Reproduction Numbers R_0, R_t for COVID-19 Infections with Gaussian Distribution of Generation Times, and of Serial Intervals including Presymptomatic Transmission

Basic and instantaneous reproduction numbers, "R" _"0" and "R" _"t" , are important metrics to assess progress of an epidemic and effectiveness of preventative interventions undertaken, and also to estimate coverage needed for vaccination. Reproduction numbers are related to the daily number of positive cases recorded by the national public health authorities, via the renewal equation. During periods of exponential growth or decay they are linked also to the rate constants by the Lotka-Euler equation. For either application, we need the distribution of generation times between primary and secondary infections. In practice, we use instead the directly observable serial interval between symptoms onset of infector and infectee. Pre-symptomatic transmission that occurs in COVID infection causes serial intervals to extend to negative values, which can be described with a Gaussian distribution. Consistent application of the two approaches requires careful attention to lower limits imposed on the distribution. Allowing Gaussian-distributed serial intervals to extend to minus infinity with the Lotka-Euler equation, as commonly is done, results in lower reproduction numbers than predicted from the discretized renewal equation. Here, we formulate the Lotka-Euler equation for Gaussian distributions including an explicit lower cut-off, and use this to explore the consequences of presymptomatic transmission for COVID-19 infections.