Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T23:50:28.660Z Has data issue: false hasContentIssue false
  • English
  • Français

The final outcome and temporal solution of a carrier-borne epidemic model

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Frank Ball  and
Damian Clancy
Frank Ball*
Affiliation:
University of Nottingham
Damian Clancy*
Affiliation:
University of Newcastle upon Tyne
*
∗Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗∗Postal address: Department of Mathematics and Statistics, University of Newcastle, Newcastle upon Tyne NE1 7RU, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a stochastic model for the spread of a carrier-borne epidemic amongst a closed homogeneously mixing population, in which a proportion 1 − π of infected susceptibles are directly removed and play no part in spreading the infection. The remaining proportion π become carriers, with an infectious period that follows an arbitrary but specified distribution. We give a construction of the epidemic process which directly exploits its probabilistic structure and use it to derive the exact joint distribution of the final size and severity of the carrier-borne epidemic, distinguishing between removed carriers and directly removed individuals. We express these results in terms of Gontcharoff polynomials. When the infectious period follows an exponential distribution, our model reduces to that of Downton (1968), for which we use our construction to derive an explicit expression for the time-dependent state probabilities.

Keywords

CARRIER-BORNE EPIDEMICSSIZE OF EPIDEMICAREA UNDER TRAJECTORY OF INFECTIVESGONTCHAROFF POLYNOMIALSTIME-DEPENDENT SOLUTION

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 2 , June 1995 , pp. 304 - 315
Copyright
Copyright © Applied Probability Trust 1995 

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.)

Footnotes

Work carried out in part while Damian Clancy was supported by an SERC research studentship at the University of Nottingham.

References

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.CrossRefGoogle Scholar
Ball, F. G. (1990) A new look at Downton's carrier-borne epidemic model. In Stochastic Processes in Epidemic Theory, ed. Gabriel, J. P., Lefèvre, C. and Picard, P., pp. 71–85. Lecture Notes in Biomathematics 86, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Ball, F. G. and Clancy, D. (1993) The final size and severity of a generalised stochastic multitype epidemic model. Adv. Appl. Prob. 25, 721–736.Google Scholar
Billard, L. (1981) Generalized two-dimensional bounded birth-and-death processes and some applications. J. Appl. Prob. 18, 335–347.Google Scholar
Booth, J. G. (1989) On the limiting behaviour of Downton's carrier epidemic in the case of a general infection mechanism. J. Appl. Prob. 26, 625–630.CrossRefGoogle Scholar
Clancy, D. (1993) Epidemic Models in Heterogeneous Populations. , University of Nottingham.Google Scholar
Denton, G. M. (1972) On Downton's carrier-born epidemic. Biometrika 59, 455–461.CrossRefGoogle Scholar
Downton, F. (1968) The ultimate size of carrier-borne epidemics. Biometrika 55, 277–289.CrossRefGoogle Scholar
Gani, J. (1967) On the general stochastic epidemic. Proc. 5th Berkeley Symp. Math. Statist. Prob. 4, 271–279.Google Scholar
Kryscio, R. J. (1972) The transition probabilities of the extended simple stochastic epidemic model and the Haskey model. J. Appl. Prob. 9, 471–485.Google Scholar
Kryscio, R. J. (1975) The transition probabilities of the general stochastic epidemic model. J. Appl. Prob. 12, 415–424.CrossRefGoogle Scholar
Lefèvre, C. (1978) The expected ultimate size of a carrier-borne epidemic. J. Appl. Prob. 15, 414–419.Google Scholar
Lefèvre, C. (1979) The area under the trajectory of carriers in a carrier-borne epidemic. J. Appl. Prob. 16, 231–241.Google Scholar
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.Google Scholar
Picard, P. (1984) Applications of martingale theory to some epidemic models, II. J. Appl. Prob. 21, 677–684.CrossRefGoogle 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
Severo, N. C. (1969a) The probabilities of some epidemic models. Biometrika 56, 197–201.CrossRefGoogle Scholar
Severo, N. C. (1969b) A recursion theorem on solving differential-difference equations and applications to some stochastic processes. J. Appl. Prob. 6, 673–681.CrossRefGoogle Scholar
Weiss, G. H. (1965) On the spread of epidemics by carriers. Biometrics 21, 481–491.Google Scholar