The ultimate size of carrier-borne epidemics

SUMMARY An epidemic model is discussed in which there is an initial introduction of a carrier or carriers into a population and further carriers may be created from the susceptibles. The probability distribution for the number of survivors is derived and approximations and numerical illustrations are given. In a recent paper Weiss (1965) has considered an epidemic in a closed population which is spread not by the infected individuals but by carriers. He supposed that one or more carriers were introduced into the population and that these carriers infected susceptible individuals until detected, the time before detection having an exponential distribution. At no time during the epidemic were any additional carriers introduced or created and the epidemic terminated as soon as all the carriers were detected or all susceptibles had been infected. Weiss's model has the mathematical advantage that it is completely soluble in terms of elementary functions; see Dietz (1966) and Downton (1967b). It is, however, unrealistic in its assumption that no new carriers can be created. The present paper discusses a model in which it is supposed that after the initial introduction of a carrier or carriers, no new carriers are introduced from outside the population, but that new carriers may be created from the susceptibles in that population. The model is appropriate for the situation where a proportion 7T of those infected contract the disease in such a mild form that their symptoms are unnoticeable even though they are capable of passing on the disease. Such subelinically infected persons would then act as carriers until detected. For such a case we would expect oT to be small; oT = O corresponds to the Weiss model. The main aim of this paper is to obtain approximate expressions for the probability distribution and moments of the number of susceptibles surviving the epidemic in that case. The ultimate behaviour of the epidemic does not, however, depend upon oT in a simple way so that while valid approximations have been obtained for small it, even if this results in a large epidemic, it turns out that these approximations may be used in certain cases even when or is not small. In particular, they may be employed even if 7t = 1, when the mathematics becomes identical with that of the Kermack-McKendrick (1927) model for the general stochastic epidemic provided attention is confined to small subcritical epidemics. The stochastic model describing the process is defined in ? 2, where an explicit expression for the probability distribution of the number of survivors is given, together with an indication of the ways in which it may be obtained. In ? 3 the deterministic analogue of the stochastic model is discussed, providing the deterministic approximation to the mean number of survivors. This approximation is valid for small epidemics in large populations. Returning to the stochastic model ? 4 shows that the survivor probabilities may be expressed in terms of