This paper applies the constant population approximation to the study of epidemics which involve more than a single type of infective. An example of this would be a situation in which both clinically infected individuals and subclinically infected individuals or carriers are present. We derive equations for the expected numbers of clinically infected individuals and carriers at any time t for the model with zero latent period and infectious periods having negative exponential distributions. From these equations we derive conditions under which a unimodal incidence curve can result, and expressions for the expected total epidemic size. The equations describing the course of an epidemic are analytically intractable except in certain simple cases, [1]. When one examines the reason for the mathematical difficulties that arise, one finds that the principal stumbling block is the assumption of a finite population of susceptibles. It can be argued, however, that in modern societies epidemics very rarely menace an entire population, and that observed epidemic sizes are usually much smaller than the total susceptible population. This is due to a variety of reasons, among which are public health measures and good communication facilities available to a modern society. In consequence it is reasonable to try to bypass some of the mathematical difficulties inherent in the theory of epidemics by assuming a susceptible population whose size does not change through the course of the epidemic. Bailey [2], Morgan [3], and Williams [4] have discussed in some detail the theory of epidemics using the constant population approximation, although such an approximation was used by several authors earlier, [5], [6], [7]. The theory developed so far has dealt mainly with epidemic processes in which there is only a single type of infective and a single type of susceptible. Recently Gart [8] has considered a model for epidemics involving more than one type of susceptible. It is the purpose of this paper to analyze the development in time of epidemics which involve more than a single type of infective individual. This Received in revised form 3 January 1967. 257 This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:20:57 UTC All use subject to http://about.jstor.org/terms 258 HUGH M. PETTIGREW AND GEORGE H. WEISS problem, in which there may be several types of infectives, is suggested by diseases in which carriers are important, [9]. In the present paper we derive equations for the expected numbers of carriers and clinically infected individuals at any time t, in the case of a zero latent period and negative exponential distributions of infectious periods. From these equations we will give the conditions under which an epidemic arises from the introduction of a bearer of the disease, i.e., the conditions under which the reporting curve increases initially. These conditions are of some theoretical interest since Bailey has shown that in the infinite population approximation to an epidemic with only one class of susceptibles and one class of infected individuals, no initial increase in the reporting curve can occur. Finally, we derive expressions; for the expected total epidemic size. Let us consider a homogeneously mixing population which, in the present approximation, can be characterized by four parameters, yi(t), y,(t), zi(t), zc(t). These are, respectively, the number of clinically infected individuals, the number of carriers, the cumulative number of removals of infectives, and the cumulative number of removals of carriers, all evaluated at time t. The probability that a single susceptible individual will become clinically infected in (t, t + dt) is assumed to be fli(y, + yc)dt, and the probability that a susceptible will become a carrier is assumed to be Ic(yi + ye)dt, where both l's are assumed to be constant and it is assumed for simplicity that carriers and infectives are equally infectious. The constancy of these p's is the principal assumption of the present theory; in the more detailed stochastic theory discussed by Bailey [1] the f's are proportional to the number of susceptibles as well. In the simplest model, we assume that the latent period is zero, i.e., that newly infected individuals are immediately capable of infecting others, and that the infectious period is a random variable with a negative exponential distribution. The rate parameter appearing in the distribution for clinically infected will be denoted by vi and that appearing in the distribution for carriers will be denoted by v,. In order to describe the stochastic process we introduce a set of probabilities p(r,,r2, r3, r4, t) defined by p(r, t) = Pr{y,(t) = r, yc(t) = r2, z(t) = r3, zc(t) = r4}. By our assumptions the p(r, t) satisfy ap = fl (r1 + r2 1)p(r, 1,r2,r3, r4, t) + fl(r, + r2 -1)p(r1,r2 1,r3, r4, t) (1) + vA(r1 + 1)p(r + 1,r2, r 1,r4)+ vc(r2 + 1)p(r, r2 + 1, r3,r4 1) [(fl + ic + vi)rl + (fl + flPc + ve)r2] p(r, t). This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:20:57 UTC All use subject to http://about.jstor.org/terms Epidemics with carriers: the large population approximation 259 The moment generating function (2) M(01, 02, 03 , 0,t) = M(O, t) = E{eo'Y'+2yYc+03zi+042C} therefore satisfies aM M = [f#,(ee I1) + flc(e02 1) + vi(ee, +03 1)] (3) aM + [fl(ee' 1) + flc(e02 1) + vc(e-02+4 1)] The interesting features of the development of the epidemic can be determined by analyzing the properties of the mean values Lpi(t) = E{yi(t)}, pc(t) = E{yc(t)} (4) ow(t) = E{z,(t)}, wc(t) = E{zc(t)}. The equations for these mean values are easily determined by equating coefficients of the O's on both sides of Equation (3). They are Pi = (/01v-i)it?+fliltc (5) Pc = flcpi+ (fc v)Pc