# A Mathematical Study of COVID-19 Spread by Vaccination Status in Virginia
## 2 Materials and Methods  
### n-Stage Vaccination Model  
We introduce the following \(n\)-stage vaccination model, where within each vaccination class \(i=0,\ldots,n\), there are susceptible individuals (\(S_{i}\)), infectious individuals (\(I_{i}\)), hospitalized individuals (\(H_{i}\)), removed individuals (\(R_{i}\)), and deceased individuals (\(D_{i}\)):  
\[\begin{CD}\text{\text@underline{i}^{th}\text@underline{\text{Vacination Class}}}\\ (i-1)^{st}\text@underline{\text{class}}@>{V_{i}^{\prime}(t)}>{}>\text@underline{ \text{Susceptible}}\\ (S_{i})@>{V_{i+1}^{\prime}(t)}>{}>(i+1)^{st}\text@underline{\text{class}} \end{CD} \tag{1}\]  
Note that \(i=0\) corresponds to unvaccinated individuals and that the removed classes \(R_{i}\) may include those who have recovered from illness but also those who have been removed by other means, such as self-isolation or quarantine. Individuals are transferred from the \((i-1)^{st}\) vaccination class to the \(i\)th vaccination class through the \(i\)th dose vaccination uptake rate curve \(V_{i}^{\prime}(t)\), which is the derivative of the cumulative \(i\)th dose uptake curve \(V_{i}(t)\). To account for there only being a finite number of vaccination classes, we set \(V_{0}^{\prime}(t)=0\), and \(V_{n+1}^{\prime}(t)=0\). That is, we do not include an incoming edge to the \(0^{th}\) vaccination class or an outgoing edge from the \(n^{th}\) vaccination class. We allow for a time-dependent baseline transmission rate for unvaccinated individuals, \(\beta(t)\), so that the transmission rate may vary, for instance, by changes in public health policies or changes in prevalence of COVID-19 variants.  
To model COVID-19 spread in the United States, we use the two-stage and three-stage vaccination models: (i) \(i=0\) for individuals who have not received any vaccine shots (unvaccinated); (ii) \(i=1\) for individuals who have received one shot of either the Pfizer or Moderna vaccine (partially vaccinated); (iii) \(i=2\) for individuals who have received two shots of either the Pfizer or Moderna vaccine, or one shot of the Johnson & Johnson vaccine (fully vaccinated); and (iv) \(i=3\) for individuals who are fully vaccinated and received a booster shot.  
The model (1) makes several assumptions: (a) there is no latency period (i.e., susceptible individuals become infectious immediately upon becoming infected); (b) both the vaccine and infection provide long-lasting immunity (i.e., there is no waning immunity or reinfection); (c) only hospitalized individuals may die; and (d) previously infected individuals will not vaccinate. We note that COVID-19 has a mean latency period of 3-7 days [27; 28], that studies suggest there is decreased immunity 6 months after becoming fully vaccinated [29; 30; 31], that an estimated 10-15% of COVID-19 deaths in the United States occur outside of the hospital [32], and that the CDC recommends vaccination for individuals who have previously been infected with COVID-19. We chose not to incorporate these effects for model simplicity; however, an extension of the \(n\)-stage vaccination model (1) which incorporates waning immunity and reinfection will be the focus of future work.