Abstract
Approaches to the estimation of the full state vector of a larger epidemiological model for the spread of Covid-19 in Sweden at the initial time instant from available data and with a simplified dynamical model are proposed and evaluated. The larger epidemiological model is based on a time-continuous Markov chain and captures the demographic composition of and the transport flows between the counties of Sweden. Its intended use is to predict the outbreak development in temporal and spatial coordinates as well as across the demographic groups. It can also support evaluations and comparisions of prospective intervention strategies in terms of, e.g., lockdown in certain areas or isolation of specific age groups. The simplified model is a discrete time-invariant linear system that has cumulative infectious incidence, infected population, asymptomatic population, exposed population, and infectious pressure as the state variables. Since the system matrix of the model depends on a number of transition rates, structural properties of the model are investigated for suitable parameter ranges. It is concluded that the model becomes unobservable for some parameter values. Two contrasting approaches to the initial state estimation are considered. One is a version of Rauch–Tung–Striebel smoother and another is based on solving a batch nonlinear optimization problem. The benefits and shortcomings of the considered estimation techniques are analyzed and compared.
Keywords: Mathematical models, initial states, linear systems, smoothing filters, Markov models, model approximation
Footnotes
This work is funded by the PhD program at the Centre for Interdisciplinary Mathematics, Uppsala University, Sweden, by the Swedish Research Council, under grant 2019-04451, and by Vinnova grant 2020-03173, "Model-based data-driven tools for the optimization of pro-active epidemiological interventions".
References
- Allen L.J. A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis. Infectious Disease Modelling. 2017;2(2):128–142. doi: 10.1016/j.idm.2017.03.001. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Aravkin, A. and Burke, J. (2012). Smoothing Dynamic Systems with State-Dependent Covariance Matrices. Proceedings of the IEEE Conference on Decision and Control, 2015. doi:10.1109/CDC.2014.7039913.
- Engblom S., Eriksson R., Widgren S. Bayesian epidemiological modeling over high-resolution network data. Epidemics. 2020;32:100399. doi: 10.1016/j.epidem.2020.100399. [DOI] [PubMed] [Google Scholar]
- Folkhälsomyndigheten (2020a). Bekräftade fall i Sverige. https://www.arcgis.com/sharing/rest/content/items/b5e7488e117749c19881cce45db13f7e/data. Online; accessed: 2020-06-05.
- Folkhälsomyndigheten (2020b). Ny fas kräver nya instatser mot covid 19. https://www.folkhalsomyndigheten.se/nyheter-och-press/nyhetsarkiv/2020/mars/ny-fas-kraver-nya-insatser-mot-covid-19/. Online; accessed: 2020-06-30.
- Giordano G., Blanchini F., Bruno R., Colaneri P., Di Filippo A., Di Matteo A., Colaneri M. Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nature Medicine. 2020:1–6. doi: 10.1038/s41591-020-0883-7. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Rauch H., Tung F., Striebel C.T. Maximum likelihood estimates of linear dynamic systems. AIAA Journal. 1965;3(8):1445–1450. [Google Scholar]
- Wang G., Zhang Y., Wang X. Maximum correntropy Rauch-Tung-Striebel smoother for nonlinear and non-gaussian systems. IEEE Transactions on Automatic Control. 2020:1. [Google Scholar]
- Widgren S., Bauer P., Eriksson R., Engblom S. SimInf: An R package for data-driven stochastic disease spread simulations. J. Stat. Softw. 2019;91(12):1–42. [Google Scholar]
- Widgren S., et al. Spatio-temporal modelling of verotoxigenic E. coli O157 in cattle in Sweden: Exploring options for control. Veterinary Res. 2018;49(78) doi: 10.1186/s13567-018-0574-2. doi: 10.1186/s13567-018-0574-2. [DOI] [PMC free article] [PubMed] [Google Scholar]