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Persistence-time estimation for some stochastic SIS epidemic models
1. | Department of Applied Mathematics and Statistics and Operations Research, University of the Basque Country UPV/EHU, Spain, Spain |
2. | Department of Dierential Equations and Numerical Analysis, University of Sevilla, Spain |
References:
[1] |
E. Allen, Modeling with Itô Stochastic Differential Equations, Springer, 2007. |
[2] |
E. Allen, L. Allen and H. Schurz, A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability, Mathematical Biosciences, 196 (2005), 14-38.
doi: 10.1016/j.mbs.2005.03.010. |
[3] |
L. Allen, An Introduction to Stochastic Processes with Applications to Biology, Person Prentice Hall, 2003. |
[4] |
L. Allen and E. Allen, A comparison of three different stochastic population models with regard to persistence time, Theoretical Population Biology, 64 (2003), 439-449.
doi: 10.1016/S0040-5809(03)00104-7. |
[5] |
L. Allen and A. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Mathematical Biosciences, 163 (2000), 1-33.
doi: 10.1016/S0025-5564(99)00047-4. |
[6] |
X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., 4 (2008), 729-796. |
[7] |
E. Becker, G. Caray and J. Oden, Finite Elements. Vol. 1, 2, 3 and 4,, Prentice-Hall, (): 1981.
|
[8] |
P. Ciarlet and J. Lions, Handbook of Numerical Analysis Vol. II, Finite Element Methods, North-Holland, 1991. |
[9] |
F. de la Hoz and F. Vadillo, A mean extinction-time estimate for a stochastic Lotka-Volterra predator-prey model, Applied Mathematics and Computation, 219 (2012), 170-179.
doi: 10.1016/j.amc.2012.05.060. |
[10] |
C. Doering, K. Sargsyan and L. Sander, Extinction times for birth-death processes: Exact results, continuum asymptotics, and the failure of the Fokker-Planck approximation, Multiscale Model. Simul., 3 (2005), 283-299.
doi: 10.1137/030602800. |
[11] |
M. Gockenbach, Understanding and Implementing the Finite Element Method, SIAM, 2006.
doi: 10.1137/1.9780898717846. |
[12] |
F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. |
[13] |
D. Higham, X. Mao and A. Stuarts, Strong convergence of euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.
doi: 10.1137/S0036142901389530. |
[14] |
C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambrigde University Press, 1987. |
[15] |
P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, 1992.
doi: 10.1007/978-3-662-12616-5. |
[16] |
R. Kryscio and C. Lefèvre, On the extinction of the S-I-S stochastic logistic epidemic, J. Appl. Prob., 26 (1989), 685-694.
doi: 10.2307/3214374. |
[17] |
M. Liu and K. Wanga, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457.
doi: 10.1016/j.jmaa.2010.09.058. |
[18] |
J. Mena-Lorca and H. Hethcote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693-716.
doi: 10.1007/BF00173264. |
[19] |
I. Nåsell, Stochastic models of some endemic infections, Mathematical Biosciences, 179 (2002), 1-19.
doi: 10.1016/S0025-5564(02)00098-6. |
[20] |
R. Norden, On the distribution of the time to extinction in the stochastic logistic population model, Adv. Appl. Prob., 14 (1982), 687-708.
doi: 10.2307/1427019. |
[21] |
L. Trefethen, Spectral Methods in MATLAB, SIAM, 2000.
doi: 10.1137/1.9780898719598. |
show all references
References:
[1] |
E. Allen, Modeling with Itô Stochastic Differential Equations, Springer, 2007. |
[2] |
E. Allen, L. Allen and H. Schurz, A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability, Mathematical Biosciences, 196 (2005), 14-38.
doi: 10.1016/j.mbs.2005.03.010. |
[3] |
L. Allen, An Introduction to Stochastic Processes with Applications to Biology, Person Prentice Hall, 2003. |
[4] |
L. Allen and E. Allen, A comparison of three different stochastic population models with regard to persistence time, Theoretical Population Biology, 64 (2003), 439-449.
doi: 10.1016/S0040-5809(03)00104-7. |
[5] |
L. Allen and A. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Mathematical Biosciences, 163 (2000), 1-33.
doi: 10.1016/S0025-5564(99)00047-4. |
[6] |
X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., 4 (2008), 729-796. |
[7] |
E. Becker, G. Caray and J. Oden, Finite Elements. Vol. 1, 2, 3 and 4,, Prentice-Hall, (): 1981.
|
[8] |
P. Ciarlet and J. Lions, Handbook of Numerical Analysis Vol. II, Finite Element Methods, North-Holland, 1991. |
[9] |
F. de la Hoz and F. Vadillo, A mean extinction-time estimate for a stochastic Lotka-Volterra predator-prey model, Applied Mathematics and Computation, 219 (2012), 170-179.
doi: 10.1016/j.amc.2012.05.060. |
[10] |
C. Doering, K. Sargsyan and L. Sander, Extinction times for birth-death processes: Exact results, continuum asymptotics, and the failure of the Fokker-Planck approximation, Multiscale Model. Simul., 3 (2005), 283-299.
doi: 10.1137/030602800. |
[11] |
M. Gockenbach, Understanding and Implementing the Finite Element Method, SIAM, 2006.
doi: 10.1137/1.9780898717846. |
[12] |
F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. |
[13] |
D. Higham, X. Mao and A. Stuarts, Strong convergence of euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.
doi: 10.1137/S0036142901389530. |
[14] |
C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambrigde University Press, 1987. |
[15] |
P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, 1992.
doi: 10.1007/978-3-662-12616-5. |
[16] |
R. Kryscio and C. Lefèvre, On the extinction of the S-I-S stochastic logistic epidemic, J. Appl. Prob., 26 (1989), 685-694.
doi: 10.2307/3214374. |
[17] |
M. Liu and K. Wanga, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457.
doi: 10.1016/j.jmaa.2010.09.058. |
[18] |
J. Mena-Lorca and H. Hethcote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693-716.
doi: 10.1007/BF00173264. |
[19] |
I. Nåsell, Stochastic models of some endemic infections, Mathematical Biosciences, 179 (2002), 1-19.
doi: 10.1016/S0025-5564(02)00098-6. |
[20] |
R. Norden, On the distribution of the time to extinction in the stochastic logistic population model, Adv. Appl. Prob., 14 (1982), 687-708.
doi: 10.2307/1427019. |
[21] |
L. Trefethen, Spectral Methods in MATLAB, SIAM, 2000.
doi: 10.1137/1.9780898719598. |
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