# A Mathematical Study of COVID-19 Spread by Vaccination Status in Virginia

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. n-Stage Vaccination Model

^{th}vaccination class or an outgoing edge from the ${n}^{th}$ vaccination class. We allow for a time-dependent baseline transmission rate for unvaccinated individuals, $\beta \left(t\right)$, so that the transmission rate may vary, for instance, by changes in public health policies or changes in prevalence of COVID-19 variants.

#### 2.2. Governing System of Differential Equations

#### 2.3. Basic and Effective Reproduction Number

#### 2.4. Data Fitting Techniques

#### 2.5. Parameter Sensitivity Analysis

## 3. Results

#### 3.1. Fitting Two-Stage Vaccination Model to Virginia Data

#### 3.2. Efficacy of Vaccination

#### 3.3. Transmissibility of Delta Variant

#### 3.4. Effect of Waning Immunity and Delta Variant on Vaccine Efficacy

#### 3.5. Parameter Sensitivity Analysis

#### 3.6. Modeling with Different Vaccine Coverage Levels

#### 3.7. Modeling with Booster Shots

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

`Vaccination_Fit.py`and

`Model_Fit.py`are available in the “Downloads” section of johnstonmd.wordpress.com (accessed on 31 January 2022).

`Vaccination_Fit.py`fits daily vaccination uptake data to a Richard’s curve (4) and

`Model_Fit.py`fits the two-stage vaccination model (2) to weekly data on COVID-19 incidence, hospitalizations, and deaths.

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**On the left, we indicate the best fit parameters for the generalized logistic curve or Richards’ curve (4) to Virginia Department of Health data [25] on vaccination uptake starting with $t=0$ corresponding to 23 January 2021. The values ${M}_{1}/N$ and ${M}_{2}/N$ correspond to the saturation proportions for the first and second doses, respectively (i.e., the number of willing/able vaccine recipients). We use the population denominator N = 8,535,519 corresponding to the population of Virginia. On the right, we display the model fit (solid) to the observed data (circles).

**Figure 2.**Plots of VDH data and the parameter-fit two-stage vaccination model (2) with piecewise-constant transmission rate $\beta \left(t\right)$ (3) and vaccination curves ${V}_{i}\left(t\right)$ (4) with the parameters from Figure 1 and Table 2. On the left, we track the time-course breakdown of the new cases, hospitalizations, and deaths by vaccination status. Individual statuses are represented as bands (vertical distance between open circles for data and solid regions for model). On the right, we track the cumulative observed and model-predicted cases, hospitalizations, and deaths over time.

**Figure 3.**Sensitivity analysis using Equation (11) on the cumulative cases of COVID-19 in Virginia (left column) and ${R}_{t}$ (the effective reproduction number, right column) with respect to vaccination parameters (${M}_{1}$, ${M}_{2}$, ${\alpha}_{1}$, and ${\alpha}_{2}$) and recovery rates (${\gamma}_{0}$, ${\gamma}_{1}$, and ${\gamma}_{2}$). Note the negative impact of the parameters on the cumulative cases, but a non-trivial impact on the effective reproduction number as population sizes change in each vaccination stage through time. Further notice the small influence that ${M}_{1}$ has on both outputs, until later in the simulation.

**Figure 4.**Simulations of the two-stage vaccination model (1) with parameters from Table 2 and differing values of overall vaccine uptake (${M}_{1}$ and ${M}_{2}$). The transmission rate during the period September 18 through 31 December 2021 is assumed to be that of the final fit period (i.e., $\beta \left(t\right)={\beta}_{8}=0.453$). On the left, forecast values through 31 December 2021 are shown for cumulative cases, hospitalizations, and deaths given current vaccination uptake, 10% more vaccination, and 10% less vaccination.

**Figure 5.**Simulations of the three-stage vaccination model (2) with $j=3$ corresponding to individuals who have received a booster shot. The parameters for the booster shot uptake curve ${V}_{3}\left(t\right)$ are shown, and the parameters for the vaccination uptake curves ${V}_{j}\left(t\right)$, $j=\{0,1,2\}$, are from Table 2. The reductions in the baseline transmission parameter for the booster shot are ${\alpha}_{3}=0.95$ (low booster) and ${\alpha}_{3}=1.00$ (high booster) [39,40], and the model runs from 23 January through 31 December 2021.

Parameter | Units | Interpretation |
---|---|---|

N | people | Population size (total) |

${N}_{i}$ | people | Population size (ith class) |

${V}_{i}\left(t\right)$ | people | Vaccination uptake curve (ith class) |

${r}_{i}$ | days${}^{-1}$ | Vaccination uptake rate (ith class) |

${M}_{i}$ | people | Maximum vaccination total (ith class) |

${M}_{i}/N$ | none | Maximum vaccination proportion (ith class) |

${\tau}_{i}$ | days | Vaccination time shift (ith class) |

${a}_{i}$ | none | Vaccination shape parameter (ith class) |

$\beta \left(t\right)$ | days${}^{-1}$ | Baseline transmission rate for unvaccinated |

${\alpha}_{i}$ | none | Reduction from baseline transmission rate (ith class) |

${\gamma}_{i}$ | days${}^{-1}$ | Removal rate (ith class) |

${h}_{i}$ | days${}^{-1}$ | Hospitalization rate (ith class) |

${\delta}_{i}$ | days${}^{-1}$ | Death rate (ith class) |

${h}_{i}/({h}_{i}+{\gamma}_{i})$ | none | Proportion infected to hospitalized (ith class) |

${\delta}_{i}/({\delta}_{i}+{\gamma}_{i})$ | none | Proportion hospitalized to deceased (ith class) |

**Table 2.**Best fitting initial conditions (left) and parameters (right) for the two-stage vaccination model (1) and corresponding system of differential Equation (2) with piece-wise constant transmission function $\beta \left(t\right)$ (3), and vaccination uptake curve ${V}_{i}\left(t\right)$ (4) with parameters from Figure 1.

Initial Conditions | Parameters | ||||
---|---|---|---|---|---|

Parameter | Value | Source | Parameter | Value | Source |

N | 8,535,519 | [25] | ${\beta}_{1}$ | $0.152777$ | fitted |

${N}_{0}$ | 8,136,306 | assumed (9) | ${\beta}_{2}$ | $0.199888$ | fitted |

${N}_{1}$ | 289,839 | assumed (9) | ${\beta}_{3}$ | $0.250908$ | fitted |

${N}_{2}$ | 109,375 | assumed (9) | ${\beta}_{4}$ | $0.240565$ | fitted |

${S}_{0}\left(0\right)$ | 7,695,938 | assumed (10) | ${\beta}_{5}$ | $0.238368$ | fitted |

${I}_{0}\left(0\right)$ | 19,557 | fitted | ${\beta}_{6}$ | $0.464551$ | fitted |

${R}_{0}\left(0\right)$ | 411,709 | fitted | ${\beta}_{7}$ | $0.538527$ | fitted |

${H}_{0}\left(0\right)$ | 823 | fitted | ${\beta}_{8}$ | $0.453217$ | fitted |

${D}_{0}\left(0\right)$ | 8279 | assumed | ${\alpha}_{1}$ | $0.559980$ | fitted |

${S}_{1}\left(0\right)$ | 281,970 | assumed (10) | ${\alpha}_{2}$ | $0.898494$ | fitted |

${I}_{1}\left(0\right)$ | 7642 | fitted | ${h}_{0}$ | $0.005768$ | fitted |

${H}_{1}\left(0\right)$ | 227 | fitted | ${h}_{1}$ | $0.002875$ | fitted |

${R}_{1}\left(0\right)$ | 0 | assumed | ${h}_{2}$ | $0.003598$ | fitted |

${D}_{1}\left(0\right)$ | 0 | assumed | ${\gamma}_{0}$ | $0.176093$ | fitted |

${S}_{2}\left(0\right)$ | 108,839 | assumed (10) | ${\gamma}_{1}$ | $0.081428$ | fitted |

${I}_{2}\left(0\right)$ | 536 | fitted | ${\gamma}_{2}$ | $0.096318$ | fitted |

${H}_{2}\left(0\right)$ | 0 | fitted | ${\delta}_{0}$ | $0.085761$ | fitted |

${R}_{2}\left(0\right)$ | 0 | assumed | ${\delta}_{1}$ | $0.048616$ | fitted |

${D}_{2}\left(0\right)$ | 0 | assumed | ${\delta}_{2}$ | $0.065626$ | fitted |

**Table 3.**Vaccination level, effective transmission rate, and the effective reproduction number for the spread of COVID-19 in Virginia over the period 23 January through 18 September 2021.

Interval | Dates | % Part Vax | % Full Vax | $\mathit{\beta}\left(\mathit{t}\right)$ | ${\mathit{R}}_{\mathit{t}}$ |
---|---|---|---|---|---|

1 | 1/23–2/21 | 4.4% | 0.5% | 0.153 | 0.786 |

2 | 2/22–3/23 | 14.0% | 6.1% | 0.200 | 0.978 |

3 | 3/24–4/22 | 26.7% | 14.7% | 0.251 | 1.106 |

4 | 4/23–5/21 | 44.3% | 29.1% | 0.241 | 0.906 |

5 | 5/22–6/20 | 53.2% | 42.7% | 0.238 | 0.762 |

6 | 6/21–7/20 | 58.2% | 50.3% | 0.464 | 1.312 |

7 | 7/21–8/19 | 60.7% | 53.7% | 0.539 | 1.404 |

8 | 8/20–9/18 | 64.2% | 56.1% | 0.453 | 1.119 |

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**MDPI and ACS Style**

Johnston, M.D.; Pell, B.; Nelson, P.
A Mathematical Study of COVID-19 Spread by Vaccination Status in Virginia. *Appl. Sci.* **2022**, *12*, 1723.
https://doi.org/10.3390/app12031723

**AMA Style**

Johnston MD, Pell B, Nelson P.
A Mathematical Study of COVID-19 Spread by Vaccination Status in Virginia. *Applied Sciences*. 2022; 12(3):1723.
https://doi.org/10.3390/app12031723

**Chicago/Turabian Style**

Johnston, Matthew D., Bruce Pell, and Patrick Nelson.
2022. "A Mathematical Study of COVID-19 Spread by Vaccination Status in Virginia" *Applied Sciences* 12, no. 3: 1723.
https://doi.org/10.3390/app12031723