Swaziland has set a national goal of eliminating malaria transmission in the very short term, which would make it the first country in sub-Saharan Africa to do so. More than half of the cases of malaria that are observed in Swaziland are caused by infections picked up by travelers while they were in other countries where the disease is much more prevalent. The other cases – people who became infected in Swaziland – are the cases that the government of Swaziland is trying to prevent.

If Swaziland is going to eliminate malaria, it will need to identify any places where the malaria parasites are still spreading throughout the population so it can target those communities with effective prevention measures. It will also need to manage the risk that infections imported from abroad may re-start transmission in places where it has been stopped.

To work out how likely it is that a malaria infection will be transmitted by mosquitoes in a particular place, researchers can look at past malaria data and calculate how many new infections are caused by each case. Reiner et al. have now produced a computer model that estimates how this number varies across Swaziland, highlighting places where the government is going to need to focus efforts to eliminate malaria. The model shows that in some rural areas near Mozambique, each individual infected with malaria is causing more than one other person to become infected. This confirms that the disease has not yet been eliminated from these areas. However, in other regions of the country, malaria rarely spreads between individuals.

The detailed regional information from the model may help public health authorities in Swaziland better target their anti-malaria resources. In large cities where most cases are imported, Reiner et al. suggest focusing resources on providing preventive treatment to travelers who plan on visiting places where malaria is spreading. However, in rural areas where malaria continues to spread, preventively treating the whole population or providing them with tools to protect them from mosquitoes might be more appropriate. Similar considerations of regional differences in the spread of malaria could also help other countries to more effectively combat the disease.

Malaria is a leading cause of childhood morbidity and mortality (Naghavi, 2015), and chronic malaria infections contribute to and complicate diagnosis and treatment of other diseases. Controlling malaria to reduce its burden has long been a top global health priority, and eradication became an official policy of the United Nations in 1955 (Mayo and Brady, 1955). The ensuing Global Malaria Eradication Campaign (GMEP, 1955–1969), directed by the World Health Organization, contributed to a massive, permanent contraction in the geographic range of malaria (Gramiccia et al., 1988; Smith, 2013), and it led to development of concepts and methods for eliminating malaria that remain useful today. A key insight was the need to use different metrics, suited to natural constraints and programmatic needs, to mark progression from endemic malaria to elimination (Hay et al., 2008). Malaria eradication is once again recognized as a global priority, and as of 2011, 36 of 99 malaria endemic countries had plans to eliminate malaria (Cotter, 2011).

Concepts and methodologies developed for the GMEP are being revisited and updated today in light of contemporary challenges and new technologies. Three useful concepts were called vulnerability (i.e., the rate of malaria importation), receptivity (i.e., the potential for ongoing local transmission), and their product, which was called malariogenic potential (i.e., the expected number of cases that could occur as a result of vulnerability and receptivity) (Pampana, 1969). Given that GMEP was predicated on the idea of universal insecticide spraying throughout at-risk areas, these notions were used primarily to evaluate the vigilance required to prevent reestablishment of transmission post-elimination rather than to make decisions about intervention selection or targeting (World Health Organization, 2012). They thus require updating in light of the different assumptions of the present campaigns. Malaria importation rates for African malaria elimination countries are much higher than those for countries that eliminated malaria in past decades due to high mobility and highly endemic neighbors (Tatem, 2010), and receptivity may be greater. Countries have multiple tools at their disposal but must determine how to deploy them optimally to achieve and sustain elimination given constrained resources. These challenges are now being faced by Swaziland, a sub-Saharan country that has made substantial progress towards eliminating malaria. Success would make it the first mainland sub-Saharan country to eliminate malaria, and so the lessons from its national experience will be relevant for the rest of the continent.

Residual transmission and elimination

A key need is for new quantitative methods and practical operational advice to guide the final stages of malaria elimination, when few local cases remain. Elimination programs have repeatedly demonstrated the critical importance of identifying areas where transmission continues in order to make the most of limited resources. The architects of the smallpox eradication strategy, for example, credit the campaign’s ultimate success to a shift from universal to targeted vaccination (Foege et al., 1971). Successful elimination of malaria during the GMEP similarly demonstrated that, in resource-constrained environments, a shift is required away from a focus on universal coverage for endemic malaria towards heightened surveillance, case investigation, identification of areas where transmission risk remains high (i.e., residual transmission foci), and highly targeted interventions (Moonen, 2010). Despite progress (Ja and Hg, 2009), little guidance is available on what specific methods to use where and which metrics are appropriate for measuring progress in today's eliminating countries.

Since 1999, Swaziland’s National Malaria Control Program reports that the incidence of malaria has declined from 2.9 to 0.07 malaria cases per 1,000 people per year (Roll Back Malaria, 2012). Between 2010 and June 2014, Swaziland confirmed only 2,129 malaria cases, with case investigation of 1,524 of them. Of these, 870 (57%) were classified as imported, with the proportion of cases likely having acquired their infection elsewhere increasing since 2010. At a national level, the decreasing ratio of local to imported malaria cases since 2010 is suggestive of an reproductive number under control, R_c, much less than 1 on average, which would mean that elimination of endemic transmission may have already been achieved or is imminent (Churcher et al., 2014).

Analysis of national trends in reporting data provides a useful measure of overall progress but potentially masks local spatial heterogeneity in transmission, and it leaves unanswered the question of how to stratify Swaziland to allocate resources most efficiently within the country. Ideally, programs would focus attention on places and at times where and when the risk of malaria transmission is highest. Programs might direct aggressive interventions, such as focal mass drug administration, towards residual endemic foci where transmission would persist even in the absence of importation, but might opt for less aggressive maintenance of reduced risk in places where transmission is driven only by continual replenishment from imported infections. Determining whether locally acquired malaria infections are the result of endemic transmission or result from transmission chains stemming from importation requires analyzing transmission at an individual level. National risk maps have been developed based on the household locations of local malaria cases (Cohen, 2013) but these maps do not describe how much transmission is likely occurring, only whether or not there is risk of any local infections in a given location as narrowly defined by the location of infected individuals. Furthermore, they do not explain the relative importance of importation as a driver of transmission and thus cannot inform intervention selection.

Improving assessment of progress and making action maps requires developing individual-based assessment of risk to link cases together, determine the magnitude of transmission as measured by R_C, and evaluate where transmission may be occurring endemically versus where it is driven only by ongoing importation. Most of the imported cases have been identified in and around the large cities of Mbabane and Manzini (Figure 1A), but proportionally fewer locally acquired cases were found in these cities (Cohen, 2013). Malaria cases imported into the major cities of Swaziland are likely only responsible for causing a few local cases (i.e., R_c in the large cities is very low). Areas outside the major cities—specifically those in the east and north of Swaziland—appear to have a higher ratio of local to imported cases, suggesting that though on average each malaria case generates less than one other case (i.e. R_c < 1), there are some focal areas where endemic transmission may continue to smolder. Measuring progress and achieving elimination requires properly characterizing and quantifying heterogeneity in this residual transmission. Here, to address these needs, we have defined vulnerability, receptivity, and malariogenic potential in quantitative terms based on a continuous point-process and developed fine-grained maps.

Figure 1

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Mapping receptivity

Elimination programs in the endgame have long been advised to categorize cases as imported, introduced (the result of first degree transmission from an imported infection), or indigenous (the result of second or more degree transmission), yet no method for making this distinction has ever been formally described (Pampana, 1969). Methods for doing so are largely based on case investigations, but such methods could be validated and augmented with genetic and computational methods for linking cases. Understanding which cases are linked to other cases is also important because it allows direct measurement of R_C, an important measure of the need for additional intervention. We developed computational methods for assessing malaria transmission (see Methods) that quantifies the most likely “parent” of each local case. We do not distinguish between potential parents in terms of their status as “imported” or “local”. In this, we are estimating causal links that can be counted in space-time to get measurements of receptivity. Further, these links build transmission chains that can be used to gain a deeper understanding of the spatial variation in R_c. This approach uses an understanding of malaria transmission dynamics to redefine “proximity” of two malaria cases as a generalized probabilistic measure based on distance and time separating two cases to evaluate potential causality. Malaria transmission directly links two individuals through two bites from the same mosquito. The time elapsed between detection of two linked cases will be bound by the serial interval (Fine, 2003) (i.e., the length of a complete malaria transmission cycle), mosquito mortality, and the timing of case detection relative to infection. These aspects of malaria ecology provide probabilistic bookends to temporal proximity, as even with detection (and assuming all detected cases are treated promptly), cases are ever more unlikely to be linked as the chance of a mosquito living long enough to link them diminishes. In space, the simple but common Gaussian diffusion-based approximation of movement approximates the combined spread of mosquitoes and humans, and is governed by a single constant. Given the use of continuous distribution functions, the algorithm was forced to always find a link even if it was inconceivable. As such, we incorporated a minimum threshold that dichotomized links into “plausible” (with a level of plausibility) and “implausible.”

By accounting for potential differences in the distribution of the serial interval for human malaria cases and timing of case identification, multiplying the temporal processes with spatial diffusion, and finally sweeping across a suite of potential diffusion constants (see Methods) we linked local cases to a “most likely” parent case. By combining “most likely” links, we arrived at a single weighted network that represents the consensus linkages (Figure 1). As was suspected by visual inspection of the spatial distribution of the data, most of the imported cases within the major cities of Swaziland did not appear to be responsible for ongoing transmission. However, both imported and locally transmitted cases in the north and east did, infrequently, initiate extended transmission chains. Elsewhere, no identified case was a likely parent, so these cases are classified as “orphans.”

Taking the output of the transmission network, the number of direct ‘offspring’ arising from each case is its estimated R_c. Using zero-inflated negative binomial regression models on a set of ecological, social and demographic covariates, likely values of R_c were extrapolated spatially from observed case locations to all of Swaziland at 100-meters squared resolution (Figure 2A). The resulting map of R_c illustrates the estimated heterogeneous distribution of current malaria receptivity within Swaziland. To the west, R_c is close to 0. Within Mbabane and Manzini, R_c is estimated at 0.08 and 0.12 respectively. Conversely, in the northeast near the Mozambican border, R_c estimates increase up to 1.70. As suspected, while endemic transmission does not appear to occur within the urban centers of Swaziland, it does not yet appear eliminated from the entirety of the country. The smoothed R_c map was found to be statistically significantly different from a flat map (p-value <2e-16) but it remained unclear the statistical significance of each pixel. Further analysis is required to assess the robustness of isolated locations where R_c appears elevated. This will be an important step to appropriately interpret these maps for the purpose of resource allocation given constraints on the number of individual locations that can be visited.

Figure 2

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Malariogenic potential

The potential for local transmission will not result in actual transmission unless malaria parasites are present. As Swaziland successfully extirpates these final foci of endemic transmission, local transmission in the country will increasingly arise only around imported malaria cases. The total number of locally acquired malaria cases in Swaziland is the product of importation (vulnerability) and onward transmission (receptivity), with the ongoing operational challenge of maintaining gains greatest in regions with both higher receptivity (i.e. R_C) and higher vulnerability (i.e., the number of malaria infections imported each year that could seed new transmission): the product of these quantities has been defined as the ‘malariogenic potential’ (Pampana, 1969). Malariogenic potential was mapped as the product of R_c and vulnerability (Figure 2c). The resulting map illustrates where locally acquired/transmitted cases are most likely to occur, and thus where resources may need to be prioritized to prevent reestablishment of malaria given the joint risks of importation and subsequent transmission.

We assessed the stability of our malariogenic potential maps as well as our other output by splitting the data into two halves (before and after July 1, 2012). The malariogenic potential maps for the two halves appear very similar (Figure 3C versus Figure 4C), as are the importation maps (Figure 3B versus 4B). There are some differences in the R_c maps (Figure 3A versus 4A), but both analyses identified regions in the northeast where R_c was larger than 1. Both analyses identified a larger ‘max’ R_c (35.36 for the first half and 3.13 for the second half) as well as a larger percent of the population living in the highest R_c category (R_c>1.4).

Figure 3

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Figure 4

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Occult local transmission

Ideally, every case of malaria in Swaziland would be detected. In reality, case detection is imperfect, and it is likely unnecessary to find every case to create circumstances that lead to elimination. In this analysis, missed cases could matter if they biased the estimates of R_c, depending on whether locally transmitted or imported cases were more likely to be missed. If a case that is missed results in local transmission and if those future cases are captured by surveillance, those future cases may appear to have no plausible cause. The identification of such “orphan” cases can help indicate places where Swaziland must work to implement or strengthen active infection detection. Within this analysis, due to the inclusion of temporal uncertainty to account for potential differences between detection times and time of onset, rare links can be formed between a case and a second case where the second case was picked up at the same time or even later. These rare links would only form if no other case that occurred earlier were close in space-time as judged by the spatio-temporal kernels. In these circumstances, due to the flexibility of the spatio-temporal kernel, a pair of cases could be identified as being the most likely parent of each other. These “loops” identify two orphaned cases that were identified close in time-space to each other but whose actual parent was not captured by surveillance.

Within this analysis, there were 22 pairs of local cases that formed loops (Figure 1A, Figure 1C, red diamonds) as well as a single case that was not linked to any other case across any of the potential parameter sets (Figure 1A, circled red diamond, Figure 1C, red circle). Prioritizing enhanced surveillance in the areas surrounding these orphan chains would help narrow uncertainty about residual transmission within Swaziland.

The majority of these loops occurred during the months with the least transmission (Figure 5), which reflects the decreased chance of any cases being detected in the month prior. The method described here yields estimates of transmissibility that can guide interventions to places where occult transmission is most likely to be happening even in the absence of knowledge of specific infected individuals. Nevertheless, the more complete the surveillance effort, the more accurate these mapping efforts will become.

Figure 5

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Asymptomatic infections are another possible explanation for the orphaned cases. Missing asymptomatic cases could either result in an overestimate or underestimate of R_c (if they are likely parents or likely offspring of other cases). For this analysis, we did not account for asymptomatic or inapparent cases. Although it was not done universally, intensive infection detection around cases resulted in very few additional infections (53/7307 between July 2014 and June 2015) consistent with the assumption that there is not a large pool of unreported infections that would greatly bias our results.

Communicable disease policies require different approaches than policies for non-communicable diseases, as each case presents both medical and public health challenges (Smith et al., 2005). For infectious diseases, reproductive numbers provide a theoretical basis for strategic planning and programmatic evaluation, such as critical vaccine coverage levels and outbreak responses (Anderson and May, 1991). Estimating individual reproductive numbers by linking up malaria infections is a special case of a method that has be used more widely for other diseases (Walker et al., 2010).

Malaria, like other diseases with an environmental component, represents a special challenge because of spatial heterogeneity in the risk of transmission (Bejon et al., 2010; Bousema et al., 2012; Bejon et al., 2014; Bousema et al., 2010). In the end phases of elimination, population-level measures become inefficient and inadequate, so as countries approach the goal of eliminating malaria, individual-based estimates of transmission must identify putative foci where transmission remains high and where resources should be targeted (Hay et al., 2008). Case counts alone do not necessarily convey information about transmission, since many of those cases may have been acquired far from where they were detected. Aggregate ratios of local to imported cases in time (or in space) alone, while more representative of overall progress, could obscure localized transmission if, for example, most cases failed to transmit but some pockets of transmission remained. This analysis, which identifies places and times where cases are most likely to be transmitted, confirms that there has been dramatic progress towards elimination overall, but it also identified substantial heterogeneity in progress within Swaziland.

The Swaziland National Malaria Control Program (NMCP) will need to manage imported malaria as long as endemic transmission continues in neighboring countries, so directing and optimizing limited resources is crucial. Combining assessments of receptivity with assessments of vulnerability provide actionable intelligence to support malaria programs in designing targeted intervention strategies in the most relevant places; for example, the NMCP may consider targeting travelers with prophylaxis in places with high vulnerability, while focal mass drug administration or other aggressive measures might be most appropriate in places with evidence of endemic transmission and low vulnerability. Our approach provides spatiotemporally relevant and resolved metrics of transmission that can be used to identify future cases as either critical or relatively unimportant for overall elimination efforts. Further, and perhaps most important, our approach can be used to stratify future control responses by differentiating between locations where elimination would be a consequence of merely decreasing effective importation versus where elimination of endemic transmission is needed through reduction in local receptivity. Predictions generated by our approach will also be useful as a baseline for in-development genetic testing and molecular typing models (Greenhouse and Smith, 2015), and will remain pertinent as a proxy for such methods in places where resources are limited to enable universal parasite typing. These methods can help Swaziland reassess its needs and remain malaria-free as surrounding countries control transmission and make further progress. Through regional elimination, economic growth, and efficient use of existing resources, malaria elimination can perhaps become as stable in Swaziland as it has been elsewhere (Smith, 2013; Chiyaka et al., 2013).

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Thank you for submitting your work entitled "Mapping residual transmission for malaria elimination" for peer review at eLife. Your submission has been favorably evaluated by Prabhat Jha (Senior editor), Mark Jit (Reviewing editor), and two reviewers, one of whom, John Drake, has agreed to reveal his identity.

The reviewers have discussed the reviews with one another and the Reviewing editor has drafted this decision to help you prepare a revised submission.

The reviewers and the Reviewing editor agreed that your paper takes a unique approach to modelling malaria transmission – that of linking cases with likely transmission pairs or groups. While similar approaches have been taken with viral respiratory illness (and likely other pathogens that we are not aware of) this appears to be the first application to a vector borne disease. It is sophisticated but not overly complicated and represents an important benchmark on the path to developing better analytics to guide and document the success of malaria elimination programs in sub-Saharan Africa. The analysis appears sound and internally consistent.

There are a few issues that we would like you to address, listed below. In addition, we normally require that modelling papers in eLife make their data openly accessible. Please could you tell us of your arrangements for this, or otherwise justify why you are not able to do so.

1) Can any indication of the temporal stability of the map of R_c be given? It is essential to know if the apparently smoldering foci are stable and therefore targets of intervention, or variable and therefore blanket control measures are all that is tractable.

2) We are not clear what view you have of the number of undetected cases or what effect that might have on their map. Presumably it is theoretically possible that all the R_cs are, in fact, several fold higher and case detection frustrates the exercise?

3) How much transmission might be asymptomatic? Swaziland was previously at higher transmission intensity and therefore many adults may be able to tolerate parasitaemia without becoming unwell.

4) When results on pairs of cases are presented in one figure then probabilities are assigned. However elsewhere and in the text the implication is made that a dichotomy has been made between cases that are linked and single cases – how have you moved from continuous probability to a dichotomy?

5) Genetic testing would be a more definitive way of distinguishing linked from unlinked cases. Can you comment on the likely utility of their modelling approach once genetic testing is available?

6) Is the "smoldering transmission" reflective of long-term transmission in the absence of imported cases, or does it require an imported case every so often to keep it smoldering?

7) Fine-grained needs a definition (pixel size? Or is it a point-process with pixelation simply for presentation purposes?).

8) There is a confluent area of high R_c in the North East. In the rest of the country there are occasional dots of high R_c scattered around. What is the possibility that these would have arisen by chance? The model implies very multiple comparisons. If a simulation was carried out where the episodes were randomly distributed through the population then how often would one see a "spike" of red colour in the map just due to chance?

9) If immigrants tend to go to the same places and at the same time (e.g. because they are all being recruited to work in the same factory) then that would increase the likelihood that the two cases would spuriously be associated in time and place despite not having a transmission cycle. Would that source of spurious linkage be accounted for in the migration data that was used?

10) Is there an issue with spatial autocorrelation (Figure 2 and zero inflated negative binomial regression)? Were the errors inspected? (In general, regression diagnostics, including differences in AIC values among models, have not been reported.)

11) What are the units in Tables 1 and 2? Were data centered and scaled prior to fitting GAMs? In my experience, this sometimes improves performance.

[…] There are a few issues that we would like you to address, listed below. In addition, we normally require that modelling papers in eLife make their data openly accessible. Please could you tell us of your arrangements for this, or otherwise justify why you are not able to do so.

1) Can any indication of the temporal stability of the map of R_c be given? It is essential to know if the apparently smoldering foci are stable and therefore targets of intervention, or variable and therefore blanket control measures are all that is tractable.

We have repeated our analysis using the first and second halves of the data (before and after July 1, 2012). There are some differences in the R_c maps, but there are also many similarities. One difficulty in interpreting the “stability” of our estimated R_c maps in this low-transmission setting is that in practice R_c will be based on observed cases. We consider the R_c map analogous to a map of where fires may occur if a match was dropped. If you only have a few matches dropped across the landscape, it will be difficult to know which areas are most flammable. We have added these follow-up analyses and some discussion of the results to the manuscript in the “Malariogenic Potential” section.

One interesting result of the stability analysis was that in both halves of the data, the max R_c by pixel was considerably higher (35.36 and 3.13 in the first and second half, respectively) as well as the percent of the population living in the highest R_c group.

2) We are not clear what view you have of the number of undetected cases or what effect that might have on their map. Presumably it is theoretically possible that all the R_cs are, in fact, several fold higher and case detection frustrates the exercise?

We have added text at the end of the “Occult Local transmission” section to further discuss the impact of undetected cases on our analyses. It is unclear that missing cases would increase R_cs. For example, consider a case (say case A) that appeared to “cause” 2 other cases (say, case B in their village 30 days later and case C in a different village a few km away 45 days later). Case A’s R_c would be estimated as 2. Now, if there was a missing case (say case D) in that other village that occurred 30 days before case C (15 days after case A), then we would say that A “caused” B, D “caused” C and both R_cs (A’s and D’s) would be 1. Missing offspring would result in increased R_cs but missing parents would result in decreased R_c.

3) How much transmission might be asymptomatic? Swaziland was previously at higher transmission intensity and therefore many adults may be able to tolerate parasitaemia without becoming unwell.

In response to the previous reviewers’ comment, we added text discussing asymptomatic. One part of this text directly addresses this question. In several circumstances active case detection was conducted around a case to assess the level of transmission and found almost no additional infections (53/7307 in 2014-2015). While this was not done universally, it is consistent with the assumption that there is not a large pool of unreported infections (asymptomatic or otherwise) for every case.

4) When results on pairs of cases are presented in one figure then probabilities are assigned. However elsewhere and in the text the implication is made that a dichotomy has been made between cases that are linked and single cases – how have you moved from continuous probability to a dichotomy?

We used a threshold to differentiate between unlikely links and spurious associations due to that fact that some case has to be “most likely” even when none are plausible (see “Identifying most likely chains of transmission” section of the Methods, paragraph two, sentence four. We have added text on lines within the main text emphasizing this in the “Mapping Receptivity” section).

5) Genetic testing would be a more definitive way of distinguishing linked from unlinked cases. Can you comment on the likely utility of their modelling approach once genetic testing is available?

We have added to our discussion on this topic in the last paragraph of the Discussion to reinforce our belief that even as genetic testing methods improve, our approach will be useful as a baseline.

6) Is the "smoldering transmission" reflective of long-term transmission in the absence of imported cases, or does it require an imported case every so often to keep it smoldering?

Once a case exists, it enters the algorithm in the exact same way if it is an “imported” case or a “local” case. As such, our model wouldn’t predict that there is need for “new blood” (or a “new strain”). We have added text in the “Mapping Receptivity” section to clarify this point.

7) Fine-grained needs a definition (pixel size? Or is it a point-process with pixelation simply for presentation purposes?).

We have added text at the end of the “Residual Transmission and Elimination” section to clarify. Our algorithm (the case matching algorithm) occurs in continuous space. Our spatial regression that converts observed R_cs into a smoothed R_c map, as well as all other smoothed maps, is a pixel-driven process. Arguably, it too could be conducted at any resolution for which covariate layers exist.

8) There is a confluent area of high R_c in the North East. In the rest of the country there are occasional dots of high R_c scattered around. What is the possibility that these would have arisen by chance? The model implies very multiple comparisons. If a simulation was carried out where the episodes were randomly distributed through the population then how often would one see a "spike" of red colour in the map just due to chance?

This is an excellent question. We are following up these analyses with a statistical analysis of our algorithm using bootstrapping. We have added text in the “Mapping Receptivity” section indicating that assessing how likely it would be to see a “spike” of red due to chance is important to appropriately interpreting these maps for the purpose of resource allocation (resource allocation is the main topic of the follow-up analysis and assessing the signal/noise ratio of these maps is key for that endeavor).

This is also tangentially discussed in the “Modeling malaria receptivity” section of the Methods in response to the second half of Comment 10 (see below). A key distinction we now emphasize further is that we are presenting an algorithm to direct operational decisions. The "model selection" procedure for smoothing the calculated R_c values into a map consisted of the comparison of 2¹²-1=4095 models. This approach is most useful when fit using recent data to understand the transmission landscape at that time point.

9) If immigrants tend to go to the same places and at the same time (e.g. because they are all being recruited to work in the same factory) then that would increase the likelihood that the two cases would spuriously be associated in time and place despite not having a transmission cycle. Would that source of spurious linkage be accounted for in the migration data that was used?

The immigration process you describe would certainly influence the related analysis of “which areas are seeded by which countries”. As far as our analyses are concerned, both of the immigrants you describe would be treated identically. However, they would not link to each other since they would both be considered “imported” cases and imported cases are assumed by our approach to never be caused by other cases. Since they would both be infectious at, for example, the same factory, then they should both “count” the same in terms of our approach.

10) Is there an issue with spatial autocorrelation (Figure 2 and zero inflated negative binomial regression)? Were the errors inspected? (In general, regression diagnostics, including differences in AIC values among models, have not been reported.)

Given the extremely low numbers of cases, accounting for spatial autocorrelation is quite difficult and will be further investigated in our follow-up analysis where we assess the concerns brought up in Comment 8.

In response to the second half of this comment, we have added most of the desired details, but emphasized that the exact values found in the “best” model are not the focus of our analysis. Using AIC, we compared 2¹²-1=4095 models. There are numerous other models that are ‘almost’ as good as the best. We have emphasized in the “Modeling malaria receptivity” section of the Methods that our regression algorithm is a middle step between linking cases together and making operational recommendations. We do not think it would be useful to report the 4095 AIC values. Our analysis is about the entire algorithm, which would be rerun at any time-point in the future given new data and which would likely result in a different “best” model linking spatial covariates to the output of the first part of our algorithm.

11) What are the units in Tables 1 and 2? Were data centered and scaled prior to fitting GAMs? In my experience, this sometimes improves performance.

We have added units to Tables 1 for the factors that have units. Table 2 does not display any values that have units. We centered and scaled the data and found the same results. Not only were the effective degrees of freedom essentially identical, but so were the fitted smooths. We have not included this analysis in the manuscript, as we did not think it was critical, but could do so if required.

Author details

1. Robert C Reiner Jr

   1. Fogarty International Center, National Institutes of Health, Bethesda, United States
   2. Department of Epidemiology and Biostatistics, Indiana University School of Public Health, Bloomington, United States

   Contribution

   RCR, Conception and design, Analysis and interpretation of data, Drafting or revising the article

   For correspondence

   rcreiner@indiana.edu

   Competing interests

   The authors declare that no competing interests exist.

2. Arnaud Le Menach

   Clinton Health Access Initiative, Boston, United States

   Contribution

   ALM, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

3. Simon Kunene

   National Malaria Control Program, Manzini, Swaziland

   Contribution

   SK, Acquisition of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

4. Nyasatu Ntshalintshali

   Clinton Health Access Initiative, Boston, United States

   Contribution

   NN, Acquisition of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

5. Michelle S Hsiang

   1. Department of Pediatrics, University of Texas Southwestern Medical Center, Dallas, United States
   2. Malaria Elimination Initiative, Global Health Group, University of California, San Francisco, San Francisco, United States
   3. Department of Pediatrics, University of California, San Francisco Benioff Children’s Hospital, United States

   Contribution

   MSH, Acquisition of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

6. T Alex Perkins

   1. Fogarty International Center, National Institutes of Health, Bethesda, United States
   2. Eck Institute for Global Health, University of Notre Dame, Notre Dame, United States
   3. Department of Biological Sciences, University of Notre Dame, Notre Dame, United States

   Contribution

   TAP, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

7. Bryan Greenhouse

   Department of Medicine, University of California, San Francisco, San Francisco, United States

   Contribution

   BG, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

8. Andrew J Tatem

   1. Fogarty International Center, National Institutes of Health, Bethesda, United States
   2. Department of Geography and Environment, University of Southampton, Southampton, United Kingdom

   Contribution

   AJT, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

9. Justin M Cohen

   Clinton Health Access Initiative, Boston, United States

   Contribution

   JMC, Conception and design, Acquisition of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

10. David L Smith

    1. Fogarty International Center, National Institutes of Health, Bethesda, United States
    2. Spatial Ecology and Epidemiology Group, Department of Zoology, University of Oxford, Oxford, United Kingdom
    3. Institute for Health Metrics and Evaluation, University of Washington, Seattle, Washington, United States
    4. Sanaria Institute for Global Health and Tropical Medicine, Rockville, Maryland, United States

    Contribution

    DLS, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

    Competing interests

    The authors declare that no competing interests exist.

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