Estimation of R(t) based on illness onset data: An analysis of 1907-1908 smallpox epidemic in Tokyo

R(t), the actual average number of secondary cases per primary case at calendar time t, is epidemiologically useful in assessing transmission dynamics in a population with varying susceptibility levels. However, a technical limitation of existing methods of estimating R(t) is the reliance on the daily number of cases with illness onset and the distribution of the serial interval, although the estimator of R(t) should be calculated as the ratio of newly infected cases at time t to the total number of potentially infectious people at the same time. Using historical data of a smallpox outbreak in Tokyo City, Japan, approximately 100 years ago, we propose a new method to compute R(t) that can be estimated using information on illness onset. Our method decomposes the mechanism of transmission into two distinct pieces of information: the frequency of secondary transmission relative to disease age and the probability density function of the incubation period. Employing a piecewise constant model, our maximum likelihood estimates of R(t) dropped below unity (0.6; 95% confidence interval: 0.5-0.7) for the period from Day 64 to Day 79, indicating that the epidemic was under control in this period. R(t) was continuously below one through the remaining days. The model prediction captured the overall observed pattern of the epidemic well. Our method is appropriate for acute infectious diseases other than smallpox for which variations in infectivity relative to disease age should be considered to correctly estimate the transmission potential, such as the ongoing global epidemic of coronavirus disease 2019 (COVID-19).