diff --git a/model_definition.md b/model_definition.md index 938f5c46..d20bddda 100644 --- a/model_definition.md +++ b/model_definition.md @@ -98,7 +98,7 @@ where $I_0$ is the initial per capita infection incident infections and $r$ is t We decompose the instantaneous reproduction number $\mathcal{R}(t)$ into two components: an _unadjusted_ instantaneous reproduction number $\mathcal{R}^\mathrm{u}(t)$ and a damping term that accounts for the effect of recent infections on the instantaneous reproduction number[^Asher2018]. -We assume that the unadjusted reproduction number $\mathcal{R}^\mathrm{u}(t)$ is a piecewise-constant function with weekly change points (i.e., if $t$ and $t'$ are days in the same week, then $\mathcal{R}^\mathrm{u}(t) = \mathcal{R}^\mathrm{u}(t')$). To account for the dependence of the unadjusted reproduction number in a given week on the previous week, we use a differenced auto-regressive process for the log-scale reproduction number. A log-scale representation is used to ensure that the reproduction number is positive and so that week-to-week changes are multiplicative rather than additive. +We assume that the unadjusted reproduction number $\mathcal{R}^\mathrm{u}(t)$ is a piecewise-constant function with weekly change points (i.e., if $t$ and $t'$ are days in the same week, then $\mathcal{R}^\mathrm{u}(t)$ = $\mathcal{R}^\mathrm{u}(t')$ ). To account for the dependence of the unadjusted reproduction number in a given week on the previous week, we use a differenced auto-regressive process for the log-scale reproduction number. A log-scale representation is used to ensure that the reproduction number is positive and so that week-to-week changes are multiplicative rather than additive. $$ \log[\mathcal{R}^\mathrm{u}(t_3)] \sim \mathrm{Normal}\left(\log[\mathcal{R}^\mathrm{u}(t_2)] + \beta \left(\log[\mathcal{R}^\mathrm{u}(t_2)] - \log[\mathcal{R}^\mathrm{u}(t_1)]\right), \sigma_r \right)