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fix rendering error #7

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Mar 13, 2024
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2 changes: 1 addition & 1 deletion model_definition.md
Original file line number Diff line number Diff line change
Expand Up @@ -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)
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