Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic

John Leventides; Costas Poulios; Georgios Alkis Tsiatsios; Maria Livada; Stavros Tsipras; Konstantinos Lefcaditis; Panagiota Sargenti; Aleka Sargenti

doi:10.3934/jdg.2021004

Article Contents

2021, Volume 8, Issue 3: 167-186. Doi: 10.3934/jdg.2021004

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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

^* Corresponding author: Costas Poulios
^* Corresponding author: Costas Poulios

Received: June 30, 2019

Revised: November 30, 2020

Early access: March 2021

Published: July 2021

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Abstract

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:

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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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Figure 2. Mitigation strategy scenarios for UK showing critical care bed requirements. Source: Ferguson et al. (2020)

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Figure 3. Feedback control for the implementation of social distancing policies

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Figure 4. The evolution of the number $ I(t) $ of infected people in the case where no mitigation measures are applied

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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

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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

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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

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Figure 8. According to WHO there are 4, 761, 559 confirmed cases. Source: WHO (2020) [17]

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Figure 9. According to John Hopkins University there are 4, 927, 487 confirmed cases. Source:John Hopkins University (2020) [19]

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Figure 10. According to Harvard and the Children hospital there are 9, 474, 948 confirmed cases. Source: Harvard (2020) [18]

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Figure 11. Daily evolution of $ b $ for 21 countries starting the 22/1/2020 and finishes the 8/5/2020

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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

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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} $

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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.

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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	$ b $ parameter
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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