INFECTIOUS-DISEASE PERSISTENCE WHEN TRANSMISSION VARIES SEASONALLY

The generation reproduction number, R-0, is the fundamental parameter of population biology. Communicable disease epidemiology has adopted R -0 as the threshold parameter, called the basic case reproduction numb er (or ratio). In deterministic models, R-0 must be greater than 1 for a pathogen to persist in its host population. Some standard methods o f estimating R-0 for an endemic disease require measures of incidence, and the theory underpinning these estimators assumes that incidence i s constant through time. When transmission varies periodically (e.g., seasonally), as it does for most pathogens, it should be possible to e xpress the criterion for long-term persistence in terms of some averag e transmission (and hence incidence) rate. A priori, there are reasons to believe that either the arithmetic mean or the geometric mean tran smission rate may be correct. By considering the problem in terms of t he real-time growth rate of the population, we are able to demonstrate formally that, to a very good approximation, the arithmetic mean tran smission rate gives the correct answer for a general class of infectio n functions. The geometric mean applies only to a highly restricted se t of cases. The appropriate threshold parameter can be calculated from the average transmission rate, and we discuss ways of doing so in the context of an endemic vector-borne disease, canine leishmaniasis. (C) 1997 Elsevier Science Inc.