Attack rate and the price of SARS-CoV-2 herd immunity in Brazil

We introduce a compartmental model with age structure to study the dynamics of the SARS-COV-2 pandemic. The contagion matrix in the model is given by the product of a probability per contact with a contact matrix explicitly taking into account the contact structure among different age groups. The probability of contagion per contact is considered as time dependent to represent non-pharmaceutical interventions, and is fitted from the time series of deaths. The approach is used to study the evolution of the COVID-19 pandemic in the main Brazilian cities and compared to two good quality serological surveys. We also discuss with some detail the case of the city of Manaus which raised special attention due to a previous report of three-quarters attack rate by the end of 2020. We discuss estimates for Manaus and all Brazilian cities with a total population of more than one million. We also estimate the attack rate with respect to the total population, in each Brazilian state by January, 1^st 2021 and May, 23 2021.