1475-2875-10-378-S7.PDF [article]

Additional file A7. Modelling PfEIR and PfR c from PfPR A7.1 Background and introduction An empirical relationship exists between the PfPR and the PfEIR; Beier et al. first assembled a dataset of paired observations and analyzed it [1] . Later, that dataset was expanded and reanalyzed by Hay et al. [2] . Both studies used linear regression to fit the log transformed PfEIR to the PfPR. Smith et al. then compared the fits of the log-linear model with various models of transmission [3] , and found
more » ... ion [3] , and found that a simple extension of the Ross-Macdonald model fit the data better than a log-linear model. The transmission model utilized an assumption from a model formulated during the Garki Project about parasite clearance when a person could be superinfected [4] , and combined it with a model of heterogeneous biting [5, 6] . The fitted parameters in the transmission model had a biological interpretation, and the fitted values were roughly consistent with directly observed estimates of the parameters: the waiting time to clear simple infections in relation to the efficiency of transmission by mosquitoes was close to observed or presumed values, and the distribution of biting rates followed a proposed Pareto rule in which 20% of the population gets 80% of the bites. This model was then analyzed to estimate the PfR 0 from 121 PfPR-PfEIR pairs. This relationship has since been used as a basis for mapping the PfR 0 [7-9]; the proposed relationship is based mainly on an a priori relationship between the PfEIR and the vectorial capacity. In fact, the underlying data reflect varying levels of vector control, so the reproductive numbers are referred to generically as PfR. The relationship has never been used to map the PfEIR, and there has been no attempt to formally quantify the uncertainty. Filion et al. raised questions about the analysis of PfPR-PfEIR relationships because of apparent extra-binomial variation; they also noted that the same kinds of empirical relationships would occur if immunity blocked some infections [10] . Smith has meanwhile argued that it would be unwise to place much confidence in the estimates of heterogeneous biting based on this sort of analysis [11] . Some additional insights arise from an analysis of the empirical relationship between the PfEIR and the P. falciparum force of infection, PfFOI; the PfFOI is much lower than predicted by estimates of the PfPR [11, 12] . Re-analysis of one particularly rich dataset suggests that immunity is not the reason why transmission appears to be highly inefficient at high transmission intensity [12] ; other studies have concluded the same thing [13] . Re-analysis of multiple datasets suggests the apparent slowing is consistent with a model in which approximately 20% of the population gets 80% of the bites, but where the degree of heterogeneity is highly variable [11, 12] . These large effects are probably masked by other large sources of error. 2 A simple and biologically plausible model with heterogeneous biting and superinfection can thus explain the main patterns in paired estimates of the PfEIR and the PfPR as well as paired estimates of the PfEIR and the PfFOI across the transmission spectrum [11, 12] . This sort of evidence should not be interpreted as a kind of proof that the model is "correct" -indeed, there may be other models that explain the data equally well -but models that are approximately consistent with the data would have to explain these patterns and would thus give similar answers. Immunity could be responsible for some additional suppression of transmission from humans to mosquitoes, for example. The transmission model with heterogeneous biting and superinfection, however, is biologically plausible, consistent with the existing patterns in PfEIR-PfPR and PfEIR-PfFOI data, and sufficiently well grounded to serve as a basis for analyzing transmission for the purposes of vector-based control across the spectrum using endemicity data. In the analysis of the original transmission models, Smith et al. introduced a correction for age using the minimum and maximum age of the sampled human population [3] . Since then, an algorithm developed on 21 highly age-stratified PfPR sets and validated on an independent set demonstrated that there is, in fact, a great deal of structure in age-PfPR data and, thus, that agecorrection is useful [14] . The PfPR in children who are at least 2 years old but younger than 10 has, moreover, been used to stratify transmission for at least fifty years [15] . Here, the PfPR -PfEIR relationship has been revisited in light of recent research, and the transmission model has been re-evaluated and used as a basis for estimating PfR 0 for the purposes of informing control. New estimates of the transmission parameters have meanwhile appeared [12], as well as the age-correction algorithm [14] . There is, therefore, a reason to update the original analysis [3, 16, 7, 8] in the service of mapping transmission at a global scale. The first step in generating maps of the PfEIR was to revisit and update the paired PfEIR and PfPR data, and use these to reconsider the empirical relationship between the PfEIR and the PfPR. The second step was to revisit the model and parameter estimates. The third step was to revisit the algorithm that generates estimates of the PfR from the PfPR. Upon revisiting this analysis once again, it is worth noting that the purpose of the study is to investigate the inverse relationship: given an estimate of the PfPR, what is the best estimate of the PfEIR? It is also worth noting that the causal relationship between the PfPR and the PfEIR is not symmetric. The PfEIR is, a priori, an important causal factor in determining the PfPR, and the PfPR is, a priori, in the causal pathway for the P. falciparum sporozoite rate (PfSR), the fraction of mosquitoes with sporozoites in their salivary glands. The PfEIR, however is the product of the sporozoite rate and the human biting rate (HBR), the number of bites by vectors, per person, per day. The PfPR is not a priori a factor that directly affects the HBR; if all the parasites could be instantaneously obliterated, mosquito ecology and blood feeding behaviour would be essentially unchanged. Moreover, the HBR can fluctuate rapidly driven by the rapid generation times and high reproductive capacity of mosquitoes. The age structure of fluctuating (A7.1) The population is stratified by the average rate of exposure, such that exposure in stratum is , and the prevalence of infection in that stratum is denoted . The dummy variable is called a biting weight. Biting weights are assumed to follow a one-parameter family of gamma distributions with mean one; the distribution is specified by one free parameter, , called the index of heterogeneous biting:    
doi:10.7916/d8xs65wb fatcat:rsdvdyu675dshgkefjwk27qiwa