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  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 & Continuous Dynamical Systems - B, 2021, 26 (10) : 5707-5722. doi: 10.3934/dcdsb.2021041
References:
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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.  Google Scholar

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A. L. Barbera and B. Spagnolo, Spatio-temporal patterns in population dynamics, Physica A, 314 (2002), 120-124.   Google Scholar

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G. BerkeD. 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.  Google Scholar

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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. Google Scholar

[6]

A. A. Dubkov and B. Spagnolo, Verhulst model with lévy white noise excitation, Eur. Phys. J. B, 65 (2008), 361-367.   Google Scholar

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L. C. Evans, An Introduction to Stochastic Differential Equations, Amer Mathematical Society, Providence, RI, 2013. doi: 10.1090/mbk/082.  Google Scholar

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A. FiasconaroB. SpagnoloA. 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.   Google Scholar

[9]

A. FiasconaroA. Ochab-MarcinekB. 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.   Google Scholar

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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. Google Scholar

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A. FiasconaroD. Valenti and B. Spagnolo, Nonmonotonic behavior of spatiotemporal pattern formation in a noisy Lotka-Volterra system, Acta Physica Polonica B, 35 (2004), 1491-1500.   Google Scholar

[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.  Google Scholar

[13]

Q. HanT. YangC. Zeng and H. Wang, Impact of time delays on stochastic resonance in an ecological system describing vegetation, Physica A, 408 (2014), 96-105.   Google Scholar

[14]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[15]

R. Lefever and W. Horsthemke, Bistability in fluctuating environments. Implications in tumor immunology, Bulletin of Mathematical Biology, 41 (1979), 469-490.   Google Scholar

[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.  Google Scholar

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D. LiW. XuC. 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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

X. MaoG. 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.  Google Scholar

[24]

X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997.  Google Scholar

[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.  Google Scholar

[26]

A. Ochab-MarcinekE. Gudowska-NowakA. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

N. PizzolatoD. P. AdornoD. Valenti and B. Spagnolo, Stochastic dynamics of leukemic cells under an intermittent targeted therapy, Theory in Biosciences, 130 (2011), 203-210.   Google Scholar

[30]

T. RooseS. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208.  doi: 10.1137/S0036144504446291.  Google Scholar

[31]

S. A. RosenbergP. 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.  Google Scholar

[32]

M. SmythD. Godfrey and J. Trapani, A fresh look at tumor immunosurveillance and immunotherapy, Nat. Immunol., 2 (2001), 293-299.  doi: 10.1038/86297.  Google Scholar

[33]

D. ValentiL. TranchinaM. BraiA. CarusoC. 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.   Google Scholar

[34]

D. ValentiL. Schimansky-GeierX. Sailer and B. Spagnolo, Moment equations for a spatially extended system of two competing species, Eur. Phys. J. B, 50 (2006), 199-203.   Google Scholar

[35]

Q. XieT. WangC. ZengX. Dong and L. Guan, Predicting fluctuations-caused regime shifts in a time delayed dynamics of an invading species, Physica A, 493 (2018), 69-83.   Google Scholar

[36]

Y. XuJ. FengJ. 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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[39]

C. Zeng and H. Wang, Noise and large time delay: Accelerated catastrophic regime shifts in ecosystems, Ecological Modelling, 233 (2012), 52-58.   Google Scholar

[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. Google Scholar

[41]

C. ZengC. ZhangJ. Zeng and H. Luo, Noises-induced regime shifts and -enhanced stability under a model of lake approaching eutrophication, Ecological complexity, 22 (2015), 102-108.   Google Scholar

[42]

J. ZengC. ZengQ. XieL. GuanX. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

A. L. Barbera and B. Spagnolo, Spatio-temporal patterns in population dynamics, Physica A, 314 (2002), 120-124.   Google Scholar

[4]

G. BerkeD. 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.  Google Scholar

[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. Google Scholar

[6]

A. A. Dubkov and B. Spagnolo, Verhulst model with lévy white noise excitation, Eur. Phys. J. B, 65 (2008), 361-367.   Google Scholar

[7]

L. C. Evans, An Introduction to Stochastic Differential Equations, Amer Mathematical Society, Providence, RI, 2013. doi: 10.1090/mbk/082.  Google Scholar

[8]

A. FiasconaroB. SpagnoloA. 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.   Google Scholar

[9]

A. FiasconaroA. Ochab-MarcinekB. 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.   Google Scholar

[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. Google Scholar

[11]

A. FiasconaroD. Valenti and B. Spagnolo, Nonmonotonic behavior of spatiotemporal pattern formation in a noisy Lotka-Volterra system, Acta Physica Polonica B, 35 (2004), 1491-1500.   Google Scholar

[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.  Google Scholar

[13]

Q. HanT. YangC. Zeng and H. Wang, Impact of time delays on stochastic resonance in an ecological system describing vegetation, Physica A, 408 (2014), 96-105.   Google Scholar

[14]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[15]

R. Lefever and W. Horsthemke, Bistability in fluctuating environments. Implications in tumor immunology, Bulletin of Mathematical Biology, 41 (1979), 469-490.   Google Scholar

[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.  Google Scholar

[17]

D. LiW. XuC. 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.  Google Scholar

[18]

D. LiW. XuY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

X. MaoG. 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.  Google Scholar

[24]

X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997.  Google Scholar

[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.  Google Scholar

[26]

A. Ochab-MarcinekE. Gudowska-NowakA. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

N. PizzolatoD. P. AdornoD. Valenti and B. Spagnolo, Stochastic dynamics of leukemic cells under an intermittent targeted therapy, Theory in Biosciences, 130 (2011), 203-210.   Google Scholar

[30]

T. RooseS. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208.  doi: 10.1137/S0036144504446291.  Google Scholar

[31]

S. A. RosenbergP. 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.  Google Scholar

[32]

M. SmythD. Godfrey and J. Trapani, A fresh look at tumor immunosurveillance and immunotherapy, Nat. Immunol., 2 (2001), 293-299.  doi: 10.1038/86297.  Google Scholar

[33]

D. ValentiL. TranchinaM. BraiA. CarusoC. 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.   Google Scholar

[34]

D. ValentiL. Schimansky-GeierX. Sailer and B. Spagnolo, Moment equations for a spatially extended system of two competing species, Eur. Phys. J. B, 50 (2006), 199-203.   Google Scholar

[35]

Q. XieT. WangC. ZengX. Dong and L. Guan, Predicting fluctuations-caused regime shifts in a time delayed dynamics of an invading species, Physica A, 493 (2018), 69-83.   Google Scholar

[36]

Y. XuJ. FengJ. 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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[39]

C. Zeng and H. Wang, Noise and large time delay: Accelerated catastrophic regime shifts in ecosystems, Ecological Modelling, 233 (2012), 52-58.   Google Scholar

[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. Google Scholar

[41]

C. ZengC. ZhangJ. Zeng and H. Luo, Noises-induced regime shifts and -enhanced stability under a model of lake approaching eutrophication, Ecological complexity, 22 (2015), 102-108.   Google Scholar

[42]

J. ZengC. ZengQ. XieL. GuanX. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

Figure 1.  The potential $ U(x) $ as a function of $ x $ for different value $ \beta $ with $ \theta = 0.25 $
Figure 2.  The steady probability distribution function of tumor cells population $ x(t) $ for model (4) with $ \theta = 0.25 $
Figure 3.  The valid regions as a function of $ \theta(t) $ and $ \beta(t) $
Figure 4.  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 $
Figure 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 $
Figure 6.  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 $
Figure 7.  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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