Unjustified Poisson assumptions lead to overconfident estimates of the effective reproductive number

Time-varying effective reproductive numbers of infectious diseases are commonly estimated using renewal equation models. In the widely applied R package EpiEstim, this approach is combined with a Poisson distributional assumption. This has been criticized on various occasions, mostly on grounds of general model realism or a desire to estimate overdispersion parameters. Here we argue that an important issue arising from the Poisson assumption is that inference about the effective reproductive number becomes overconfident in presence of overdispersion. By how much standard errors are underestimated follows in a straightforward manner from theory on generalized linear models. We therefore recommend to replace the Poisson assumption by quasi-Poisson or negative binomial extensions, and contrast their respective properties. We illustrate our arguments in three examples on Ebola, pandemic influenza and COVID-19.