Vaccine efficacy and effectiveness (VE) is generally estimated as one minus some measure of relative risk (RR) in the vaccinated group compared to the unvaccinated group Halloran et al.:
\[
\text{VE} = 1 - \text{RR}
\]
RR measures may include relative infection probability given exposure, hazard ratio, or the ratio of cumulative incidence (or attack rate). I will explain different effects of the vaccine using the cumulative incidence (\(C\)) measure, as outlined in the Study design for dependent happenings.
In population \(A\), a fraction, \(f\), of the population was vaccinated (denoted as \(1\)) and the rest, \(1-f\), of the Population A remains unvaccinated (denoted as \(0\)). In Population \(B\), no one was vaccinated. I will use \(C_{A0}\) and \(C_{A1}\) to denote cumulative incidence, \(C\), in the unvaccinated and vaccinated individuals of Population \(A\). Similarly, \(C_{B0}\) will denote the cumulative incidence in the unvaccinated population \(B\).
Vaccine effectiveness (VE) can be divided into four different measures: direct, indirect, total, and overall VE, which are denoted as DVE, IVE, TVE, and OVE, respectively.
I will illustrate the different metrics of VE using simple \(SEIR\) model simulations.
The model, seir_2grp, includes two subgroups, which will be used to differentiate vaccinated and unvaccinated populations.
A fraction \(f\) of population A received the vaccine, with vaccine efficacy denoted as VE. Vaccination is implemented using an all-or-nothing approach, meaning a fraction of the vaccine recipients, \(Nf\text{VE}\), are completely protected from infection. Here, \(N\), \(f\), and VE represent the population size, the fraction of the population that is vaccinated, and vaccine efficacy, respectively. In the SEIR model, we can mimic the vaccination scenario by setting the initial population size for the \(R\) compartment to \(Nf\text{VE}\).
