Bounding the generation time distribution uncertainty on <i>R</i> ₀ estimation from exponential growth rates

The basic reproduction number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> is one of the main parameters determining the spreading of an epidemic in a population of susceptible individuals. Wallinga and Lipsitch proposed a method for estimating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> using the Euler-Lotka equation, which requires the Laplace transform of the generation interval distribution. The determination of the generation time distribution is challenging, as the generation time is not directly observable. We prove upper and lower bounds on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> using only the first few moments of the generation interval distributions and study the sensitivity of the bounds to these parameters. The bounds do not require the exact shape of the generation interval distribution and give robust estimates of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> relationship.