Reproduction number for an age of infection structured model

We study the basic reproduction number (R0) in an epidemic model where infected individuals are initially asymptomatic and structured by the time since infection. At the beginning of an epidemic outbreak the computation of R0 relies on limited data based mostly on symptomatic cases, since asymptomatic infected individuals are not detected by the surveillance system. R0 has been widely used as an indicator to assess the dissemination of infectious diseases. Asymptomatic individuals are assumed to either become symptomatic after a fixed period of time or they are removed (recovery or disease-related death). We determine R0 understood as the expected secondary symptomatic cases produced by a symptomatic primary case through a chain of asymptomatic infections. R0 is computed directly by interpreting the model ingredients and also using a more systematic approach based on the next-generation operator. Reported Covid-19 cases data during the first wave of the pandemic in Spain are used to fit the model and obtain both values of R0 before and after the severe lockdown imposed in March 2020. The results confirm that SARS-CoV-2 was expanding within the population before the lockdown whereas the virus spreading was controlled two weeks after the lockdown. In memoriam Carles Perelló, emeritus professor at UAB, recently deceased and scientific father/grandfather of the authors.