Italy and its health care system are currently under great pressure due to the SARS-CoV-2 pandemic. Here we attempt to provide an approach to the medium-term prediction of the epidemic’s progress.
Richard’s GLM. Given that the daily (and publicly available) indicators are counts, it’s reasonable to assume a Poisson or, more likely, Negative Binomial distribution. In both cases, we assume that the expected value for the random variable at day t has the following form (if you want to see how it works look here)
E[Y]) = b+(T-b)/[1+10(h(p-t))]s
where b is the lower asymptote, T is the upper asymptote, h is the slope, p is a location parameter governing the first derivative’s maximum, and s is an asymmetry parameter that affects near which asymptote maximum growth occurs.
An extended version of this model has been considered, where the peak is allowed to persist in time through a soft-thresholding operator. This extension will be treated separately in a future technical document. To include covariates and constraints, the parameters mentioned above have been reparametrized accordingly. For example, the parametrization T-b = r and log(r) = a+bx allows us to get rid of the T>b constraint and allows us to evaluate the effect of the covariate x on T.
The likelihood function is maximized using a quasi-Newton numerical approximation.
Standard errors and confidence intervals have been obtained, as usual, by numerical approximations of the information matrix. The information matrix was also used to check for local identifiability.
In the following days, the R code that was used to implement the aforementioned model will be made available. A notice will be posted on the StatGroup-19 page, and this document will be updated with a link to the code.
Data. The data is official data shared by the Protezione Civile, which has been disaggregated by region. Given that the data on positive daily cases is now heavily distorted due to an excessive selection of cases with severe symptoms (and with swab tests 1 to 10 days old), we focus on counts of intensive care unit (ICU) hospitalizations.
Results. Using the number of ICU hospitalizations as the main indicator, it is expected that Lombardy will reach the epidemic peak, i.e. the onset of the first derivative’s maximum, starting on 26/03 (22/03 - 30/3). Assuming a 7-10-day lag period between the peak number of ICU hospitalizations and the peak number of positive recorded cases, it is reasonable to say that the peak of positive cases in Lombardy goes back to 18/3 (12/03 – 23/03). For all other regions (and on a national level) the inflection point, always referred to the considered data, is estimated around 25/03. This is an unreliable estimate, as it’s impossible to quantify the effects of the current restrictive measures, which will not be observable before the end of March.
StatGroup-19[1]
(Translation by Gabriele Fabozzi)
[1] StatGroup-19: Fabio Divino (Università del Molise), Alessio Farcomeni (Università di Roma Tor Vergata), Giovanna Jona Lasinio (Università di Roma La Sapienza), Gianfranco Lovison (Università di Palermo), Antonello Maruotti (LUMSA, Roma). Special thanks to Dr. Gabriele Sene (Banca D’Italia).
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