×

Stochastic epidemic models: a survey. (English) Zbl 1188.92031

Summary: This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic (relying on a large community) properties are presented. The purpose of modelling is illustrated by studying effects of vaccination and also in terms of inference procedures for important parameters, such as the basic reproduction number and the critical vaccination coverage. Several generalizations towards realism, e.g., multitype and household epidemic models, are also presented, as is a model for endemic diseases.

MSC:

92D30 Epidemiology
60J85 Applications of branching processes
92C60 Medical epidemiology
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anderson, R. M.; May, R. M., Infectious Diseases of Humans; Dynamic and Control (1991), Oxford University Press: Oxford University Press Oxford
[2] Andersson, H.; Britton, T., Stochastic Epidemic Models and their Statistical Analysis. Stochastic Epidemic Models and their Statistical Analysis, Springer Lecture Notes in Statistics (2000), Springer Verlag: Springer Verlag New York · Zbl 0951.92021
[3] von Bahr, B.; Martin-Löf, A., Threshold limit theorems for some epidemic processes, Adv. Appl. Prob., 12, 319 (1980) · Zbl 0425.60074
[4] Bailey, N. T.J., The Mathematical Theory of Infectious Diseases and its Applications (1975), Griffin: Griffin London · Zbl 0115.37202
[5] Ball, F. G., The threshold behaviour of epidemic models, J. Appl. Prob., 20, 227 (1983) · Zbl 0519.92023
[6] Ball, F. G., A unified approach to the distribution of total size and total area under the trajectory of the infectives in epidemic models, Adv. Appl. Prob., 18, 289 (1986) · Zbl 0606.92018
[7] Ball, F. G.; Clancy, D., The final size and severity of a generalised stochastic multitype epidemic model, Adv. Appl. Prob., 25, 721 (1993) · Zbl 0798.92025
[8] Ball, F. G.; Donnelly, P., Strong approximations for epidemic models, Stoch. Proc. Appl., 55, 1 (1995) · Zbl 0823.92024
[9] Ball, F. G.; Mollison, D.; Scalia-Tomba, G., Epidemics with two levels of mixing, Ann. Appl. Prob., 7, 46 (1997) · Zbl 0909.92028
[10] Barbour, A. D., The duration of the closed stochastic epidemic, Biometrika, 62, 477 (1975) · Zbl 0307.92014
[11] Bartlett, M. S., Some evolutionary stochastic processes, J. Roy. Stat. Soc. B, 11, 211 (1949) · Zbl 0037.08503
[12] Becker, N. G., Analysis of Infectious Disease Data (1989), Chapman and Hall: Chapman and Hall London
[13] Becker, N. G.; Britton, T.; O’Neill, P. D., Estimating vaccine effects from studies of outbreaks in household pairs, Stat. Med., 25, 1079 (2006)
[14] D. Bernoulli, Essai d’une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l’inoculation pour la prévenir. Mém., Math. Phys. Acad. Roy. Sci. Paris, pp. 1-45, 1760.; D. Bernoulli, Essai d’une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l’inoculation pour la prévenir. Mém., Math. Phys. Acad. Roy. Sci. Paris, pp. 1-45, 1760.
[15] Bollobás, B., Random Graphs (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0997.05049
[16] Britton, T.; Janson, S.; Martin-Löf, A., Graphs with specified degree distributions, simple epidemics and local vaccination strategies, Adv. Appl. Prob., 39, 922 (2007) · Zbl 1134.60007
[17] Cauchemez, S.; Valleron, A.; Boëlle, P.; Flahault, A.; Ferguson, N., Estimating the impact of school closure on influenza transmission from sentinel data, Nature, 452, 750 (2008)
[18] Conlan, A. J.K.; Grenfell, B. T., Seasonality and the persistence and invasion of measles, Proc. Roy. Soc. Lond. B, 274, 1614, 1133 (2007)
[19] Cox, C., The delta method, (Armitage, P.; Colton, T., Encyclopedia of Biostatistics (1998), Wiley: Wiley Chichester)
[20] Daley, D. J.; Gani, J., Epidemic Modelling: An Introduction (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0922.92022
[21] Diekmann, O.; Heesterbeek, J. A.P., Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (2000), Wiley: Wiley Chichester · Zbl 0997.92505
[22] van Doorn, E. A., Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes, Adv. Appl. Prob., 23, 683 (1991) · Zbl 0736.60076
[23] Fraser, C.; Donnelly, C. A.; Cauchemez, S.; Hanage, W. P.; Van Kerkhove, M. D.; Hollingsworth, T. D.; Griffin, J.; Baggaley, R. F.; Jenkins, H. E.; Lyons, E. J., Pandemic potential of a strain of influenza A (H1N1): early findings, Science, 324, 1557 (2009)
[24] Haccou, P.; Jagers, P.; Vatutin, V. A., Branching Processes: Variation, Growth, and Extinction of Populations (2005), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1118.92001
[25] Halloran, M. E.; Longini, I. M.; Struchiner, C. J., Design and Analysis of Vaccine Studies (2010), Springer · Zbl 1269.62075
[26] Halloran, M. E.; Préziosi, M-P.; Chu, H., Estimating vaccine efficacy from secondary attack rates, J. Am. Stat. Assoc., 98, 38 (2003) · Zbl 1047.62108
[27] Heesterbeek, J. A.P., A brief history of R0 and a recipe for its calculation, Acta Biotheoret., 50, 189 (2002)
[28] Jagers, P., Branching Processes with Biological Applications (1975), Wiley: Wiley London · Zbl 0356.60039
[29] Keeling, M.; Rohani, P., Modeling Infectious Diseases in Humans and Animals (2008), Princeton University Press: Princeton University Press Princeton · Zbl 1279.92038
[30] Kendall, D. G., Deterministic and stochastic epidemics in closed populations, Proc. Thirs. Berkeley Symp. Math. Statist. Prob., 4, 149 (1956) · Zbl 0070.15101
[31] Kermack, W. O.; McKendrick, A. G., A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115, 700 (1927) · JFM 53.0517.01
[32] Mollison, D., Spatial contact models for ecological and epidemic spread (with Discussion), J. Roy. Stat. Soc. B, 39, 283 (1977) · Zbl 0374.60110
[33] Newman, M. E.J., The structure and function of complex networks, SIAM Rev., 45, 167 (2003) · Zbl 1029.68010
[34] Nåsell, I., On the time to extinction in recurrent epidemics, J. Roy. Stat. Soc. B., 61, 309 (1999) · Zbl 0917.92023
[35] O’Neill, P. D., A tutorial introduction to Bayesian inference for stochastic epidemic models using Markov chain Monte Carlo methods Math, Bioscience, 180, 103 (2002) · Zbl 1021.62094
[36] Picard, P.; Lefèvre, C., A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes, Adv. Appl. Prob., 22, 269 (1990) · Zbl 0719.92021
[37] Ross, R., The Prevention of Malaria (1911), Murray: Murray London
[38] Scalia-Tomba, G., Asymptotic final size distribution for some chain-binomial processes, Adv. Appl. Prob., 17, 477 (1985) · Zbl 0581.92023
[39] G. Scalia-Tomba, On the asymptotic final size distribution of epidemics in heterogeneous populations, in: J.-P. Gabriel, C. Lefèvre, P. Picard (Eds.), Stochastic Processes in Epidemic Theory, Springer Lecture Notes in Biomath., vol. 86, pp. 189-196, 1990.; G. Scalia-Tomba, On the asymptotic final size distribution of epidemics in heterogeneous populations, in: J.-P. Gabriel, C. Lefèvre, P. Picard (Eds.), Stochastic Processes in Epidemic Theory, Springer Lecture Notes in Biomath., vol. 86, pp. 189-196, 1990. · Zbl 0777.92019
[40] Sellke, T., On the asymptotic distribution of the size of a stochastic epidemic, J. Appl. Prob., 20, 390 (1983) · Zbl 0526.92024
[41] Y. Yang, J.D. Sugimoto, M.E. Halloran, N.E. Basta, D.L. Chao, L. Matrajt, G. Potter, E. Kenah, I.M. Longini, Jr., The Transmissibility and Control of Pandemic Influenza A (H1N1) Virus, Science, 2009, doi:10.1126/science.1177373.; Y. Yang, J.D. Sugimoto, M.E. Halloran, N.E. Basta, D.L. Chao, L. Matrajt, G. Potter, E. Kenah, I.M. Longini, Jr., The Transmissibility and Control of Pandemic Influenza A (H1N1) Virus, Science, 2009, doi:10.1126/science.1177373.
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.