ as a function of $x$ for different value $\beta$ with $\theta = 0.25$
The steady probability distribution function of tumor cells population $x(t)$ for model (4) with $\theta = 0.25$
The valid regions as a function of $\theta(t)$ and $\beta(t)$
Solutions of extinction of tumor cells for $(a):\sigma^2(t) = 0.02+0.004\sin t$, $\beta(t) = 3+\sin t$; $(b):\sigma^2(t) = 1+0.8\sin t$, $\beta(t) = 3+\sin t$; $(c):\sigma^2(t) = 1+0.8\sin t$, $\beta(t) = 6+\sin t$, with the initial value $x_0 = 0.5$
Solutions of strong persistence in the mean of tumor cells for $(d):\sigma^2(t) = 0.002+0.001\sin t$, $\beta(t) = 0.97+0.009\sin t$, with the initial value $x_0 = 0.5$
Solutions of strong persistence in the mean of tumor for $(e):\sigma^2(t) = 0.02+0.002\sin t$, $\beta(t) = 0.8+0.008\sin t$, with the initial value $x_0 = 0.5$
Mean time to extinction (MET) as a function of the noise intensity for different values of $\beta$ at $\theta = 0.25$
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