Figure 1.
Visualization of the SIR model with random $ \beta $ and $ \gamma $ in time $ t $ . As initial values, S(0) = 999, I(0) = 1 and R(0) = 0 were chosen
- Previous Article
- JDG Home
- This Issue
-
Next Article
Generalized intransitive dice II: Partition constructions
doi: 10.3934/jdg.2021004
Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic
| 1. | Department of Economics, Faculty of Economics and Political Sciences, National and Kapodistrian University of Athens, Greece |
| 2. | University of Cagliari, Italy |
We utilize systems theory in the study of the implementation of non pharmaceutical strategies for the mitigation of the COVID-19 pandemic. We present two models. The first one is a model of predictive control with receding horizon and discontinuous actions of unknown costs for the implementation of adaptive triggering policies during the disease. This model is based on a periodic assessment of the peak of the pandemic (and, thus, of the health care demand) utilizing the latest data about the transmission and recovery rate of the disease. Consequently, the model seems to be suitable for discontinuous, non-mechanical (i.e. human) actions with unknown effectiveness, like those applied in the case of COVID-19. Secondly, we consider a feedback control problem in order to contain the pandemic at the capacity of the NHS (National Health System). As input parameter we consider the value $ p $ that reflects the intensity-effectiveness of the measures applied and as output the predicted maximum of infected people to be treated by NHS. The feedback control regulates $ p $ so that the number of infected people is manageable. Based on this approach, we address the following questions: (a) the limits of improvement of this approach; (b) the effectiveness of this approach; (c) the time horizon and timing of the application.
Keywords:
Covid-19 pandemic,
SIR model,
systems theory,
adaptive control,
feedback control,
non-pharmaceutical intervention methods.
Mathematics Subject Classification:
Primary: 93B52, 93C40, 92D30.
Citation:
John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti.
Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, doi: 10.3934/jdg.2021004
![]()
References:
| [1] |
N. Azevedo, D. Pinheiro and G.-W. Weber,
Dynamic programming for a Markov-switching jump-diffusion, J. Comp. Appl. Math., 267 (2014), 1-19.
doi: 10.1016/j.cam.2014.01.021. |
| [2] |
G. Baskozos, G. Galanis and C. Di Guilmi, A Behavioural SIR Model and its Implications for Physical Distancing, Centre for research in Economic theory and its applications (CRETA), Department of economics, Univerity of Warwick, 2020. Google Scholar |
| [3] |
Y.-C. Chen, P.-E. Lu, C.-S. Chang and T.-H. Liu, A Time-dependent SIR model for COVID-19 with Undetectable Infected Persons, Institute of Communications Engineering National Tsing Hua University Hsinchu 30013, Taiwan, R.O.C. 2020.
doi: 10.1109/TNSE.2020.3024723. |
| [4] |
N. M. Ferguson, D. Laydon, G. Nedjati-Gilani et al., Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, 2020. Google Scholar |
| [5] |
G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, Prentice Hall PTR, Upper Saddle River, NJ. Google Scholar |
| [6] |
Q. Lin, S. Zhao and D. Gao et al.,
A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan, China with individual reaction and governmental action, International Journal of Infectious Diseases, 93 (2020), 211-216.
doi: 10.1016/j.ijid.2020.02.058. |
| [7] | M. Newman, Netwroks: An Introduction, Oxford University Press, 2010. Google Scholar |
| [8] |
B. Øksendal, A. Sulem and T. Zhang,
Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Advances in Applied Probability, 43 (2011), 572-596.
doi: 10.1239/aap/1308662493. |
| [9] |
C. S. Pedamallu, L. Ozdamar, G.-W. Weber and E. Kropat, A system dynamics model to study the importance of infrastructure facilities on quality of primary education system in developing countries, AIP Conference Proceedings 1239, 321 (2010).
doi: 10.1063/1.3459767. |
| [10] |
K. Prem, Y. Liu, T. W. Russell et al., The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: A modelling study, The Lancet Public Health, 5 (2020), E261-E270.
doi: 10.1016/S2468-2667(20)30073-6. |
| [11] |
E. Savku, N. Azevedo and G.-W. Weber, Optimal control of stochastic hybrid models in the framework of regime switches, Modeling, Dynamics, Optimization and Bioeconomics II (eds. A. Pinto, D. Zilberman), Springer Proceedings in Mathematics & Statistics, vol 195, Springer, (2017), 371–387.
doi: 10.1007/978-3-319-55236-1_18. |
| [12] |
G.-W. Weber, O. Defterli, S. Z. Alparslan Gök and E. Kropat,
Modeling, inference and optimization of regulatory networks based on time series data, European J. Oper. Res., 211 (2011), 1-14.
doi: 10.1016/j.ejor.2010.06.038. |
| [13] |
H. Weiss, The SIR model and the foundations of public health, MATerial Matemá Matics, (2013), 17 pp. Google Scholar |
| [14] |
E. W. Weisstein, Least Squares Fitting, Available from: From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/LeastSquaresFitting.html Google Scholar |
| [15] |
https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases# Google Scholar |
| [16] | |
| [17] | |
| [18] | |
| [19] |
https://www.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299423467b48e9ecf6 Google Scholar |
show all references
References:
| [1] |
N. Azevedo, D. Pinheiro and G.-W. Weber,
Dynamic programming for a Markov-switching jump-diffusion, J. Comp. Appl. Math., 267 (2014), 1-19.
doi: 10.1016/j.cam.2014.01.021. |
| [2] |
G. Baskozos, G. Galanis and C. Di Guilmi, A Behavioural SIR Model and its Implications for Physical Distancing, Centre for research in Economic theory and its applications (CRETA), Department of economics, Univerity of Warwick, 2020. Google Scholar |
| [3] |
Y.-C. Chen, P.-E. Lu, C.-S. Chang and T.-H. Liu, A Time-dependent SIR model for COVID-19 with Undetectable Infected Persons, Institute of Communications Engineering National Tsing Hua University Hsinchu 30013, Taiwan, R.O.C. 2020.
doi: 10.1109/TNSE.2020.3024723. |
| [4] |
N. M. Ferguson, D. Laydon, G. Nedjati-Gilani et al., Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, 2020. Google Scholar |
| [5] |
G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, Prentice Hall PTR, Upper Saddle River, NJ. Google Scholar |
| [6] |
Q. Lin, S. Zhao and D. Gao et al.,
A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan, China with individual reaction and governmental action, International Journal of Infectious Diseases, 93 (2020), 211-216.
doi: 10.1016/j.ijid.2020.02.058. |
| [7] | M. Newman, Netwroks: An Introduction, Oxford University Press, 2010. Google Scholar |
| [8] |
B. Øksendal, A. Sulem and T. Zhang,
Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Advances in Applied Probability, 43 (2011), 572-596.
doi: 10.1239/aap/1308662493. |
| [9] |
C. S. Pedamallu, L. Ozdamar, G.-W. Weber and E. Kropat, A system dynamics model to study the importance of infrastructure facilities on quality of primary education system in developing countries, AIP Conference Proceedings 1239, 321 (2010).
doi: 10.1063/1.3459767. |
| [10] |
K. Prem, Y. Liu, T. W. Russell et al., The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: A modelling study, The Lancet Public Health, 5 (2020), E261-E270.
doi: 10.1016/S2468-2667(20)30073-6. |
| [11] |
E. Savku, N. Azevedo and G.-W. Weber, Optimal control of stochastic hybrid models in the framework of regime switches, Modeling, Dynamics, Optimization and Bioeconomics II (eds. A. Pinto, D. Zilberman), Springer Proceedings in Mathematics & Statistics, vol 195, Springer, (2017), 371–387.
doi: 10.1007/978-3-319-55236-1_18. |
| [12] |
G.-W. Weber, O. Defterli, S. Z. Alparslan Gök and E. Kropat,
Modeling, inference and optimization of regulatory networks based on time series data, European J. Oper. Res., 211 (2011), 1-14.
doi: 10.1016/j.ejor.2010.06.038. |
| [13] |
H. Weiss, The SIR model and the foundations of public health, MATerial Matemá Matics, (2013), 17 pp. Google Scholar |
| [14] |
E. W. Weisstein, Least Squares Fitting, Available from: From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/LeastSquaresFitting.html Google Scholar |
| [15] |
https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases# Google Scholar |
| [16] | |
| [17] | |
| [18] | |
| [19] |
https://www.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299423467b48e9ecf6 Google Scholar |
Figure 2.
Mitigation strategy scenarios for UK showing critical care bed requirements. Source: Ferguson et al. (2020)
Figure 3.
Feedback control for the implementation of social distancing policies
Figure 4.
The evolution of the number $ I(t) $ of infected people in the case where no mitigation measures are applied
Figure 5.
The evolution of the number $ I(t) $ of infected people (on the left) and of the parameter $ p $ (on the right) in the case where a feedback control scenario with $ \lambda = 0.0025 $ is applied
Figure 6.
The evolution of the number $ I(t) $ of infected people (on the left) and of the parameter $ p $ (on the right) in the case where a feedback control scenario $ \lambda = 0.002 $ is applied
Figure 7.
The evolution of the number $ I(t) $ of infected people (on the left) and of the parameter $ p $ (on the right) in the case where a feedback control scenario $ \lambda = 0.002 $ is applied with a delay of $ T = 2 $ unit times
Figure 11.
Daily evolution of $ b $ for 21 countries starting the 22/1/2020 and finishes the 8/5/2020
Figure 12.
Estimation of progress for the average $ \beta(t) $ in time $ t $ . The black dots are the average values of $ \beta $ and the red line is the polynomial best fit line
Figure 13.
Daily evolution of the average $ I_{\max} $ and the $ I(t) $ from 4 countries as an indicative trend, starting the 22/1/2020 and finishes the 8/5/2020. The blue line represent the $ I(t) $ and the red the $ I_{\max} $
Table 1.
Summary of NPI interventions considered
| Label | Policy | Description |
| CI | Case isolation in the home | Symptomatic cases stay at home for 7 days, reducing non-household contacts by 75% for this period. Household contacts remain unchanged. Assume 70% of household comply with the policy. |
| HQ | Voluntary home quarantine | Following identification of a symptomatic case in the household, all household members remain at home for 14 days. Household contact rates double during this quarantine period, contacts in the community reduce by 75%. Assume 50% of household comply with the policy. |
| SDO | Social distancing of those over 70 years of age | Reduce contacts by 50% in workplaces, increase household contacts by 25% and reduce other contacts by 75%. Assume 75% compliance with policy. |
| SD | Social distancing of entire population | All households reduce contact outside household, school or workplace by 75%. School contact rates unchanged, workplace contact rates reduced by 25%. Household contact rates assumed to increase by 25%. |
| PC | Closure of schools and universities | Closure of all schools, 25% of universities remain open. Household contact rates for student families increase by 50% during closure. Contacts in the community increase by 25% during closure. |
| Label | Policy | Description |
| CI | Case isolation in the home | Symptomatic cases stay at home for 7 days, reducing non-household contacts by 75% for this period. Household contacts remain unchanged. Assume 70% of household comply with the policy. |
| HQ | Voluntary home quarantine | Following identification of a symptomatic case in the household, all household members remain at home for 14 days. Household contact rates double during this quarantine period, contacts in the community reduce by 75%. Assume 50% of household comply with the policy. |
| SDO | Social distancing of those over 70 years of age | Reduce contacts by 50% in workplaces, increase household contacts by 25% and reduce other contacts by 75%. Assume 75% compliance with policy. |
| SD | Social distancing of entire population | All households reduce contact outside household, school or workplace by 75%. School contact rates unchanged, workplace contact rates reduced by 25%. Household contact rates assumed to increase by 25%. |
| PC | Closure of schools and universities | Closure of all schools, 25% of universities remain open. Household contact rates for student families increase by 50% during closure. Contacts in the community increase by 25% during closure. |
Table 2.
Sample of twenty-one countries from the 267 countries and big cities. For China the city of Beijing was chosen. The $ b $ parameter was estimated by using Equation (18)
| COUNTRY | |
| SWISS | 0.257158556 |
| BRAZIL | 0.233817403 |
| ITALY | 0.203539698 |
| NORWAY | 0.197869991 |
| SPAIN | 0.19306215 |
| EGYPT | 0.19012821 |
| USA | 0.189437334 |
| PAKISTAN | 0.184307553 |
| BELGIUM | 0.178032037 |
| ETHIOPIA | 0.167117704 |
| RUSSIA | 0.160845659 |
| FRANCE | 0.153958202 |
| UK | 0.153852808 |
| GERMANY | 0.153678417 |
| SOUTH KOREA | 0.148665135 |
| SWEDEN | 0.141608744 |
| BULGARIA | 0.141558434 |
| CYPRUS | 0.132396806 |
| GREECE | 0.120185025 |
| ALBANIA | 0.109905162 |
| CHINA | 0.053298278 |
| COUNTRY | |
| SWISS | 0.257158556 |
| BRAZIL | 0.233817403 |
| ITALY | 0.203539698 |
| NORWAY | 0.197869991 |
| SPAIN | 0.19306215 |
| EGYPT | 0.19012821 |
| USA | 0.189437334 |
| PAKISTAN | 0.184307553 |
| BELGIUM | 0.178032037 |
| ETHIOPIA | 0.167117704 |
| RUSSIA | 0.160845659 |
| FRANCE | 0.153958202 |
| UK | 0.153852808 |
| GERMANY | 0.153678417 |
| SOUTH KOREA | 0.148665135 |
| SWEDEN | 0.141608744 |
| BULGARIA | 0.141558434 |
| CYPRUS | 0.132396806 |
| GREECE | 0.120185025 |
| ALBANIA | 0.109905162 |
| CHINA | 0.053298278 |
| [1] |
Jorge Rebaza. On a model of COVID-19 dynamics. Electronic Research Archive, 2021, 29 (2) : 2129-2140. doi: 10.3934/era.2020108 |
| [2] |
João M. Lemos, Fernando Machado, Nuno Nogueira, Luís Rato, Manuel Rijo. Adaptive and non-adaptive model predictive control of an irrigation channel. Networks & Heterogeneous Media, 2009, 4 (2) : 303-324. doi: 10.3934/nhm.2009.4.303 |
| [3] |
Zi Sang, Zhipeng Qiu, Xiefei Yan, Yun Zou. Assessing the effect of non-pharmaceutical interventions on containing an emerging disease. Mathematical Biosciences & Engineering, 2012, 9 (1) : 147-164. doi: 10.3934/mbe.2012.9.147 |
| [4] |
James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 445-462. doi: 10.3934/naco.2013.3.445 |
| [5] |
Notice Ringa, Chris T. Bauch. Spatially-implicit modelling of disease-behaviour interactions in the context of non-pharmaceutical interventions. Mathematical Biosciences & Engineering, 2018, 15 (2) : 461-483. doi: 10.3934/mbe.2018021 |
| [6] |
Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861 |
| [7] |
Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063 |
| [8] |
M. Predescu, R. Levins, T. Awerbuch-Friedlander. Analysis of a nonlinear system for community intervention in mosquito control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 605-622. doi: 10.3934/dcdsb.2006.6.605 |
| [9] |
Sanling Yuan, Yongli Song, Junhui Li. Oscillations in a plasmid turbidostat model with delayed feedback control. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 893-914. doi: 10.3934/dcdsb.2011.15.893 |
| [10] |
Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653 |
| [11] |
Qiying Hu, Chen Xu, Wuyi Yue. A unified model for state feedback of discrete event systems II: Control synthesis problems. Journal of Industrial & Management Optimization, 2008, 4 (4) : 713-726. doi: 10.3934/jimo.2008.4.713 |
| [12] |
Changzhi Wu, Kok Lay Teo, Volker Rehbock. Optimal control of piecewise affine systems with piecewise affine state feedback. Journal of Industrial & Management Optimization, 2009, 5 (4) : 737-747. doi: 10.3934/jimo.2009.5.737 |
| [13] |
V. Rehbock, K.L. Teo, L.S. Jennings. Suboptimal feedback control for a class of nonlinear systems using spline interpolation. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 223-236. doi: 10.3934/dcds.1995.1.223 |
| [14] |
Cătălin-George Lefter, Elena-Alexandra Melnig. Feedback stabilization with one simultaneous control for systems of parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 777-787. doi: 10.3934/mcrf.2018034 |
| [15] |
Magdi S. Mahmoud. Output feedback overlapping control design of interconnected systems with input saturation. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 127-151. doi: 10.3934/naco.2016004 |
| [16] |
Stephan Gerster, Michael Herty. Discretized feedback control for systems of linearized hyperbolic balance laws. Mathematical Control & Related Fields, 2019, 9 (3) : 517-539. doi: 10.3934/mcrf.2019024 |
| [17] |
Carsten Collon, Joachim Rudolph, Frank Woittennek. Invariant feedback design for control systems with lie symmetries - A kinematic car example. Conference Publications, 2011, 2011 (Special) : 312-321. doi: 10.3934/proc.2011.2011.312 |
| [18] |
Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control & Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019 |
| [19] |
Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 |
| [20] |
Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]