9.1 Calculating the incremental cost of interventions compared to a ‘no intervention’ scenario

We can reduce the search space by eliminating some tests or modifications from consideration before running the binary integer program. To do so, we leverage the fact that a test or modification that is not preferred over ‘no intervention’ (i.e., objective function cost of intervention alone is greater than the objective function cost of ‘no intervention’) cannot be part of the optimal portfolio for a specific donor group. This is only true for the tailored portfolio problem. However, if a test or modification was not preferred over ‘no intervention’ in any donor group then it cannot be part of the optimal portfolio even when additional constraints are added, as with the universal portfolio problem.

We will let \(\gamma_{ji}^{test}\) denote the cost of using only test \(j\) (no other tests or modifications) compared to using no interventions in donor group \(i\). Similarly, we let \(\gamma_{li}^{mod}\) denote the cost of using only modification \(l\) compared to using no interventions. These quantities can be computed as:

\[ \begin{align} \gamma^\text{test}_{j^*, i^*} =& \mathbb{E}[\mathcal{C}(z_{i^*} = 1, z_{\neg i^*} = 0, a_{j^*} = 1, a_{\neg j^*} = 0, \textbf{M} = \textbf{0}, \textbf{n} = \textbf{1})] - \\ &\mathbb{E}[\mathcal{C}(z_{i^*} = 1, z_{\neg i^*} = 0, \textbf{A} = \textbf{0}, \textbf{M} = \textbf{0}, \textbf{n} = \textbf{1})] \\ \gamma^\text{mod}_{l^*, i^*} =& \mathbb{E}[\mathcal{C}(z_{i^*} = 1, z_{\neg i^*} = 0, m_{l^*} = 1, m_{\neg l^*} = 0, \textbf{A} = \textbf{0}, \textbf{n} = \textbf{1})] - \\ &\mathbb{E}[\mathcal{C}(z_i^* = 1, z_{\neg i^*} = 0, \textbf{A} = \textbf{0}, \textbf{M} = \textbf{0}, \textbf{n} = \textbf{1})] \\ \end{align} \]

The expected cost components of these expressions can be simplified as:

\[ \begin{align} \mathbb{E}[\mathcal{C}(z_{i^*} = 1, z_{\neg i^*} = 0, \textbf{A} = \textbf{0}, \textbf{M} = \textbf{0}, \textbf{n} = \textbf{1})] &= \textbf{w} + \textbf{p}_{*i^*} \textbf{c}\\ \mathbb{E}[\mathcal{C}(z_i^* = 1, z_{\neg i^*} = 0, a_j^* = 1, a_{\neg j^*} = 0, \textbf{M} = \textbf{0}, \textbf{n} = \textbf{1})] &= \textbf{w} + a_{j^*} \phi_{j^*} +\\ &(1-s^{i^*j^*}) ((\textbf{1} - \textbf{r}_{*j^*}) \circ \textbf{p}_{*i^*} ) \textbf{c} + \\ &gs^{i^*j^*}\\ \mathbb{E}[\mathcal{C}(z_{i^*} = 1, z_{\neg i^*} = 0, \textbf{A} = \textbf{0}, m_{l^*} = 1, m_{\neg l^*} = 0, \textbf{n} = \textbf{1})] &= w + m_{l^*} \phi_{l^*} + ( \textbf{h}_{{*l^*}} \circ \textbf{p}_{*i^*})\textbf{c} \end{align} \]

where \(\textbf{p}_{*i^*}\), \(\textbf{r}_{*j^*}\), and \(\textbf{h}_{*l^*}\) indicate the \(i^*\)th, \(j^*\)th, and \(l^*\)th rows of \(\textbf{P}\), \(\textbf{R}\), and \(\textbf{H}\), respectively. The term \(s^{i^*j^*}\) is the percent of donations removed by testing when only test \(j^*\) is used in donor group \(i^*\) and is computed as \(s^{i^*j^*} = \det \text{diag} [(\textbf{q}_{*j^*}(1 - \textbf{p}_{*i^*}) + (1-\textbf{r}_{*j^*}) \textbf{p}_{*i^*} )]\).

Using these, we can now calculate \(\gamma^\text{test}_{ji}\) and \(\gamma^\text{mod}_{li}\):

\[ \begin{align} \gamma^\text{test}_{j, i} (j^*,i^*) &= \textbf{w} + \phi_{j^*} + (1-s^{i^*j^*}) ((\textbf{1} - \textbf{r}_{*j^*}) \circ \textbf{p}_{*i^*} ) \textbf{c} + gs^{i^*j^*} - (\textbf{w} + \textbf{p}_{*i^*} \textbf{c})\\ &= \phi_{j^*} + gs^{i^*j^*} + (1-s^{i^*j^*}) ((\textbf{1} - \textbf{r}_{*j^*}) \circ \textbf{p}_{*i^*} ) \textbf{c} - \textbf{p}_{*i^*} \textbf{c}\\ &=\phi_{j^*} + gs^{i^*j^*} - (1-s^{i^*j^*}) (\textbf{r}_{*j^*} \circ \textbf{p}_{*i^*} ) \textbf{c}\\ \gamma^\text{mod}_{li} (l^*,i^*) &= w + \phi_{l^*} + ( \textbf{h}_{{*l^*}} \circ \textbf{p}_{*i^*})\textbf{c} - (\textbf{w} + \textbf{p}_{*i^*} \textbf{c})\\ &= \phi_{l^*} - ((\textbf{1} - \textbf{h}_{{*l^*}}) \circ \textbf{p}_{*i^*})\textbf{c} \end{align} \]

In some situations, a policymaker needs to decide whether to pursue development of a novel test or modification (for example, this was the case when the nucleic acid test for Zika was developed in 2016). One can determine whether it will ever be cost-effective to develop such a test or modification by making pessimistic assumptions about the risk of disease in the donor populations and optimistic assumptions about the efficacy and costs of an intervention. For a given test (with index \(\hat{j}\)) or a given modification (with index \(\hat{l}\)) that influences risk for only one disease (with index \(\hat{k}\)), the following is a necessary but not sufficient condition for use of the intervention to be preferred to using no intervention:

\[ \begin{align} \boldsymbol{\phi}_\hat{j} < \textbf{c}_\hat{k} \max_i [p_{i \hat{k}}] \quad &\text{for tests,}\\ \boldsymbol{\psi}_\hat{l} < \textbf{c}_\hat{k} \max_i [p_{i \hat{k}}] \quad &\text{for modifications.} \end{align} \]

The quantity \(\textbf{c}_\hat{k} \max_i [p_{i \hat{k}}]\) is the value of the intervention assuming it eliminates all risk of infection for disease \(\hat{k}\) in the donor group with the highest risk, without incurring any additional costs (e.g., replacing donations that test positive). If the per-donation cost of a test (\(\boldsymbol{\phi}_\hat{j}\)) or modification (\(\boldsymbol{\psi}_\hat{l}\)) is not below that quantity, the intervention can be removed from consideration.

For interventions that reduce the risk for multiple diseases we can use a generalization requiring for the inequality to hold for all donor groups (unless the analyst identifies one donor group which has the highest risk for all diseases influenced by the intervention). The conditions for interventions influencing risk for multiple diseases are:

\[ \begin{align} \boldsymbol{\phi}_{\hat{j}} < [\textbf{P}(\textbf{c} \circ \boldsymbol{\omega}_{\hat{j}})]_i \quad \forall i \quad &\text{for tests,}\\ \boldsymbol{\psi}_{\hat{l}} < [\textbf{P}(\textbf{c} \circ \boldsymbol{\omega}_{\hat{l}})]_i \quad \forall i \quad &\text{for modifications.} \end{align} \] where \(\boldsymbol{\omega}_{\hat{j}} = \textbf{1}_{r_{\hat{j}k > 0}}\) indicates the diseases a given test can detect and \(\boldsymbol{\omega}_{\hat{l}} = \textbf{1}_{h_{\hat{l}k > 0}}\) indicates the diseases for which the modification reduces risk.

While these conditions were derived for eliminating interventions from consideration with the optimal portfolio model, they can be applied on their own to eliminate tests and modifications from consideration.