SHAR and effective SIR models: from dengue fever toy models to a COVID-19 fully parametrized SHARUCD framework
DOI:
https://doi.org/10.5614/cbms.2020.3.1.6Keywords:
COVID-19, Bayes factor, master equation, Fokker-Planck equations, epidemiological growth factorsAbstract
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.
References
Filipe, R., Stollenwerk, N., Mateus, L., Ghaffari, P., Halstead, S., Aguiar, M., Effective infection rate in SIR-type models from models with symptomatic and asymptomatic infection, Proceedings of the 16th International Conference on Mathematical Methods in Science and Engineering - CMMSE 2015, C'adiz, Spain, ISBN: 978-84-608-6082-2, edited by Jesus Vigo et al., 2016.
Stollenwerk, N., Filipe, R., Mateus, L., Ghaffari,P., Kooi, B., Halstead, S., and Aguiar, M., Effective parameters, likelihoods and Bayesian model selection in application to epidemiological models: from SHAR to effective SIR models, Proceedings of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2017, C'adiz, Spain, ISBN: 978-84-617-8694-7, edited by Jesus Vigo et al., 2017.
Aguiar, M., Ballesteros, S., Kooi, B.W., and Stollenwerk, N., The role of seasonality and import in a minimalistic multi-strain dengue model capturing differences between primary and secondary infections: complex dynamics and its implications for data analysis, Journal of Theoretical Biology, 289, 181-196, 2011.
Kooi, W. B., Aguiar, M., and Stollenwerk, N., Bifurcation analysis of a family of multi-strain epidemiology models, Journal of Computational and Applied Mathematics, 252, 148-158, 2013.
Kooi, W.B., Aguiar, M., and Stollenwerk, N., Analysis of an asymmetric two-strain dengue model, Mathematical Biosciences, 248, 128-139, 2014.
Aguiar, M., Paul, R., Sakuntabhai, A., and Stollenwerk, N., Are we modeling the correct data set? Minimizing false predictions for dengue fever in Thailand, Epidemiology and Infection, 142, 2447-2459, 2014.
Stollenwerk, N., and Jansen, V.A.A., Meningitis, pathogenicity near criticality: the epidemiology of meningococcal disease as a model for accidental pathogens, Journal of Theoretical Biology, 222, 347-359, 2003.
Stollenwerk, N., Maiden, M.C.J., and Jansen, V.A.A., Diversity in pathogenicity can cause outbreaks of menigococcal disease, Proc. Natl. Acad. Sci. USA 101, 10229-10234, 2004.
Stollenwerk, N., and Jansen, V., Population Biology and Criticality: From critical birth-death processes to self-organized criticality in mutation pathogen systems, (Imperial College Press, World Scientific, London), 2011.
Stollenwerk, N., and Jansen, V.A.A., Evolution towards criticality in an epidemiological model for meningococcal disease. Physics Letters A 317, 87-96, 2003.
Poll'an, M., et al., Prevalence of SARS-VoV-2 in Spain (ENE-COVID): a nationwide, population-based seroepidemiological study. The Lancet, published online July 6, 2020. https://doi.org/10.1016/S0140-6736(20)31483-5.
Aguiar, M., Mill'an Ortundo, E., Bidaurrazaga Van-Dierdonck, J., Mar, J., and Stollenwerk, N., Modelling COVID-19 in the Basque Country: from introduction to control measure response, medRxiv preprint, 2020. https://doi.org/10.1101/2020.05.10.20086504.
Aguiar, M., Van-Dierdonck, J. B., Stollenwerk, N., Reproduction ratio and growth rates: measures for an unfolding pandemic, medRxiv preprint and accepted for publication in PloS One, 2020. https://doi.org/10.1101/2020.05.18.20105528
Mateus, L., Stollenwerk, N., and Zambrini, J.C., Stochastic Models in Population Biology: From Dynamic Noise to Bayesian Description and Model Comparison for Given Data Sets, Int. Journal. Computer Math., 90, 2161-2173, 2013.
Lavezzo, E., et al., Suppression of COVID-19 outbreak in the municpality of V'o, Italy, medRxiv preprint, 2020. https://doi.org/10.1101/2020.04.17.20053157.
van Kampen, N. G., Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1992.
Gardiner, C. W., Handbook of stochastic methods, Springer, New York, 1985.
Honerkamp, J., Stochastic Dynamical Systems: Concepts, Numerical Methods and Data Analysis, VCH Publishers, Heidelberg, New York, 1993.
MacKay, D., Information theory, inference, and learning algorithms, Cambridge University Press, Cambridge, 2005.
Sivia, D.S., Data analysis: A Bayesian tutorial, Oxford University Press, 1996.
Stollenwerk, N., Aguiar, M., Ballesteros, S., Boto, J., Kooi, W. B., and Mateus, L., Dynamic noise, chaos and parameter estimation
in population biology, Interface, Focus, 2, 156-169, 2012.
van Noort, S., and Stollenwerk, N., From dynamical processes to likelihood functions: an epidemiological application to influenza, in Proceedings of 8th Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2008, ISBN 978-84-612-1982-7, 2008.
Aguiar, M., and Stollenwerk, N., A new chaotic attractor in a basic multi-strain epidemiological model with temporary crossimmunity, arXiv:0704.3174v1 [nlin.CD], 2007. Available at http://arxive.org.
Aguiar, M., Kooi, B.W., and Stollenwerk, N., Epidemiology of dengue fever: A model with temporary cross-immunity and possible secondary infection shows bifurcations and chaotic behaviour in wide parameter regions, Math. Model. Nat. Phenom., 3, pp. 48-70, 2008.
Aguiar, M., Stollenwerk, N., and Kooi, B.W., Torus bifurcations, isolas and chaotic attractors in a simple dengue model with ADE and temporary cross immunity, Int. Journal of Computer Math., 86, pp. 1867-1877, 2009.
Toni, T., Welch, D., Strelkowa, N., Ipsen, A., and Stumpf, M. P. H., Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems, J. R. Soc. Interface, 6, 187-202, 2009.
Sun, L., Lee, C., and Hoeting, J.A., Parameter inference and model selection in deterministic and stochastic dynamical models via approximate Bayesian computation: modeling a wildlife epidemic, arXiv:1409.7715v2, 2015.
Stollenwerk, N., Drepper, F., and Siegel, H., Testing nonlinear stochastic models on phytoplankton biomass time series, Ecological Modelling, 144, 261-277, 2001.
Halstead, S.B., Immune enhancement of viral infection. Progress in Allergy, 31, 301-364, 1982.
Halstead, S.B., Neutralization and antibody-dependent enhancement of dengue viruses, Advances in Virus Research, 60, 421-467, 2003.
Duong, V., Lambrechts, L., Paul, R. E., Ly, S., Laya, R. S., Long, K. C., Huy, R., Tarantola, A., Scott, T. W., Sakuntabhai, A., and Buchy, P., Asymptomatic humans transmit dengue virus to mosquitoes, Proc. Nat. Acad. Science, 112, 14688-14693, 2015.
Stollenwerk, N., Mateus, L., Rocha, F., Skwara, U., Ghaffari, P., Aguiar, M., Prediction and predictability in population biology: Noise and chaos, Math. Model. Nat. Phenom., 10, 142-164, 2015.
Aguiar, M., Stollenwerk, N., and Halstead, S., The impact of the newly licensed dengue vaccine in endemic countries, accepted for publication in " PLOS Neclected Tropical Diseases" , published online December 21, 2016.
Aguiar, M., Stollenwerk, N., and Halstead, S., The risks behind Dengvaxia recommendation, The Lancet Infectious Diseases, 16, 882-883, 2016.
Aguiar, M., Halstead, S., and Stollenwerk, N., Consider stopping dengvaxia administration without immunological screening, accepted for publication in " Expert Review of Vaccines" , published online December 23, 2016, http://dx.doi.org/10.1080/14760584. 2017.1276831.
Aguiar, M., Stollenwerk, N., and Kooi, W. B., Scaling of stochasticity in dengue hemorrhagic fever epidemics, Math. Model. Nat. Phenom., 7, 1-11, 2012.
Aguiar, M., Kooi, W. B., Rocha, F., Ghaffari, P., and Stollenwerk, N., How much complexity is needed to describe the fluctuations observed in dengue hemorrhagic fever incidence data?, Ecological Complexity, 16, 31-40, 2013.
Gillespie, D.T., A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, Journal of Computational Physics, 22, 403-434, 1976.
Gillespie, D.T., Monte Carlo simulation of random walks with residence time dependent transition probability rates, Journal of Computational Physics, 28, 395-407, 1978.
Gang, Hu, Stationary solution of master equations in the large-system-size limit, Physical Review A 36 5782-5790, 1987.
Billings, L., Mier-y-Teran-Romero, L., Lindley, B., and Schwartz, I., Intervention-based stochastic disease eradication, 2013. arXiv:1303.5614v1.
Stollenwerk, N, Fuentes Sommer, P., Kooi, B., Mateus, L., Ghaffari, P., and Aguiar, M., Hopf and torus bifurcations, torus destruction and chaos in population biology, Ecological Complexity, 30, 91-99, 2017.
Ghaffari, P., Jansen, V., and Stollenwerk, N., Evolution towards critical fluctuations in a system of accidental pathogens, Numerical Analysis and Applied Mathematics ICNAAM 2011, Chalkidiki, AIP Conf. Proc., 1389, 1263-1266, 2011. doi: 10.1063/1.3637837. Copyright 2011 American Institute of Physics 978-0-7354-0956-9).
Jansen, V.A.A., Stollenwerk, N., Jensen, H.J., Ramsay, M.E., Edmunds, W.J., and Rhodes, C.J., Measles outbreaks in a population with declining vaccine uptake, Science, 301, 804, 2003.
Allen, E.J., Allen L.J.S, Arciniega, A., Greenwood, P., Construction of equivalent stochastic differential equation models, Stochastic Analysis and Application, 26, 274-297, 2008.
Stollenwerk, N., and Briggs, K.M., Master equation solution of a plant disease model, Physics Letters, A 274, 84-91, 2000.
World Health Organization, Emergencies preparedness, response, Novel Coronavirus - China, 2020. Retrieved from https://www.who.int/csr/don/12-january-2020-novel-coronavirus-china/en/
World Health Organization, WHO announces COVID-19 outbreak a pandemic, 2020. Retrieved from http://www.euro.who.int/en/health-topics/health-emergencies/coronavirus-covid-19/news/news/2020/3/who-announces-covid-19-outbreak-a-pandemic
World Health Organization, Coronavirus disease (COVID-2019) situation reports,2020. https://www.who.int/docs/default-source/coronaviruse/situation-reports/20200420-sitrep-91-covid-19.pdf?sfvrsn=fcf0670b 4
Governo Italiano Presidenza del Consiglio dei Ministri, March 9th, 2020. Retrieved from http://www.governo.it/it/articolo/firmato-il-dpcm-9-marzo-2020/14276
Ministerio de la Presidencia, Relaciones con las Cortes y Memoria Democr'atica, March 14th, 2020. Retrieved from http://noticias.juridicas.com/base datos/Laboral/661797-rd-463-2020-de-14-mar-estado-de-alarma-para-la-gestion-de-la-situacion-de.html
Ministerio de la Presidencia, Relaciones con las Cortes y Memoria Democr'atica, March 27, 2020. Retrieved from http://noticias.juridicas.com/base datos/Admin/662751-real-decreto-476-2020-de-27-de-marzo-por-el-que-se-prorroga-el-estado-de.html
Ministerio de la Presidencia, Relaciones con las Cortes y Memoria Democr'atica, March 29, 2020. Retrieved from http://noticias.juridicas.com/base datos/Laboral/662759-rd-ley-10-2020-de-29-mar-regulacion-de-un-permiso-retribuido-recuperable.html
Bolet'nn oficial: BOPV (Pa'ns Vasco), DECRETO 6/2020, de 13 de marzo, del Lehendakari, Retrieved from https://www.legegunea.euskadi.eus//eli/es-pv/d/2020/03/13/6/dof/spa/html/x59-contfich/es/
Ministerio de la Presidencia, Relaciones con las Cortes y Memoria Democr'atica, April 10th, 2020. Retrieved from https://www.boe.es/eli/es/rd/2020/04/10/487
Ministerio de la Presidencia, Relaciones con las Cortes y Memoria Democr'atica, April 10th, 2020. Retrieved from https://www.boe.es/boe/dias/2020/04/25/pdfs/BOE-A-2020-4665.pdf
Anderson, R. M., Heesterbeek, H., Klinkenberg, D., D'eirdre Hollingsworth, How will country-based mitigation measures influence the course of the COVID-19 epidemic?, The Lancet, 2020. doi: https://doi.org/10.1016/S0140-6736(20)30567-5
Imperial College COVID-19 Response Team, Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries, March 30th, 2020. Retrieved from https://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/covid-19/report-13-europe-npi-impact/
Kissler, S. M., et al., Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period, Science, 14 Apr 2020. eabb5793, DOI: 10.1126/science.abb5793
Anastassopoulou, C., Russo, L., Tsakris, A. and Siettos, C., Data-based analysis, modelling and forecasting of the COVID-19 outbreak, PLoS One, 15, e0230405, 2020.
Michaud, J., Kates, J., and Levitt, L., COVID-19 Models: Can They Tell Us What We Want to Know?, April 2020. Retrieved from https://www.kff.org/coronavirus-policy-watch/covid-19-models/
Giordano, G., et al., Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nature Medicine, 2020. doi: https://doi.org/10.1038/s41591-020-0883-7
Yi, Y.,, Lagniton, P.N.P., Ye, S., Li, E., Xu, R-H., COVID-19: what has been learned and to be learned about the novel coronavirus disease, Int. J. Biol. Sci., 16(10), 1753-1766, 2020.
Kramer, M., et al., Epidemiological data from the nCoV-2019 Outbreak: Early Descriptions from Publicly Available Data, 2020.Retrieved from https://virological.org/t/epidemiological-data-from-the-ncov-2019-outbreak-early-descriptions-from-publicly-available-data/337
Wu. P., et al., Real-time tentative assessment of the epidemiological characteristics of novel coronavirus infections in Wuhan, China, as at 22 January 2020. Euro Surveill., 25, 2000044, 2020.
Wei-jie Guan et al. Clinical Characteristics of Coronavirus Disease 2019 in China, New England Medicine Journal, 382, 1708-1720, 2020.
Lauer, S. A., et al., The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application, Annals of Internal Medicine, 2020. doi: https://doi.org/10.7326/M20-0504.
Liu, Y., et al., Viral dynamics in mild and severe cases of COVID-19, The Lancet Infectious Diseases, 2020. doi: https://doi.org/
1016/S1473-3099(20)30232-2.
