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      Ross, Macdonald, and a Theory for the Dynamics and Control of Mosquito-Transmitted Pathogens

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          Abstract

          Ronald Ross and George Macdonald are credited with developing a mathematical model of mosquito-borne pathogen transmission. A systematic historical review suggests that several mathematicians and scientists contributed to development of the Ross-Macdonald model over a period of 70 years. Ross developed two different mathematical models, Macdonald a third, and various “Ross-Macdonald” mathematical models exist. Ross-Macdonald models are best defined by a consensus set of assumptions. The mathematical model is just one part of a theory for the dynamics and control of mosquito-transmitted pathogens that also includes epidemiological and entomological concepts and metrics for measuring transmission. All the basic elements of the theory had fallen into place by the end of the Global Malaria Eradication Programme (GMEP, 1955–1969) with the concept of vectorial capacity, methods for measuring key components of transmission by mosquitoes, and a quantitative theory of vector control. The Ross-Macdonald theory has since played a central role in development of research on mosquito-borne pathogen transmission and the development of strategies for mosquito-borne disease prevention.

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          A World Malaria Map: Plasmodium falciparum Endemicity in 2007

          Introduction Maps are essential for all aspects of the coordination of malaria control [1]. In an international policy environment where the malaria control community has been challenged to rethink the plausibility of malaria elimination [2–4], malaria cartography will become an increasingly important tool for planning, implementing, and measuring the impact of malaria interventions worldwide. The last global map of P. falciparum endemicity was published in 1968 [5]. In common with all previous maps of the global distribution of malaria [6–10], and to a large extent those that followed [11–16], the map (i) suffered from an incomplete description of the input data used; (ii) defined contours of “risk” using subjective and poorly explained expert-opinion rules; and (iii) provided no quantification of the uncertainty around predictions. Here we describe the generation of a new global map of malaria endemicity that overcomes these major deficiencies. Geographic Scope of the Modelling The global spatial limits of P. falciparum malaria transmission have been mapped recently by triangulating nationally reported case incidence data, other medical intelligence, and biological rules of transmission exclusion, derived from temperature and aridity limits to the bionomics of locally dominant Anopheles vectors [17,18]. The results of this exercise stratified the world into three classes: the spatial representation of no risk, unstable risk (P. falciparum annual parasite incidence [PfAPI] 5% to 100 km2] and small [>25 km2] polygons [58]; removing those surveys that could not be, or were only geo-positioned imprecisely; and removing those that could not be temporally disaggregated into independent surveys or for which the date was unknown), 7,991 PfPR surveys remained (Figure S1.2 in Protocol S1). All PfPR data were then age-standardized to the 2- to 10-y age range before mapping using an algorithm based on catalytic conversion models first adapted for malaria by Pull and Grab [59]. This algorithm was found to perform best out of a set of candidate standardization procedures and is described in detail elsewhere (Protocol S1.3) [51]. The final dataset was stratified into three major global regions (Figure 1): the Americas; Africa, Yemen, and Saudi Arabia (Africa+); and Central and South and East Asia (CSE Asia) (Protocol S1.4). This division allowed these biogeographically, entomologically, and epidemiologically distinct regions [8,16] to be considered separately, whilst retaining sufficient data in each region for meaningful analysis. These global divisions were further supported by observing the distinct spatial structure of the PfPR2−10 data in each region, illustrated by their semi-variograms (Figure S1.1 in Protocol S1). Malaria transmission-specific approaches to mapping urban, peri-urban, and rural extents were developed, the rationale for which is described in detail elsewhere (Protocol S2) [50]. In brief, all urban extents (UEs) defined by the Global Rural Urban Mapping Project (GRUMP) alpha version UE mask (GRUMP UE) [60,61] were identified at 1 × 1 km spatial resolution (Protocol S2.1) [50]. Within these extents, those areas containing population densities greater than 1,000 people per km2 according to the Gridded Population of the World version 3 population density surface [60,61] were then mapped [48]. All surveys were then assigned as either urban (Gridded Population of the World version 3 ≥ 1,000 km2 within GRUMP UE), peri-urban (Gridded Population of the World version 3 5% to 5% to 5 to 5% to 5 to 5 to < 40%) and 0.345 billion under conditions of high risk (PfPR2−10 ≥ 40%) (Figure 7; Table 4). In the areas of intermediate risk, mathematical modelling suggests that by taking ITNs to scale, the interruption of P. falciparum malaria transmission might be achieved, whereas in the high transmission areas, malaria transmission will be more intractable and require aggressive control with suites of additional and complementary interventions [19,55]. Statistical Implementation and Model Validation The modelling procedure presented here represents a large scale implementation of modern Bayesian geostatistical techniques and incorporates a number of novel components. The incorporation of an age-standardization model has allowed the coherent assimilation of survey data obtained across a wide variety of surveyed age ranges whilst acknowledging the uncertainty introduced by this additional source of variation. Likewise, the use of a fully spatiotemporal random field has allowed surveys from as early as 1985 to be incorporated in the prediction of contemporary P. falciparum endemicity in a statistically and epidemiologically plausible framework. MBG techniques are exceptionally computationally demanding even for small prediction problems. To our knowledge this is the first time these procedures have been applied to any disease at the global scale. This computational burden has also imposed a number of restrictions on the modelling procedure that may have improved predictive capability. In particular, the current model adopts a single mean and covariance function within each global region, representing an assumption of second-order stationarity within each. Approximations to nonstationary random fields adopted in smaller scale studies [32,76] represent possible refinements to the current model, but were considered computationally infeasible globally. Assessment of the various validation statistics revealed that the model performed satisfactorily for each of the three performance aspects: predicting PfPR2−10 point values and endemicity class, and providing realistic measures of prediction uncertainty. Given the highly variable nature of P. falciparum endemicity over even short distances, an overall correlation of 0.82 between the model predictions and validation data, and an average absolute error magnitude of 9.75% PfPR2−10 represents an unexpected level of precision. Certain aspects of the uncertainty measures output by the model are suboptimal: in particular, the tendency to underestimate slightly the probability of PfPR2−10 taking very low values. Nevertheless, given the multitude of sources of uncertainty that are captured and propagated though the modelling framework, the resulting uncertainty predictions represent a rich source of information in the generation of output products for decision makers. The model was fitted using MCMC [77,78]. MCMC is an extremely powerful algorithm, and is the only general-purpose, computationally tractable algorithm available for many Bayesian problems. However, it is an approximate algorithm. No fail-proof method for estimating its error is available, but using a heuristic method (Protocol S1.3) we estimated that our “Monte Carlo error” is unimportant relative to the uncertainty in our actual posterior distributions. The information contained in the maps presented here and the associated uncertainty varies across a range of geographical scales. The large-scale variation in endemicity described between regions and countries is unambiguous, robustly quantified, and of direct use to global planners. As progressively finer scales are considered, however, the utility of these maps for local malaria control managers diminishes although this is heavily dependent on the local availability and density of survey points. The appropriate threshold and metric of uncertainty will vary enormously for different end users and applications of the maps. As a rule-of-thumb, however, it is suggested that the differentiation in endemicity between areas smaller than the first administrative level would be inappropriate for most countries. Examination of the frequency distributions for all-year and 2007 input PfPR2−10 data, and for the predicted PfPR2−10 surface, revealed a number of important features. Firstly, 2007 data from all three regions displayed substantially smaller median and maximum values and were more positively skewed than data from all years considered together (compare Figure 8A and 8B). Secondly, there were marked differences in all regions between the distribution of 2007 data values and the distribution of values from the predicted PfPR2−10 surface (compare Figure 8B and 8C). Specifically, the latter distributions had larger medians, were less positively skewed, and for the Americas and Africa+ had substantially smaller maximum values. The overall shift towards higher PfPR2−10 in the predicted surfaces can be attributed to the spatial clustering of the survey locations. It must always be remembered that the set of surveys collated represents an opportunistic sample driven by the motivations and constraints of a multitude of individuals, organizations, and governments. Visual examination of this set reveals a considerably larger proportion located in lower endemicity regions than would be the case in a spatially random sample and, as such, summary statistics of these raw data display a substantial bias. By predicting endemicity over a continuous surface, the MBG process compensated implicitly for this clustering in the output maps and the resulting frequency distribution was not biased in the same way. The MBG process makes predictions at unsampled locations using linear combinations of survey data. For this reason, the resulting surfaces are inevitably smoother than the raw data from which they are predicted. One feature of this smoothing process is that the range of extreme high and low values in the predicted surface is likely to be smaller than that displayed by the input data. This explains why the frequency distributions for the predicted PfPR2−10 surface cover substantially smaller ranges of values than those of the input data. An important implication of this smoothing effect is that the predicted surface provides a more robust prediction of endemicity at larger scales but is less able to represent faithfully the short-scale variations occurring over very short distances. Using Environmental Covariates to Make Continuous Maps The extreme limiting effects of climate covariates have been incorporated comprehensively in the definition of the stable and unstable limits of P. falciparum malaria transmission described above [18]. There is an illusory attraction in the further use of environmental covariates to increase complexity and improve predictive accuracy in MBG endemicity mapping. This is because such analyses are based on the assumption that the contemporary distribution and endemicity of malaria approximates its fundamental niche [79,80]. This assumption is unfounded because the global distribution of malaria has contracted substantially [18] since its hypothesised maximum distribution circa 1900 [14]. Moreover, it is not known to what extent the environmental determinants of the remaining distribution reflect this fundamental niche, how these relationships might vary spatially, and therefore, what artefacts might be introduced by their inclusion in the analyses. In addition, it is not trivial to obtain “adequate” environmental covariates at a global level with the required spatial and temporal fidelity [63,81]. Finally, the degree to which these relations would be further obscured by ongoing and spatially variable intervention efforts is also unquantified. An increasing body of evidence points to these intervention effects being substantial, to have accelerated in the post 2000 period, and to represent a spatial mosaic of influence that would act to confound substantially any modelled relationships [82–90]. Unsurprisingly, no statistical support was found for the inclusion of a range of climate [62] and remotely sensed [63] environmental covariates (Protocol S1.7). In eschewing the use of environmental covariates in this analysis framework, the output maps are determined only by the input survey data and the assumptions of the modelling. This choice ensures a maximally parsimonious baseline, against which future changes may be audited. Potential Geostatistical Improvements In embracing the MBG approach, the rationale for excluding surveys with a sample size below 50 is diminished, as the uncertainty in relation to the population sampled is explicitly modelled by the technique (Protocol S3). This exclusion rule was devised at a time before MBG could be applied at a global scale and will be revised in future iterations of the map. The spatial resolution with which these MBG techniques could be reasonably implemented on a computer cluster was on a 5 × 5 km grid. The entire process took an average of one month at this spatial resolution and has been estimated to take one year to run on a 1 × 1 km spatial grid. There are no plans to increase the spatial resolution of the output maps at the global scale because they are robust for the regional planning purposes for which they are intended. For smaller areas, such as PfPR data rich countries where higher spatial resolution maps may be desirable to support national control plans, however, MBG outputs to 1 × 1 km grids can be considered [33]. Moreover, at these national scales, the fidelity of the geo-positioning of the input PfPR survey data may have an important influence on the uncertainty of the predictions, so procedures that can help incorporate these effects into the modelling may also need to be investigated [91–93]. In this study, the uncertainty likely to be contributed by geo-positioning errors was thought to be trivial in relation to the scales of spatial variation in observed endemicity and given the global scale of model outputs. We were not able to improve the age-correction model's predictive performance by modelling the age-dependent sensitivities of microscopy and rapid diagnostic tests separately or by modelling diagnostic specificity. The accuracy in the determination of PfPR by microscopy or rapid diagnostic tests were assumed to be equivalent in these analyses, but the sensitivity of the diagnostic technique [94–98] could be included into a future iterations of this MBG framework. No solution could be found to applying these MBG techniques across large tracts of ocean (for example in the Caribbean, Madagascar, and the Indonesian archipelago), given the global distribution of the PfPR data and the lack of data in some regions (Figure 1). Potential biogeographical influences on malaria transmission on islands are ignored by these analyses. Future map iterations would ideally have sufficient data to treat islands separately or sufficient information on the distribution of Anopheles vectors to help inform the predictions [56]. We have incorporated the ability for the analyses to be cognisant of secular trends in the PfPR data and of annual variations in transmission. This map does not provide a full description of seasonal malaria dynamics [99–101], however, and further information on the global variation of malaria seasonality might inform future map iterations. The Road Ahead: Public Domain and Dynamic Maps These mapped surfaces are made available in the public domain with the publication of this article. The underlying data used in their predictions are due for public release in 2009 [1], and the online infrastructure to host this service is under development. The MAP team anticipate providing annual updates of this P. falciparum global malaria endemicity map and the accompanying PfPR database. Annual updates will also be required to reflect the changing spatial limits of stable and unstable P. falciparum malaria transmission [18] in order to define accurately the limits within which endemicity predictions need to be made. If the international community is successful in rolling back malaria, informed decisions will need to be made about the temporal discontinuity between the spatial limits of P. falciparum malaria transmission (defined, where possible, by the average PfAPI in the three most recently recorded years [18]) and the endemicity data (PfPR collected since 1985). It is obvious that the predicted map represents a snapshot of the year 2007 from a malaria endemicity that changes through time. No degree of statistical sophistication can circumvent the fact that additional data will increase the fidelity of the map, by either increasing the spatial resolution of the malariometric surveys or updating an existing survey location with more recent information. The methods have been devised specifically so that these surfaces can be updated rapidly. The predominantly univariate approach adopted also means changes in future maps' iterations can be attributed reliably to finding more data in areas of high uncertainty (changes in space) or to changes brought about through intervention success or disease recession (changes in time), rather than any temporal and spatial mix of the relationship of the PfPR2−10 data and the environmental covariates. We encourage the submission of additional existing data to improve the map in areas where we have least spatial accuracy, and new data to sustain future production of updated contemporary maps. Current areas of highest uncertainty are indicated to a good approximation by the inverse of the class prediction probability (Figure 5), although future work is aimed at refining this information. Therefore, an immediate priority is to generate regional maps showing the optimal location of new surveys that would need to be implemented to maximally reduce the variance in the existing endemicity surface for the minimum cost. These solutions are substantially more involved than the list of areas with highest variance provided here because (i) each new survey will change the structure of the spatial variance and affect the optimal location of the next survey; (ii) both the number and spatial distribution of surveys will affect the outcome and require multiple simulations to converge on optimal solutions; and (iii) potential survey locations will need to be weighted appropriately by the distribution of the human population. Immediate MAP Goals The initial focus of the MAP has been P. falciparum [1] due to its global epidemiological significance [102] and its better prospects for control and local elimination [19]. We have not yet addressed the significant problem of P. vivax burden [103] despite its increasingly recognised clinical importance [104–106], but have archived over 2,500 P. vivax parasite rate surveys with which to start this process. Another immediate goal is in refining global burden of disease estimates for P. falciparum (both morbidity [102] and mortality [48,107,108]) to support global estimation of antimalarial intervention and commodity needs. The statistical methods used in this analysis will allow the next iteration of burden estimates to represent more holistically and robustly the uncertainty in predictions. In the medium term, combinations of these global endemicity maps with forthcoming maps of the distribution of the dominant Anopheles vectors of human malaria [56] should empower malaria control managers to make more informed decisions regarding interventions appropriate to the bionomics of their local suite of vectors. In the long term we hope to not only monitor and evaluate progress with these maps, but to increase our ability to model future malaria endemicity and support objective assessment of where in the world it might be possible to eliminate malaria. Conclusions The state of the P. falciparum malaria world in 2007 represents an enormous opportunity for the international community to act [109,110], but these actions remain considerably under-resourced [111]. Regardless of whether nations champion sustained, intensive control or reach for the higher ambition of malaria elimination [2–4,74,112–114], the intermediate intervention paths are similar [19]. This cartographic resource will help countries determine their needs and serve as a baseline to monitor and evaluate progress towards interventional goals. We wish to continue to work alongside individuals, countries, and regions to improve future iterations of this map and document hopefully these intervention successes. Supporting Information Alternative Language Text S1 Translation of the Article into French by Frédéric Piel and Stéphanie Loute (1.04 MB DOC) Click here for additional data file. Alternative Language Text S2 Translation of the Article into Chinese by Robert Li (438 KB DOC) Click here for additional data file. Alternative Language Text S3 Translation of the Article into Indonesian by Iqbal R.F. Elyazar and Siti Nurlela (1.08 MB DOC) Click here for additional data file. Alternative Language Text S4 Translation of the Article into Vietnamese by Bui H. Manh (549 KB DOC) Click here for additional data file. Alternative Language Text S5 Translation of the Article into Spanish by Carlos A. Guerra (796 KB DOC) Click here for additional data file. Protocol S1 The PfPR Malariometric Survey Database S1.1 Summary of Data Search and Data Abstraction Procedures S1.2 Data Exclusion Rules S1.3 Age-Standardisation S1.4 Semi-Variograms of PfPR2−10 Data by Region S1.5 Geostatistical Filter for the Detection of Extreme Outliers S1.6 Malariometric Survey Data Summary and Descriptive Statistics S1.7 Relationships with Environmental Covariates (3.4 MB DOC) Click here for additional data file. Protocol S2 Demographic Databases and Procedures S2.1 Parasite Rate Survey Urban/Peri-Urban/Rural Classification Rules S2.2 Urban/Peri-Urban/Rural Status and Prevalence S2.3 GRUMP alpha Human Population Surface S2.4 PAR Derivation (2.5 MB DOC) Click here for additional data file. Protocol S3 Model Based Geostatistical Procedures S3.1 Overview of the Statistical Model S3.2 Prior Specification S3.3 Age-Standardization S3.4 Implementation Details S3.5 Overview of Map Generation (23 MB DOC) Click here for additional data file. Protocol S4 Model Validation Procedures S4.1 Creation of the Validation Sets S4.2 Procedures for Testing Model Performance S4.3 Additional Results (26 MB DOC) Click here for additional data file.
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            Diseases of humans and their domestic mammals: pathogen characteristics, host range and the risk of emergence.

            Pathogens that can be transmitted between different host species are of fundamental interest and importance from public health, conservation and economic perspectives, yet systematic quantification of these pathogens is lacking. Here, pathogen characteristics, host range and risk factors determining disease emergence were analysed by constructing a database of disease-causing pathogens of humans and domestic mammals. The database consisted of 1415 pathogens causing disease in humans, 616 in livestock and 374 in domestic carnivores. Multihost pathogens were very prevalent among human pathogens (61.6%) and even more so among domestic mammal pathogens (livestock 77.3%, carnivores 90.0%). Pathogens able to infect human, domestic and wildlife hosts contained a similar proportion of disease-causing pathogens for all three host groups. One hundred and ninety-six pathogens were associated with emerging diseases, 175 in humans, 29 in livestock and 12 in domestic carnivores. Across all these groups, helminths and fungi were relatively unlikely to emerge whereas viruses, particularly RNA viruses, were highly likely to emerge. The ability of a pathogen to infect multiple hosts, particularly hosts in other taxonomic orders or wildlife, were also risk factors for emergence in human and livestock pathogens. There is clearly a need to understand the dynamics of infectious diseases in complex multihost communities in order to mitigate disease threats to public health, livestock economies and wildlife.
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              Statics and dynamics of malaria infection in Anopheles mosquitoes

              The classic formulae in malaria epidemiology are reviewed that relate entomological parameters to malaria transmission, including mosquito survivorship and age-at-infection, the stability index (S), the human blood index (HBI), proportion of infected mosquitoes, the sporozoite rate, the entomological inoculation rate (EIR), vectorial capacity (C) and the basic reproductive number (R 0). The synthesis emphasizes the relationships among classic formulae and reformulates a simple dynamic model for the proportion of infected humans. The classic formulae are related to formulae from cyclical feeding models, and some inconsistencies are noted. The classic formulae are used to to illustrate how malaria control reduces malaria transmission and show that increased mosquito mortality has an effect even larger than was proposed by Macdonald in the 1950's.
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                Author and article information

                Contributors
                Role: Editor
                Journal
                PLoS Pathog
                PLoS Pathog
                plos
                plospath
                PLoS Pathogens
                Public Library of Science (San Francisco, USA )
                1553-7366
                1553-7374
                April 2012
                April 2012
                5 April 2012
                : 8
                : 4
                Affiliations
                [1 ]Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health, Baltimore, Maryland, United States of America
                [2 ]Malaria Research Institute, Johns Hopkins Bloomberg School of Public Health, Baltimore, Maryland, United States of America
                [3 ]Fogarty International Center, National Institutes of Health, Bethesda, Maryland, United States of America
                [4 ]Spatial Ecology and Epidemiology Group, Department of Zoology, Oxford University, Oxford, United Kingdom
                [5 ]Center for Vectorborne Diseases, University of California, Davis, California, United States of America
                [6 ]Department of Pathology, Microbiology, and Immunology, School of Veterinary Medicine, University of California, Davis, California, United States of America
                [7 ]Department of Entomology, University of California, Davis, California, United States of America
                International Centre for Genetic Engineering and Biotechnology, India
                Author notes
                Article
                PPATHOGENS-D-11-02708
                10.1371/journal.ppat.1002588
                3320609
                22496640
                This is an open-access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.
                Page count
                Pages: 13
                Categories
                Review
                Biology
                Population Biology
                Population Dynamics
                Population Modeling

                Infectious disease & Microbiology

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