A practical generation interval-based approach to inferring the strength of epidemics from their speed

Sang Woo Park; David Champredon; Joshua S. Weitz; Jonathan Dushoff

The primary goal of this paper is to further explore and explain the r-R relationship. We explore the qualitative relationship between generation time, initial rate of spread r, and initial reproductive number R using means, variance measures and gamma approximations. We show that summarizing generation-interval distributions using moments provides biological explanations that unify previous findings. We further discuss the generality of the gamma approximation and provide examples of its relevance in using realistic epidemiological parameters from previous outbreaks. By doing so, we shed light on the underpinnings of the relationship between r and R, and on the factors underlying its robustness and its practical use when data on generation intervals is limited or hard to obtain. In this section, we introduce an analytical framewok, and recapitulate previous work relating growth rate r and reproductive number R. These two quantities are linked by the generation-interval distribution, which describes the interval between the time an individual becomes infected and the time that they infect another individual. In particular, if R is known, a shorter generation interval means a faster epidemic (larger r). Conversely (and perhaps counter-intuitively), if r is known, then shorter disease generations imply a lower value of R, because more individual generations are required to realize the same population spread of disease [11, 32] (see Fig. 1 ). The generation-interval distribution can be defined using a renewalequation approach. A wide range of disease models can be described using the model [15, 10, 36, 1, 44, 37] :