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November  2015, 20(9): 2933-2947. doi: 10.3934/dcdsb.2015.20.2933

Persistence-time estimation for some stochastic SIS epidemic models

Francisco de la Hoz 1, , Anna Doubova 2, and  Fernando Vadillo 1,

1. 

Department of Applied Mathematics and Statistics and Operations Research, University of the Basque Country UPV/EHU, Spain, Spain
2. 

Department of Di erential Equations and Numerical Analysis, University of Sevilla, Spain

Received  July 2014 Revised  July 2015 Published  September 2015

In this paper, we study two stochastic SIS epidemic models: the first one with a constant population size, and the second one with a death factor. We analyze persistence and extinction behaviors for these models. The persistence time depends on the initial population size and satisfies a stationary backward Kolmogorov differential equation, which is a linear second-order partial differential equation with variable degenerate coefficients. We solve this equation numerically using a classical finite element method. We give computational evidence that the importance of understanding the dynamics of both the deterministic and the stochastic epidemic models is due to the numerical approximations to the mean persistence time. This can give more information about the model and may perhaps explain strange behaviors, such as the differences between the deterministic model and the stochastic one for long times.
Keywords: Persistence time, finite element method., stochastic differential equations, population models.
Mathematics Subject Classification: 60H10, 92B05, 65N3.
Citation: Francisco de la Hoz, Anna Doubova, Fernando Vadillo. Persistence-time estimation for some stochastic SIS epidemic models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2933-2947. doi: 10.3934/dcdsb.2015.20.2933
References:
[1]

E. Allen, Modeling with Itô Stochastic Differential Equations, Springer, 2007.  Google Scholar

[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.  Google Scholar

[3]

L. Allen, An Introduction to Stochastic Processes with Applications to Biology, Person Prentice Hall, 2003. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

E. Becker, G. Caray and J. Oden, Finite Elements. Vol. 1, 2, 3 and 4,, Prentice-Hall, (): 1981.   Google Scholar

[8]

P. Ciarlet and J. Lions, Handbook of Numerical Analysis Vol. II, Finite Element Methods, North-Holland, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Gockenbach, Understanding and Implementing the Finite Element Method, SIAM, 2006. doi: 10.1137/1.9780898717846.  Google Scholar

[12]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  Google Scholar

[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.  Google Scholar

[14]

C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambrigde University Press, 1987.  Google Scholar

[15]

P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

I. Nåsell, Stochastic models of some endemic infections, Mathematical Biosciences, 179 (2002), 1-19. doi: 10.1016/S0025-5564(02)00098-6.  Google Scholar

[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.  Google Scholar

[21]

L. Trefethen, Spectral Methods in MATLAB, SIAM, 2000. doi: 10.1137/1.9780898719598.  Google Scholar

show all references

References:
[1]

E. Allen, Modeling with Itô Stochastic Differential Equations, Springer, 2007.  Google Scholar

[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.  Google Scholar

[3]

L. Allen, An Introduction to Stochastic Processes with Applications to Biology, Person Prentice Hall, 2003. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

E. Becker, G. Caray and J. Oden, Finite Elements. Vol. 1, 2, 3 and 4,, Prentice-Hall, (): 1981.   Google Scholar

[8]

P. Ciarlet and J. Lions, Handbook of Numerical Analysis Vol. II, Finite Element Methods, North-Holland, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Gockenbach, Understanding and Implementing the Finite Element Method, SIAM, 2006. doi: 10.1137/1.9780898717846.  Google Scholar

[12]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  Google Scholar

[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.  Google Scholar

[14]

C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambrigde University Press, 1987.  Google Scholar

[15]

P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

I. Nåsell, Stochastic models of some endemic infections, Mathematical Biosciences, 179 (2002), 1-19. doi: 10.1016/S0025-5564(02)00098-6.  Google Scholar

[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.  Google Scholar

[21]

L. Trefethen, Spectral Methods in MATLAB, SIAM, 2000. doi: 10.1137/1.9780898719598.  Google Scholar

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