Infectious disease persistence when transmission varies seasonally

The generation reproduction number, R0, is the fundamental parameter of population biology. Communicable disease epidemiology has adopted R0 as the threshold parameter, called the basic case reproduction number (or ratio). In deterministic models, R0 must be greater than 1 for a pathogen to persist in its host population. Some standard methods of estimating R0 for an endemic disease require measures of incidence, and the theory underpinning these estimators assumes that incidence is constant through time. When transmission varies periodically (e.g., seasonally), as it does for most pathogens, it should be possible to express the criterion for long-term persistence in terms of some average transmission (and hence incidence) rate. A priori, there are reasons to believe that either the arithmetic mean or the geometric mean transmission rate may be correct. By considering the problem in terms of the real-time growth rate of the population, we are able to demonstrate formally that, to a very good approximation, the arithmetic mean transmission rate gives the correct answer for a general class of infection functions. The geometric mean applies only to a highly restricted set 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.