# To Freeze or Not to Freeze? Epidemic Prevention and Control in the DSGE Model Using an Agent-Based Epidemic Component

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## Abstract

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## 1. Introduction

- Should we freeze an economy in order to decrease the pace of SARS-CoV-2 transmission?
- What should the scale and composition of an efficient lockdown policy look like?

## 2. Literature Review

## 3. COVID-19 Dynamics—The ABM Approach

- an $M\times T$ matrix $\mathit{H}$, which recorded the health status of each agent in a society after each iteration
- an $M\times T$ matrix $\mathit{W}$, which recorded the productivity of each individual in a society after each iteration
- an $M\times T$ matrix $\mathit{A}$, which recorded the age of each individual in a society after each iteration
- an $M\times 2T$ matrix $\mathit{X}$, which recorded the location of each individual on the map after each iteration (x- & y-coordinates)
- an $M\times 4$ matrix $\mathit{F}$, which recorded the full data set

#### 3.1. Cases for Healthy Individuals

#### 3.2. Cases for Infected Individuals

#### 3.3. Cases for Treated or Infected Individuals in Isolation

#### 3.4. Cases for Healthy Individuals in Preventive Quarantine

- the productivity of the society
- the number of infected citizens by age
- the number of healthy individuals by age
- the number of agents undergoing treatment by age
- the number of individuals in preventive quarantine by age
- the number of deceased by age.

## 4. Potential Epidemic Scenarios

#### 4.1. Scenario 1: The Persistent Spread of the Pandemic under Mild Restrictions

#### 4.2. Scenario 2: The Spread of Pandemic under Mobility Restrictions

#### 4.3. Scenario 3: The Spread of Epidemic under Gradual Preventive Restrictions

- local lockdowns, that is, for specific areas of a country
- moderate mobility restrictions in public transport
- limiting the number of people participating in assemblies and meetings
- an emphasis on remote work in selected sectors of the economy, where this remote work did not reduce the overall productivity of the sectors
- hybrid preventive measures in the education sector

#### 4.4. Scenario 4: The Persistent Spread of Pandemic without Restrictions

## 5. Macroeconomic Consequences of Pandemics—The DSGE Approach

## 6. COVID-19 Prevention and Control Schemes—Efficiency Comparison

## 7. Policy Implications

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ABM | Agent-Based Modelling |

AR(1) | Autoregressive model of order 1 |

COVID-19 | Coronavirus Disease 2019 |

COVID-ABS | Coronavirus Disease 2019 Agent-Based Simulation |

CSO | Central Statistical Office |

DSGE | Dynamic Stochastic General Equilibrium |

FOC | First Order Condition |

GUS | Central Statistical Office |

pp. | percentage points |

SARS-CoV-2 | Severe acute respiratory syndrome coronavirus 2 |

SIR | Susceptible-Infectious-Recovered model |

SEIR | Susceptible-Exposed-Infected-Recovered model |

## References

- Eichenbaum, M.; Rebelo, S.; Trabandt, M. The Macroeconomics of Epidemics. NBER WP
**2020**, 26882, 1–35. [Google Scholar] [CrossRef] - Mihailov, A. Quantifying the Macroeconomic Effects of the COVID 19 Lockdown: Comparative Simulations of the Estimated Galí Smets Wouters Model. Univ. Read. Discuss. Pap.
**2020**, 7, 1–32. [Google Scholar] - Guerrieri, V.; Lorenzoni, G.; Straub, L.; Werning, I. Macroeconomic Implications of COVID-19: Can Negative Supply Shocks Cause Demand Shortages? NBER WP
**2020**, 26918. [Google Scholar] [CrossRef] - Yanga, Y.; Zhangb, H.; Chenc, X. Coronavirus pandemic and tourism: Dynamic stochastic general equilibrium modeling of infectious disease outbreak. Ann. Tour. Res.
**2020**, 83, 2–6. [Google Scholar] [CrossRef] [PubMed] - Bayraktar, E.; Cohen, A.; Nellis, A. A Macroeconomic SIR Model for COVID-19. medRxiv Prepr.
**2020**, 1–23. [Google Scholar] [CrossRef] - Brotherhood, L.; Kircher, P.; Santos, C.; Tertilt, M. An Economic Model of the COVID-19 Epidemic: The Importance of Testing and Age-Specific Policies. IZA Discuss. Pap.
**2020**, 13265, 1–68. [Google Scholar] - Toda, A. Susceptible-Infected-Recovered (SIR) Dynamics of Covid-19 and Economic Impact. Covid Econ. Vetted Real-Time Pap. (CEPR)
**2020**, 1, 43–63. [Google Scholar] - Hunter, E.; Mac Nameeb, B.; Kellehera, J. A Taxonomy for Agent-Based Models in Human Infectious Disease Epidemiology. J. Artif. Soc.
**2017**, 20, 3. [Google Scholar] [CrossRef][Green Version] - Ajelli, M.; Gonçalves, B.; Balcan, D.; Colizza, V.; Hu, H.; Ramasco, J.; Merler, S.; Vespignani, A. Comparing large-scale computational approaches to epidemic modeling: Agent-based versus structured metapopulation models. BMC Infect. Dis.
**2010**, 10, 190. [Google Scholar] [CrossRef][Green Version] - Dunham, J. An agent-based spatially explicit epidemiological model in MASON. J. Artif. Soc.
**2005**, 9, 1. [Google Scholar] - Cuevas, E. An agent-based model to evaluate the COVID-19 transmission risks in facilities. Comput. Biol. Med.
**2020**, 121, 1–12. [Google Scholar] [CrossRef] [PubMed] - Silva, P.; Batista, P.; Lima, H.; Alves, M.; Guimarães, F.; Silva, R. COVID-ABS: An agent-based model of COVID-19 epidemic to simulate health and economic effects of social distancing interventions. Chaos Soliton Fract.
**2020**, 139, 1–12. [Google Scholar] [CrossRef] [PubMed] - Shamil, M.S.; Farheen, F.; Ibtehaz, N.; Khan, I.; Rahman, M. An Agent Based Modeling of COVID-19: Validation, Analysis and Recommendations. medRxiv Prepr.
**2020**, 1–17. [Google Scholar] [CrossRef] - Hoertel, N.; Blachier, M.; Blanco, C.; Olfson, M.; Massetti, M.; Sánchez Rico, M.; Limosin, F.; Leleu, H. A stochastic agent-based model of the SARS-CoV-2 epidemic in France. Nat. Med.
**2020**, 26, 1417–1421. [Google Scholar] [CrossRef] [PubMed] - Wallentin, G.; Kaziyeva, D. Reibersdorfer-Adelsberger, E. COVID-19 Intervention Scenarios for a Long-term Disease Management. Int. J. Health Policy Manag.
**2020**, 1–9, in press. [Google Scholar] [CrossRef] - Currie, C.; Fowler, J.; Kotiadis, K.; Monks, T.; Onggo, B.; Robertson, D.; Tako, A. How simulation modelling can help reduce the impact of COVID-19. J. Simul.
**2020**, 14, 83–97. [Google Scholar] [CrossRef][Green Version] - Bertozzi, A.; Franco, E.; Mohler, G.; Short, M.; Sledge, D. The challenges of modeling and forecasting the spread of COVID-19. Proc. Natl. Acad. Sci. USA
**2020**, 117, 16732–16738. [Google Scholar] [CrossRef] - Klôh, V.; Silva, G.; Ferro, M.; Araújo, E.; Barros de Melo, C.; Pinho de Andrade Lima, J.R.; Rademaker Martins, E. The virus and socioeconomic inequality: An agent-based model to simulate and assess the impact of interventions to reduce the spread of COVID-19 in Rio de Janeiro. Braz. J. Health Rev.
**2020**, 3, 647–3673. [Google Scholar] [CrossRef] - Maziarz, M.; Zach, M. Agent-based modelling for SARS-CoV-2 epidemic prediction and intervention assessment: A methodological appraisal. J. Eval. Clin.
**2020**, 26, 1352–1360. [Google Scholar] [CrossRef] - Kano, T.; Yasui, K.; Mikami, T.; Asally, M.; Ishiguro, A. An Agent-Based Model for Interrelation Between COVID-19 Outbreak and Economic Activities. Available online: https://arxiv.org/abs/2007.11988 (accessed on 29 October 2020).
- Brottier, I. COVID 19: The Good, the Bad and the Agent Based Model. Available online: https://www.anylogic.com/blog/covid-19-the-good-the-bad-and-the-agent-based-model/ (accessed on 29 October 2020).
- Adam, D. Special Report: The Simulations Driving the World’S Response to COVID-19. How Epidemiologists Rushed to Model the Coronavirus Pandemic. Available online: https://www.nature.com/articles/d41586-020-01003-6 (accessed on 29 October 2020).
- Wolfram, C. Agent-Based Network Models for COVID-19. Available online: https://www.wolframcloud.com/obj/covid-19/Published/Agent-Based-Networks-Models-for-COVID-19.nb (accessed on 20 October 2020).
- Jordà, O.; Singh, S.R.; Taylor, A.M. Longer-run Economic Consequences of Pandemics. NBER WP
**2020**, 26934, 3–20. [Google Scholar] [CrossRef] - Wolfram, S. Cellular automata as models of complexity. Nature
**1984**, 311, 419–424. [Google Scholar] [CrossRef] - Ilachinski, A. Cellular Automata. A Discrete Universe, 1st ed.; World Scientific Publishing Co. Pte. Ltd.: London, UK, 2001; pp. 1–462. [Google Scholar]
- Galí, J. Monetary Policy, Inflation, and the Business Cycle. An Introduction to the New Keynesian Framework and Its Applications, 1st ed.; Princeton University Press: Princeton, NJ, USA, 2015; pp. 2–224. [Google Scholar]
- Christiano, L.; Eichenbaum, M.; Evans, C. Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy. J. Polit. Econ.
**2005**, 113, 1–45. [Google Scholar] [CrossRef][Green Version] - Galí, J. Unemployment Fluctuations and Stabilization Policies. A New Keynesian Perspective, 1st ed.; MIT Press: Cambridge, MA, USA, 2011; pp. 2–120. [Google Scholar]
- Galí, J.; Smets, F.; Wouters, R. Unemployment in an Estimated New Keynesian Model. In NBER Macroeconomics Annual 2011, 1st ed.; Acemoglu, D., Woodford, M., Eds.; Chicago University Press: Chicago, IL, USA, 2012; Volume 26, pp. 329–360. [Google Scholar]
- King, R.G.; Plosser, C.I.; Rebelo, S.T. Production, growth and business cycles: I. The basic neoclassical model. J. Monet. Econ.
**1988**, 21, 195–232. [Google Scholar] [CrossRef] - Smets, F.; Wouters, R. Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach. Am. Econ. Rev.
**2007**, 97, 586–606. [Google Scholar] [CrossRef][Green Version] - Abel, A. Asset Prices under Habit Formation and Catching up with the Jonses. Am. Econ. Rev.
**1990**, 80, 38–42. [Google Scholar] - Christiano, L.; Trabandt, M.; Walentin, K. Involuntary Unemployment and the Business Cycle. Rev. Econ. Dyn.
**2020**, in press. [Google Scholar] [CrossRef] - Calvo, G. Staggered prices in a utility-maximising framework. J. Monet. Econ.
**1983**, 12, 383–398. [Google Scholar] [CrossRef] - Merz, M. Search in the labor market and the real business cycle. J. Monet. Econ.
**1995**, 36, 269–300. [Google Scholar] [CrossRef] - Armington, P. A Theory of Demand for Products Distinguished by Place of Production. IMF Staff Pap.
**1969**, 16, 159–178. [Google Scholar] [CrossRef] - Green, H. Aggregation in Economic Analysis, 1st ed.; Princeton University Press: Princeton, NJ, USA, 1964; pp. 2–140. [Google Scholar]
- Dixit, A.; Stiglitz, J. Monopolistic Competition and Optimum Product Diversity. Am. Econ. Rev.
**1977**, 67, 297–308. [Google Scholar] [CrossRef] - Christiano, L.; Trabandt, M.; Walentin, K. DSGE Models for Monetary Policy Analysis. In Handbooks in Economics. Monetary Economics, 1st ed.; Friedman, B., Woodford, M., Eds.; North-Holland: Amsterdam, The Netherlands, 2011; Volume 3A, pp. 285–367. [Google Scholar]
- Hurtado, S. DSGE models and the Lucas critique. Econ. Model.
**2014**, 44, S12–S19. [Google Scholar] [CrossRef][Green Version] - Kaszowska, J.; Santos, J.; Pablo-Martí, F. Assessment of Policies Using the ’Core’ and ’Periphery’ Macroeconomic Models in the Post-Crisis Environment. Argum. Oeconomica
**2019**, 1, 185–212. [Google Scholar] [CrossRef]

**Figure 2.**Scenario 1: Spatial-temporal spread of the coronavirus in a society. States: Healthy (h), Infected (i), Treated (l), Preventive quarantine (k), Deceased (d).

**Figure 3.**Scenario 1: Changes in agents’ productivity over time during the COVID-19 pandemic (people of working age only).

**Figure 4.**Scenario 1: Changes in agents’ productivity over time during the COVID-19 pandemic (including people in the pre-productive age and those who were retired and had by definition zero productivity).

**Figure 6.**Scenario 2: Spatial-temporal spread of the coronavirus in a society (for first sub-scenario*). States: Healthy (h), Infected (i), Treated (l), Preventive quarantine (k), Deceased (d); * See robustness checks in Section 6 for a further explanation.

**Figure 7.**Scenario 2: Changes in individuals’ productivity over time during the COVID-19 pandemic for first sub-scenario (people of working age only). For further explanation see pp. 8–9.

**Figure 9.**Scenario 3: Spatial-temporal spread of the coronavirus in a society. States: Healthy (h), Infected (i), Treated (l), Preventive quarantine (k), Deceased (d).

**Figure 10.**Scenario 3: Changes in individuals’ productivity over time during the COVID-19 pandemic (people of working age only). For further explanation see pp. 8–9.

**Figure 12.**Scenario 4: Spatial-temporal spread of the coronavirus in a society. States: Healthy (h), Infected (i), Treated (l), Preventive quarantine (k), Deceased (d).

**Figure 13.**Scenario 4: Changes in individuals’ productivity over time during the COVID-19 pandemic (people of working age only). For further explanation see pp. 8–9.

**Figure 16.**Conditional forecasts of the major macroeconomic indicators under different COVID-19 prevention and control schemes.

**Figure 18.**Conditional forecasts of the major macroeconomic indicators under Scenario 2–robustness tests.

Initial Conditions | Explanation | Restr. |
---|---|---|

T | Number of iterations (weeks). | $\ge 0$ |

${s}_{t}^{Ind}$ | Health status of the individual at time $t=0$ (1–healthy, 2–infected, 3–treated, 4–healthy individual in preventive quarantine, 5–deceased) | Int $\in \{1,2,3,4,5\}$ |

${\left(Age\right)}_{t}^{Ind}$ | Age of an individual at time $t=0$ | |

${N}^{Ind}$ | Number of individuals at time $t=0$ | Int $\ge 0$ |

${K}^{Ind}$ | Number of infected individuals at time $t=0$ (including asymptomatically infected) | Int $\ge 0$ |

${S}_{t}\times {S}_{t}$ | Dimensions of the grid at time t* | Int $\ge 0$ |

${\left(Ag\right)}_{t}^{1}$ | Share of citizens of pre-working age at time t | $\in \langle 0,1\rangle $ |

${\left(Ag\right)}_{t}^{2}$ | Share of citizens of working age at time t | $\in \langle 0,1\rangle $ |

${\left(Ag\right)}_{t}^{3}$ | Share of retired individuals at time t | $\in \langle 0,1\rangle $ |

${\left(Wp\right)}_{t}^{Ind}$ | Productivity of an individual at time $t=0$ (if healthy) | $=1$ |

${\left(Wp\right)}_{t}^{av\_inf}$ | The productivity of an individual when infected at time t (the decline in productivity was estimated based on empirical data) | $\in \langle 0,1\rangle $ |

${\left(Wp\right)}_{t}^{av\_q}$ | The productivity of an individual who is healthy and in quarantine at time t (the decline in productivity was estimated based on empirical data) | $\in \langle 0,1\rangle $ |

${\left(Wp\right)}_{t}^{av\_t}$ | The productivity of an individual when treated or who is infected and in quarantine at time t (the decline in productivity was estimated based on empirical data) | $\in \langle 0,1\rangle $ |

Parameter | Explanation | Restr. |
---|---|---|

${\left(Pr\right)}_{t}^{12}$ | The probability that a healthy individual (1) will become infected (2) at time t | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{14}$ | The probability that a healthy individual (1) will be in quarantine (although she is healthy) (4) at time t | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{15}$ | The probability that a healthy individual (1) will become infected and will die almost instantly (within week) (5) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{21}$ | The probability that an infected individual (2) will become healthy (1) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{23}$ | The probability that an infected individual (2) will be treated in a hospital or will stay in quarantine (3) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{25}$ | The probability that an infected individual (2) dies (5) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{31}$ | The probability that an infected individual in a hospital or quarantine (3) gets better (1) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{35}$ | The probability that an infected individual in a hospital or quarantine (3) dies (5) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{41}$ | The probability that a healthy individual in quarantine (4) will end the quarantine, that is, is healthy (1) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{43}$ | The probability that a healthy individual in quarantine (4) will become infected during the quarantine and she is still in quarantine (but now is already infected) (3) at time t | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{45}$ | The probability that a healthy individual in quarantine (4) dies (5) | $\in (0,1)$ |

Variable | Explanation | Restr. |
---|---|---|

${\left(Pr\right)}_{t}^{13}$ | The probability that a healthy individual (1) will become treated in the hospital (or isolation) after becoming infected (3) at time t | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{42}$ | The probability that a healthy individual in quarantine (4) will become infected at the end of her quarantine (2) | $\in (0,1)$ |

p | Temporal variable (threshold probability 1) | $\in (0,1)$ |

q | Temporal variable (threshold probability 2) | $\in (0,1)$ |

r | Temporal variable (threshold probability 3) | $\in (0,1)$ |

${s}_{t}^{Ind}$ | Health status of the individual at time $t>0$ (1–healthy, 2–infected, 3–treated, 4–healthy individual in preventive quarantine, 5–deceased) | Int $\in \{1,2,3,4,5\}$ |

${\left(Age\right)}_{t}^{Ind}$ | Age of an individual at time $t>0$ | $\ge 0$ |

${\left(Wp\right)}_{t}^{Ind}$ | Productivity of an individual at time $t>0$ | $\in \langle 0,1\rangle $ |

Notation | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 |
---|---|---|---|---|

T | 104 | 104 | 104 | 104 |

${N}^{Ind}$ | 10,000 | 10,000 | 10,000 | 10,000 |

${K}^{Ind}$ | 150 | 150 | 150 | 150 |

${S}_{t}\times {S}_{t}$ | $100\times 100$ for all t | Dynamic adjustment | Dynamic adjustment | $100\times 100$ for all t |

${\left(Ag\right)}_{t}^{1}$ | 0.181 | 0.181 | 0.181 | 0.181 |

${\left(Ag\right)}_{t}^{2}$ | 0.219 | 0.219 | 0.219 | 0.219 |

${\left(Ag\right)}_{t}^{3}$ | 0.6 | 0.6 | 0.6 | 0.6 |

${\left(Wp\right)}_{t}^{av\_h}$ | 1 for all t | Dynamic adjustment | Dynamic adjustment | 1 for all t |

${\left(Wp\right)}_{t}^{av\_inf}$ | 0.9 | 0.9 | 0.9 | 0.9 |

${\left(Wp\right)}_{t}^{av\_q}$ | 0.8 | 0.8 | 0.8 | – |

${\left(Wp\right)}_{t}^{av\_t}$ | 0.3 | 0.3 | 0.3 | 0.3 |

${\left(Pr\right)}_{t}^{12}$ | 0.03 | 0.03 | Dynamic adjustment | 0.2 |

${\left(Pr\right)}_{t}^{13}$ | 0.1 | 0.1 | Dynamic adjustment | 0 |

${\left(Pr\right)}_{t}^{15}$ | 0.00002 | 0.00002 | Dynamic adjustment | 0.00002 |

${\left(Pr\right)}_{t}^{21}$ | 0.6998 | 0.6998 | Dynamic adjustment | 0.6998 |

${\left(Pr\right)}_{t}^{24}$ | 0.2 | 0.2 | Dynamic adjustment | 0.2 |

${\left(Pr\right)}_{t}^{25}$ | 0.0002 | 0.0002 | Dynamic adjustment | 0.005 |

${\left(Pr\right)}_{t}^{41}$ | 0.6 | 0.6 | Dynamic adjustment | – |

${\left(Pr\right)}_{t}^{43}$ | 0.1 | 0.1 | Dynamic adjustment | – |

${\left(Pr\right)}_{t}^{45}$ | 0.0002 | 0.0002 | Dynamic adjustment | – |

${\left(Pr\right)}_{t}^{31}$ | 0.7 | 0.7 | Dynamic adjustment | 0.7 |

${\left(Pr\right)}_{t}^{35}$ | 0.0002 | 0.0002 | Dynamic adjustment | 0.002 |

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**MDPI and ACS Style**

Kaszowska-Mojsa, J.; Włodarczyk, P. To Freeze or Not to Freeze? Epidemic Prevention and Control in the DSGE Model Using an Agent-Based Epidemic Component. *Entropy* **2020**, *22*, 1345.
https://doi.org/10.3390/e22121345

**AMA Style**

Kaszowska-Mojsa J, Włodarczyk P. To Freeze or Not to Freeze? Epidemic Prevention and Control in the DSGE Model Using an Agent-Based Epidemic Component. *Entropy*. 2020; 22(12):1345.
https://doi.org/10.3390/e22121345

**Chicago/Turabian Style**

Kaszowska-Mojsa, Jagoda, and Przemysław Włodarczyk. 2020. "To Freeze or Not to Freeze? Epidemic Prevention and Control in the DSGE Model Using an Agent-Based Epidemic Component" *Entropy* 22, no. 12: 1345.
https://doi.org/10.3390/e22121345