In jesse-smith/covidscreen: COVID-19 Asymtomatic Screening Strategy Assessment

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

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

jesse-smith/covidscreen documentation built on June 15, 2022, 7:46 p.m.