×

Stochastic epidemics: The expected duration of the endemic period in higher dimensional models. (English) Zbl 0930.92021

Summary: A method is presented to approximate the long-term stochastic dynamics of an epidemic modelled by state variables denoting the various classes of the population such as in SIR and SEIR models. The modelling includes epidemics in populations at different locations with migration between these populations. A logistic stochastic process for the total infectious population is formulated; it fits the long-term stochastic behaviour of the total infectious population in the full model. A good approximation is obtained if only the dynamics near the equilibria is fit.

MSC:

92D30 Epidemiology
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60K99 Special processes
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] K. Dietz, Some problems in the theory of infectious disease transmission and control, in: D. Mollison (Eds.), Epidemic Models, Cambridge University, Cambridge, 1995, p. 3; K. Dietz, Some problems in the theory of infectious disease transmission and control, in: D. Mollison (Eds.), Epidemic Models, Cambridge University, Cambridge, 1995, p. 3 · Zbl 0850.92045
[2] N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Griffin, London, 1975, p. 143; N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Griffin, London, 1975, p. 143
[3] N.S. Goel, N. Richter-Dyn, Stochastic Models in Biology, Academic Press, New York, 1974; N.S. Goel, N. Richter-Dyn, Stochastic Models in Biology, Academic Press, New York, 1974
[4] M.F. Neuts, J.-M. Li, An algorithmic study of S-I-R stochastic epidemic models, in: C.C. Heyde, Yu.V. Prohorov, R. Pyke, S.T. Rachev (Eds.), Athens Conference on Applied Probability and Times Series, vol. I: Applied Probability, Springer Lecture Notes in Statistics, vol. 114, 1996, p. 295; M.F. Neuts, J.-M. Li, An algorithmic study of S-I-R stochastic epidemic models, in: C.C. Heyde, Yu.V. Prohorov, R. Pyke, S.T. Rachev (Eds.), Athens Conference on Applied Probability and Times Series, vol. I: Applied Probability, Springer Lecture Notes in Statistics, vol. 114, 1996, p. 295 · Zbl 0857.92013
[5] Van Herwaarden, O. A.; Grasman, J., Stochastic epidemics: major outbreaks and the duration of the endemic period, J. Math. Biol., 33, 581 (1995) · Zbl 0830.92024
[6] Van Herwaarden, O. A., Stochastic epidemics: the probability of extinction of an infectious disease at the end of a major outbreak, J. Math. Biol., 35, 793 (1997) · Zbl 0877.92024
[7] Grasman, J., The expected extinction time of a population within a system of interacting biological populations, Bull. Math. Biol., 58, 555 (1996) · Zbl 0852.92024
[8] Roozen, H., An asymptotic solution to a two-dimensional exit problem arising in population dynamics, SIAM J. Appl. Math., 49, 1793 (1989) · Zbl 0697.92022
[9] H. Roozen, Analysis of the exit problem for randomly perturbed dynamical systems in applications, PhD thesis, Wageningen Agriculture University, The Netherlands, 1990; H. Roozen, Analysis of the exit problem for randomly perturbed dynamical systems in applications, PhD thesis, Wageningen Agriculture University, The Netherlands, 1990
[10] Grasman, J.; HilleRisLambers, R., On local extinction in a metapopulation, Ecol. Modelling, 103, 71 (1997)
[11] C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer, Berlin, 1983; C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer, Berlin, 1983 · Zbl 0515.60002
[12] M. Eichner, K.P. Hadeler, K. Dietz, Stochastic models for the eradication of Poliomyelitis: Minimum population size for polio virus persistence, in: V. Isham, G. Medley (Eds.), Models for Infectious Human Diseases, Cambridge University, cambridge, 1996; M. Eichner, K.P. Hadeler, K. Dietz, Stochastic models for the eradication of Poliomyelitis: Minimum population size for polio virus persistence, in: V. Isham, G. Medley (Eds.), Models for Infectious Human Diseases, Cambridge University, cambridge, 1996 · Zbl 1076.92500
[13] Bartlett, M. S., The critical community size of measles in the United States, J. Roy. Stat. Soc. A, 123, 37 (1960)
[14] Chan, M.-S.; Jeger, M. J., An analytical model of plant virus disease dynamics with roguing and replanting, J. Appl. Ecol., 31, 413 (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.