Epidemic Models: Thresholds and Population Regulation

The complexity of epidemic processes is often obscured by the relatively simple way in which transmission is characterized in some epidemic models. A large part of current deterministic epidemic theory extends the S (susceptible host density), I (infective host density), and N (total host population density) differential equation model first proposed by Ross (cf. Ross 1916; Ross and Hudson 1917). Most of these models predict that the density of susceptibles (S) must exceed a threshold (NT) before an infectious disease can become epidemic (for a review of the theory, see Serfling 1952; Bailey 1975). Here we show that a seemingly minor change in the way transmission is modeled can significantly alter conclusions as to whether such a threshold exists and to whether an infectious disease can regulate its host population. In a recent series of papers Anderson and May (Anderson 1978, 1979a, 1979b, 1980, 1982; Anderson and May 1978, 1979, 1980, 1981, 1982; May and Anderson 1978, 1979) focused considerable attention, using Ross-type differential equation models, on the role of infectious diseases in the observed dynamics of host populations including invertebrate hosts. They examined the role of parasitesbroadly defined to include viruses, bacteria, protozoans and helminths-in biological control (Anderson 1979a, 1982; Anderson and May 1980), and they determined measures necessary for the control and eradication of pathogens, especially viral diseases such as rabies, whooping cough, and measles (Anderson et al. 1981; Anderson and May 1982). In most cases, their models predict the existence of a threshold NT> 0. This threshold is a consequence of the assumption that the rate of disease transmission is proportional to the number of random encounters between infectives (of density I) and susceptibles (of density S) in the population, i.e., the transmission rate is P3SI, where I is the transmission parameter. More complex characterizations of transmissions are common and have been discussed at length (Anderson 1979b, 1980, 1982; Bailey 1975; Anderson and May 1981; Yorke et al. 1979). Generally, the rate of disease transmission will depend on the rate at which propagules are produced by the infected individuals within a population, the dynamics of propagule transfer (e.g., vector dynamics), and the rate at which propagules invade susceptible hosts. For our purposes we assume that propagule production is a linear function of the number of infectives and either that the vector density remains sufficiently high so that transmission is essentially independent of their dynamics or that vectors are not required for transmission (e.g., venereal diseases). Then, if we assume that the rate at which propagules invade susceptible hosts is dependent on the proportion of individuals at risk in the host