Jointly estimating transmissibility and prior immunity from epidemic time series

Infectious disease time series are often used to estimate a pathogen's basic reproduction number, $R_0$. However, fits of epidemic models to time series conflate pathogen transmissibility with pre-existing population immunity, so only the *effective* reproduction number, $R_{eff}$, can be inferred. This composite parameter is the product of the underlying $R_0$ and the pre-epidemic susceptible fraction, $x^-$. We show that a conservation law associated with epidemic momentum---prevalence weighted by potential to infect---makes it possible to disentangle transmissibility from prior immunity and to infer $R_0$ and $x^-$ separately from a single epidemic time series. We test the methodology using stochastic epidemic simulations, and illustrate the approach with a reappraisal of influenza transmissibility during the 1918 pandemic, estimating rather than assuming the degree of prior population immunity. For the autumn wave in Philadelphia, USA, we find $R_0\approx2.7$ and $x^-\approx0.8$, implying that about 20\% of the population was already immune before that wave, plausibly as a result of infection during the spring 1918 herald wave.