9.3 Methods for mapping prevalence to the optimal policy

When donor groups are identical except for the number of donors ($n_i$) and the prevalence of each TTI ($p_{ik}$), a $k$-dimensional function $f(\textbf{p})$ mapping the prevalence by TTI to the optimal policy can be defined as follows:

$$(\textbf{m}, \textbf{a}, z)^* = f(\textbf{p}), \quad \text{where} \text{ } f(\textbf{p}) = {\arg\min}_{\textbf{m}, \textbf{a}, z} \mathbb{E}[\mathcal{C}(z, \textbf{m}, \textbf{a}, \textbf{p}) ].$$

For notational convenience, we define a policy $\pi$ as a set of decision variables $\pi = (z, \textbf{m}, \textbf{a})$. For a given policy and a single donor group, expected cost as a function of prevalence is:

$$\mathbb{E}[\mathcal{C}(\textbf{p}) \mid \pi] = (1- z)d + z \Big[w + \textbf{a}^\top\boldsymbol{\phi} + \textbf{m}^\top \boldsymbol{\psi} + \big(1- s_2\big)\big(\textbf{v}_1 \circ \textbf{v}_3 \circ \textbf{p} \big)^\top \textbf{c} + gs_2(\textbf{a},\textbf{p}) \Big],$$

where vectors $\textbf{v}_1$ and $\textbf{v}_3$ are functions of $\pi$ and the constant $s_2$ is a function of both $\pi$ and $\textbf{p}$. By grouping terms that are constant in $\textbf{p}$, this can be rewritten as

$$\mathbb{E}[\mathcal{C}(\textbf{p}) \mid \pi] = \alpha(\pi) + \big(1- s_2(\textbf{a},\textbf{p})\big) \beta(\pi) \textbf{p}+ gs_2(\textbf{a},\textbf{p}) \Big],$$

where $\alpha(\pi) = (1- z)d + z \Big[w + \textbf{a}^\top\boldsymbol{\phi} + \textbf{m}^\top \boldsymbol{\psi} \Big]$ and $\beta(\pi) = z (\textbf{v}_1 \circ \textbf{v}_3 \circ \textbf{c} )^\top$.

For a policy $\pi$, we define its ‘policy region’ as the $k$-dimensional set of prevalence values for which $\mathbb{E}[\mathcal{C}(\textbf{p}) \mid \pi] = \min_{\pi'} \mathbb{E}[\mathcal{C}(\textbf{p}) \mid \pi']$. For two policies $\pi_1$ and $\pi_2$, the $k$-dimensional ‘pairwise decision boundary’ (PDB) between their respective policy regions is the solution to $\mathbb{E}[\mathcal{C}(\textbf{p}) \mid \pi_1] = \mathbb{E}[\mathcal{C}(\textbf{p}) \mid \pi_2]$. Policy regions can be identified through the following procedure:

1. Solve for the PDB for each unique pairing of policies ($0.5T(T-1)$ pairs)

2. Divide the prevalence space into mutually exclusive and collectively exhaustive PBD-regions, indexed by $g$, and identify $\Lambda_g$, the optimal policy for each region, by evaluating any point in the region ($\Lambda_g = {\arg\min}_{\Lambda} \mathbb{E}[\mathcal{C}(\textbf{p} \mid \Lambda) ]$ for any $\textbf{p}$ in PDB region $g$).

3. Aggregate PBDs for which $\Lambda_g = \Lambda_t$ into policy region $\Lambda_t$.

Alternatively, decision boundaries could be approximated through grid search or other search algorithms.