On a Perturbation Method for the Theory of EpidemicsAuthor(s): George H. WeissSource: Advances in Applied Probability, Vol. 3, No. 2 (Autumn, 1971), pp. 218-220Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426163 .

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218

References

BAILEY, N. T. J. (1968) A perturbation approach to the simple stochastic epidemic in a large population. Biometrika 55, 199-209.

DANIELS, H. E. (1960) Approximate solutions of Green's type for univariate stochastic processes. J. R. Statist. Soc. B 22, 376-401.

WEISS, G. H. (1970) On a perturbation method for the theory of epidemics. (WHO Symposium).

WHITTLE, P. (1957) On the use of the normal approximation in the treatment of stochastic processes. J. R. Statist. Soc. B 19, 268-281.

GEORGE H. WEISS

On a perturbation method for the theory of epidemics

GEORGE H. WEISS, National Institutes of Health, Bethesda, Md.

The theory of epidemics has been formulated as a deterministic theory and as a stochastic theory. There has been little work, except for a recent paper by Bailey [1], on the transition between the two formulations. In this paper we present an alternate formulation to that of Bailey that leads to a perturbation method more easily extendible to higher orders.

Let us consider the simple epidemic model of Bailey [2] in which the total

population size is N and the number of susceptibles is s(t). If I denotes an infected and S denotes a susceptible then a schematic representation of the model is I + Sa- 2S, and the deterministic equation is

ds (1) -d = - oas(N - s).

If we define dimensionless quantities C(t) and z by s(t) = N?(t) and z = %Nt, then the solution to (1) is equivalent to

(2) ((z) = C(0) /[(0) + (1 - ((O))e'].

In the stochastic theory one defines a set of probabilities {pn(t)} by

(3) p&(t) = Pr {s(t) = n}

which satisfy

(4) dp,, a(n + 1)(N - n - )p+,,1

- an(N - n)p,. (4) dt

Detailed solutions to this set of equations have been given by Bailey [2] and Williams [3].

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On a perturbation method for the theory of epidemics 219

In order to develop a perturbation theory we introduce a set of moments

{v,(t)) by

(5) Vr(t) = N-' n'pn(t). n=O

These moments satisfy the equations

(6) = r(vr+ 1

- v,)+N-r (_-

1)j r+ NJ(v,-j-1

- v,_j), dj=o + 2V)

where z has been introduced after (1). So far the analysis is exact. We now assume that the moments v,(z) can be expanded in inverse powers of N:

(1•) V(2) (7) v,(Z) =v~?)(z) N + N2 ...

and consider 1/N to be the perturbation parameter. The initial conditions are chosen to be the following:

(8) vo)(0) = ('(0), v5(0) = 0, j ? 1.

If the expansion of (7) is substituted into (6) and the coefficients of 1 /N are set equal to zero, we find that the v(D) satisfy equations of the form

dv•) (9) r(v"?+

- )

+ G(J)(r),

where

(10) G0o'(z) = 0,

G•')(z) - r(r - 1)

-vO). Gr2(2) =(v -r

Equation (9) can easily be solved by introducing the generating functions

00 Sr

U(s, 2) = Z v,(X•r) (11) r=1 r

oo Sr H(s, z) = E G,(z) -

r 1 r

The resulting equations have the solutions

U(O)(s, z) = In 1(z)s (12)

U()(s, z) = [H()(1 + (s - 1)e(- x),x) - UtJ(x)]dx, j? 1.

The zero order terms are therefore

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220

(13) v(z = ),

and these are equivalent to the deterministic results. Higher order results can be obtained from (12) in a straightforward fashion. The technique outlined here can be extended to discuss the Kermack-McKendrick model although detailed results have not been given so far.

References [1] BAILEY, N. T. J. (1968) A perturbation approach to the simple stochastic epidemic

in a large population. Biometrika 55, 199-209. [2] BAILEY, N. T. J. (1950) A simple stochastic epidemic. Biometrika 37, 193-202. [3] WILLIAMS, G. T. (1965) The simple stochastic epidemic curve for large populations

of susceptibles. Biometrika 52, 571-579.

GEORGE H. WEISS AND MENACHEM DISHON

Asymptotic behavior of a generalization of Bailey's simple epidemic

GEORGE H. WEISS, National Institutes of Health, Bethesda, Md. MENACHEM DISHON, Weizmann Institute of Science, Rehovot, Israel

It has been shown that for many epidemic models, the stochastic theory leads to essentially the same results as the deterministic theory provided that one identifies mean values with the functions calculated from the deterministic differential equations (cf. [1]). If one considers a generalization of Bailey's simple epidemic for a fixed population of size N, represented schematically by I + S - 21, 1 + S, where I refers to an infected, S refers to a susceptible, and a and p are appropriate rate constants, then it is evident that at time t = 00o, the expected number of infected individuals must be zero provided that P > 0. If x(t) denotes the number of infected at time t, then the deterministic model is summarized by

(1) dx (1)= ax(N - x) - fix.

Letting z = at, p = /3a/, we can write the solution to this equation

(N - p)x(O)e(N-P)T N - p + x(O)(e(N-Pr) - 1)

with the asymptotic solution

(3) x(oo) = 0 for N;< p,

= N - p for N>p.

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