An Algebraic Proof of the Threshold Theorem for the General Stochastic EpidemicAuthor(s): Trevor WilliamsSource: Advances in Applied Probability, Vol. 3, No. 2 (Autumn, 1971), p. 223Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426166 .

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An algebraic proof of the threshold theorem for the general stochastic epidemic

An algebraic proof of the threshold theorem

for the general stochastic epidemic

TREVOR WILLIAMS, University of Bristol

Bailey (1957) has considered the general stochastic epidemic with relative removal rate p, one initial infective and n susceptibles. His Equation (5.51) may be re-cast as

(1) PkA

where the jf satisfy the recurrence relationship

(2)k= ( (1 x)+fk(X)= , (j 0).

The precise expressions for the fk appear to be rather complicated, but it turns out that fk is proportional to xk(x - 0). Indeed, if we write 0 = p/n for the relative removal rate per susceptible, it may be shown, after a certain amount of algebra, that

(2k)! Ok+1 (3) Pk k!(k + 1)! (1 +)2k+1' (n ).

One readily verifies that these probabilities sum to 0 (0 < 1) and to 1 (0 > 1); the missing probability of 1 - 0 in the first instance corresponding to the probability of a 'true' epidemic, running its course through the entire popula- tion.

Reference BAILEY, N. T. J. (1957) The Mathematical Theory of Epidemics. Griffin, London.

223

that the distribution of individual terms changes and depends on chance. Nevertheless, it can also be treated with the asymptotic technique.

References [1] BAILEY, N. T. J. (1957) The Mathematical Theory of Epidemics. Griffin, London. [2] NAGAEV, A. N. AND STARTSEV, A. N. (1970) Asymptotic analysis of a stochastic

epidemic model. Teor. Verojat. Primen. 15, 97-105.

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