# Mitigation Strategies for COVID-19: Lessons from the K-SEIR Model Calibrated to the Observable Data

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## Abstract

**:**

## 1. Introduction

#### 1.1. Opening Remarks

#### 1.2. Pandemics and Their Mitigation

#### 1.3. Epidemiology: Successes and Failures

- Susceptible–infected–removed (SIR) and susceptible–infected–susceptible (SIS) models and their extensions, such as susceptible–exposed–infected–removed (SEIR) models;
- Agent-based models;
- Network-based models;
- Simple curve-fitting models that continually attract new enthusiasts despite being largely futile.

#### 1.4. Our Methodology

#### 1.5. The Structure of This Paper

## 2. COVID-19

#### 2.1. What Is COVID-19?

#### 2.2. Government Responses to COVID-19

- Social distancing for the general population.
- Quarantine (14 days) of individuals returning from infected areas.
- Lockdown/shelter-in-place/stay-at-home of areas with widespread infections.
- Self-isolation of individuals who have tested positive or are suspected to be infected.

#### 2.3. The Great Shutdown

#### 2.4. The Need for Exit Strategies

## 3. The Standard SIR and SEIR Models

## 4. Data and Its Analysis

#### 4.1. Available Data

- Confirmed infections (C);
- Deaths (D);
- Recoveries (R).

#### 4.2. Problems with the Data

#### 4.3. Case Fatality Rate

#### 4.4. Example: The Swine Flu

#### 4.5. Adjusted Fatality Rate for COVID-19

#### 4.5.1. Case Study: The Diamond Princess

#### 4.5.2. Case Study: Iceland

#### 4.6. Reproductive Numbers and Their Estimation

#### 4.7. Key COVID-19 Characteristics

## 5. The K-SEIR Model

#### 5.1. K-SEIR—SEIR for Several Interacting Groups

#### 5.2. Contacts between Groups

#### 5.3. The Nonlinear Effects

#### 5.4. Variation of Susceptibility

#### 5.5. Stationary State and Herd Immunity

#### 5.6. The Impact of Variability on the Herd Immunity

#### 5.7. Finite ICU Capacity and Its Implications

#### 5.8. Description of the Lockdown

## 6. Lives vs. Lives

## 7. Delay Difference Equations

## 8. Calibration

#### 8.1. Calibration of the SIR Model

#### 8.2. Calibration of the 1-SEIR Model

#### 8.3. Calibration of the 2-SEIR Model

## 9. Lockdown Strategies—Pros and Cons

- Can early quarantine save lives?
- Did the actual quarantine save lives?
- Is there any benefit in imposing a quarantine on the low-risk population?

## 10. Learning from the Swedish Experience

## 11. Discussion

## 12. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | One should not confuse quarantining of the sick with the sheltering in place of the healthy, which we discuss next. |

2 | Riots during the lockdown of Moscow during the plague of 1770–1772 come to mind. |

3 | Less charitably, one can characterize this model as a catastrophic failure, primarily due to its inability to differentiate between a hypothetical worst-case scenario and a realistic one. Moreover, it requires no effort to arrive at the worst-case scenario conclusions via back-of-the-envelope calculations. |

4 | The UAE and Bahrain have conducted even more extensive testing, however they are currently still experiencing a large number of new cases. For this reason, we consider the statistics of Iceland and Faeroe as closer to final. |

5 | Unfortunately, the data tend to be so polluted that proper pre-processing is needed. |

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**Figure 1.**Transmissibility vs. fatality rates of various diseases. At this point, these variables are still uncertain for COVID-19. Source: Thomas-Rüddel et al. (2021).

**Figure 2.**Eighty-eight countries have instituted controls, covering a combined 6.17 billion inhabitants (approx. 79.09). The median start date was 22 March 2020. California was the first state in the U.S. to impose a stay-at-home order on 19 March 2020.

**Figure 3.**As of 1 June 2020, the U.S. had 40 million unemployed. Job losses over the past 8 weeks erased all jobs created over the past decade. Source: The United States Department of Labor.

**Figure 4.**Government interventions have a disproportionate adverse effect on minorities, the working class, and the poor. Source: Adams-Prassl et al. (2020).

**Figure 8.**Time series of confirmed infections, deaths, and recovered, in logarithmic scale. In the U.S., the number of recovered cases is barely above the number of deaths, likely due to limited testing.

**Figure 9.**After more than 12 weeks since the WHO declared COVID-19 a pandemic, we still do not have accurate statistics. Only a few countries have conducted well-designed statistical experiments to estimate the true values of C, D, and R.

**Figure 10.**Flawed data led to a massive overestimation of the fatality rate of H1N1. Eleven years later, data collection is still a challenge. Source: CDC.

**Figure 11.**Countries that have administered tests to a broader portion of the population tend to report lower case fatality rates. This is consistent with the fact that $\theta <<1$. Source: https://ourworldindata.org/.

**Figure 12.**Number of Diamond Princess passengers and equivalent U.S. distribution. Source: Russell et al. (2020).

**Figure 13.**Bayesian estimate of ${\widehat{d}}_{\theta}=0.42\%$ for COVID-19, with $95\%$ confidence bands of $[0.375\%,0.75\%]$. This is significantly lower than the WHO’s estimate of $3.14\%$.

**Figure 14.**The distribution of ${\mathfrak{R}}_{0}$ and ${\mathfrak{R}}_{1}$ across 159 countries shows a steep decline in reproductive numbers following interventions.

**Figure 15.**Spain and Italy have economies that rely heavily on tourism. The case of Spain was particularly alarming, with cases almost doubling every 2 days. Government-mandated lockdowns successfully curbed the spread of the disease. France and Germany also experienced exponential growth. Their governments were able to tame the spread of the disease without resorting to drastic lockdowns, like Spain and Italy. Before the government intervened, cases grew in the U.S. exponentially, at a growth rate of 0.29. Even after intervention, it took 10 days for cases to fall below the double-every-3-days line. Benefits are not instantaneous. U.K.’s COVID-19 growth rate of ${r}_{0}=0.12$ is in line with other European countries.

**Figure 16.**The herd-immunity level as a function of $\alpha $ for several representative values of $\rho $. It is clear that variability affects it in a profound way.

**Figure 17.**Calibration of the SIR model to the initial phase of the disease in Italy and the U.K. (

**a**) Mortallity-Observations vs. Model—ITA (

**b**) Mortallity-Observations vs. Model—GBR.

**Figure 18.**Calibration of the SEIR model to the hospitalization and death statistics in NYC. (

**a**) Hospitalizations (

**b**) Mortality.

**Figure 19.**Calibration of the 2-SEIR model to the hospitalization and death statistics in NYC for the LR and HR groups.

**Figure 20.**Hospitalization and mortality, assuming that the quarantine is imposed three weeks earlier than in reality.

**Figure 22.**Hospitalization and mortality assuming that there is no quarantine for the low-risk population and there is quarantine for the high-risk population.

**Figure 23.**Mortality in Sweden: (

**a**) daily mortality for 2015–2021; (

**b**) number of death for 1970–2021; (

**c**) death rate for 1970–2021. Source: Statistics Sweden.

**Table 1.**Recent pandemics and their characteristics. Source: Centres for Disease Control and Prevention (2020).

Name\Params | Period | Place of Origin | Death Toll | Cause |
---|---|---|---|---|

COVID-19 | 2019–present | China | >0.4 mil | SARS-CoV-2 |

HIV/AIDS | 1981–present | DRC | >36 mil | HIV |

The swine flu | 2009–2009 | Mexico | >0.5 mil | H1N1 |

The HK flu | 1968–1969 | HK | >1 mil | H3N2 |

The Asian flu | 1956–1958 | China | >2 mil | H2N2 |

The Spanish flu | 1918–1920 | Unknown | >20 mil | H1N1 |

The Russian flu | 1889–1890 | Turkestan | >1 mil | H3N8 or H2N2 |

Cholera | 1852–1860 | India | >1 mil | Vibrio Cholerae |

**Table 2.**Constant characteristic parameter values for COVID-19; source Khalili et al. (2020); Kucharski et al. (2020); Li et al. (2020); Linton et al. (2020); Russell et al. (2020); Salje et al. (2020); Verity et al. (2020); Woelfel et al. (2020).

Age | 0–49 Years | 50–64 Years | ≥65 Years |
---|---|---|---|

Pre-existing immunity | None | None | None |

Percentage of transmission occurring prior to symptom onset | 40% | 40% | 40% |

Time from exposure to symptom onset (mean) | ~6 days | ~6 days | ~6 days |

Time between symptom onsets in an individual and a second individual infected by the first (mean) | ~6 days | ~6 days | ~6 days |

Mean number of days from symptom onset to hospitalization (standard deviation) | 6.9 (5.0) days | 7.2 (5.3) days | 6.2 (5.7) days |

Mean number of days of hospitalization without admittance to ICU (standard deviation) | 3.9 (3.7) days | 4.9 (4.3) days | 6.3 (5.1) days |

Mean number of days of hospitalization with admittance to ICU (standard deviation) | 9.5 (7.2) days | 10.5 (7.0) days | 10.0 (6.8) days |

Percent admitted to ICU among those hospitalized | 21.9% | 29.2% | 26.8% |

Percent on ventilators among those in ICU | 72.1% | 77.6% | 75.5% |

Mean number of days on ventilators (standard deviation) | 5.5 (5.3) days | 5.5 (5.3) days | 5.5 (5.3) days |

Mean number of days from symptom onset to death (standard deviation) | 14.9 (7.7) days | 15.3 (8.1) days | 12.9 (7.6) days |

Parameters | Scenario | 0–49 Years | 50–64 Years | ≥65 Years | Overall |
---|---|---|---|---|---|

$\begin{array}{c}{R}_{0}\end{array}$ | $\begin{array}{c}1\\ 2\\ 3\\ 4\\ 5\end{array}$ | $\begin{array}{c}2.0\\ 2.0\\ 3.0\\ 3.0\\ 2.5\end{array}$ | $\begin{array}{c}2.0\\ 2.0\\ 3.0\\ 3.0\\ 2.5\end{array}$ | $\begin{array}{c}2.0\\ 2.0\\ 3.0\\ 3.0\\ 2.5\end{array}$ | $\begin{array}{c}2.0\\ 2.0\\ 3.0\\ 3.0\\ 2.5\end{array}$ |

$\begin{array}{c}\mathrm{Symptomatic}\phantom{\rule{4.pt}{0ex}}\mathrm{Case}\\ \mathrm{Fatality}\phantom{\rule{4.pt}{0ex}}\mathrm{Ratio}\end{array}$ | $\begin{array}{c}1\\ 2\\ 3\\ 4\\ 5\end{array}$ | $\begin{array}{c}0.02\%\\ 0.02\%\\ 0.10\%\\ 0.10\%\\ 0.05\%\end{array}$ | $\begin{array}{c}0.10\%\\ 0.10\%\\ 0.60\%\\ 0.60\%\\ 0.20\%\end{array}$ | $\begin{array}{c}0.60\%\\ 0.60\%\\ 3.20\%\\ 3.20\%\\ 1.30\%\end{array}$ | $\begin{array}{c}0.20\%\\ 0.20\%\\ 1.00\%\\ 1.00\%\\ 0.40\%\end{array}$ |

$\begin{array}{c}\mathrm{Symptomatic}\phantom{\rule{4.pt}{0ex}}\mathrm{Case}\\ \mathrm{Hospitalization}\phantom{\rule{4.pt}{0ex}}\mathrm{Ratio}\end{array}$ | $\begin{array}{c}1\\ 2\\ 3\\ 4\\ 5\end{array}$ | $\begin{array}{c}1.30\%\\ 1.30\%\\ 2.60\%\\ 2.60\%\\ 1.70\%\end{array}$ | $\begin{array}{c}3.60\%\\ 3.60\%\\ 5.70\%\\ 5.70\%\\ 4.50\%\end{array}$ | $\begin{array}{c}5.20\%\\ 5.20\%\\ 10.00\%\\ 10.00\%\\ 7.40\%\end{array}$ | $\begin{array}{c}2.80\%\\ 2.80\%\\ 4.10\%\\ 4.10\%\\ 3.40\%\end{array}$ |

$\begin{array}{c}\mathrm{Percent}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{asymptomatic}\phantom{\rule{4.pt}{0ex}}\\ \mathrm{infections}\end{array}$ | $\begin{array}{c}1\\ 2\\ 3\\ 4\\ 5\end{array}$ | $\begin{array}{c}20\%\\ 50\%\\ 20\%\\ 50\%\\ 35\%\end{array}$ | $\begin{array}{c}20\%\\ 50\%\\ 20\%\\ 50\%\\ 35\%\end{array}$ | $\begin{array}{c}20\%\\ 50\%\\ 20\%\\ 50\%\\ 35\%\end{array}$ | $\begin{array}{c}20\%\\ 50\%\\ 20\%\\ 50\%\\ 35\%\end{array}$ |

$\begin{array}{c}\mathrm{Relative}\phantom{\rule{4.pt}{0ex}}\mathrm{infectiousness}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\\ \mathrm{asymptomatic}\phantom{\rule{4.pt}{0ex}}\mathrm{individuals}\end{array}$ | $\begin{array}{c}1\\ 2\\ 3\\ 4\\ 5\end{array}$ | $\begin{array}{c}50\%\\ 100\%\\ 50\%\\ 100\%\\ 100\%\end{array}$ | $\begin{array}{c}50\%\\ 100\%\\ 50\%\\ 100\%\\ 100\%\end{array}$ | $\begin{array}{c}50\%\\ 100\%\\ 50\%\\ 100\%\\ 100\%\end{array}$ | $\begin{array}{c}50\%\\ 100\%\\ 50\%\\ 100\%\\ 100\%\end{array}$ |

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**MDPI and ACS Style**

Lipton, A.; de Prado, M.L. Mitigation Strategies for COVID-19: Lessons from the K-SEIR Model Calibrated to the Observable Data. *J. Risk Financial Manag.* **2022**, *15*, 248.
https://doi.org/10.3390/jrfm15060248

**AMA Style**

Lipton A, de Prado ML. Mitigation Strategies for COVID-19: Lessons from the K-SEIR Model Calibrated to the Observable Data. *Journal of Risk and Financial Management*. 2022; 15(6):248.
https://doi.org/10.3390/jrfm15060248

**Chicago/Turabian Style**

Lipton, Alexander, and Marcos Lopez de Prado. 2022. "Mitigation Strategies for COVID-19: Lessons from the K-SEIR Model Calibrated to the Observable Data" *Journal of Risk and Financial Management* 15, no. 6: 248.
https://doi.org/10.3390/jrfm15060248