The SARS-CoV-2 epidemic outbreak: a review of plausible scenarios of containment and mitigation for Mexico

Manuel Adrian Acuna-Zegarra; Andreu Comas-Garcia; Esteban Hernandez-Vargas; Mario Santana-Cibrian; Jorge X. Velasco-Hernandez

where π(a) is the probability density function (pdf) of a Uniform(0,1) distribution, π(r) is the pdf of a Gamma(shape=2, scale=1), π(K) is the pdf of a Uniform(K min , K max ), and π(α) is the pdf of a Gamma(shape=2, scale=0.1). To select the prior for parameter r, we consider that previous estimation of r are close to 0.3 [15] , and a Gamma(2,1) represent a weekly informative prior as it allows for a wide range of values of r. Also, there is no available prior information regarding the final size of the outbreak K. This is a critical parameter in the model and, in order to avoid bias, we assume a uniform prior over K min and K max . To set these last to values, we consider that the minimum number of confirmed cases is the current number of observed cases Y (t n ) times nine, i.e. K min = y n * 9. The reasoning behind this number is that it has been suggested that the observed cases are under reported by a factor of nine (REF). To set the upper bound for K, we consider a fraction of the total population K max = N * 0.01, where N is the population size of Mexico. This fraction was determined base on the observations of other countries such as Italy where the proportion of infected represents one of the worst case scenarios considering its population size. Then, the posterior distribution of the parameters of interest is π(θ|y 1 , . . . , y n ) ∝ π(y 1 , . . . , y n |θ)π(θ), and it does not have an analytical form since the likelihood function depends on the solution of the Richards model, which must be approximated numerically. We analyze the posterior distribution using an MCMC algorithm that does not require tuning called t-walk s [25] . This algorithm generates samples from the posterior distribution that can be used to estimate marginal posterior densities, mean, variance, quantiles, etc. We refer the reader to [26] for more details on MCMC methods and to [27] for an introduction to Bayesian inference with differential equations. We run the t-walk for 100,000 iterations, discard the first 25000 and use 1,000 samples to generate estimates of the parameters and short-term predictions for the next 14 days.