From 17dc265683e85fc1c09bac658ad78dcd870642bb Mon Sep 17 00:00:00 2001 From: Paige Miller Date: Tue, 3 Dec 2024 20:53:14 +0000 Subject: [PATCH] equations --- docs/approach.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs/approach.md b/docs/approach.md index 903cf4e..f3d0432 100644 --- a/docs/approach.md +++ b/docs/approach.md @@ -31,16 +31,16 @@ If all $\gamma_i = \gamma$ and we assume SIR dynamics. The basic reproductive number $R_0$ is calculated as the dominant eigenvalue of $R$. -This model incorporates vaccination by recalculating the distribution of susceptible individuals in each group $S_{i}^{vax}$ (assuming all or nothing vaccination, with vaccine efficacy given by $ve$): +This model incorporates vaccination by recalculating the distribution of susceptible individuals in each group $S_{i}^{vax}$ (assuming all or nothing vaccination, with vaccine efficacy given by $ve$ and the proportion of $i$ vacinated is $v_i$): $$ \mathbf{S_{i}^{vax}} = S_{i}^{initial} * (1 - v_{i} * ve) $$ -Then $R_ij$ with vaccination factored in is given by +So that $S_i^{vax}$ is the population $i$ that is still susceptible post vaccination administration. Then $R_ij$ with vaccination factored in is given by $$ -\mathbf{R}_{ij}^{vax} = \frac{\beta_{ij} S_{i}^{vax}}{\gamma N} +\mathbf{R}_{ij}^{vax} = \mathbf{R}_{ij} \frac{S_{i}^{vax}}{N_i} $$