A carrier-borne epidemic model, in which the carrier removal process is influenced by the number of detected infected individuals, is considered. The distribution of the total size of the epidemic is found and expressions for the expected ultimate size of the epidemic and the expected area under the trajectory of carriers are given.

An alternative approach to the derivation of a general class of results in carrier-borne epidemic theory is presented. Some insight can be gained into the basic structure of carrier-borne epidemics and the method indicates which other problems of a similar type could yield to analysis.

In this article we examine the behavior of Mm(τ), the number of susceptibles remaining uninfected at time τ in a carrier-borne epidemic as the initial number of susceptibles m tends to infinity for fixed initial number of carriers n. We show that as m → ∞ the proportion of susceptibles process {Mm(τ)/m : ≧ 0} converges weakly to a limiting process. The limiting process has sample functions almost surely of the form exp (– h(τ)), where the form of h(τ) is determined by the removal times of the carriers.

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.

In this paper, the cost of the carrier-borne epidemic is considered. The definition of duration, as used by Weiss (1965) and subsequent authors, is generalised and the probability distribution for the number of located carriers is obtained. One component of cost, namely the area generated by the trajectory of carriers, is examined and its probability density function derived. The expected area generated is then shown to be proportional to the expected number of carriers located during the epidemic, a result which has an analogue in the general stochastic epidemic.

The deterministic and stochastic models developed by Bartlett (1956) to describe the initial spatial spread of an epidemic such as measles are extended to host-vector and carrier-borne epidemics such as malaria and typhoid in which more than one type of individual is involved. If the epidemic does not die out, then the behaviour predicted by the deterministic model and an examination of the mean distributions in the stochastic model is a wave of infection spreading out from the initial source of infection.