Asymptotic analysis of the general stochastic epidemic with variable infectious periods

Journal of Applied Probability

Volumn 38, Issue 1, 2001, Pages 18-35

  Startsev, A N ^a  

^a S YU YUNUSOV INSTITUTE OF THE CHEMISTRY OF PLANT SUBSTANCES   (Uzbekistan)

Author keywords

Epidemics; Limit theorems; Non Markovian model; Size of epidemic; Variable infectious period

Indexed keywords

EID: 0035533768     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1239/jap/996986640     Document Type: Article

References (31)

1. BAILEY, N. T. J. (1975). The Mathematical Theory of Infectious Diseases and its Applications, 2nd edn. Griffin, London.
2. BALL, F. (1986). A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv. Appl. Prob. 18, 289-310.
3. BALL, F. G. AND BARBOUR, A. D. (1990). Poisson approximation for some epidemic models. J. Appl. Prob. 27, 479-490.
4. BORISOV, I. S. AND BOROVKOV, A. A. (1987). Second-order approximation of random polygonal lines in the Donsker-Prokhorov invariance principle. Theory Prob. Appl. 31, 179-202.
5. DALEY, D. J. (1990). The size of epidemics with variable infectious periods. Res. Rept SMS-012-90, Australian National University.
6. DALEY, D. J. AND GANI, J. (1999). Epidemic Modelling: an Introduction (Cambridge Studies Math. Biol. 15). Cambridge University Press.
7. DANIELS, H. E. (1967). The distribution of the total size of an epidemic. In Proc. 5th. Berkeley Symp. Math. Stat. Prob. University of California Press, Berkeley, pp. 281-293.
8. FELLER, W. (1971). An Introduction to Probability Theory and its Application, Vol. 2, 2nd edn. John Wiley, New York.
9. GANI, J. (1965). On the partial differential equation of epidemic theory. I. Biometrika 52, 617-622.
10. GANI, J. (1967). On the general stochastic epidemic. In Proc. 5th Berkeley Symp. Math. Statist. Prob. University of California Press, Berkeley, pp. 271-279.
11. LEFÈVRE, C. AND PICARD, P. (1990). A non-standard family of polynomials and the final size distribution of Reed-Frost epidemic processes. Adv. Appl. Prob. 22, 25-48.
12. LEFÈVRE, C. AND UTEV, S. (1995). Poisson approximation for the final state of a generalized epidemic process. Ann. Prob. 23, 1139-1162.
13. LUDWIG, D. (1974). Stochastic Population Theories (Lecture Notes Biomath. 3). Springer, Berlin.
14. MARTIN-LÖF, A. (1986). Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265-282.
15. MARTIN-LÖF, A. (1998). The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Prob. 35, 671-682.
16. NAGAEV, A. V. AND STARTSEV, A. N. (1970). The asymptotic analysis of a stochastic model of an epidemic. Theory Prob. Appl. 15, 98-107.
17. PICARD, P. AND LEFÈVRE, C. (1990). A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes. Adv. Appl. Prob. 22, 262-294.
18. SCALIA-TOMBA, G. (1985). Asymptotic final-size distribution for some chain-binomial processes. Adv. Appl. Prob. 17, 477-495.
19. SCALIA-TOMBA, G. (1990). On the asymptotic final size distribution of epidemic in heterogeneous population. In Stochastic Processes in Epidemic Theory (Lecture Notes Biomath. 86), eds J.-P. Gabriel, C. Lefèvre and P. Picard. Springer, New York.
20. SELLKE, T. (1983). On the asymptotical distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390-394.
21. SISKIND, V. (1965). Solution of the general stochastic epidemic theory. Biometrika 52, 617-622.
22. STARTSEV, A. N. (1971). Limit theorems for the epidemic size in the general stochastic model. In Random Processes and Statistical Decision. Fan, Tashkent, pp. 60-73 (in Russian).
23. STARTSEV, A. N. (1988). On approximation conditions of the distribution of the maximum sums of independent random variables. Ann. Acad. Sci. Fennicae, Ser. A I. Math. 13, 269-275.
24. STARTSEV, A. N. (1994). On distribution of the first passage time for a class of a two-dimensional Markov random walk. Uzbek Math. J. 4, 60-66.
25. STARTSEV, A. N. (1997). On epidemic size distribution in a non-Markovian model. Theory Prob. Appl. 41, 730-740.
26. STARTSEV, A. N. AND CHAǏ, Z. S. (1987). Limit theorems for the epidemic size in a generalized probability model. In Probability Models and Mathematical Statistics, ed. S. Kh. Sirazhdinov. Fan, Tashkent, pp. 92-105 (in Russian).
27. VON BAHR, B. AND MARTIN-LÖF, A. (1980). Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319-349.
28. WANG, J. S. (1977). Gaussian approximation of some closed stochastic epidemic models. J. Appl. Prob. 14, 221-231.
29. WATSON, R. (1980). On the size distribution for some epidemic models. J. Appl. Prob. 17, 912-921.
30. WEISS, G. H. (1965). On the spread of epidemics by carriers. Biometrics 21, 481-490.
31. WHITTLE, P. (1955). The outcome of stochastic epidemic. Biometrika 42, 116-122.