# Modeling of the Long-Term Epidemic Dynamics of COVID-19 in the United States

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. T-SIR Model

#### 2.2. Data Sources and Processing

#### 2.2.1. Epidemiological Data

#### 2.2.2. Geographical and Demographic Data

#### 2.2.3. Mobility Data

#### 2.3. Fitting

_{,}and $\gamma $, are required to be determined by fitting the model with the cumulative data, where the corresponding dynamic reproduction numbers ${R}_{1}={\beta}_{1}/\gamma $ and ${R}_{2}={\beta}_{2}/\gamma $. We fitted our T-SIR model using the grid search of parameters ${R}_{1}$, ${R}_{2}$, ${\sigma}_{1}$, ${\sigma}_{2}$, and $\gamma $ in reasonable ranges. The ranges for ${R}_{x}$, ${\sigma}_{x}$

_{,}and $\gamma $ were set to $\left[0.1,5\right]$, $\left[0,0.3\right]$, and $\left[0,0.3\right]$ with steps sized 0.1, 0.01, and 0.01, respectively [25]. The parameters of our T-SIR model, ${R}_{1}$, ${R}_{2}$, ${\sigma}_{1}$, ${\sigma}_{2}$, and $\gamma $, were then automatically determined through the best fit between the predicted data of T-SIR model and the reported COVID-19 data according to the mean squared error (MSE), where the three times, ${t}_{0}$, ${t}_{1}$, and ${t}_{2}$

_{,}in Equation (2) were also automatically determined during the model fitting.

## 3. Results

#### 3.1. COVID-19 Epidemics of the 50 States in the US

#### 3.1.1. Times of COVID-19 Outbreaks

#### 3.1.2. Properties of COVID-19 Outbreaks

#### 3.1.3. Demographic Impact on COVID-19 Outbreaks

#### 3.2. Modeling the COVID-19 Dynamics of the US States

#### 3.2.1. Fitting of the T-SIR Model

#### 3.2.2. Implications of T-SIR Model Parameters

_{,}and $\gamma $, for the 50 states of the US, where ${\beta}_{1}$ and ${\sigma}_{1}$ are for the first outbreak of COVID-19 occurring in March–April or June–August, and ${\beta}_{2}$ and ${\sigma}_{2}$ are for the second outbreak occurring in October–January. These epidemic parameters gave a quantitative description of the COVID-19 transmission dynamics of different states and will enable the investigation of the impact of state geographic and demographic data on the spread of the disease. Such investigations will provide a deep understanding of the epidemic dynamics of COVID-19 in different states and help the government create the corresponding measures to prevent the spread of COVID-19. As such, we have investigated the relationship between the epidemic parameters and the demographic data for the 50 US states. Specifically, the epidemic parameters for fitting the model using respective state geographic and demographic data were compiled. Pairwise comparisons were made by examining their Pearson correlations. A significant correlation between two different variables corresponds to an important linear relationship between the two factors.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The daily confirmed cases (upper row) and corresponding new cases (lower row) of COVID-19 for three selected states: Missouri (

**a**), New Jersey (

**b**), and Arizona (

**c**), in the US that represent three categories (one outbreak; two outbreaks, with first outbreak in March–April; and two outbreaks, with first outbreak in June–August, respectively) of transmission dynamics in the US, where the dashed yellow lines indicate the epidemic outbreak peaks.

**Figure 2.**The color-coded maps of the US states. (

**a**) The states with two epidemic outbreaks occurring in March–April and October–January are highlighted in green, and those with two epidemic outbreaks occurring in June–August and October–January are highlighted in yellow. (

**b**) Protests across the US by the number of cities and towns with rallies or protests in a state, the data for which were taken from https://www.usatoday.com/in-depth/graphics/2020/06/03/map-protests-wake-george-floyds-death/5310149002. (

**c**) Population densities and (

**e**) populations of the states with two epidemic outbreaks occurring in March–April and October–January are indicated by a red star, and those with two epidemic outbreaks occurring in June–August and October–January are indicated by a yellow star. (

**d**) The population densities vs. areas for the 50 states in the US. (

**f**) The populations vs. areas for the 50 states in the US.

**Figure 3.**The fitting of our T-SIR model to the daily confirmed cases (upper row) and corresponding new cases (lower row) of COVID-19 for three selected states: Colorado (

**a**), New Jersey (

**b**), and New York (

**c**) in the US that represent three categories of transmission dynamics in the US, where the dashed lines indicate the predicted data of T-SIR model. Data were smoothed using a Savitzky–Golay filter for daily new cases [46], where the light-blue shade indicates the standard deviations of reported data.

**Figure 4.**The pairwise relationships and their correlation coefficients of three epidemic parameters, $R$, $\beta $, and $\sigma $, of the T-SIR model for the first outbreak (

**a**–

**c**) and second outbreak (

**d**–

**f**), where the solid lines are the linear fittings of the data.

**Figure 5.**The intervention parameter $\sigma $ vs. population density (

**a**,

**b**), population (

**c**,

**d**), and social mobility (

**e**,

**f**) for the 50 states in the US for the first and second outbreaks, respectively. The solid lines are the linear fittings of the data and the corresponding correlation coefficients are also shown.

**Figure 6.**The comparison between the three parameters of the T-SIR model for the first outbreaks (blue) and second outbreaks (green) of the 50 US states. (

**a**) Reproduction numbers $R$. (

**b**) Transmission rates $\beta $. (

**c**) Intervention parameters $\sigma $.

**Table 1.**The ranked lists of the 50 states in the US according to their population densities, populations, and areas, where the states with two epidemic outbreaks occurring in March–April and October–January are highlighted in green, those with two epidemic outbreaks occurring in June–August and October–January are highlighted in yellow, and those with only one epidemic outbreak occurring in October–January are not highlighted.

Population Density | Population | Area | ||||||
---|---|---|---|---|---|---|---|---|

Rank | State | Density (/km^{2}) | Rank | State | Population | Rank | State | Area (km^{2}) |

1 | New Jersey | 393.18 | 1 | California | 39,368,078 | 1 | Alaska | 1,723,337 |

2 | Rhode Island | 264.22 | 2 | Texas | 29,360,759 | 2 | Texas | 695,662 |

3 | Massachusetts | 252.18 | 3 | Florida | 21,733,312 | 3 | California | 423,967 |

4 | Connecticut | 247.75 | 4 | New York | 19,336,776 | 4 | Montana | 380,831 |

5 | Maryland | 188.47 | 5 | Pennsylvania | 12,783,254 | 5 | New Mexico | 314,917 |

6 | Delaware | 153.09 | 6 | Illinois | 12,587,530 | 6 | Arizona | 295,234 |

7 | New York | 136.85 | 7 | Ohio | 11,693,217 | 7 | Nevada | 286,380 |

8 | Florida | 127.61 | 8 | Georgia | 10,710,017 | 8 | Colorado | 269,601 |

9 | Pennsylvania | 107.17 | 9 | North Carolina | 10,600,823 | 9 | Oregon | 254,799 |

10 | Ohio | 100.72 | 10 | Michigan | 9,966,555 | 10 | Wyoming | 253,335 |

11 | California | 92.86 | 11 | New Jersey | 8,882,371 | 11 | Michigan | 250,487 |

12 | Illinois | 83.92 | 12 | Virginia | 8,590,563 | 12 | Minnesota | 225,163 |

13 | Virginia | 77.54 | 13 | Washington | 7,693,612 | 13 | Utah | 219,882 |

14 | North Carolina | 76.05 | 14 | Arizona | 7,421,401 | 14 | Idaho | 216,443 |

15 | Indiana | 71.61 | 15 | Massachusetts | 6,893,574 | 15 | Kansas | 213,100 |

16 | Georgia | 69.59 | 16 | Tennessee | 6,886,834 | 16 | Nebraska | 200,330 |

17 | Tennessee | 63.09 | 17 | Indiana | 6,754,953 | 17 | South Dakota | 199,729 |

18 | South Carolina | 62.92 | 18 | Missouri | 6,151,548 | 18 | Washington | 184,661 |

19 | New Hampshire | 56.43 | 19 | Maryland | 6,055,802 | 19 | North Dakota | 183,108 |

20 | Hawaii | 49.69 | 20 | Wisconsin | 5,832,655 | 20 | Oklahoma | 181,037 |

21 | Kentucky | 42.78 | 21 | Colorado | 5,807,719 | 21 | Missouri | 180,540 |

22 | Texas | 42.21 | 22 | Minnesota | 5,657,342 | 22 | Florida | 170,312 |

23 | Washington | 41.66 | 23 | South Carolina | 5,218,040 | 23 | Wisconsin | 169,635 |

24 | Michigan | 39.79 | 24 | Alabama | 4,921,532 | 24 | Georgia | 153,910 |

25 | Alabama | 36.25 | 25 | Louisiana | 4,645,318 | 25 | Illinois | 149,995 |

26 | Wisconsin | 34.38 | 26 | Kentucky | 4,477,251 | 26 | Iowa | 145,746 |

27 | Louisiana | 34.24 | 27 | Oregon | 4,241,507 | 27 | New York | 141,297 |

28 | Missouri | 34.07 | 28 | Oklahoma | 3,980,783 | 28 | North Carolina | 139,391 |

29 | West Virginia | 28.44 | 29 | Connecticut | 3,557,006 | 29 | Arkansas | 137,732 |

30 | Arizona | 25.14 | 30 | Utah | 3,249,879 | 30 | Alabama | 135,767 |

31 | Minnesota | 25.13 | 31 | Iowa | 3,163,561 | 31 | Louisiana | 135,659 |

32 | Vermont | 25.03 | 32 | Nevada | 3,138,259 | 32 | Mississippi | 125,438 |

33 | Mississippi | 23.65 | 33 | Arkansas | 3,030,522 | 33 | Pennsylvania | 119,280 |

34 | Arkansas | 22.00 | 34 | Mississippi | 2,966,786 | 34 | Ohio | 116,098 |

35 | Oklahoma | 21.99 | 35 | Kansas | 2,913,805 | 35 | Virginia | 110,787 |

36 | Iowa | 21.71 | 36 | New Mexico | 2,106,319 | 36 | Tennessee | 109,153 |

37 | Colorado | 21.54 | 37 | Nebraska | 1,937,552 | 37 | Kentucky | 104,656 |

38 | Oregon | 16.65 | 38 | Idaho | 1,826,913 | 38 | Indiana | 94,326 |

39 | Utah | 14.78 | 39 | West Virginia | 1,784,787 | 39 | Maine | 91,633 |

40 | Maine | 14.73 | 40 | Hawaii | 1,407,006 | 40 | South Carolina | 82,933 |

41 | Kansas | 13.67 | 41 | New Hampshire | 1,366,275 | 41 | West Virginia | 62,756 |

42 | Nevada | 10.96 | 42 | Maine | 1,350,141 | 42 | Maryland | 32,131 |

43 | Nebraska | 9.67 | 43 | Montana | 1,080,577 | 43 | Hawaii | 28,313 |

44 | Idaho | 8.44 | 44 | Rhode Island | 1,057,125 | 44 | Massachusetts | 27,336 |

45 | New Mexico | 6.69 | 45 | Delaware | 986,809 | 45 | Vermont | 24,906 |

46 | South Dakota | 4.47 | 46 | South Dakota | 892,717 | 46 | New Hampshire | 24,214 |

47 | North Dakota | 4.18 | 47 | North Dakota | 765,309 | 47 | New Jersey | 22,591 |

48 | Montana | 2.84 | 48 | Alaska | 731,158 | 48 | Connecticut | 14,357 |

49 | Wyoming | 2.30 | 49 | Vermont | 623,347 | 49 | Delaware | 6,446 |

50 | Alaska | 0.42 | 50 | Wyoming | 582,328 | 50 | Rhode Island | 4,001 |

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**MDPI and ACS Style**

Huang, D.; Tao, H.; Wu, Q.; Huang, S.-Y.; Xiao, Y.
Modeling of the Long-Term Epidemic Dynamics of COVID-19 in the United States. *Int. J. Environ. Res. Public Health* **2021**, *18*, 7594.
https://doi.org/10.3390/ijerph18147594

**AMA Style**

Huang D, Tao H, Wu Q, Huang S-Y, Xiao Y.
Modeling of the Long-Term Epidemic Dynamics of COVID-19 in the United States. *International Journal of Environmental Research and Public Health*. 2021; 18(14):7594.
https://doi.org/10.3390/ijerph18147594

**Chicago/Turabian Style**

Huang, Derek, Huanyu Tao, Qilong Wu, Sheng-You Huang, and Yi Xiao.
2021. "Modeling of the Long-Term Epidemic Dynamics of COVID-19 in the United States" *International Journal of Environmental Research and Public Health* 18, no. 14: 7594.
https://doi.org/10.3390/ijerph18147594