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.