Integral model of COVID-19 spread and mitigation in UK: identification of transmission rate
Abstract
The integral model with finite memory is employed to analyze the timeline of COVID-19 epidemic in the United Kingdom and government actions to mitigate it. The model uses a realistic infection distribution. The time-varying transmission rate is determined from Volterra integral equation of the first kind. The authors construct and justify an efficient regularization algorithm for finding the transmission rate. The model and algorithm are approbated on the UK data with several waves of COVID-19 and demonstrate a remarkable resemblance between real and simulated dynamics. The timing of government preventive measures and their impact on the epidemic dynamics are discussed.
Keyword : integral epidemiologic models, COVID-19 mitigation, Volterra integral equations, ill-posed problems, regularizing algorithm
How to Cite
Hritonenko, N., Satsky, C., & Yatsenko, Y. (2022). Integral model of COVID-19 spread and mitigation in UK: identification of transmission rate. Mathematical Modelling and Analysis, 27(4), 573–589. https://doi.org/10.3846/mma.2022.15708
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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W.O. Kermack and A.G. McKendrick. A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. Lond. A, 115:700–721, 1927. https://doi.org/10.1098/rspa.1927.0118
P.K. Lamm. A survey of regularization methods for first-kind Volterra equations. In D. Colton et al(Ed.), Surveys on Solution Methods for Inverse Problems, pp. 53–82. Springer, Wien, New York, 2000. https://doi.org/10.1007/978-3-7091-6296-5_4
P. Linz. The solution of Volterra equations of the first kind in the presence of large uncertainties. In P. Linz(Ed.), Treatment of Integral Equations by Numerical Methods, pp. 123–130. Academic Press, London, 1982.
A.L. Lloyd. Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics. Theoretic Population Biology, 60:59–71, 2001. https://doi.org/10.1006/tpbi.2001.1525
M. Martcheva. An Introduction to Mathematical Epidemiology. Springer, Massachusetts, 2015. https://doi.org/10.1007/978-1-4899-7612-3
OurWorldinData.org. Daily confirmed COVID-19 deaths, rolling 3-day average. Our World in Data, 2021. Available from Internet: https://ourworldindata.org/grapher/daily-covid-deaths-7-day?tab=chart. Accessed 29 March 2022
V.O. Sergeev. Regularization of Volterra equations of the first kind. Soviet Math. Dokl, 12:501–505, 1971.
J.J.A. van Kampen, D. van de Vijver and P.L.A. Fraaij. Duration and key determinants of infectious virus shedding in hospitalized patients with coronavirus disease-2019 (COVID-19). Nature Communications, 12:267, 2021. https://doi.org/10.1038/s41467-020-20568-4
V. Volterra. Theory of Functionals and of Integral and Integro-differential Equations. Blackie & Son Limited, London, 1930.
F.E. Alvarez, D. Argente and F. Lippi. A simple planning problem for COVID19 lockdown. Working Paper 26981. National Bureau of Economic Research, Cambridge, 2020. https://doi.org/10.3386/w26981
S. Anita. Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic Publishers, Dordrecht, 2000.
J. Arino and S. Portet. A simple model for COVID-19. Infectious Disease Modelling, 5:309–315, 2020. https://doi.org/10.1016/j.idm.2020.04.002
A. Atkeson. What will be the economic impact of COVID-19 in the US? Rough Estimates of Disease Scenarios. Working Paper 26867. National Bureau of Economic Research, Cambridge, 2020. https://doi.org/10.3386/w26867
C.T.H. Baker. Methods for Volterra equations of first kind. In C.T.H. Baker(Ed.), Numerical Solution of Integral Equations, pp. 162–174. Clarendon Press, Oxford, 1974.
B. Barlow. R number for UK below 1 for first time since August. BBC News, 2020. Available from Internet: https://www.bbc.com/news/health-55105285. Accessed 4 July 2021
A. Bergman, Y. Sella, P. Agre and A. Casadevall. Oscillations in U.S. COVID-19 incidence and mortality data reflect diagnostic and reporting factors. mSystems, 5(4):e00544–20, 2020. https://doi.org/10.1128/mSystems.00544-20
F. Brauer, S. Castillo-Chavez and Z. Feng. Mathematical Models in Epidemiology. Springer, New York, 2019. https://doi.org/10.1007/978-1-4939-9828-9
D. Breda, O. Diekmann, W. de Graaf, A. Pugliese and R. Vermiglio. On the formulation of epidemic models (an appraisal of Kermack and McKendrick). J. Biol. Dynamics, 6(2):103–117, 2012. https://doi.org/10.1080/17513758.2012.716454
C. Corduneanu. Integral Equations and Applications. Cambridge University Press, Cambridge, 1991. https://doi.org/10.1017/CBO9780511569395
A. Cori, N.M. Ferguson, C. Fraser and S. Cauchemez. A new framework and software to estimate time-varying reproduction numbers during epidemics. Amer. J. Epidemiology, 178(9):1505–1512, 2013. https://doi.org/10.1093/aje/kwt133
J. Fernandez-Villaverde and C.J. Jones. Estimating and simulating a SIRD model of COVID-19 for many countries, states, and cities. Working Paper 27128. National Bureau of Economic Research, Cambridge, 2020. https://doi.org/10.3386/w27128
H.W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42(4):599–653, 2000. https://doi.org/10.1137/S0036144500371907
N. Hritonenko, V. Hritonenko and Y. Yatsenko. A review of epidemiologic models: from SIR to distributed delays. An. Stiint. Univ. Al. I. Cuza Iasi. Mat., 66(2):237–249, 2020.
N. Hritonenko, O. Yatsenko and Y. Yatsenko. Model with transmission delays for COVID-19 control: Theory and empirical assessment. J. Public Econ Theory, 2021. https://doi.org/10.1111/jpet.12554
N. Hritonenko and Y. Yatsenko. Optimization of harvesting age in integral agedependent model of population dynamics. Math Biosciences, 195:154–167, 2005. https://doi.org/10.1016/j.mbs.2005.03.001
N. Hritonenko and Y. Yatsenko. Bifurcations in nonlinear integral models of biological systems. Internat. J. Systems Science, 38:389–399, 2007. https://doi.org/10.1080/00207720701245544
N. Hritonenko and Y. Yatsenko. Mathematical Modeling in Economics, Ecology and the Environment, Second Ed. Springer, Massachusetts, 2013. https://doi.org/10.1007/978-1-4614-9311-2
N. Hritonenko and Y. Yatsenko. Nonlinear integral models with delays: recent developments and applications. J. King Saud University - Science, 32(1):726– 731, 2020. https://doi.org/10.1016/j.jksus.2018.11.001
M. Iannelli and F. Milner. The Basic Approach to Age-Structured Population Dynamics. Models, Methods and Numerics. Springer, Massachusetts, 2017. https://doi.org/10.1007/978-94-024-1146-1
W.O. Kermack and A.G. McKendrick. A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. Lond. A, 115:700–721, 1927. https://doi.org/10.1098/rspa.1927.0118
P.K. Lamm. A survey of regularization methods for first-kind Volterra equations. In D. Colton et al(Ed.), Surveys on Solution Methods for Inverse Problems, pp. 53–82. Springer, Wien, New York, 2000. https://doi.org/10.1007/978-3-7091-6296-5_4
P. Linz. The solution of Volterra equations of the first kind in the presence of large uncertainties. In P. Linz(Ed.), Treatment of Integral Equations by Numerical Methods, pp. 123–130. Academic Press, London, 1982.
A.L. Lloyd. Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics. Theoretic Population Biology, 60:59–71, 2001. https://doi.org/10.1006/tpbi.2001.1525
M. Martcheva. An Introduction to Mathematical Epidemiology. Springer, Massachusetts, 2015. https://doi.org/10.1007/978-1-4899-7612-3
OurWorldinData.org. Daily confirmed COVID-19 deaths, rolling 3-day average. Our World in Data, 2021. Available from Internet: https://ourworldindata.org/grapher/daily-covid-deaths-7-day?tab=chart. Accessed 29 March 2022
V.O. Sergeev. Regularization of Volterra equations of the first kind. Soviet Math. Dokl, 12:501–505, 1971.
J.J.A. van Kampen, D. van de Vijver and P.L.A. Fraaij. Duration and key determinants of infectious virus shedding in hospitalized patients with coronavirus disease-2019 (COVID-19). Nature Communications, 12:267, 2021. https://doi.org/10.1038/s41467-020-20568-4
V. Volterra. Theory of Functionals and of Integral and Integro-differential Equations. Blackie & Son Limited, London, 1930.