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
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2021, 8(3): 167-186.
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 and Games,
2021, 8
(3)
: 167-186.
doi: 10.3934/jdg.2021004
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References:
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N. Azevedo, D. Pinheiro and G.-W. Weber,
Dynamic programming for a Markov-switching jump-diffusion, J. Comp. Appl. Math., 267 (2014), 1-19.
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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. |
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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. |
| [5] |
G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, Prentice Hall PTR, Upper Saddle River, NJ. |
| [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.
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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. |
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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. |
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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. |
| [14] |
E. W. Weisstein, Least Squares Fitting, Available from: From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/LeastSquaresFitting.html |
| [15] |
https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases# |
| [16] | |
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| [18] | |
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https://www.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299423467b48e9ecf6 |
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. |
| [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. |
| [5] |
G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, Prentice Hall PTR, Upper Saddle River, NJ. |
| [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.
|
| [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. |
| [14] |
E. W. Weisstein, Least Squares Fitting, Available from: From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/LeastSquaresFitting.html |
| [15] |
https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases# |
| [16] | |
| [17] | |
| [18] | |
| [19] |
https://www.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299423467b48e9ecf6 |
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 10.
According to Harvard and the Children hospital there are 9, 474, 948 confirmed cases. Source: Harvard (2020) [18]
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 |
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