The Distribution of the Total Size of an Epidemic
Abstract
This paper examines in some detail the distribution of the total number ofcases in an epidemic of the general stochastic type for a closed population. Theassumed model is that of Bartlett [2] and McKendrick [11] which Bailey [1]used to study the stochastic analogue of the deterministic threshold theorem(Kermack and McKendrick [10], D. G. Kendall [9]). Bailey obtained recurrencerelations from which the required probabilities were computed numerically.His calculations revealed a gradual transition from J-shaped distributions containingonly small epidemics for population sizes below the threshold, to U-shapeddistributions containing either large or small epidemics but practicallyno epidemics of intermediate size when the threshold is exceeded. There is alsoan interesting transitional form of distribution near the threshold value. In an attempt to understand what motivates an epidemic to behave in thisway, Whittle [13] and Kendall [9] constructed different models approximatingto the one used by Bailey but easier to handle analytically. Both explainedBailey's results in terms of an initial birth and death process where extinctionis certain in the first case and not certain in the second. This work is summarized,with additional references, in the book by Bailey [2]. In a paper presented atthis Symposium, Gani [7] develops some recent work by Siskind [12] andhimself [6] on a method of obtaining time dependent solutions of the epidemicequations. For the limiting case considered here he shows how the probabilitiescan be computed by successive multiplication of matrices.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1967
- Accession Number
- AD1027557
Entities
People
- H. E. Daniels
Organizations
- University of Birmingham