# A Mathematical Study of COVID-19 Spread by Vaccination Status in Virginia
## 3 Results  
### Fitting Two-Stage Vaccination Model to Virginia Data  
We fit the vaccination uptake curve \(V_{i}(t)\) (4) to the cumulative vaccination data from the Commonwealth of Virginia [26] using the methods described in Section 2.4. We used data points starting on 23 January and ending on 2 October and used linear interpolation to fill in missing data. The best fitting parameters and curves \(V_{1}(t)\) and \(V_{2}(t)\) are shown in Figure 1.  
We then incorporated these parameters into the vaccination uptake curve (4) and used the methods outlined in Section 2.4 to fit the two-stage vaccination model (2) to the time-course data set VDH-COVID-19-PublicUseDataset-Cases-by-Vaccination-Status (report date 8 October 2021 [25]). This data set is publicly available through the VDH Data Portal and breaks down new weekly COVID-19 infections, hospitalizations, and deaths by vaccination status (unvaccinated, partially vaccinated, and fully vaccinated). We used data from 23 January through 11 September 2021. We used the piecewise-constant transmission rate function \(\beta(t)\) (3) with \(L=238\) and \(m=8\) so that \(L/m=29.75\). This allows the transmission rate to change on a roughly monthly basis. The fit initial conditions and parameters can be found in Table 2. Plots are displayed in Figure 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{5}{c}{**Initial Conditions**} & \multicolumn{5}{c}{**Parameters**} \\ \hline
**Parameter** & **Value** & **Source** & **Parameter** & **Value** & **Source** \\ \hline \(N\) & 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 \\ \(S_{0}(0)\) & 7,695,938 & assumed (10) & \(\beta_{5}\) & 0.238368 & fitted \\ \(l_{0}(0)\) & 19,557 & fitted & \(\beta_{6}\) & 0.464551 & fitted \\ \(R_{0}(0)\) & 411,709 & fitted & \(\beta_{7}\) & 0.538527 & fitted \\ \(H_{0}(0)\) & 823 & fitted & \(\beta_{8}\) & 0.453217 & fitted \\ \(D_{0}(0)\) & 8279 & assumed & \(\alpha_{1}\) & 0.559980 & fitted \\ \(S_{1}(0)\) & 281,970 & assumed (10) & \(a_{2}\) & 0.898494 & fitted \\ \(I_{1}(0)\) & 7642 & fitted & \(l_{0}\) & 0.005768 & fitted \\ \(H_{1}(0)\) & 227 & fitted & \(h_{1}\) & 0.002875 & fitted \\ \(R_{1}(0)\) & 0 & assumed & \(h_{2}\) & 0.003598 & fitted \\ \(D_{1}(0)\) & 0 & assumed & \(\gamma_{0}\) & 0.176093 & fitted \\ \(S_{2}(0)\) & 108,839 & assumed (10) & \(\gamma_{1}\) & 0.081428 & fitted \\ \(I_{2}(0)\) & 536 & fitted & \(\gamma_{2}\) & 0.096318 & fitted \\ \(H_{2}(0)\) & 0 & fitted & \(\delta_{0}\) & 0.085761 & fitted \\ \(R_{2}(0)\) & 0 & assumed & \(\delta_{1}\) & 0.048616 & fitted \\ \(D_{2}(0)\) & 0 & assumed & \(\delta_{2}\) & 0.065626 & fitted \\ \hline \hline \end{tabular}
\end{table}
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 \(\beta(t)\) (3), and vaccination uptake curve \(V_{l}(t)\) (4) with parameters from Figure 1.  
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).