Predicting changing malaria risk after expanded insecticide-treated net coverage in Africa

The Roll Back Malaria (RBM) partnership has established goals for protecting vulnerable populations with locally appropriate vector control. In many places, these goals will be achieved by the mass distribution of insecticide treated bednets (ITNs). Mathematical models can forecast an ITN-driven realignment of malaria endemicity, defined by the Plasmodium falciparum parasite rate (PfPR) in children, to predict PfPR endpoints and appropriate program timelines for this change in Africa. The relative ease of measuring PfPR and its widespread use make it particularly suitable for monitoring and evaluation. This theory provides a method for context-dependent evaluation of ITN programs and a basis for setting rational ITN coverage targets over the next decade.

Given estimates of these six parameters, we can predict the effect size achieved by ITNs as they scale up over time, φ(t). Four species have been characterized 1,5 . The benchmark results depend on one particular set of parameters and the parameters for one vector (see Table S1). That vector was chosen as the benchmark because it was an important African vector, and also because the effect size was approximately equal to the geometric mean response of two An.
gambiae species from different places (Figure 1b). The assessment of uncertainty is based on the distributions in Table S1, and the associated variability in effect size is illustrated in Figure S1.

Malaria Transmission Model
The following summarizes previously published descriptions and analysis of a malaria transmission model [2][3][4]6 . The notation and parameters are described in those publications, and tables of the terms and parameters are described in Box S1 and Box S2.
Steady States: The following model describes a population where PfPR is stratified by exposure. PfEIR (denoted E in equations) defines the population average exposure, but each population stratum has a distribution of biting weights, ω, such that the malaria exposure within a stratum is ωE. Clearance follows the assumption of Dietz,et al. 7 . The prevalence of malaria in a risk stratum is X ω , and the dynamics are described by the equation: It is assumed that ω is Gamma distributed with a mean of 1 and a variance α, so that exposure is also Gamma distributed with a mean E and a squared coefficient of variation of α. The population prevalence integrates over all risk strata: At the steady state:X This equation fits the relationship between PfEIR and PfPR in children for b/r ≈ 0.45/yr and α ≈ 4.2 8 .
The dynamics of infections in mosquitoes generally follow the classical as-sumptions 6 , except that the rate of infection is modified to consider heterogeneous biting 3 . The probability that a mosquito becomes infected after biting a human is defined by the formula: Following Smith,et al. 3 : If no estimate of E is available, then it can be inferred fromX using Eq. 3: which gives a relationship between R 0 andX: The formula is mainly a function of the PfPR and the degree of heterogeneous biting (α). Other parameters that modify the expression are the baseline infectivity c 0 , and the stability index (Box S2).
The benchmark approximation was made by assuming that that c(ωE) = c 0 , so that:X This formula tends to underestimate R 0 if there is transmission blocking immunity. To estimate PfR 0 from PfPR for some assumption about transmission blocking immunity, we use Eq. 6, along with Eq. 7 to getX as a function of onlȳ X, and the atomic parameters b/r (which always appear together) and c 0 = 0.5 and the index S = 1.
The functions can be used to describe changes in the steady state relationships under malaria control. Given a reduction in vectorial capacity as a function of ITN coverage, V (φ), we can predict the change in steady state under malaria control, starting from a population without malaria control, using the formulas: The functions describing the algorithm for predicting the steady state PfPR in the presence and absence of control (X(φ) andX 0 , respectively) are given on top of the directional arrow. The functions can be inverted, and this also makes it possible to predict the changes in PfPR at the steady state starting from one level of malaria control (φ ) and improving (or relaxing) to another φ: R-code is freely available upon request, and a demonstration version is available at http://www.map.ox.ac.uk.
Timelines: To compute the timelines, a different set of equations is required; these have described in detail elsewhere, and are repeated here 4 . Minor differences between the steady state of these equations and of the equation above are introduced because the risk strata are subdivided into a finite number of compartments. The equations are analogous, in the sense that they make the same assumptions about the biology, and they have the same steady states in a limiting case on the mesh of compartments.
Let j subscripts denote a subpopulation with biting weight ω j that comprises a fraction W j of the whole population. Let x m,j denote the fraction of that subpopulation with a given MOI, m. Thus, m x m,j = 1, for all j. The changes in the proportion uninfected within the j th population stratum is: Here, we have taken ρ m = rm, and h j = bE. The parasite rate is defined to be The probability that a mosquito becomes infected after biting a human, denoted X and called net infectivity, is given by the formula: The dynamic of infections in mosquitoes follow the same logic as above 3 , such that PfEIR is given by the equation: The biting weights were given by a Gamma-like distribution, as described elsewhere 4 . Simulated timelines were produced by letting the equilibrium come to a specified equilibrium determined by the vectorial capacity, then instituting control. ITN coverage was simulated by changing the vectorial capacity, according to a model.

A. Sensitivity Analysis
Variability in the ITN effect size is realted to vector bionomics, as described elsewhere 1 . Another source of heterogeneity in the outcome the translation from PfPR back to PfEIR, which is largely ascribed to differences in the degree of biting heterogeneity. Taken together, baseline endemicity and uncertainty about heterogeneous biting, immunity, and vector bionomics suggest highly unpredictable endpoints after reaching universal coverage, as prescribed by RBM. Monitoring and evaluation across the transmission spectrum and across the range of dominant vector species should aim to establish context-specific expectations and goals. Figure 2b is based on the parameter values and distributions reported in Table S1, we have also generated a predicted effect size distribution ( Figure S1).
The relationship between the PfPR, the PfEIR, and the PfR 0 is strongly affected by the degree of heterogeneous biting, described by α 4, 9 , and illustrated in Figure 1a With variability in vector bionomics and heterogeneous biting, the variable outcome is shown in Figure S3. Differences between the benchmark prediction and some actual outcome can also occur because of other reasons: Our analysis also assumes that there has been no major change in any other important factors, such as drought or contemporary changes in other modes of malaria control, such as a change in drug policy. If there has been a change, these must be taken into account when assessing the effects that are attributable to ITNs.

Table S3
We simulated changes in PfPR for an ITN program that scaled-up ITN effective coverage from 0 to a specified target level at the end of five years. We extended the simulations so that after five years, the maximum level of ITN effective coverage was sustained indefinitely. Each column represents a different target level of ITN coverage, and each row represents a different baseline PfPR in 5% increments. The numbers report the number of years elapsed, counting from the start of the program, before PfPR is within 1% of its endpoint.  Figure S1 The variability in the predicted effect size achieved with 60% effective coverage, or 80% coverage and 75% use. These represent 10,000 monte carlo sample from the distributions reported in Table S1, the log effect size is Gamma distributed with shape parameter, a ≈ 13.4 and scale paramater s ≈ 0.14. The variability in outcomes achieved when considering both the variability in heteroeneous biting and effect size, using the distributions reported in Table S1.