Derivation of Incidence by Vaccination

Jesse Smith

1/10/2022

Short-Hand

\[ \begin{align} p_i &= \mathbf{P}(I=1) \\ p_v &= \mathbf{P}(V=1) \\ p_u &= \mathbf{P}(V=0) \\ p_{i \mid v} &= \mathbf{P}(I=1 \mid V=1) \\ p_{i \mid u} &= \mathbf{P}(I=1 \mid V=0) \\ e_v &= 1 - \frac{p_{i \mid v}}{p_{i \mid u}} \end{align} \]

Total Incidence as Weighted Sum of Group Incidence

\[ p_i = p_v \cdot p_{i|v} + p_u \cdot p_{i|u} \]

Proportion Unvaccinated as a Function of Proportion Vaccinated

\[ p_u = 1-p_v \]

Unvaccinated Incidence as a Function of Vaccinated Incidence

\[ \begin{align} e_v &= 1 - \frac{p_{i \mid v}}{p_{i \mid u}} \\ e_v-1 &= - \frac{p_{i \mid v}}{p_{i \mid u}} \\ 1-e_v &= \frac{p_{i \mid v}}{p_{i \mid u}} \\ p_{i \mid u} &= \frac{p_{i \mid v}}{1-e_v} \end{align} \]

Derive Incidence in Vaccinated Group

\[ \begin{align} p_i &= p_v \cdot p_{i|v} + p_u \cdot p_{i|u} \\ p_i &= p_v \cdot p_{i|v} + (1-p_v) \cdot \frac{p_{i \mid v}}{1-e_v} \\ p_i &= p_{i \mid v} (p_v + \frac{1-p_v}{1-e_v}) \\ p_{i \mid v} &= p_i \frac{1}{p_v (1 - \frac{1}{1-e_v}) + \frac{1}{1-e_v}} \\ p_{i \mid v} &= p_i \frac{1}{-p_v \frac{e_v}{1-e_v} + \frac{1}{1-e_v}} \\ p_{i \mid v} &= p_i \frac{1}{\frac{1}{1-e_v} (1 - p_v \cdot e_v)} \\ p_{i \mid v} &= p_i \frac{1}{\frac{1}{1-e_v}} \frac{1}{1 - p_v \cdot e_v} \\ p_{i \mid v} &= p_i (1-e_v)(\frac{1}{1 - p_v \cdot e_v}) \\ p_{i \mid v} &= p_i \frac{1-e_v}{1 - p_v \cdot e_v} \end{align} \]

Derive Incidence in Unvaccinated Group

\[ \begin{align} p_{i \mid u} &= \frac{p_{i \mid v}}{1-e_v} \\ p_{i \mid u} &= p_i \frac{1-e_v}{1 - p_v \cdot e_v} \frac{1}{1-e_v} \\ p_{i \mid u} &= p_i \frac{1}{1 - p_v \cdot e_v} \end{align} \]