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Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system
A comparison between random and stochastic modeling for a SIR model
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain |
In this article, a random and a stochastic version of a SIR nonautonomous model previously introduced in [
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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