  Fitting Susceptible-Infected-Recovered (SIR) models to incidence data is
problematic when not all infected individuals are reported. Assuming an
underlying SIR model with general but known distribution for the time to
recovery, this paper derives the implied differential-integral equations for
observed incidence data when a fixed fraction of newly infected individuals are
not observed. The parameters of the resulting system of differential equations
are identifiable. Using these differential equations, we develop a stochastic
model for the conditional distribution of current disease incidence given the
entire past history of reported cases. We estimate the model parameters using
Bayesian Markov Chain Monte-Carlo sampling of the posterior distribution. We
use our model to estimate the transmission rate and fraction of asymptomatic
individuals for the current Coronavirus 2019 outbreak in eight American
Countries: the United States of America, Brazil, Mexico, Argentina, Chile,
Colombia, Peru, and Panama, from January 2020 to May 2021. Our analysis reveals
that consistently, about 40-60% of the infections were not observed in the
American outbreaks. The two exception are Mexico and Peru, with acute
under-reporting in Mexico.
