March  2016, 21(2): 721-736. doi: 10.3934/dcdsb.2016.21.721

Stochastic epidemic models driven by stochastic algorithms with constant step

1. 

Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China

Received  December 2014 Revised  July 2015 Published  December 2015

In this paper we propose the stochastic epidemic model that relates directly to the deterministic counterpart and reveal close connections between these two models. Under the classic assumptions, the sample path of process eventually converges to the disease-free equilibrium, even though the corresponding deterministic flow converges to an endemic equilibrium. From the fact that disease can occur sporadically, we adjust the stochastic model slightly by introducing a stochastic incidence and establish precise connections between the long-run behavior of the discrete stochastic process and its deterministic flow approximation for large populations.
Citation: Lifeng Chen, Jifa Jiang. Stochastic epidemic models driven by stochastic algorithms with constant step. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 721-736. doi: 10.3934/dcdsb.2016.21.721
References:
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M. Benaïm, Recursive algorithms, urn processes and chaining number of chain recurrent sets, Ergodic Theory Dynam. Systems, 18 (1998), 53-87. doi: 10.1017/S0143385798097557.

[2]

M. Benaïm, Dynamics of stochastic approximation algorithms, Séminaire de Probabilités, XXXIII, 1-68, Lecture Notes in Math., 1709, Springer, Berlin, 1999. doi: 10.1007/BFb0096509.

[3]

M. Benaïm and M. W. Hirsch, Stochastic approximation algorithms with constant step size whose average is cooperative, Ann. Appl. Probab., 9 (1999), 216-241. doi: 10.1214/aoap/1029962603.

[4]

M. Benaïm and M. W. Hirsch, Differential and stochastic epidemic models, Fields Inst. Commun., 21 (1999), 31-44.

[5]

R. Bowen, $\omega$-limit sets of Axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339. doi: 10.1016/0022-0396(75)90065-0.

[6]

D. L. Burkholder, B. J. Davis and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, Proc. 6th Berkeley Symp. Math. Statis. and Probab., 2 (1972), 223-240.

[7]

C. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, 1978.

[8]

J. C. Fort and G. Pagès, Asymptotic behavior of a Markovian stochastic algorithm with constant step, SIAM J. Control Optim., 37 (1999), 1456-1482. doi: 10.1137/S0363012997328610.

[9]

H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.

[10]

O. Hernández-Lerma and J. B. Lasserre, Markov Chains and Invariant Probabilities, Birkhäuser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-8024-4.

[11]

M. Y. Li and C. Xu, Limit threshold theorems for stochastic epidemic models, preprint, 2014.

[12]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.

[13]

A. Lajmanovich and J. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.

show all references

References:
[1]

M. Benaïm, Recursive algorithms, urn processes and chaining number of chain recurrent sets, Ergodic Theory Dynam. Systems, 18 (1998), 53-87. doi: 10.1017/S0143385798097557.

[2]

M. Benaïm, Dynamics of stochastic approximation algorithms, Séminaire de Probabilités, XXXIII, 1-68, Lecture Notes in Math., 1709, Springer, Berlin, 1999. doi: 10.1007/BFb0096509.

[3]

M. Benaïm and M. W. Hirsch, Stochastic approximation algorithms with constant step size whose average is cooperative, Ann. Appl. Probab., 9 (1999), 216-241. doi: 10.1214/aoap/1029962603.

[4]

M. Benaïm and M. W. Hirsch, Differential and stochastic epidemic models, Fields Inst. Commun., 21 (1999), 31-44.

[5]

R. Bowen, $\omega$-limit sets of Axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339. doi: 10.1016/0022-0396(75)90065-0.

[6]

D. L. Burkholder, B. J. Davis and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, Proc. 6th Berkeley Symp. Math. Statis. and Probab., 2 (1972), 223-240.

[7]

C. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, 1978.

[8]

J. C. Fort and G. Pagès, Asymptotic behavior of a Markovian stochastic algorithm with constant step, SIAM J. Control Optim., 37 (1999), 1456-1482. doi: 10.1137/S0363012997328610.

[9]

H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.

[10]

O. Hernández-Lerma and J. B. Lasserre, Markov Chains and Invariant Probabilities, Birkhäuser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-8024-4.

[11]

M. Y. Li and C. Xu, Limit threshold theorems for stochastic epidemic models, preprint, 2014.

[12]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.

[13]

A. Lajmanovich and J. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.

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