Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T02:02:58.010Z Has data issue: false hasContentIssue false
  • English
  • Français

Outcomes of epidemic models with general infection and removal rate functions at certain stopping times

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Damian Clancy
Damian Clancy*
Affiliation:
University of Liverpool
*
∗Postal address: Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, UK. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper aims to show how certain known martingales for epidemic models may be derived using general techniques from the theory of stochastic integration, and hence to extend the allowable infection and removal rate functions of the model as far as possible. Denoting by x, y the numbers of susceptible and infective individuals in the population, then we assume that new infections occur at rate βxyxy and infectives are removed at rate γxyy, where the ratio βxy / γxy can be written in the form q(x+y) / xp(x) for appropriate functions p,q. Under this condition, we find equations giving the distribution of the number of susceptibles remaining in the population at appropriately defined stopping times. Using results on Abel–Gontcharoff pseudopolynomials we also derive an expression for the expectation of any function of the number of susceptibles at these times, as well as considering certain integrals over the course of the epidemic. Finally, some simple examples are given to illustrate our results.

Keywords

Epidemicssize of epidemicseverity of epidemicmartingalesAbel–Gontcharoff pseudopolynomials

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 799 - 813
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abakuks, A. (1973). An optimal isolation policy for an epidemic. J. Appl. Prob. 10, 247–262.CrossRefGoogle Scholar
Abakuks, A. (1974). Optimal immunisation policies for epidemics. Adv. Appl. Prob. 6, 494–511.Google Scholar
Bailey, N. T. J. (1975). The Mathematical Theory of Infectious Diseases and its Applications, 2nd edn. Griffin, London.Google Scholar
Ball, F. G. (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.Google Scholar
Ball, F. G., and O'Neill, P. D. (1993). A modification of the general stochastic epidemic motivated by AIDS modelling. Adv. Appl. Prob. 25, 39–62.Google Scholar
Downton, F. (1968). The ultimate size of carrier-borne epidemics. Biometrika 55, 277–289.Google Scholar
Gani, J., and Jerwood, D. (1972). The cost of a general stochastic epidemic. J. Appl. Prob. 9, 257–269.Google Scholar
Gleissner, W. (1988). The spread of epidemics. Appl. Math. Comput. 27, 167–171.Google Scholar
Lefèvre, C., and Picard, P. (1996). Abelian-type expansions and non-linear death processes (II). Adv. Appl. Prob. 28, 877–894.Google Scholar
McNeil, D. R. (1972). On the simple stochastic epidemic. Biometrika 59, 494–497.CrossRefGoogle Scholar
O'Neill, P. D. (1997). An epidemic model with removal-dependent infection rate. Ann. Appl. Prob. 7, 90–109.Google Scholar
Picard, P. (1980). Applications of martingale theory to some epidemic models. J. Appl. Prob. 17, 583–599.Google Scholar
Picard, P. (1984). Applications of martingale theory to some epidemic models, II. J. Appl. Prob. 21, 677–684.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
Picard, P. and Lefèvre, C. (1996). First crossing of basic counting processes with lower non-linear boundaries: a unified approach through pseudopolynomials (I). Adv. Appl. Prob. 28, 853–876.Google Scholar
Rogers, L. C. G., and Williams, D. (1986). Diffusions, Markov Processes, and Martingales Volume 2: Itô Calculus. Wiley, New York.Google Scholar
Saunders, I. W. (1980a). A model for myxomatosis. Math. Biosci. 48, 1–15.Google Scholar
Saunders, I. W. (1980b). An approximate maximum likelihood estimator for chain binomial models. Austral. J. Statist. 22, 307–316.Google Scholar
Severo, N. C. (1969). Generalizations of some stochastic epidemic models. Math. Biosci. 4, 395–402.CrossRefGoogle Scholar
Watson, R., and Yip, P. (1993). A bivariate counting process. J. Appl. Prob. 30, 353–364.Google Scholar
Whittle, P. (1955). The outcome of a stochastic epidemic–-a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar