SHAR and effective SIR models: from dengue fever toy models to a COVID-19 fully parametrized SHARUCD framework


  • Maira Aguiar 1) Dipartimento di Matematica, Universit`a degli Studi di Trento, Via Sommarive, 14 – 38123 Povo (Trento), Italy 2) Basque Center for Applied Mathematics (BCAM), Alameda Mazarredo, 14, 48009 Bilbao, Spain 3) Ikerbasque, Basque Foundation for Science, Bilbao, Spain
  • Nico Stollenwerk 1) Dipartimento di Matematica, Universit`a degli Studi di Trento, Via Sommarive, 14 – 38123 Povo (Trento), Italy 2) Center for Mathematics, Fundamental Applications and Operations Research, Lisbon University, Portugal



COVID-19, Bayes factor, master equation, Fokker-Planck equations, epidemiological growth factors


We review basic models of severe/hospitalized and mild/asymptomatic infection spreading (with classes of susceptibles S, hopsitalized H, asymptomatic A and recovered R, hence SHAR-models) and develop the notion of comparing different models on the same data set as exemplified in the comparison of SHAR models with effective SIR models, where only the H-class of the SHAR model is taken into account in the SIR model. This is done via the so-called Bayes factor. A simpler pair of models with analytical expressions up to the Bayes factor will be briefly mentioned as well. The notions developed with respect to dengue fever epidemiology will then be used to analyze recently becoming available data on coronavirus disease 2019, COVID-19, where models can be fully parametrized including hospital admission and more extensions like intensive care unit (ICU) admission and deceased, always with a close look on as simple as possible models but not simpler, as exercised in Ocham’s razor and analyzed by e.g. the Bayes factor. We present the resulting models of SHAR-type with additional classes of ICU admissions U, and deceased D, and for data analysis of cumulative disease data, also accounting the cumulative classes C, in the so-called SHARUCD framework. Besides a first basic version, SHARUCD model 1, we investigate also in detail a refined version, SHARUCD model 2, which could be achieved by a closer analysis of available data only obtained after the exponential growth phase of the epidemic, when lockdown control measures showed effects. Namely, the ICU admissions turned out to be more in synchrony with the hospitalized than with e.g. the deceased cases, such that we could adjust the transitions so that ICU admissions are modeled like hospitalizations in model 2, and not like recovery or disease induced death as assumed in model 1, explaining much better the empirical data, specially after the effects of the lockdown became visible. Special attention will be given here, for the first time, to the initial phase of the COVID-19 epidemics, before all variables entered into the exponential phase, and its interplay between asymptomatic and severe hospitalized cases, always in close check with the SIR-limiting case. Such improved understanding of the initial phase will help in the future analysis of re-emergent outbreaks of COVID-19, likely to happen in the next or a subsequent respiratory disease season in autumn or winter.

Author Biography

Maira Aguiar, 1) Dipartimento di Matematica, Universit`a degli Studi di Trento, Via Sommarive, 14 – 38123 Povo (Trento), Italy 2) Basque Center for Applied Mathematics (BCAM), Alameda Mazarredo, 14, 48009 Bilbao, Spain 3) Ikerbasque, Basque Foundation for Science, Bilbao, Spain


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