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1 On an Epideic in a Stratified Population Author(s): R. K. Watson Source: Journal of Applied Probability, Vol. 9, No. 3 (Sep., 1972), pp Published by: Applied Probability Trust Stable URL: Accessed: 07/12/ :28 Your use of the JSTOR archive indicates your acceptance of JSTOR's Ters and Conditions of Use, available at JSTOR's Ters and Conditions of Use provides, in part, that unless you have obtained prior perission, you ay not download an entire issue of a journal or ultiple copies of articles, and you ay use content in the JSTOR archive only for your personal, non-coercial use. Please contact the publisher regarding any further use of this work. Publisher contact inforation ay be obtained at Each copy of any part of a JSTOR transission ust contain the sae copyright notice that appears on the screen or printed page of such transission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use inforation technology and tools to increase productivity and facilitate new fors of scholarship. For ore inforation about JSTOR, please contact support@jstor.org. Applied Probability Trust is collaborating with JSTOR to digitize, preserve and extend access to Journal of Applied Probability.

2 J. Appl. Prob. 9, (1972) Printed in Israel? Applied Probability Trust 1972 ON AN EPIDEMIC IN A STRATIFIED POPULATION R. K. WATSON, University of Melbourne Abstract Most epideic odels previously studied have assued a hoogeneously ixing population. Instead of aking this assuption, a population divided into classes is considered; and it is assued that the degree of ixing between classes is less than that within classes. The stochastic odel in this for is intractable and approxiations are ade, yielding results in reasonable agreeent with siulation trials. STOCHASTIC EPIDEMIC MODEL; NONHOMOGENEOUS MIXING; DETERMINISTIC APPROXIMATION; STOCHASTIC APPROXIMATION; SIMULATION 1. Introduction Of the assuptions coonly used in continuous infection odels, the least likely to be even approxiately true in large populations, is that of hoogeneous ixing. In this paper, we investigate a odel for the spread of infection aongst a population which is divided into classes, such that the individuals of each class ix hoogeneously aongst theselves, but ix to a lesser degree with individuals of other classes. A class could be thought of either as a group of friends or associates, or as a collection of individuals in a certain region of the counity. An assuption of this kind appears to be a quite realistic substitute for the hoogeneous ixing assuption. That such a odel is appropriate is also suggested by observed epideic behaviour. It is known that epideics in large populations can often be broken down into saller outbreaks which are in general not in phase, and which interact with each other to soe extent, as envisaged in the present odel. Siple odels of this kind have been put forward by Haskey (1957), who studied the case of a siple epideic (i.e., no reoval) in two classes, and Rushton and Mautner (1955) who studied the case of a deterinistic siple epideic in equivalent classes. Also, Bailey (1957) ade ention of the possibility of Received 24 May Research supported by a C.S.I.R.O. studentship. 659

3 660 R. K. WATSON such a odel, and Bartlett (1957) siulated a recurrent epideic odel using a siilar assuption. Although the priary purpose of this odel is to provide an alternative to the hoogeneous ixing assuption, non-hoogeneous behaviour with respect to the infection ay also be taken into account. Thus, for exaple, this ay be useful in dealing with the case of a counity which includes a region, such as a slu area, the individuals of which, for soe reason or other, are ore prone to infection. Further, by suitable aendent of the interpretation given to the paraeters, the odel can be used to describe another iportant situation: that of a population consisting of a nuber of distinct types of individuals exhibiting differing susceptibility and rates of recovery. The division of the population into age groups is a possible classification for which the odel is applicable. As is perhaps to be expected, the resulting stochastic odel presents as yet insurountable difficulties, but nevertheless, soe useful results can be obtained by eans of approxiating processes. 2. Definition of the odel We consider a population of size N, divided into distinct classes or subpopulations, C, of size N, (1 < r <i ). The essential features of the process insofar as infection and reoval are concerned are the sae as for the general stochastic epideic. We ake the following definitions: X, = nuber of susceptibles in C,, X = E X,; Y, = nuber of infectives in C,, Y = E Y,; Z, = nuber of reoved cases in C,, Z = I Z,; flr = infection rate in C, due to infectives in C,, (1 < r, s < ); Yr, = reoval rate in C,, (1 r? ). The process is then defined by the _ transition rates: rate of infection in C, (X, -+ X,-1, Y,-+ Y,. + 1) = X, rsy,; rate of reoval in C, (Y, -+ Y, - 1,,IM Z, -+ Z, + 1) = y,y,. The initial conditions are (1) (X,,Y,,Z,) = (n,,a,, 0); and it is usually assued that the outbreak is started in one class, C, say, so that only a, is non-zero. The final state of the process is given by (X,, Y,,Z,) = (n,- r,o,,, + a,), with 0, = (,/n, = the severity of the outbreak in C,. In the case of a population subdivided into a nuber of classes such that the ebers of each class ix hoogeneously aongst theselves, but to a

4 On an epideic in a stratified population 661 lesser degree with individuals of other classes, the fl,, can be expressed in ters of ore eaningful paraeters as follows: where flrs = fprsn,/ns, i, = infection rate in C,, and p,, = degree of ixing between C, and Cs. There are certain constraints on the p,, inherent in their eaning: Psr =Prs, p,, = 1, O p,r, 1. It should be noted that p,, = 1 gives hoogeneous ixing, while p,, = 0 gives copletely separate populations. Thus, the process under consideration is bounded by these two known extrees. We will consider in soe detail the case of equivalent classes, for which we have (2) Nr= N,, =,,, = (r s), = q (r s), where 0? q?_ 1. This odel corresponds to the general stochastic epideic except in regard to the hoogeneous ixing assuption-we have a unifor population divided into classes. 3. Deterinistic approxiation for the severity The deterinistic approxiations are obtained as the solutions of the differential equations which result when the second oents are neglected fro the equations satisfied by the stochastic eans. They do not reflect the detailed behaviour of the process, particularly in regard to extinctions, and are best regarded as approxiations to the eans. The deterinistic approxiations then, which we denote by (Xd, Yrd,Zrd) satisfy the following equations: (d3) dx"d dt s=1 - Xrd C rsysd, dy, _ (5) dzrd dzd YrYrd, with initial conditions (1), and final state (Xrd, Yrd, Zrd) = (nr - ~rd, 0, 'rd + a,); with,d = ~rd/n, = the deterinistic approxiation to the severity of the outbreak in C,.

5 662 R. K. WATSON We now derive equations for the rd,. Fro (5) we obtain (6) dt I - d M s1 Zsd rsysd Then, we derive fro (3) and (6), using initial conditions (1), (7) Xrd = nrexp - 'Zsd So, letting t -+ oo in (7) we are led to (8) log(1 (8) log(l- ord)+ I nsflrs asd + asflrs 0. s=1 l s s=1 l Thus, we have a set of equations in unknowns for the rda. No explicit solution is available, but in any particular case, solutions ay be found by an iterative procedure. For exaple, if o ) is the estiate of ord after the ithiteration, then we define,?e+x) 1 - p e r-1 nsflrs,(i+1m) nsfirsd - asfl (i) s=1 Ys sr s =1 Ys with do) = 1, say. Such a ethod is easily prograed, and high accuracy can be obtained quite quickly-for exaple, with = 50, four decial accuracy is obtained in less than ten seconds on an IBM 7044 achine. In the case of equivalent classes, the Equations (8) for the ard becoe log(1 - rd) + -1'(N -ar.[)rd + q asd + fly-1[ar + q as = 0. s~r I L s~r If we neglect the ar, then 0rd = a, where ad satisfies (9) log(1- a) + QN0Py- lad = 0, with Q = 1 + (- 1)q. 4. The equivalent classes odel For this odel, defined by (2), since X, < No, we have Pr{Y-- Y+ 1 in (t, t + At)} = Pr(Y-- Y- 1 in (t,t + At)) = r=l s=l s flrsxryat + o(at)? QflNoYAt + o(at), 1 yyat + o(at) = yyat + o(at). r=1 So, the linear birth and death process, Y,(t), having birth rate QflNo and death rate y, is an upper bounding process which approxiates Y(t) in the early stages.

6 On an epideic in a stratified population 663 It is known, e.g., Karlin ((1966), ch. 7), that Y,(t) becoes extinct with probability equal to Po = in{1,(y/pqno)y"u()}; and this corresponds to early extinction of Y(t), i.e., a inor outbreak. Otherwise, Yu(t) explodes, corresponding to a ajor outbreak. Thus, as ight be expected fro (9), we find a threshold such that a ajor outbreak is possible only if fqno > y. But, the nature of this outbreak has yet to be specified. It is to be expected that if q is large, the outbreak will include a ajority of the classes, while if q is sall, only a few of the classes will be affected; assuing the outbreak to be initiated fro only one of the classes. So, we now envisage three types of outbreak: localized outbreak - inor outbreak in initially infected class; restricted outbreak - ajor outbreak in a few classes; generalized outbreak - ajor outbreak in ost classes. Which one of these types of outbreak will be exhibited is dependent on the paraeters of the odel. Figure 1 indicates the probability of the three types of outbreak, as a function of 0 = PNo/ly. Probability localized (1 - Po)PG generalized Po restricted Figure 1 The probabilities of the three types of outbreak, as functions of 0 = /Noly

7 664 R. K. WATSON Here, 01 and 02 are functions of, q. Fro the expression for Po, we see that 01(, q) = 1/Q = 1/(1 + q(-1)). Further we expect that 02-01,, PG 1 except when q is sall. We approxiate to PG, 02 in this case below. The class to class spread of infection, when q is sall, ay be approxiated by a odel siilar to one considered by Daley (1967). This odel ay be described as follows. A class, once infected, reains infective for a rando positive tie, at the end of which the infection is transitted to a nuber of other classes, each of which have a probability p of being contacted. A susceptible class becoes infective with probability n after contact, while any infective or reoved class contacted is unaffected. Here, we apply the following interpretations. (i) One class contacts another if an infective in the one infects at least one susceptible in the other. (ii) A class is susceptible if there has been no ajor outbreak in the class. (iii) A class is infected if a ajor outbreak occurs in the class. (iv) A class is reoved if a ajor outbreak has occurred in the class. Corresponding to the original odel, it is assued that initially there is only one infective, in class C1 say. For this odel, Daley's results are odified to the following. (i) Threshold. A generalized outbreak is possible only if ( - 1)pn > 1. (ii) Severity. If there is a generalized outbreak the nuber of classes affected is approxiated by?g = 1 + (-l1)o, where ag satisfies log(1 - ar) + (- l)pprac = 0. (iii) Probability of generalized outbreak. The probability of generalized outbreak is (1-Po)PR, where PG is the larger solution of 1-x = 1 - pnx"-, 1; 0<x=< and P0 = in{1, y/?no). All that is required then, is to find expressions for n, p in ters of the paraeters of the odel. Exact expressions have not been derived, but reasonable and siple approxiations are easily found for the case when q is sall, and oderately large, (say > 10), so that distinction between restricted and generalized outbreaks is possible. If Lis the nuber of individuals in a given class infected fro another infected class, then, assuing q sufficiently sall, L has a distribution which is approxiately negative binoial. Thus, if L* is such that

8 On an epideic in a stratified population 665 Pr{L* = j} =.) ( + qno (-' 1 + q (j >0), where C is the size of the outbreak in the infected class; then L* is an upper approxiation for L. If, instead of the rando variable C, we use the deterinistic approxiation 'd, the solution of the equation log(1 - (d/no)) + Qfld- 1 = 0, then we find the following approxiations: p = Pr{L > 0} (1 + qflno0y-)- 7c = Pr{ajor outbreak IL > 0} ' {1 - (1 - qq-' + qflnol)-d}. - Note that 02 is deterined as the solution of ( - 1)p(8)nt(8)= 1, which gives, using the above approxiations q-l {(1 + 1/( -2))lfCd - 1 Two nuerical exaples are given in Tables 1 and 2 to illustrate the variation of p,, CG', PG with q. TABLE 1 = 10, No = 100, y = 0.06, P = qcd P PG

9 666 R. K. WATSON TABLE 2 = 50, No = 50, y = fl= q d P C r PG Further, a series of siulations of the epideic process has been done for populations of size 1000 and 2500 with various values of, No, q, and p =y/fl; and the results obtained are in reasonable agreeent with the theoretical predictions. Details of the siulation series ay be obtained on application to the author. References BAILEY, N. T. J. (1957) The Matheatical Theory of Epideics. Griffin, London; Hafner, New York. BARTLETr, M. S. (1957) Measles periodicity and counity size. J. Roy. Statist. Soc. A 120, DALEY, D. J. (1967) Soe aspects of Markov chains in queueing theory and epideiology. Ph. D. thesis, University of Cabridge. HASKEY, H. W. (1957) Stochastic cross-infection between two otherwise isolated groups. Bioetrika 44, KARLIN, S. (1966) A First Course in Stochastic Processes. Acadeic Press, New York. RUSHTON, S. AND MAUTNER, A. J. (1955) The deterinistic odel of a siple epideic for ore than one counity. Bioetrika 42,