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

TL;DR

This paper develops a mathematical framework to accurately estimate COVID-19 reproduction numbers considering Gaussian-distributed serial intervals with presymptomatic transmission, highlighting the importance of proper distribution bounds.

Contribution

It introduces a formulation of the Lotka-Euler equation for Gaussian serial intervals with a lower cut-off, improving reproduction number estimates in COVID-19 modeling.

Findings

Gaussian serial intervals can extend to negative values due to presymptomatic transmission.

Ignoring the lower bound in Gaussian serial interval distributions leads to underestimated reproduction numbers.

The new formulation provides more accurate estimates of R0 and Rt in COVID-19 epidemiology.

Abstract

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.