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January  2017, 16(1): 151-162. doi: 10.3934/cpaa.2017007

A comparison between random and stochastic modeling for a SIR model

Tomás Caraballo , and  Renato Colucci

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain

E-mail address: caraball@us.es

Received  June 2016 Revised  August 2016 Published  November 2016

Fund Project: Partially supported by FEDER and Ministerio de Economía y Competitividad under grant MTM2015-63723-P and Junta de Andalucía under Proyecto de Excelencia P12-FQM-1492.

In this article, a random and a stochastic version of a SIR nonautonomous model previously introduced in [19] is considered. In particular, the existence of a random attractor is proved for the random model and the persistence of the disease is analyzed as well. In the stochastic case, we consider some environmental effect on the model, in fact, we assume that one of the coefficients of the system is affected by some stochastic perturbation, and analyze the asymptotic behavior of the solutions. The paper is concluded with a comparison between the two different modeling strategies.

Keywords: Epidemiology, stochastic differential equations, random dynamical systems.
Mathematics Subject Classification: Primary: 34C11, 34F05; Secondary: 60H10.
Citation: Tomás Caraballo, Renato Colucci. A comparison between random and stochastic modeling for a SIR model. Communications on Pure and Applied Analysis, 2017, 16 (1) : 151-162. doi: 10.3934/cpaa.2017007
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

Y. Asai and P. E. Kloeden, Numerical schemes for random ODEs via stochastic differential equations, Commun. Appl. Analysis, 17 (2013), 521-528.

[3]

F. Brauer and C. Castillo-Chavez, Mathematical Models for Communicable Diseases, Society for Industrial and Applied Mathematics, Series: CBMS-NSF Regional Conference Series in Applied Mathematics (Book 84), December 28,2012.

[4]

T. Britton, Stochastic epidemic models: A survey, Mathematical Biosciences, 225 (2010), 24-35. doi: 10.1016/j.mbs.2010.01.006.

[5]

J. L. Brooks and I. D. Stanley, Predation, body size, and composition of plankton, Science, 150.3692 (1965), 23-35.

[6]

Y. Cai, X. Wang, W. Wang and M. Zhao, Stochastic Dynamics of an SIRS Epidemic Model with Ratio-Dependent Incidence Rate, Abstract and Applied Analysis, vol. 2013, Article ID 172631, 11 pages. doi: 10.1155/2013/172631.

[7]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.

[8]

T. Caraballo, R. Colucci and X. Han, Predation with indirect effects in fluctuating environments, Nonlinear Dynamics, 84 (2016), 115-126. doi: 10.1007/s11071-015-2238-3.

[9]

T. Caraballo, X. Han and P. E. Kloeden, Chemostats with time-dependent inputs and wall growth, Applied Mathematics and Information Sciences, 9 (2015), 2283-2296.

[10]

T. Caraballo, X. Han and P. E. Kloeden, Chemostats with random inputs and wall growth, Mathematical Methods in the Applied Sciences, 38 (2015), 3538-3550. doi: 10.1002/mma.3437.

[11]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.

[12]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 6 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.

[13]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0.

[14]

J. Cresson, M. Efendiev and S. Sonner, On the positivity of solutions of systems of stochastic PDEs, ZAMM, 93 (2013), 414-422. doi: 10.1002/zamm.201100167.

[15]

F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stochastics Stochastics Rep., 59 (1996), 21-45.

[16]

C. Jia, D. Jianga, Q. Yanga and N. Shia, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131.

[17]

D. Jiang, J. Yua, C. Ji and N. Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Mathematical and Computer Modelling, 54 (2011), 221-232. doi: 10.1016/j.mcm.2011.02.004.

[18]

D. Jiang, C. Ji, N. Shi and J. Yu, The long time behavior of DI SIR epidemic model with stochastic perturbation, J. Math. Anal. Appl., 372 (2010), 162-180. doi: 10.1016/j.jmaa.2010.06.003.

[19]

P. E. Kloeden and V. S. Kozyakin, The dynamics of epidemiological systems with nonautonomous and random coefficients, MESA: Mathematics in Engineering, Science and Aerospace, 2 (2011).

[20]

P. E. Kloeden and C. Pötzsche, Nonautonomous bifurcation scenarios in SIR models, Mathematical Methods in the Applied Sciences, 38 (2015), 3495-3518. doi: 10.1002/mma.3433.

[21]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[22]

Y. Lin, D. Jiang and P. Xia, Long-time behavior of a stochastic SIR model, Applied Mathematics and Computation, 236 (2014), 1-9. doi: 10.1016/j.amc.2014.03.035.

[23]

K. Rohde, Nonequilibrium Ecology, Cambridge University Press, 2005.

[24]

B. Schmalfuß, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998), 91-113. doi: 10.1006/jmaa.1998.6008.

[25]

H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.

[26]

K. Xu, Z. Zhou and H. Zhao, Dynamical analysis of a parasite-host model within fluctuating environment, Mathematical Problems in Engineering, 2016 (2016), Article ID 2972956, 12 pages. doi: 10.1155/2016/2972956.

[27]

Q. Yang, D. Jiang, N. 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.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

Y. Asai and P. E. Kloeden, Numerical schemes for random ODEs via stochastic differential equations, Commun. Appl. Analysis, 17 (2013), 521-528.

[3]

F. Brauer and C. Castillo-Chavez, Mathematical Models for Communicable Diseases, Society for Industrial and Applied Mathematics, Series: CBMS-NSF Regional Conference Series in Applied Mathematics (Book 84), December 28,2012.

[4]

T. Britton, Stochastic epidemic models: A survey, Mathematical Biosciences, 225 (2010), 24-35. doi: 10.1016/j.mbs.2010.01.006.

[5]

J. L. Brooks and I. D. Stanley, Predation, body size, and composition of plankton, Science, 150.3692 (1965), 23-35.

[6]

Y. Cai, X. Wang, W. Wang and M. Zhao, Stochastic Dynamics of an SIRS Epidemic Model with Ratio-Dependent Incidence Rate, Abstract and Applied Analysis, vol. 2013, Article ID 172631, 11 pages. doi: 10.1155/2013/172631.

[7]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.

[8]

T. Caraballo, R. Colucci and X. Han, Predation with indirect effects in fluctuating environments, Nonlinear Dynamics, 84 (2016), 115-126. doi: 10.1007/s11071-015-2238-3.

[9]

T. Caraballo, X. Han and P. E. Kloeden, Chemostats with time-dependent inputs and wall growth, Applied Mathematics and Information Sciences, 9 (2015), 2283-2296.

[10]

T. Caraballo, X. Han and P. E. Kloeden, Chemostats with random inputs and wall growth, Mathematical Methods in the Applied Sciences, 38 (2015), 3538-3550. doi: 10.1002/mma.3437.

[11]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.

[12]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 6 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.

[13]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0.

[14]

J. Cresson, M. Efendiev and S. Sonner, On the positivity of solutions of systems of stochastic PDEs, ZAMM, 93 (2013), 414-422. doi: 10.1002/zamm.201100167.

[15]

F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stochastics Stochastics Rep., 59 (1996), 21-45.

[16]

C. Jia, D. Jianga, Q. Yanga and N. Shia, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131.

[17]

D. Jiang, J. Yua, C. Ji and N. Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Mathematical and Computer Modelling, 54 (2011), 221-232. doi: 10.1016/j.mcm.2011.02.004.

[18]

D. Jiang, C. Ji, N. Shi and J. Yu, The long time behavior of DI SIR epidemic model with stochastic perturbation, J. Math. Anal. Appl., 372 (2010), 162-180. doi: 10.1016/j.jmaa.2010.06.003.

[19]

P. E. Kloeden and V. S. Kozyakin, The dynamics of epidemiological systems with nonautonomous and random coefficients, MESA: Mathematics in Engineering, Science and Aerospace, 2 (2011).

[20]

P. E. Kloeden and C. Pötzsche, Nonautonomous bifurcation scenarios in SIR models, Mathematical Methods in the Applied Sciences, 38 (2015), 3495-3518. doi: 10.1002/mma.3433.

[21]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[22]

Y. Lin, D. Jiang and P. Xia, Long-time behavior of a stochastic SIR model, Applied Mathematics and Computation, 236 (2014), 1-9. doi: 10.1016/j.amc.2014.03.035.

[23]

K. Rohde, Nonequilibrium Ecology, Cambridge University Press, 2005.

[24]

B. Schmalfuß, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998), 91-113. doi: 10.1006/jmaa.1998.6008.

[25]

H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.

[26]

K. Xu, Z. Zhou and H. Zhao, Dynamical analysis of a parasite-host model within fluctuating environment, Mathematical Problems in Engineering, 2016 (2016), Article ID 2972956, 12 pages. doi: 10.1155/2016/2972956.

[27]

Q. Yang, D. Jiang, N. 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.

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