An interactive explainer
This explainer examines several non-linearities in malaria transmission and control.
Insecticide-treated nets protect users directly by blocking infectious bites, and protect users and non-users indirectly by lowering mosquito survival and density and shrinking the infectious reservoir.
Direct protection scales with personal use; the indirect effect scales with community coverage and depends on vector behaviour, insecticide efficacy, resistance, net durability and spatial clustering. Pyrethroid resistance erodes the insecticidal benefits but leaves the physical barrier intact.
Ross and Macdonald's model expresses the basic reproduction number R₀, the average number of new human infections caused by one infected person dropped into a fully susceptible population. If R₀ > 1 malaria spreads; if R₀ < 1 it is eliminated.
A treated net acts on the two highlighted terms at once: it reduces bites (lowering the biting rate a) and kills mosquitoes (lowering daily survival p). Biting enters squared (a2) and survival as pn, so a modest change in either produces a much larger change in transmission. Baseline transmission also spans a wide range: estimated R₀ runs from about 1 to over 3,000 across African settings1.
Mosquitoes still alive each day. Only the shaded tail past day 10 can transmit, and lowering daily survival shrinks it sharply.
When a mosquito bites an infected person it does not become infectious straight away. The parasite must develop inside it for about n ≈ 10 days: the extrinsic incubation period. Only a mosquito that survives those 10 days can transmit. Lifespans are roughly exponential: each day a mosquito survives with probability p, so its chance of still being alive after n days is pn. Because that is a power, a small change in daily survival is amplified substantially over 10 days.
Lowering daily survival from 0.90 to 0.80 reduces survival to day 10 from about 35% to 11%, roughly a three-fold reduction. This exponential sensitivity is the source of the non-linear bed-net effect on transmission, and it carries through each subsequent step.
The relationship between the entomological inoculation rate (EIR) and parasite prevalence is strongly non-linear. At low EIR, small changes produce large changes in prevalence; at high EIR, PfPR saturates, so large differences in EIR map to small differences in prevalence. Cross-site analyses find PfPR tracks roughly the logarithm of annual EIR5: some sites with annual EIR below 5 still exceed 40% PfPR, while high-EIR sites cluster near a ceiling.
Two features sustain this. Biting is heterogeneous (a minority of people receive most infectious bites), and infections last around six months, so prevalence integrates exposure over time rather than tracking it instantly6.
A minimal prevalence model captures the shape (the curve plotted in the model below is the fuller Griffin equilibrium8,9, which reproduces the same saturation through the mechanisms noted here):
where λ is the force of infection (related to EIR) and r the clearance rate. At low λ prevalence rises sharply; at high λ most people are already infected, so it approaches a ceiling. In practice the ceiling is reinforced by superinfection (a further bite does not change a positive survey), long infection duration, heterogeneous biting, and immunity effects on parasite density and detectability.
The interactive model below reads clinical incidence from the same Griffin-model equilibrium8,9 that produces prevalence, so the two curves are internally consistent. The PfPR–incidence relationship has been characterised in detail from field data and models.7 Incidence data are sparser and less standardised than prevalence surveys, and the relationship depends on age and immunity rather than being a single universal curve.
At low transmission, incidence scales roughly linearly with prevalence. As transmission rises, prevalence saturates while clinical episodes continue through repeated infection and superinfection, producing convexity, most strongly in young children. Older children and adults, with years of cumulative exposure, acquire immunity to clinical disease, so their incidence plateaus at a low level as transmission rises, while young children (not yet protected) keep accumulating episodes. The age pattern shifts with it: below about 10% PfPR much of the burden falls on adults; at higher transmission it concentrates in children. After a recent decline in transmission, young children carry a larger share, because older groups retain immunity acquired under previously higher transmission.
The driver is exposure-acquired immunity. Clinical incidence decomposes as:
PfPR reflects the proportion infected; clinical incidence depends on how often people are infected and how often infection becomes febrile. The second term falls with accumulated exposure and is age- and history-dependent: a setting that has long sat at 30% PfPR differs from one that has just fallen from 70%. PfPR is an equilibrium marker of infection; clinical incidence is a dynamic outcome shaped by recent exposure.
A toy model to trace the impact of non-linearities
Set the no-net baseline transmission, then adjust coverage. The top row is the general relationship between transmission, prevalence and clinical incidence; the bottom row is the specific bed-net cascade, where higher coverage lowers EIR and moves the operating point along those relationships.
General relationships — how transmission intensity maps to prevalence and to clinical disease (independent of the intervention)
1 PfPR vs EIR
Parasite prevalence rises steeply at low EIR then plateaus. Line: Griffin-model equilibrium.8,9 Points: field observations (PfPR in children <15).6 Log EIR axis.
2 Incidence vs EIR
All-age clinical incidence against EIR. It rises then plateaus as prevalence saturates. Log EIR axis.
— all ages ▮ under-5 ▮ 5–15y ▮ >15y
3 Incidence vs PfPR
The same incidence plotted against prevalence rather than EIR. The shaded bands split the total by age group.
— all ages ▮ under-5 ▮ 5–15y ▮ >15y
Bed-net cascade — the specific intervention: coverage lowers EIR, which moves prevalence and incidence along the curves above
4 EIR vs coverage
Direct protection lowers the biting rate (a2) and insecticidal killing reduces mosquito survival (pn), so EIR falls more than proportionally.
5 Incidence vs coverage
All-age clinical incidence against bed-net coverage (the full cascade). The shaded bands show the age split shifting as transmission falls.
— all ages ▮ under-5 ▮ 5–15y ▮ >15y
Trial benchmark: the Cochrane ITN review found roughly 50% fewer uncomplicated episodes in children versus no nets10. The model's under-5 readout runs higher than this across the plotted range (about 80–95% at 80% coverage), because it assumes an idealised net (full potency, no resistance, decay or residual transmission) and reports a per-child incidence-rate reduction rather than a measured trial-arm effect. The two are not directly comparable.
The cascade composes equilibrium relationships: it shows roughly where a setting would settle at each coverage level, not the dynamic trajectory after nets are deployed. The PfPR → incidence step in particular reflects long-run immunity and exposure history, so "moving along the curve" with coverage is a simplification, not a post-intervention prediction.
Each step is non-linear in a different way, and they compound. The EIR–coverage curve falls steeply then flattens (biting enters squared, survival exponentially). The PfPR–EIR curve saturates, so the same EIR reduction produces less prevalence change when baseline transmission is high. Clinical incidence carries that saturation through: it rises with EIR then plateaus, and its age composition (the shaded bands) shifts from adults towards young children as transmission rises.
Baseline transmission sets the response. At high baseline EIR the operating point sits on the PfPR plateau, so a large transmission reduction changes prevalence only slightly. Clinical incidence is also less responsive at high transmission, but not to the same degree: young-child incidence can still fall appreciably even when prevalence barely shifts, because episodes accumulate through repeated infection. At low baseline EIR the same coverage produces a much larger prevalence and case response. The proportional effect of a given coverage therefore increases as transmission falls.
A small prevalence change is not the same as no effect. At high transmission a real reduction in biting can be obscured by a saturated prevalence figure while still lowering clinical disease. The conclusion depends on which outcome is measured.
Methods. An illustrative model showing the shape of these relationships, not a specific setting. Not to be used for decision making. The no-net baseline EIR is set by the slider and coverage is capped at 80%.
validation/validate_cascade.R.References.