Measuring malaria endemicity from intense to interrupted transmission

Summary The quantification of malaria transmission for the classification of malaria risk has long been a concern for epidemiologists. During the era of the Global Malaria Eradication Programme, measurements of malaria endemicity were institutionalised by their incorporation into rules outlining defined action points for malaria control programmes. We review the historical development of these indices and their contemporary relevance. This is at a time when many malaria-endemic countries are scaling-up their malaria control activities and reconsidering their prospects for elimination. These considerations are also important to an international community that has recently been challenged to revaluate the prospects for malaria eradication.


Mathematical modelling
For more than a century of malaria control, the modelling framework has been central to devising epidemiologically informed intervention strategies and the means by which to measure their impact. 1,2 George Macdonald (1903-67), more than any other malariologist, used models to influence policy during the Global Malaria Eradication Programme, 3 which led to their further development and testing. [4][5][6][7][8][9][10] The models have since been further refined 9,[11][12][13][14][15] and predict relations between malaria metrics that have implications for the classification of malaria risk. These implications are explored below.
The basic Ross-Macdonald model of malaria transmission 4,16-19 using the notation commonly used by malariologists is as follows: where m is the ratio of anophelines to human beings (number of anophelines/number of human beings), a the human biting rate (number of bites on a human being per anopheline per day), b the transmission efficiency of infected anopheline to human being, c the transmission efficiency of infected human being to anopheline, p the proportion of anophelines surviving 1 day, n the duration of sporogony (days)-the process of parasite development occurring in the anopheles mosquito that follows sexual union of gametes and ends with the formation of infective sporozoites-and r the rate of recovery of the human being from infection (days), so that 1/r is the human infectious period. Put together correctly, these parameters define the classic, steady-state version of the basic reproductive number for malaria, R 0 , the expected number of hosts who would be infected by a single infectious person, introduced into an otherwise naive population, after one generation of the parasite. The R 0 is thus a threshold concept; if <1 a disease will die out and if >1 it will increase. The greater the positive value the more rapid its rate of increase. The R 0 concept has been used to help classify malaria endemicity 20 and this scheme has additionally been incorporated into figure 1 and figure 2 of the main text. These models are applicable to all malaria and generalised to a hypothetical anopheles vector. They are considered specifically with respect to Plasmodium falciparum transmission below.
Previous work [13][14][15] has shown that the Ross-Macdonald model predicts well-defined three-way relations between PfPR (P falciparum parasite rate), PfEIR (ε) (P falciparum entomological inoculation rate), and PfR 0 (P falciparum basic reproductive number) at the steady state: All notation is the same as in the first Ross-Macdonald equation, but for simplicity we express the stability index, s, rather than a/-lnp. If a control programme reduces PfR 0 uniformly by a factor, F, then P falciparum malaria will be eliminated everywhere that R 0 <F, or where (note the formulae give daily ε if the units of r are days): For example, if F=10 (a 90% reduction in transmission), then malaria will be eliminated wherever PfPR<80%, or equivalently a PfEIR<10 (assuming b=0·8, 1/r=200, c=0·5, and s=1). Obviously, these formulae ignore imported malaria, heterogeneous biting, and many other factors known to be of importance for malaria transmission but are nevertheless useful to structure thinking.

Mathematical models and the relations between malaria metrics
The prediction of the ubiquitous Ross-Macdonald formulations 4, [16][17][18][19] can be compared graphically with more recent revisions 14 to show the relation between the hostbased measures of prevalence (the PfPR) and the vectorbased PfEIR, the number of P falciparum infective bites per human being, per unit time (webfigure 1). 21,22 The relation between PfPR and PfEIR predicted by the Ross-Macdonald theory shows that PfPR is very sensitive to small increases in PfEIR at low transmission intensity and insensitive where transmission is high (webfigure 1). The PfPR is therefore an excellent measure of malaria endemicity before the relation saturates at holoendemic transmission levels, where theoretically, PfEIR would be a more valuable guide, although for mainly logistic reasons (see main text) it is infrequently sampled. 21,22 This PfPR-PfEIR relation has also been corroborated empirically 21 reveal the importance of incorporating heterogeneity into malaria transmission models. 14,15 Heterogeneity in transmission (irrespective of cause) intensifies the PfPR-PfEIR relation at low endemicity (since bites are more focused on infected and infectious individuals: superspreaders), but this effect becomes increasingly moderated with higher prevalence (these same bites are concentrated on already infected individuals: superabsorbers; webfigure 1). There is little argument therefore that malariometric surveys remain an appropriate choice for assessing the impact of control measures. It is also clear that, under both modelling frameworks, any value of PfPR≤10% is significantly below a PfEIR of one infectious bite per year (webfigure 1) and thus constitutes a low and operationally uniform transmission stratum. The decision about when to stop malariometric surveys and move to surveillance is not informed by theory but is guided mainly by the point where the sample sizes required become operationally prohibitive (see main text).
It is possible to examine the modelled expectation of the relation between PfPR and the basic reproductive number, R 0 , for P falciparum malaria, in the same way that we have detailed the relation between PfPR and PfEIR (webfigure 2). The PfR 0 is the perfect metric for evaluating control and elimination feasibility, because it was derived from theory for specifically that purpose 26 (see above) and is discussed in the following section.

Mathematical models and the feasibility of control
Since the time of the Global Malaria Eradication Programme, malaria control programmes have become increasingly plural in the range of interventions that they consider 27 and are generally more cognisant of the fact that their optimal mix should be stratified by ecological, entomological, and sociological settings. [28][29][30] Despite the diversity of interventions available, those which are routinely taken to scale are few and the deployment of insecticide-treated bednets is currently the most widespread. [31][32][33][34][35][36] It has been shown by many field trials 37 and recent theory 38 that a ten-fold reduction in PfEIR can be achieved with a 45-75% coverage of bednets (ownership multiplied by use), depending on the local vector species. What effect might this order of reduction have on our malaria metrics and how does this relate to classically defined endemicity levels?
The immediate impact of insecticide-treated bednets on PfR 0 are directly proportional to the immediate reductions in PfEIR and to the baseline PfPR. PfR 0 is directly proportional to vectorial capacity, V. Mathematical theory predicts that PfEIR will change in two phases after starting vector control: dPfR 0 / dt≈dPfEIR/dt=dV/dt*PfPR+VdPfPR/dt. 38 The largest and most immediate reductions in PfEIR are caused by reductions in the vector populations and changes in their age composition, as measured by vectorial capacity: but on fast time scales, dPfPR/dt≈0. Thus, the immediate changes in PfEIR are caused by changes in vectorial capacity scaled by PfPR, which translate directly into changes in PfR 0 . Later, PfEIR will fall even further because of feedback with PfEIR reducing the proportion of people who are infectious to mosquitoes. In places where PfR 0 is high, the slower changes are also small because PfPR is relatively insensitive to the initial changes in PfEIR. Where PfR 0 is low enough and transmission is interrupted by vector control, elimination may follow after a few years. These principles apply to any sustained reductions in vector populations achieved through vector control, whether through insecticide-treated bednets, indoor residual spraying, or some other means. The actual reductions achieved through vector control vary, depending on the vector and the coverage achieved. 38   mark the prevalence levels that transition between the canonical endemicity classes. 23 The dotted lines (red) show the PfPR level that can be completely controlled with a ten-fold reduction in PfR 0 . These relations are modelled simplifications because control effects are not instantaneous and will take time to manifest as changes in community prevalence caused by the duration of P falciparum infections 25 and other factors (see text), but they are useful guides.
insecticide-treated bednets and indoor residual spraying have achieved factor of ten reductions, 39,40 although much larger decreases were achieved during the Global Malaria Eradication Programme. [41][42][43][44] The first important observation is that the classic PfPRbased divisions of malaria endemicity 23 have no clear relation with PfR 0 and thus to the feasibility of control or elimination. It is clear from these considerations that there is a crucial need to investigate retrospectively, through the plethora of documented control and "eradication" attempts, the performance of mathematical models at predicting the success of interventions in highly endemic areas. Second, the Ross-Macdonald models show that almost the full PfPR prevalence spectrum (0-90%) is spanned in a range of R 0 <10 (webfigure 2). The theoretical inferences one might draw from these models on the ease of control are difficult to reconcile with historical examples of very determined, comprehensive, and mixed intervention strategies that did not achieve an interruption of transmission in highly endemic areas. [41][42][43][44] In other words, it seems unlikely that such vigorous control didn't achieve a ten-fold reduction in PfR 0 , which the Ross-Macdonald models predict should end transmission even at holoendemic levels. The more recent Smith models 14 that include transmission heterogeneity indicate that the R 0 range is more plausibly extended across the PfPR transmission spectrum, so that holoendemic malaria with PfPR≥75% equates to PfR 0 ≥150 (webfigure 2). The differences predicted by these models obviously impact considerably on any evaluation of the feasibility of malaria control by endemic level. Even if the more conservative modern framework 14 is adopted and a ten-fold reduction in PfR 0 assumed, all populations with a natural baseline PfPR≤40% would be able to eliminate malaria by the scaling-up of insecticide-treated bednets (webfigure 2).
These mathematical models therefore offer further scope for guided hypotheses on the feasibility of control and/or elimination, when combined with additional evidence on the impact of existing interventions. These are expanded further in the main text.