# Immunity in the ABM-DSGE Framework for Preventing and Controlling Epidemics—Validation of Results

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## Abstract

**:**

## 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. Updated ABM Component for Studying the Dynamics of the COVID-19 Pandemic

- Cases for Healthy Individuals

- Cases for Infected Individuals

- Cases for Treated or Infected Individuals in Isolation

- Cases for Healthy Individuals in Preventive Quarantine

- Cases for Recovered Individuals

- Cases for Vaccinated Individuals

- Healthy individuals by age for each iteration;
- Infected agents by age for each iteration;
- Recovered agents by age for each iteration;
- Vaccinated agents by age for each iteration;
- Individuals receiving treatment by age for each iteration;
- Agents in preventive quarantine by age for each iteration;
- Agents deceased by age for each iteration.

## 4. Validation of Scenarios in Connection with the Introduction of Vaccination

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 |

Notation | Scenario 1.1 | Scenario 1.2 | Scenario 1.3 |
---|---|---|---|

T | 104 | 104 | 104 |

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

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

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

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

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

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

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

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

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

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

${\left(Pr\right)}_{t}^{14}$ | 0.1 | 0.1 | 0.1 |

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

${\left(Pr\right)}_{t}^{17}$ | 0.05 | 0.05 | 0.3 |

${\left(Pr\right)}_{t}^{23}$ | 0.2 | 0.2 | 0.2 |

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

${\left(Pr\right)}_{t}^{26}$ | 0.6998 | 0.6998 | 0.6998 |

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

${\left(Pr\right)}_{t}^{36}$ | 0.7 | 0.7 | 0.7 |

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

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

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

${\left(Pr\right)}_{t}^{46}$ | 0.06 | 0.06 | 0.06 |

${\left(Pr\right)}_{t}^{47}$ | 0.06 | 0.06 | 0.06 |

${\left(Pr\right)}_{t}^{62}$ | 0.01 | 0.01 | 0.01 |

${\left(Pr\right)}_{t}^{63}$ | 0.0005 | 0.005 | 0.005 |

${\left(Pr\right)}_{t}^{64}$ | 0.05 | 0.05 | 0.05 |

${\left(Pr\right)}_{t}^{65}$ | 0.00001 | 0.00001 | 0.00001 |

${\left(Pr\right)}_{t}^{67}$ | 0.009 | 0.1 | 0.2 |

${\left(Pr\right)}_{t}^{72}$ | 0.009 | 0.005 | 0.005 |

${\left(Pr\right)}_{t}^{73}$ | 0.00045 | 0.00025 | 0.00025 |

${\left(Pr\right)}_{t}^{74}$ | 0.05 | 0.05 | 0.05 |

${\left(Pr\right)}_{t}^{75}$ | 0.00001 | 0.00001 | 0.00001 |

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

## 6. COVID-19 Prevention and Control Schemes—What Does the Vaccination Change?

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ABM | Agent-Based Modelling |

COVID-19 | Coronavirus Disease 2019 |

DSGE | Dynamic Stochastic General Equilibrium |

pp. | percentage points |

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

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**Figure 1.**State transition probabilities in the agent-based epidemic component. Health status: 1—healthy (h), 2—infected (i), 3—treated (t), 4—healthy individuals in preventive quarantine (q), 5—deceased (d), 6—recovered (r), 7—vaccinated (v) ${P}^{ij}$—transition probability between states i and j, see Table 2 and Table 3.

**Figure 5.**Aggregate labour productivity under the different COVID-19 prevention and control schemes. Please note that this figure is similar to the one that was published in [1] in November 2020. This figure enables the results for the scenarios that were analysed in 2021 to be compared with those from 2020.

**Figure 6.**Aggregate labour productivity under the different COVID-19 vaccination schemes. Vaccination Scenario 1 is (1.1); Vaccination Scenario 2 is (1.2) and Vaccination Scenario 3 is (1.3).

**Figure 7.**The major macroeconomic indicators under the different COVID-19 prevention and control schemes (conditional forecasts using the DSGE model). Please note that this figure is similar to the one that was published in [1] in November 2020. However, the capital accumulation process was recalibrated in the DSGE model as is explained in Section 5. This figure enables the results for scenarios analysed in 2021 to be compared with those from 2020.

**Figure 8.**The major macroeconomic indicators under the different COVID-19 vaccination schemes (conditional forecasts using the DSGE model). Vaccination Scenario 1 is (1.1); Vaccination Scenario 2 is (1.2) and Vaccination Scenario 3 is (1.3).

Initial Conditions | Explanation | Restrictions |
---|---|---|

T | Number of time steps (weeks) | ≥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; 6—recovered, 7—vaccinated) | Int $\in \{1,2,3,4,5,6,7\}$ |

${\left(Age\right)}_{t}^{Ind}$ | 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 $\ge 0$ |

${\lambda}^{max}$ | The parameter corresponding to the maximum number of vaccinated persons in the iteration (week) | 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 $ |

${\left(Wp\right)}_{t}^{av\_h}$ | The productivity of an individual when healthy at time t (it was assumed to be equal to one) | $\in \langle 0,1\rangle $ |

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

${\left(Wp\right)}_{t}^{av\_r}$ | The productivity of an individual after recovery 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 $ |

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

${\left(Wp\right)}_{t}^{av\_v}$ | The productivity of an individual who has been vaccinated at time t (it was assumed to be equal to one) | $\in \langle 0,1\rangle $ |

_{t}= S.

Parameter | Explanation | Restrictions |
---|---|---|

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

${\left(Pr\right)}_{t}^{14}$ | The probability that a healthy agent (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 agent (1) will become infected and will die almost instantly (within week) (5) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{17}$ | The probability that the healthy agent (1) will be vaccinated (7) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{26}$ | The probability that an infected agent (2) will become healthy (will recover) (6) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{23}$ | The probability that an infected agent (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 agent (2) dies (5) | $\in (0,1)$ |

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

${\left(Pr\right)}_{t}^{36}$ | The probability that an infected agent in a hospital or quarantine (3) gets better (6) (recovers) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{41}$ | The probability that a healthy agent 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 agent 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 agent in quarantine (4) dies (5) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{46}$ | The probability that a healthy agent in quarantine (4) was not infected and returned to the state “recovered” (6) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{47}$ | The probability that a healthy agent in quarantine (4) was not infected and returned to the state “vaccinated” (7) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{61}$ | The probability that the recovered agent (6) will get infected (1) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{64}$ | The probability that the recovered agent (6) will go to the quarantine (4) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{65}$ | The probability that the recovered agent (6) will die (5) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{67}$ | The probability that the recovered agent (6) will get vaccinated (7) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{72}$ | The probability that the vaccinated agent (7) will get infected (2) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{74}$ | The probability that the vaccinated agent (7) will go to the quarantine (4) | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{75}$ | The probability that the vaccinated agent (7) will die (5) | $\in (0,1)$ |

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

${\left(Pr\right)}_{t}^{13}$ | The probability that a healthy agent (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 agent in quarantine (4) will become infected at the end of her quarantine at time t | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{63}$ | The probability that a recovered agent (6) will be hospitalised (3) at time t | $\in (0,1)$ |

${\left(Pr\right)}_{t}^{73}$ | The probability that a vaccinated agent (7) will be hospitalised (3) at time t | $\in (0,1)$ |

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

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

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

z | New temporal variable that defines a threshold probability 4 | $\in (0,1)$ |

${s}_{t}^{Ind}$ | Health status of the agent at time $t>0$ (1—healthy, 2—infected, 3—treated, 4—healthy individual in preventive quarantine, 5—deceased, 6—recovered, 7—vaccinated) | Int $\in \{1,2,3,4,5,6,7\}$ |

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

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

Variable | Description | Calibrated Values |
---|---|---|

$\mathcal{A}$ | Elasticity of output towards the changes of labour | 0.25 |

$\phi $ | Reverse of the labour supply elasticity | 5 |

${\u03f5}_{w}$ | Elasticity of substitution between types of labour | 4.52 |

${\u03f5}_{p}$ | Elasticity of substitution between types of goods | 9 |

${\theta}_{w}$ | Calvo index of wage rigidity | 0.9807 |

${\theta}_{p}$ | Calvo index of price rigidity | 0.9807 |

$\beta $ | Discount factor | 0.9996 |

$\delta $ | Capital depreciation rate | 0.0175 |

${\varphi}_{k}$ | Capital adjustment costs’ scaling parameter | 12 |

h | Habit persistence parameter | 0.9 |

${\rho}_{a}$ | Autoregressive parameter of the technological shock | 0.99 |

${\rho}_{\chi}$ | Autoregressive parameter of the labour supply shock | 0.99 |

${\rho}_{a}$ | Autoregressive parameter of the technological shock | 0.99 |

${\rho}_{\chi}$ | Autoregressive parameter of the labour productivity shock | 0.99 |

${\rho}_{M}$ | Autoregressive parameter of the monetary policy shock | 0.965 |

${\varphi}_{\pi}$ | Central bank’s reaction to the deviation of inflation from its steady state value | 0.115 |

${\varphi}_{y}$ | Central bank’s reaction to the deviation of output gap from its steady state value | 0.0096 |

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**MDPI and ACS Style**

Kaszowska-Mojsa, J.; Włodarczyk, P.; Szymańska, A. Immunity in the ABM-DSGE Framework for Preventing and Controlling Epidemics—Validation of Results. *Entropy* **2022**, *24*, 126.
https://doi.org/10.3390/e24010126

**AMA Style**

Kaszowska-Mojsa J, Włodarczyk P, Szymańska A. Immunity in the ABM-DSGE Framework for Preventing and Controlling Epidemics—Validation of Results. *Entropy*. 2022; 24(1):126.
https://doi.org/10.3390/e24010126

**Chicago/Turabian Style**

Kaszowska-Mojsa, Jagoda, Przemysław Włodarczyk, and Agata Szymańska. 2022. "Immunity in the ABM-DSGE Framework for Preventing and Controlling Epidemics—Validation of Results" *Entropy* 24, no. 1: 126.
https://doi.org/10.3390/e24010126