# A Mathematical Study of COVID-19 Spread by Vaccination Status in Virginia
## 1 Introduction  
In December 2019, a novel and highly-contagious coronavirus SARS-CoV-2 was identified in Wuhan, China. The associated disease, COVID-19, causes fever, muscles aches, fatigue, and shortness of breath and has been associated with a significant mortality rate, especially among older and immunocompromised individuals [1]. In the early stages of the pandemic, mitigation measures focused on non-pharmaceutical interventions including travel/border restrictions, social distancing/masking campaigns, work-from-home orders, school closures, and lockdowns. After the Emergency Use Authorization of effective vaccines in the United States at the end of 2020 (Pfizer-BioNTech 11 December 2020; Moderna, 18 December 2020; Johnson & Johnson, 27 February 2021), the response focused on vaccination drives and awareness campaigns. While these efforts have averted the worst-case forecast scenarios, COVID-19 continues to spread widely due in part to the more transmissible B.1.617.2 (Delta) variant, which became the dominant COVID-19 strain in the United States in July 2021 [2].  
Mathematical modeling has played a vital role in public policy decision-making since the beginning of the pandemic. SIR-type compartmental models [3] have been used to estimate key epidemiological parameters of infectious disease spread, such as the transmission rate and basic reproduction number [4; 5; 6]; to evaluate the pathways of spread, such as spread through asymptomatic carriers [7; 8]; and to forecast disease dynamics under a variety of non-pharmaceutical intervention strategies, such as lockdowns, contact tracing, and face masking campaigns [9; 10; 11; 12; 13; 14]. More recent research has focused on aspects of pharmaceutical interventions, such as comparing COVID-19 vaccination strategies (e.g., which segment of society to vaccinate first) and post-vaccination re-opening scenarios [15; 16; 17; 18]. While these and other studies incorporate vaccination into their modeling frameworks, time-course differential equation models have not been widely used to assess the real-world efficacy of vaccination itself. Rather, studies on vaccine efficacy have been limited to controlled clinical trials [19; 20; 21] and analysis of data aggregates [22; 23; 24].  
In this paper, we introduce a novel \(n\)-stage vaccination model that partitions a population according to their vaccination status. This model allows us to concisely assess and forecast the spread of an infectious disease as the vaccination level of a population changes. We fit the model to COVID-19 incidence data from the Virginia Department of Health (VDH) [25] and vaccination uptake data from Our World in Data [26] to estimate (a) the real-world efficacy of vaccination in preventing COVID-19 infections, hospitalizations, and deaths, and (b) the transmissibility of the B.1.617.2 (Delta) variant relative to previously dominant strains. We also conduct a parameter sensitivity analysis and present forecasts to what the spread of COVID-19 could look like through 31 December 2021 given different vaccination coverage levels and with the incorporation of a booster shot.