October 2018, 15(5): 1077-1098. doi: 10.3934/mbe.2018048

Stochastic dynamics and survival analysis of a cell population model with random perturbations

Department of Mathematics and Statistics, Grant MacEwan University, Edmonton, AB T5J 4S2, Canada

* Corresponding author: Cristina Anton

Received  March 22, 2017 Accepted  April 06, 2018 Published  May 2018

Fund Project: The first author is supported by NSRC grant DDG-2015-00041

We consider a model based on the logistic equation and linear kinetics to study the effect of toxicants with various initial concentrations on a cell population. To account for parameter uncertainties, in our model the coefficients of the linear and the quadratic terms of the logistic equation are affected by noise. We show that the stochastic model has a unique positive solution and we find conditions for extinction and persistence of the cell population. In case of persistence we find the stationary distribution. The analytical results are confirmed by Monte Carlo simulations.

Citation: Cristina Anton, Alan Yong. Stochastic dynamics and survival analysis of a cell population model with random perturbations. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1077-1098. doi: 10.3934/mbe.2018048
References:
[1]

C. AntonJ. DengY. WongY. ZhangW. ZhangS. GabosD. Huang and C. Jin, Modeling and simulation for toxicity assessment, Math. BioSci. Eng., 14 (2017), 581-606. doi: 10.3934/mbe.2017034.

[2]

G. K. Basak and R. Bhattcharya, Stability in distribution for a class of singular diffusions, Ann. Prob., 20 (1992), 312-321. doi: 10.1214/aop/1176989928.

[3]

A. Friedman, Stochastic Differential Equations and Applications, Dover, New York, 2006.

[4]

A. GreyD. GreenhalghL. HuX. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM. J. Appl. Math., 71 (2011), 876-902. doi: 10.1137/10081856X.

[5]

T. HallamC. Clark and G. Jordan, Effects of toxicants on populations: A qualitative approach Ⅱ. First order kinetics, J. Math. Biology, 18 (1983), 25-37. doi: 10.1007/BF00275908.

[6]

R. Z. Hasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, 2012, 2nd ed. doi: 10.1007/978-3-642-23280-0.

[7]

J. He and K. Wang, The survival analysis for a population in a polluted environment, Nonlinear Analysis: Real World Applications, 10 (2009), 1555-1571. doi: 10.1016/j.nonrwa.2008.01.027.

[8]

C. JiD. JiangN. Shi and D. O'Regan, Existence, uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation, Math. Methods in the Appl. Sciences, 30 (2007), 77-89. doi: 10.1002/mma.778.

[9]

D. Jiang and N. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172. doi: 10.1016/j.jmaa.2004.08.027.

[10]

D. JiangN. Shi and X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597. doi: 10.1016/j.jmaa.2007.08.014.

[11]

J. JiaoW. Long and L. Chen, A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin, Nonlinear Analysis: Real World Applications, 10 (2009), 3073-3081. doi: 10.1016/j.nonrwa.2008.10.007.

[12]

P. Kloeden and E. Platen, Numerical Solutions of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[13]

Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2004. doi: 10.1007/978-1-4471-3866-2.

[14]

V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol. Ⅰ, Academic Press, New York, 1969.

[15]

M. Liu and K. Wang, Survival analysis of stochastic single-species population models in polluted environments, Ecological Modelling, 220 (2009), 1347-1357.

[16]

M. LiuK. Wang and Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969-2012. doi: 10.1007/s11538-010-9569-5.

[17]

X. Mao, Stochastic Differential Equations and Applications, Woodhead Pubilshing, Philadelphia, 2011, 2nd ed. doi: 10.1533/9780857099402.

[18]

X. MaoG. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in population dynamics, Markov Proc. and Their Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0.

[19]

X. MaoS. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0.

[20]

T. PanB. HuangW. ZhangS. GabosD. Huang and V. Devendran, Cytotoxicity assessment based on the AUC50 using multi-concentration time-dependent cellular response curves, Anal. Chim. Acta, 764 (2013), 44-52.

[21]

S. Pinheiro, On a logistic growth model with predation and power-type diffusion coefficient: Ⅰ. Existence of solutions and extinction criteria, Math. Meth. Appl. Sci., 38 (2015), 4912-4930. doi: 10.1002/mma.3413.

[22]

S. Resnik, A Probability Path, Birkhauser, Boston, 1999.

[23]

Z. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physica A, 451 (2016), 507-518. doi: 10.1016/j.physa.2016.01.084.

[24]

F. Wei and L. Chen, Psychological effect on single-species population models in a polluted environment, Math. Biosci., 290 (2017), 22-30. doi: 10.1016/j.mbs.2017.05.011.

[25]

F. WeiS. Geritz and J. Cai, A stochastic single-species population model with partial pollution tolerance in a polluted environment, Appl. Math. Letters, 63 (2017), 130-136. doi: 10.1016/j.aml.2016.07.026.

[26]

Z. XiS. KhareA. CheungB. HuangT. PanW. ZhangF. IbrahimC. Jin and S. Gabos, Mode of action classification of chemicals using multi-concentration time-dependent cellular response profiles, Comp. Biol. Chem., 49 (2014), 23-35. doi: 10.1016/j.compbiolchem.2013.12.004.

[27]

J. XingL. ZhuS. Gabos and L. Xie, Microelectronic cell sensor assay for detection of cytotoxicity and prediction of acute toxicity, Toxicology in Vitro, 20 (2006), 995-1004. doi: 10.1016/j.tiv.2005.12.008.

[28]

Q. YangD. JiangN. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248-271. doi: 10.1016/j.jmaa.2011.11.072.

[29]

Q. Yang and X. Mao, Stochastic dynamics of SIRS epidemic models with random perturbation, Math. BioSci. Eng., 11 (2014), 1003-1025. doi: 10.3934/mbe.2014.11.1003.

[30]

Y. Zhang, Y. Wong, J. Deng, C. Anton, J. Deng, S. Gabos, W. Zhang, D. Huang and C. Jin, Machine learning algorithms for mode-of-action classification in toxicity assessment, BioData Mining, 9 (2016), p19. doi: 10.1186/s13040-016-0098-0.

show all references

References:
[1]

C. AntonJ. DengY. WongY. ZhangW. ZhangS. GabosD. Huang and C. Jin, Modeling and simulation for toxicity assessment, Math. BioSci. Eng., 14 (2017), 581-606. doi: 10.3934/mbe.2017034.

[2]

G. K. Basak and R. Bhattcharya, Stability in distribution for a class of singular diffusions, Ann. Prob., 20 (1992), 312-321. doi: 10.1214/aop/1176989928.

[3]

A. Friedman, Stochastic Differential Equations and Applications, Dover, New York, 2006.

[4]

A. GreyD. GreenhalghL. HuX. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM. J. Appl. Math., 71 (2011), 876-902. doi: 10.1137/10081856X.

[5]

T. HallamC. Clark and G. Jordan, Effects of toxicants on populations: A qualitative approach Ⅱ. First order kinetics, J. Math. Biology, 18 (1983), 25-37. doi: 10.1007/BF00275908.

[6]

R. Z. Hasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, 2012, 2nd ed. doi: 10.1007/978-3-642-23280-0.

[7]

J. He and K. Wang, The survival analysis for a population in a polluted environment, Nonlinear Analysis: Real World Applications, 10 (2009), 1555-1571. doi: 10.1016/j.nonrwa.2008.01.027.

[8]

C. JiD. JiangN. Shi and D. O'Regan, Existence, uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation, Math. Methods in the Appl. Sciences, 30 (2007), 77-89. doi: 10.1002/mma.778.

[9]

D. Jiang and N. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172. doi: 10.1016/j.jmaa.2004.08.027.

[10]

D. JiangN. Shi and X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597. doi: 10.1016/j.jmaa.2007.08.014.

[11]

J. JiaoW. Long and L. Chen, A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin, Nonlinear Analysis: Real World Applications, 10 (2009), 3073-3081. doi: 10.1016/j.nonrwa.2008.10.007.

[12]

P. Kloeden and E. Platen, Numerical Solutions of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[13]

Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2004. doi: 10.1007/978-1-4471-3866-2.

[14]

V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol. Ⅰ, Academic Press, New York, 1969.

[15]

M. Liu and K. Wang, Survival analysis of stochastic single-species population models in polluted environments, Ecological Modelling, 220 (2009), 1347-1357.

[16]

M. LiuK. Wang and Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969-2012. doi: 10.1007/s11538-010-9569-5.

[17]

X. Mao, Stochastic Differential Equations and Applications, Woodhead Pubilshing, Philadelphia, 2011, 2nd ed. doi: 10.1533/9780857099402.

[18]

X. MaoG. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in population dynamics, Markov Proc. and Their Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0.

[19]

X. MaoS. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0.

[20]

T. PanB. HuangW. ZhangS. GabosD. Huang and V. Devendran, Cytotoxicity assessment based on the AUC50 using multi-concentration time-dependent cellular response curves, Anal. Chim. Acta, 764 (2013), 44-52.

[21]

S. Pinheiro, On a logistic growth model with predation and power-type diffusion coefficient: Ⅰ. Existence of solutions and extinction criteria, Math. Meth. Appl. Sci., 38 (2015), 4912-4930. doi: 10.1002/mma.3413.

[22]

S. Resnik, A Probability Path, Birkhauser, Boston, 1999.

[23]

Z. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physica A, 451 (2016), 507-518. doi: 10.1016/j.physa.2016.01.084.

[24]

F. Wei and L. Chen, Psychological effect on single-species population models in a polluted environment, Math. Biosci., 290 (2017), 22-30. doi: 10.1016/j.mbs.2017.05.011.

[25]

F. WeiS. Geritz and J. Cai, A stochastic single-species population model with partial pollution tolerance in a polluted environment, Appl. Math. Letters, 63 (2017), 130-136. doi: 10.1016/j.aml.2016.07.026.

[26]

Z. XiS. KhareA. CheungB. HuangT. PanW. ZhangF. IbrahimC. Jin and S. Gabos, Mode of action classification of chemicals using multi-concentration time-dependent cellular response profiles, Comp. Biol. Chem., 49 (2014), 23-35. doi: 10.1016/j.compbiolchem.2013.12.004.

[27]

J. XingL. ZhuS. Gabos and L. Xie, Microelectronic cell sensor assay for detection of cytotoxicity and prediction of acute toxicity, Toxicology in Vitro, 20 (2006), 995-1004. doi: 10.1016/j.tiv.2005.12.008.

[28]

Q. YangD. JiangN. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248-271. doi: 10.1016/j.jmaa.2011.11.072.

[29]

Q. Yang and X. Mao, Stochastic dynamics of SIRS epidemic models with random perturbation, Math. BioSci. Eng., 11 (2014), 1003-1025. doi: 10.3934/mbe.2014.11.1003.

[30]

Y. Zhang, Y. Wong, J. Deng, C. Anton, J. Deng, S. Gabos, W. Zhang, D. Huang and C. Jin, Machine learning algorithms for mode-of-action classification in toxicity assessment, BioData Mining, 9 (2016), p19. doi: 10.1186/s13040-016-0098-0.

Figure 1.  TCRCs for monastrol
Figure 3.  Trajectories corresponding to initial values $n(0) = 2.5$, $C_o(0) = 0$, $\sigma_1 = 0$, $\sigma_2 = 0.002$: blue "- -" line deterministic model, $C_e(0) = 380$; red "-" line stochastic model, $C_e(0) = 380$; green "-.-" line stochastic model, $C_e(0) = 379$
Figure 2.  Trajectories corresponding to initial values $n(0) = 2.5$, $C_o(0) = 0$, $\sigma_1 = 0.01$, $\sigma_2 = 0$: blue "- -" line deterministic model, $C_e(0) = 380$; red "-" line stochastic model, $C_e(0) = 380$; green "-.-" line stochastic model, $C_e(0) = 375$
Figure 4.  Histograms of the values of n(t) for the last iteration from 10 000 runs (a) and (c) and for the last 4 000 000 samples out of 5 000 000 sample of a single run (b) and (d)
Figure 5.  Histograms for the last 4 000 000 samples of a single run of 5 000 000 iterations and corresponding density functions a. $\sigma_2 = 0.001$ b. $\sigma_2 = 0.01$ c. $\sigma_2 = 0.1$ d. $\sigma_2 = 0.15$
Figure 6.  Histograms for the last 4 000 000 samples of a single run of 5 000 000 iterations and corresponding Gamma density functions a. $\sigma_1 = 0.001$ b. $\sigma_1 = 0.01$ c. $\sigma_1 = 0.1$ d. $\sigma_1 = 0.5$
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