On the time to extinction in recurrent epidemics

Journal of the Royal Statistical Society. Series B: Statistical Methodology

Volumn 61, Issue 2, 1999, Pages 309-330

  Nåsell, Ingemar ^a  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

Author keywords

Critical community size; Persistence threshold; Quasi stationary distribution; Stochastic fade out; Time to extinction

Indexed keywords

EID: 0033475339     PISSN: 13697412     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9868.00178     Document Type: Article

References (37)

1. Anderson, R. M. and May, R. M. (1991) Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press.
2. Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and Its Applications. London: Griffin.
3. Barbour, A. D. (1972) The principle of the diffusion of arbitrary constants. J. Appl. Probab., 9, 519-541.
4. -(1976) Quasi-stationary distributions in Markov population processes. Adv. Appl. Probab., 8, 296-314.
5. Bartlett, M. S. (1956) Deterministic and stochastic models for recurrent epidemics. In Proc. 3rd Berkeley Symp. Mathematical Statistics and Probability, vol. 4, pp. 81-109. Berkeley: University of California Press.
6. -(1957) Measles periodicity and community size (with discussion). J. R. Statist. Soc. A, 120, 48-70.
7. -(1960a) Stochastic Population Models in Ecology and Epidemiology. London; Methuen.
8. -(1960b) Some stochastic models in ecology and epidemiology. In Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling (eds I. Olkin, S. G. Ghurye, W. Hoeffding, W. G. Madow and H. B. Mann), pp. 89-96. Stanford: Stanford University Press.
9. -(1960c) The critical community size for measles in the United States. J. R. Statist. Soc. A, 123, 37-44.
10. Darroch, J. N. and Seneta, E. (1967) On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probab., 4, 192-196.
11. Dietz, K. (1975) Transmission and control of arbovirus diseases. In Epidemiology (eds D. Ludwig and K. L. Cooke), pp. 104-121. Philadelphia: Society for Industrial and Applied Mathematics.
12. -(1995) Some problems in the theory of infectious disease transmission and control. In Epidemic Models: Their Structure and Relation to Data (ed. D. Mollison), pp. 3-16. Cambridge: Cambridge University Press.
13. Eichner, M., Hadeler, K. P. and Dietz, K. (1996) Stochastic models for the eradication of poliomyelitis: minimum population size for polio virus persistence. In Models for Infectious Human Diseases, Their Structure and Relation to Data (eds V. Isham and G. Medley), pp. 315-327. Cambridge: Cambridge University Press.
14. Ellner, S., Gallant, A. R. and Theiler, J. (1995) Detecting nonlinearity and chaos in epidemic data. In Epidemic Models: Their Structure and Relation to Data (ed. D. Mollison), pp. 229-247. Cambridge: Cambridge University Press.
15. Gardiner, C. W. (1985) Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Berlin: Springer.
16. Hamer, W. H. (1906) Epidemic disease in England - the evidence of variability and of persistence of type. Lancet, 733-739.
17. van Herwaarden, O. A. and Grasman, J. (1995) Stochastic epidemics: major outbreaks and the duration of the endemic period. J. Math. Biol., 33, 581-601.
18. Hethcote, H. W. (1974) Asymptotic behavior and stability in epidemic models. Lect. Notes Biomath., 2.
19. -(1976) Qualitative analysis of communicable disease models. Math. Biosci., 28, 335-356.
20. Keeling, M. J. and Grenfell, B. T. (1997) Disease extinction and community size: modelling the persistence of measles. Science, 275, 65-67.
21. Kryscio, R. J. and Lefèvre, C. (1989) On the extinction of the SIS stochastic logistic epidemic. J. Appl. Probab., 26, 685-694.
22. Kurtz, T. G. (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab., 8, 344-356.
23. Lotka, A. J. (1923) Martini's equations for the epidemiology of immunising diseases. Nature, 111, 633-634.
24. -(1956) Elements of Mathematical Biology. New York: Dover Publications.
25. Martini, E. (1921) Berechnungen und Beobachtungen zur Epidemiologie und Bekämpfung der Malaria. Hamburg: Gente.
26. Nåsell, I. (1995) The threshold concept in stochastic epidemic and endemic models. In Epidemic Models: Their Structure and Relation to Data (ed. D. Mollison), pp. 71-83. Cambridge: Cambridge University Press.
27. -(1996) The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Probab., 28, 895-932.
28. Olsen, L. F. and Schaffer, W. M. (1990) Chaos versus noisy periodicity: hypotheses for childhood epidemics. Science, 249, 499-504.
29. Pollett, P. K. and Stewart, D. E. (1994) An efficient procedure for computing quasi-stationary distributions of Markov chains with sparse transition structure. Adv. Appl. Probab., 26, 68-79.
30. Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992) Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd edn. Cambridge: Cambridge University Press.
31. Ridler-Rowe, C. J. (1967) On a stochastic model of an epidemic. J. Appl. Probab., 4, 19-31.
32. Schenzle, D. (1984) An age-structured model of pre- and post-vaccination measles transmission. IMA J. Math. Appl. Biol. Med., 1, 171-178.
33. Schenzle, D. and Dietz, K. (1987) Critical population sizes for endemic virus transmission. In Räumliche Persistenz und Diffusion von Krankheiten. Heidelberg: Heidelberger Geographische Arbeiten.
34. Soper, H. E. (1929) The interpretation of periodicity in disease prevalence (with discussion). J. R. Statist. Soc., 92, 34-73.
35. Stewart, D. E. and Bebbington, M. S. (1996) An iterative aggregation-disaggregation procedure for modelling the long-term behaviour of continuous-time evanescent random processes. J. Statist. Computn Simuln, 56, 77-95.
36. Stirzaker, D. R. (1975) A perturbation method for the stochastic recurrent epidemic. J. Inst. Math. Applic., 15, 135-150.
37. Sugihara, G. and May, R. M. (1990) Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature, 344, 734-741.