Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-12-01T00:15:35.410Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Epidemic Models in Structured Populations Featuring Dynamic Vaccination and Isolation

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Frank Ball ,
Philip D. O'Neill  and
James Pike
Frank Ball*
Affiliation:
University of Nottingham
Philip D. O'Neill*
Affiliation:
University of Nottingham
James Pike*
Affiliation:
University of Nottingham
*
∗Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a stochastic model for the spread of an SEIR (susceptible → exposed → infective → removed) epidemic among a population of individuals partitioned into households. The model incorporates both vaccination and isolation in response to the detection of cases. When the infectious period is exponential, we derive an explicit formula for a threshold parameter, and analytic results that enable computation of the probability of the epidemic taking off. These quantities are found to be independent of the exposure period distribution. An approximation for the expected final size of an epidemic that takes off is obtained, evaluated numerically, and found to be reasonably accurate in large populations. When the infectious period is not exponential, but has an increasing hazard rate, we obtain stochastic comparison results in the case where the exposure period is fixed. Our main result shows that as the exposure period increases, both the severity of the epidemic in a single household and the threshold parameter decrease, under certain assumptions concerning isolation. Corresponding results for infectious periods with decreasing hazard rates are also derived.

Keywords

Epidemicstochastic epidemicvaccinationbioterrorismcouplingfinal outcome

MSC classification

Secondary: 92D30: Epidemiology 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 571 - 585
Copyright
Copyright © Applied Probability Trust 2007 

References

Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis (Lecture Notes Statist. 151). Springer, New York.Google Scholar
Ball, F. G. (1996). Threshold behaviour in stochastic epidemics among households. In Athens Conf. Appl. Prob. Time Ser., Vol. I (Lecture Notes Statist. 114), eds Heyde, C. C. et al., Springer, Berlin, pp. 253–266.CrossRefGoogle Scholar
Ball, F. G. (1999). Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math. Biosci. 156, 41–67.Google Scholar
Ball, F. G., Mollison, D. and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Prob. 7, 46–89.Google Scholar
Ball, F. G., O'Neill, P. D. and Pike, J. (2005). Stochastic epidemic models in structured populations featuring dynamic vaccination and isolation. Tech. Rep. 05-04, School of Mathematical Sciences, University of Nottingham.Google Scholar
Becker, N. G. and Starczak, D. N. (1998). The effect of random vaccine response on the vaccination coverage required to prevent epidemics. Math. Biosci. 154, 117–135.Google Scholar
Becker, N. G., Glass, K., Li, Z. and Aldis, G. K. (2005). Controlling emerging infectious diseases like SARS. Math. Biosci. 193, 205–221.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Eubank, S. et al. (2004). Modelling disease outbreaks in realistic urban social networks. Nature 492, 180–184.Google Scholar
Ferguson, N. M. et al. (2005). Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature 437, 209–214.Google Scholar
Halloran, M., Longini, I. M. Jr., Nizam, A. and Yang, Y. (2002). Containing bioterrorist smallpox. Science 298, 1428–1432.Google Scholar
Kaplan, E., Craft, D. and Wein, L. (2002). Emergency response to a smallpox attack: the case for mass vaccination. Proc. Nat. Acad. Sci. USA 99, 10935–10940.Google Scholar
Longini, I. M., Halloran, M. E., Nizam, A. and Yang, Y. (2004). Containing pandemic influenza with antiviral agents. Amer. J. Epidemiol. 159, 623–633.Google Scholar
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, 269–294.Google Scholar
Sellke, T. (1983). On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.Google Scholar
Watson, R. K. (1980). A useful random time-scale transformation for the standard epidemic model. J. Appl. Prob. 17, 324–332.Google Scholar