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6. Acknowledgments and contribution :

Maíra Aguiar, BCAM – Ikerbasque, MTB research line leader & UNITN Marie Curie Fellow and Nico Stollenwerk, University of Trento, have conceived of the study, developed and analyzed the models.

Nicole Cusimano, Post doctoral researcher of Mathematical and Theoretical Biology group (MTB). Nicole has produced the animated graphics for the SHARUCD model short predictions for the 5 dynamical variables, PCR tested positives, hospitalizations, ICUs, deceased and recovered and the animated graphic for the combined growth rates.

Marina Echeverría Ferrero, PhD Student (Predoc Severo Ochoa 2018) of Mathematical and Theoretical Biology group. Marina has collected the data and produced the bar chart race for the confirmed COVID cases number of deceased cases round the globe.

Francisco P. Rodrigues, student at Instituto Superior Técnico, University of Lisbon, Portugal, has developed the SIR simulation program.

Eduardo Millán, Osakidetza Basque Health Service, has collected for the Basque Country and prepared the data sets used for the SHARUCD modeling exercise.

We thank the huge efforts of the whole COVID-19 BMTF, specially to Joseba Bidaurrazaga Van-Dierdonck, Public Health, Basque Health Department, and Adolfo Morais Ezquerro, Vice Minister of Universities and Research of the Basque Government, for the fruitful discussions.

7. References:

[1] World Health Organization. Emergencies preparedness, response. Novel Coronavirus – China. Retrieved from

[2] World Health Organization. WHO announces COVID-19 outbreak a pandemic. Retrieved from

[3] World Health Organization. Coronavirus disease (COVID-2019) Weekly epidemiological update - 15 December. Retrieved from

[4] Maíra Aguiar, Nico Stollenwerk and BobW. Kooi. (2012). Modeling Infectious Diseases Dynamics: Dengue Fever, a Case Study. Chapter In book: Epidemiology Insights. DOI: 10.5772/31920

[5] FredBrauer. (2005). The Kermack–McKendrick epidemic model revisited. Mathematical Biosciences, 198(2), 119-131.

[6] van Kampen, N. G. (1992). Stochastic Processes in Physics and Chemistry, ISBN 978-0-44452-965-7. (North-Holland, Amsterdam).

[7] Gillespie, D.T. (1976). A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. Journal of Computational Physics, 22, 403-434, ISSN 0021-9991.

[8] Gillespie, D.T. (1978). Monte Carlo simulation of random walks with residence time dependent transition probability rates. Journal of Computational Physics, 28, 395-407, ISSN 0021-9991.

[9] Stollenwerk, N., \& Jansen, V. (2010). Population biology and criticality, ISBN 978-1-84816-401-7. (Imperial College Press, London).

[10] Aguiar, M., Ortuondo, E.M., Bidaurrazaga Van-Dierdonck, J. et al. Modelling COVID 19 in the Basque Country from introduction to control measure response. Sci Rep 10, 17306 (2020).

[11] Aguiar, M., Van-Dierdonck, J. B. & Stollenwerk, N. Reproduction ratio and growth rates: measures for an unfolding pandemic. PLoS ONE 15(7), e0236620 (2020).

[12] Aguiar, M. & Stollenwerk, N. Condition-specific mortality risk can explain differences in COVID-19 case fatalit ratios around the globe. J. Public Health. (2020) 

[13]Aguiar, M. & Stollenwerk, N. SHAR and effective SIR models: from dengue fever toy models to a COVID-19 fully parametrized SHARUCD framework. Commun. Biomath. Sci. 3(1), 60–89 (2020).

[14] World Health Organization. Coronavirus disease (COVID-2019). Weekly Epidemiological Update Coronavirus disease 2019 (COVID-19) 20 October 2020. Retrieved from