An interactive explainer
Malaria interventions are rarely deployed on their own. A new tool is usually added to a setting that already has others in place. Because the interventions act on a shared and non-linear pathway to clinical cases, their effects do not simply add up. The impact credited to any one intervention, and so its apparent cost-effectiveness, then depends on the order in which the interventions' impacts are considered.
When two or more malaria interventions overlap multiplicatively, each removing a fraction of the cases the others leave, the reduction they achieve jointly is smaller than the sum of the reductions each achieves on its own. A case prevented by one tool cannot be prevented again by another, and each tool acts on the cases that remain after the others. Modelled evaluations of intervention combinations show this diminishing return: adding a further tool to an already well-controlled setting averts fewer cases than the same tool would avert on its own.2
Cost-effectiveness is usually assessed at the margin: an intervention is credited with the cases it averts over and above what is already in place. This incremental approach is standard and, for the decision it is built for, appropriate.3 But because combined impact saturates, the incremental impact of a given intervention is not a fixed property of that intervention. It depends on the backdrop it is measured against.
Impact is not linear with effort. If each unit of control averted the same number of cases, the order of counting would not matter. But where the interventions overlap multiplicatively, each acting on the cases the others leave, impact saturates: an intervention assessed first acts against the full burden and is credited with greater impact than if it were assessed later, acting against a shrinking remainder of the burden. The interventions together avert the same total whichever way they are counted; what changes is how much of that total is assigned to each.
This direction holds when the interventions overlap multiplicatively, each acting on a shared pool of remaining cases. It is not universal. Where two or more interventions reduce transmission itself, successive reductions move the setting along the saturating curve from transmission intensity to clinical cases (the subject of the non-linearities explainer). At high baseline transmission the setting starts on the plateau of that curve, where the first reduction averts little and a later one, acting on a steeper part, can avert more: combined impact then exceeds the sum of the separate impacts, and the intervention counted first is credited with less, not more. The toy model below has a single transmission-reducing intervention, so it does not encounter this reversal.
A toy model of cases averted by three combined controls
Three interventions, A, B and C, act together. Their effect sizes are fixed and their costs are equal here; the only thing to change is the order in which they are introduced. Set the transmission level, then reorder the interventions. The staircase shows the cases averted credited to each control as it is added, and the combined total at the top does not move when only the order changes.
A reduces transmission and also gives some direct protection; B and C give direct protection only. Their effect sizes are fixed; all three cost the same, so the cost per case averted depends only on how many cases each is credited with.
| Intervention | Reduces transmission | Direct protection | Cost |
|---|---|---|---|
| A | yes | yes | $1 |
| B | no | yes | $1 |
| C | no | yes | $1 |
B and C are both direct protection but differ in strength; the effect sizes shown are illustrative.
Cases averted, credited as each control is added
Each coloured step is the impact credited to that control when it is added on top of the ones before it. The top of the staircase is the combined total, which does not change when the controls are reordered.
Cost per case averted, in this order (fixed scale)
Every intervention costs the same, so the one credited with fewer cases has the higher cost per case averted.
A new intervention is almost always evaluated as an addition to the existing package, so it is credited with only the cases it averts on top of that package. Where control is already good, that residual is small, and the cost per case averted of a genuinely effective new tool can look high. The same tool assessed against an untreated setting, as a first intervention, would look far more cost-effective. Neither figure is wrong; they answer different questions.
Two consequences follow. First, a newer intervention is systematically credited against a fuller backdrop than an older one, so a like-for-like ranking of their cost per case averted can favour whichever was assessed first. In the toy, the same control is credited with far fewer cases when introduced last than when assessed on its own, so at equal cost it looks much less cost-effective on top of the others. Second, cost-effectiveness estimates carried across settings or studies are only comparable when the backdrop is the same. The backdrop is part of the estimate, not a detail outside it.
The same arithmetic applies wherever a shared impact is split, not only across tools. Where funding is fragmented across separate funders or programmes, each paying for part of the same control effort, attributing impact to any one funder meets the identical problem: a funder's credited impact depends on what the others fund and on the order the contributions are counted, so the same contribution can look highly cost-effective assessed on its own and marginal assessed on top of the rest. A fixed joint impact cannot be divided cleanly between siloed funders; the same choices, and caveats, apply as for individual interventions.
There is no single correct way to divide credit between combined interventions; the appropriate choice depends on the question being asked. Some of the approaches in use:
The total is fixed; only the split moves. Change the order control and the combined cases averted at the top of the staircase does not change. What changes is how that fixed total is divided between the three controls. Crediting one more always credits the others less.
In this toy, going first is worth more than going last. Each control is credited with more the earlier it is introduced, because it is then measured against a fuller burden; introduced later, it is measured only against what the others have already removed. The same control can be credited several times more when it goes first than when it goes last. Which control has the largest step also depends on how strong each is, not on position alone: push the transmission slider to its highest, where reducing transmission averts little, and a direct-protection control added later can be credited more than A added first.
Equal cost, unequal cost-effectiveness. All three interventions cost the same, so their cost per case averted is set entirely by the cases they are credited with. The same control assessed later is credited with fewer cases, so its cost per case averted rises, though nothing about it has changed.
Cost-effectiveness comparisons need a shared backdrop. Because the backdrop sets the incremental impact, cost per case averted is only comparable between interventions measured against the same starting point. A ranking that mixes backdrops can mislead.
Methods. A deliberately simple, illustrative toy model showing the shape of how attributed impact and cost-effectiveness depend on the order in which combined interventions are counted; not a specific setting, and not to be used for decision making. The intervention effects and costs are fixed, arbitrary, illustrative choices.
References.