# A Mathematical Study of COVID-19 Spread by Vaccination Status in Virginia
## 2 Materials and Methods
### Data Fitting Techniques  
We fit the \(n\)-stage vaccination model (2) with the vaccination uptake curve (4) to data in two steps. We first fit the vaccination uptake curve \(V_{i}(t)\) (4) to daily vaccination uptake data by minimizing the sum of squared error. We then incorporate these parameters into the vaccination uptake curve \(\frac{dV_{i}}{dt}\) in the system of differential Equation (2) and minimize a customized error function. For both steps, we utilize a nonlinear least squares fitter written in Python. We now describe the customized error function for fitting to (2). We let \(i\in\{0,1,2\}\) indicate the number of vaccine shots (\(i=0\) unvaccinated, \(i=1\) partially vaccinated, and \(i=2\) fully vaccinated), \(j\in\{I,H,D\}\) indicate the type of individual (infected, hospitalized, or deceased), and \(k=1,\ldots,T\) denote the time point (in weeks). We let \(\mathbf{x}=[x_{i,j,k}]\) denote the cumulative number of COVID-19 cases in vaccination class \(i\) of type \(j\) and in week \(k\), and \(\mathbf{y}=[y_{i,j,k}]\) denote the weekly increase in that class from the previous week (i.e., \(y_{i,j,k}=x_{i,j,k}-x_{i,j,k-1}\)). We define \(\theta\) to be the set of all parameters to be \begin{table}
\begin{tabular}{c l l} \hline \hline
**Parameter** & **Units** & **Interpretation** \\ \hline \(N\) & people & Population size (total) \\ \(N_{i}\) & people & Population size (\(i\)th class) \\ \(V_{i}(t)\) & people & Vaccination uptake curve (\(i\)th class) \\ \(r_{i}\) & days\({}^{-1}\) & Vaccination uptake rate (\(i\)th class) \\ \(M_{i}\) & people & Maximum vaccination total (\(i\)th class) \\ \(M_{i}/N\) & none & Maximum vaccination proportion (\(i\)th class) \\ \(\tau_{i}\) & days & Vaccination time shift (\(i\)th class) \\ \(a_{i}\) & none & Vaccination shape parameter (\(i\)th class) \\ \(\beta(t)\) & days\({}^{-1}\) & Baseline transmission rate for unvaccinated \\ \(\alpha_{i}\) & none & Reduction from baseline transmission rate (\(i\)th class) \\ \(\gamma_{i}\) & days\({}^{-1}\) & Removal rate (\(i\)th class) \\ \(h_{i}\) & days\({}^{-1}\) & Hospitalization rate (\(i\)th class) \\ \(\delta_{i}\) & days\({}^{-1}\) & Death rate (\(i\)th class) \\ \(h_{i}/(h_{i}+\gamma_{i})\) & none & Proportion infected to hospitalized (\(i\)th class) \\ \(\delta_{i}/(\delta_{i}+\gamma_{i})\) & none & Proportion hospitalized to deceased (\(i\)th class) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters and rate functions for the \(n\)-stage vaccination model (1) and corresponding system of ordinary differential Equation (2) with the \(i\)th vaccination dose uptake curve \(V_{i}(t)\) (4). fit and \(\hat{\mathbf{x}}(t;\theta)=[x_{i,j}(t;\theta)]\) to denote the model simulation of (2) for the cumulative cases in vaccination class \(i\) of type \(j\) and for parameter set \(\theta\). We utilize the following normalized error function, which incorporates both the weekly cumulative and rate numbers:  
\[E(\mathbf{x},\hat{\mathbf{x}}(t;\theta))=\sum_{k=1}^{T}\sum_{ \begin{subarray}{c}i\in\{0,1,2\}\\ j\in\{I,H,D\}\end{subarray} }\left(\frac{\left(x_{i,j,k}-\hat{x}_{i,j}(t_{k}; \theta)\right)^{2}}{\mu_{j}}+\frac{\left(y_{i,j,k}-\hat{x}_{i,j}^{\prime}(t_{k };\theta)\right)^{2}}{v_{j}}\right) \tag{7}\]  
where \(\mu_{j}\) and \(v_{j}\), \(j\in\{I,H,D\}\), are the normalization factors  
\[\mu_{j}=\sum_{ \begin{subarray}{c}k=1\\ i\in\{0,1,2\}\end{subarray} }^{T}x_{i,j,k}^{2}\;\;\;\text{and}\;\;\;v_{j}=\sum_ { \begin{subarray}{c}k=1\\ i\in\{0,1,2\}\end{subarray} }^{T}y_{i,j,k}^{2}. \tag{8}\]  
Note that the normalizations (8) in the error function (7) equalize the weight of the cumulative and weekly rate of increase in infections, hospitalizations, and deaths.