Philippa Kaur

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Critical transition theory tells us that, as exogenous parameters drive the system towards a bifurcation and the emergence of a new equilibrium, we can see evidence of the upcoming transition through decreasing resistance to each oscillation peak15, like a ball rolling around in a cup whose walls are becoming less and less steep1,6. We find evidence of this critical slowing down within the year-to-year deceleration of rust growth rates just prior to their annual peak, or λ, our formal slowing down measure. This is coupled with a delay in the initiation month (takeoff) of the oscillation itself. The oscillations collapse in 2019 as the system apparently switches to a more benign non-epidemic state.In the mid-elevation zone of southwestern Mexico where we conducted our study, the general expectation from climate change, in addition to increased temperature, is reduced rainfall19,20, a pattern broadly in evidence for the past five years (Fig. S2). With regard to the coffee rust disease, this precipitation trend is most closely associated with the change in λ, and reflects the obligate range of associated moisture and temperature conditions required for the rust to flourish9,17. On the other hand, rust-resistant replanting has been occurring throughout coffee farms in much of Mesoamerica since the rust outbreak in 2012–201321,22. This could limit plant-to-plant spread by reducing opportunities for direct contact with an infected plant, as well as decrease the environmental spore load by reducing the contributing pool of infected plants in the broader region. On our site, the slowing down patterns of both λ and the rust takeoff point were associated with more rust-resistant replanting on a local level, while the significant linear delay in the year-to-year rust takeoff over the study period may reflect larger-scale effects emerging from increased resistance across the farm.Our preliminary treatment of April as the annual rust season initiation point corresponded generally with a seasonal pattern of increasing rainfall. This approach might be assumed by a manager experiencing the system in real time without knowledge of how the rust season will play out that year. Such an assumption works well in the first two seasons of our data, where the rust increase and rainy season both appear to begin in April. However, this association decouples in subsequent years. Evidently, the critical slowing down we initially estimated in Fig. 1B is partitioned into two components: one that imposes a lag on the initiation of the disease, and another that imposes a decelerating approach to the peak rust intensity, necessitating a joint model of the initiation month and the critical slowing down warning signal (λ). That the takeoff had a significant delaying trend over the study period suggests that the initiation of the rust season can also be seen as a measure of critical slowing down, in that a slower approach to the equilibrium point (the seasonal rust peak) will likely be reflected in a failure to even recognize the increase when it is still very small. Thus, the conspicuous change in takeoff point over time (Fig. 2A, B) itself suggests a critical slowing down and the negative values of λ in Fig. 1B may be mainly a reflection of a lag in takeoff time. The overall increasing lag in rust takeoff each year could mean that the transmission factor itself may be exhibiting a critical slowing down as the critical transition is approached, due to the progressive influence of parameters outside of the present analysis. Given the basic biology of the disease10, this trend may stem from a year-to-year secular decline in the environmental spore load causing initiation of the disease season to be deterred by a small amount each year.The evident relationship between critical slowing down and reduced rainfall is suggestive of a connection to climate change, while a slow secular increase in the proportion of rust-resistant varieties could have inhibited local or regional spread, leading to an effective “herd immunity” to the disease. The joint operation of reduced rainfall and fewer susceptible plants raises the possibility of dual bifurcations and hysteretic zones in the dynamic landscape, which we illustrate in a qualitative fashion (generalized to climate and management) in Fig. 4a. The initial outbreak, which was unexpected in light of the historically low, but persistent, levels of rust in the region7,8, was likely a critical transition precipitated by interactions between local and regional processes14. As foregrounded in Fig. 4b, we propose that a possible parameter driving the system to this initial bifurcation was recent increases in precipitation, as evidenced in local rainfall records (Fig. S2) and regional trends19. This could have propelled the system past a hysteretic phase space to where seasonal conditions dictated that the system jump to a high rust intensity equilibrium, represented conceptually in Fig. 4a by the dotted trajectory leading to the upper surface of the landscape that corresponds to an epidemic state of the rust. Likewise, in the years following the outbreak, our findings suggest that, while the system still tracked precipitation, progressive replanting of resistant varieties emerged as another parameter axis (management) that drove the system through a second critical transition back to a low rust equilibrium (Fig. 4c). Although the dynamical landscape in Fig. 4 is a qualitative representation, the trajectory along the upper surface helps to visualize how two exogenous forces, operating separately, both contributed to the critical transition we observed in our data. The trajectory we propose brings attention to the interesting possibility that the main driver of slowing down shifted from climate (precipitation) to management (replanting), leading to the second critical transition. Indeed, though the average yearly replanting rate remained roughly the same year-to-year (Fig. S1), we note that much of the cutting prior to the collapse in 2019 seemed to be concentrated in April that year (Fig. 1D), accompanied presumably by a similarly timed replanting campaign.Figure 4Envisioning the combination of climate and management effects in a joint hysteretic framing, stemming from gradual change in both forcing parameters. (A) Relative positions of rust intensity for each year are illustrated in their approximate positions with red arrows (other trajectories could be imagined based on the data presented herein). Inset plots provide a conceptualized view of the dynamics between the two forcing variables on rust intensity: (B) management (in our case, resistant variety replanting) and (C) climate (precipitation). In the inset plots, arrows indicate directions of change; solid and dashed lines indicate stable and unstable equilibria, respectively.Full size imageRecent work on critical transitions suggests that perturbances driving a system to transition are more realistically not distinct or isolated, and that the stochastic and deterministic elements of the system can therefore be entangled or even interdependent4. Likewise, we find that the variability in the environmental covariates of monthly rainfall and resistant variety replanting better explained patterns in λ than a linear trend leading up to the transition, as represented by the year variable. The correlation between the quadrat grouping offset estimates from the λ and takeoff components of the multivariate model also suggest that slowing down and delayed takeoff were associated at the individual quadrat level (Fig. 3C). Accounting for this spatial effect, these two components do not appear to be correlated by year (Fig. 3D). This suggests that the shared variability between these two indicators reflects variability in spatial environment within the plot rather than idiosyncratic effects of unique years. Besides the direct effect of resistant varieties, local stochasticity and spread dynamics may also play a role. Local growing conditions, such as variability in shade from overstory trees, can affect dispersal through rainfall splash and wind23,24. Additional management factors may also play a role, such as the vegetation structure and the presence of paths25, as well as the physical relationship between coffee plants26.Our observations of the rust dynamics themselves allow us to detect the general signals anticipating a critical transition, though the drivers may emerge from a complex system of dialectical interactions that must be considered in their whole7,27. The concept of critical slowing down thus may lend itself to application across coffee-growing regions, where predicted effects of climate change and other geographic conditions may differ9,19. Since the emergence of the rust outbreak, recommendations and protocols have been published for monitoring rust levels, potentially providing managers with regular data in changes in rust intensity for many areas9,28. As the resilience of a system can be interpreted through measuring critical slowing down prior to catastrophe2, as well as, in our case, the “exit time” from an undesirable regime4, we demonstrate that such concepts may be applied to this monitoring data to gain some insight into the system’s status. Future studies could explore signs of critical slowing down across coffee-growing regions and management systems to see how these signals predict significant changes and respond to local drivers, potentially adding to the vocabulary of agroecological management.In sum, it is clear that both a lag in takeoff point for the seasonal oscillation and the rate of approach to the peak each year seem to conspire to produce a critical slowing down, strong evidence that the decline in the disease in 2019–2020 is indeed a critical transition, regardless of the underlying mechanism. While our model suggests that two exogenous forces, rainfall and resistant variety replanting, may be driving the slowing down in our case, the underlying dynamical landscape is likely not unique to our site. More generally, the phenomenon of multivariate bifurcations leading to subsequent critical transitions (e.g., Fig. 4) is perhaps more common than thought29. Examinations of critical transitions should therefore consider the larger dynamical landscape for the possibility of subsequent transitions. More

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Species composition of stone wall treesFamilies and genera of stone wall treesThere were 96 stone wall trees belonging to 6 genera and 5 families in Macao. Among them, Moraceae and Ficus appeared the most frequently, both reaching 85 times, accounting for 88.5% (Table 1). It showed that Moraceae, a kind of tropical distribution family, was dominant in the stone wall trees communities, which meant that stone wall trees species in Macao appeared distinctly tropical nature18.Table 1 Frequency of occurrence of stone wall Trees in different families and genera.Full size tableSpecies of stone wall treesThere were 16 species of the stone wall trees in Macao including Bridelia tomentosa, Celtis sinensis, Eriobotrya japonica, Ficus altissima, F. benjamina, F. elastica, F. hispida, F. microcarpa, F. pandurata, F. subpisocarpa, F. tinctoria subsp. gibbosa, F. rumphii, F. variegata, F. virens, Leucaena leucocephala, and Trema cannabina (Fig. 2).Figure 216 species of stone wall trees in Macao (photo was taken by Professor Qin Xingsheng).Full size imageBased on the frequency of occurrence of various tree species, the frequency was concentrated in the range of 1–5%. Among them, Ficus microcarpa had the highest frequency, reaching 58 times, with a frequency of 60.4% (Fig. 3). This tree species is robust, adaptable and fast growing, which is the main population of Ficus19.Figure 3Frequency distribution of stone wall tree species in Macao.Full size imageStone wall trees in the historic center of MacaoThe historic center of Macao, covering an area of about 2.8 km2, is the heartland of Macao’s historical and cultural heritage, which plays a significant role in the cultural heritage around the world18. The historic center of Macao provides valuable historical and cultural resources that enable Macao to transform into a world tourism center20.A total of 14 plots were located in the historic Center of Macao (Fig. 4), with 45 stone wall trees, accounting for 47.9% of the total number of trees in the survey. Among them, Jardim Luís de Camões has the largest number of 9 stone wall trees. The park, built in the mid-eighteenth century, is one of the oldest gardens in Macao and has the largest number of old trees in Macao. The park had provided good time and environmental conditions for the growth of stone wall trees.Figure 4(a) Schematic diagram of distribution and number of stone wall trees in the historic Center of Macao. (b) Schematic diagram of historic center of Macao. (URL of the Macao map: https://www.d-maps.com/m/asia/china/macau/macau02.gif).Full size imageAccording to Decree No. 56/84/M of the Macao Special Administrative Region Government Printing Department, immovable property that represents the creation of man, or the development of nature or technology and has cultural significance is considered tangible cultural property. The occurrence of the stone wall tree was inextricably linked to ancient wall-building techniques of that time, which was of great significance for the study of the technological development and ecological landscape of the historic center of Macao. The concept of “historic urban landscape” was proposed by Zhang Song20, who argued that cities were organisms in continuous evolution, emphasizing respect for the interrelationship between natural and man-made environments. The stone wall trees in the historic center of Macao have been associated with the local culture and ecology tightly and should be preserved as important urban landscape.Symbiotic relationship between tree and stone wallsAs shown in the table below (Table 2), it was found that most of the stone wall trees had root systems that were not only superficially attached to the wall but also extended to the top or bottom of the wall. In particular, Ficus spp. whose strong root system could closely mosaic with the wall, thus forming a strong symbiosis.Table 2 The relationship between the root system of the stone wall tree and the wall.Full size tableStone walls can imitate the traditional nature-accommodating features to permit spontaneous establishment of a diverse plant assemblage. Besides vegetative diversities in terms of species composition, growth form and biomass structure, stone walls can support a mass collection of urban wildlife and provide various ecosystem service. It is highly recommended that modern urban design be created to embrace stone wall landscape as an integral part of naturalistic or ecological design.Vision for the establishment of the stone wall tree trail system in the historic of MacaoThe traditional street environment in the Macao Peninsula is a kind of distinctive urban landscape, which can highlight the specificity and value of the urban context. The combination of the stone wall trees and walls, together with the traditional streets, form a spatial urban landscape. Starting from the location of the stone wall tree landscape, the dots and lines are prospective to promote the establishment of a comprehensive stone wall tree landscape trail system (Fig. 5), so that the public can make use of the existing biological resources to have a better understanding of the land on which they live.Figure 5Schematic diagram of the stone wall trees trail system on the Macao Peninsula (URL of the Macao map: https://www.d-maps.com/m/asia/china/macau/macau02.gif and the finished map is created by Meisi Chen through the Photoshop CS6 and Arc GIS 10.2).Full size imageSince 2012, the Macao Government has been implementing the “Strolling along Macao Street” project, which aims at studying and exploring the history and culture of the streets of Macao through an in-depth cultural tourism route and promoting it to different levels of society. The establishment of the stone wall tree trail system can rely on this project to raise the public’s awareness of the protection and cultural identity of the stone wall tree landscape through a variety of ways. For example, route design competition, photography competition and exhibition, recruitment of “Stonewall Tree Protection Ambassadors” and other forms of participation, so that the public could complete the “role change” in the high degree of such participation—from “onlookers” to “bystanders”.Survey results of associated plant speciesSpecies composition and occurrence of frequencyThe survey showed that there were 101 species of stone wall tree associated plants in Macao, under 88 genera and 51 families. Most associated plants belonged to Euphorbiaceae, Compositae, and Araceae.There were 85 species with a frequency of 1–5 times, accounting for 84.2% of total species. A total of 11 species appeared 11–15 times, accounting for 4.0% (Fig. 6). There were a total of 4 species that appeared more than 15 times. They were Cocculus orbiculatus, Pteris cretica, Paederia scandens, and Pyrrosia adnascens. Most of the associated species appeared only 1–5 times, indicating that most plants were selective and accidental for the growth conditions of stone wall sites.Figure 6Occurrence frequency in various species of associated plants.Full size imageLife form compositionHerbaceous plants with 37 species, accounting the percentage of 52.3% (Fig. 7), were dominant in the associated plant species because the seeds of herbaceous plants are lighter and can be propagated to the wall surface by wind force.Figure 7Life form of associated plants with stone wall trees in Macao.Full size imageSimilarity analysis of the associated plants in MacaoIn order to compare the similarity of associated plant species in different environment, the surveyed sample sites for this study were divided into three categories: motorized lanes, non-motorized lanes, and park habitats (Table 3). According to Jaccard’s similarity principle, Sj is extremely dissimilar when it is 0.00–0.25, and the analysis showed that the similarity of companion plant species in all three habitats was extremely dissimilar. Therefore, it indicated that the companion plants in different habitats had obvious diversity and uniqueness.Table 3 Jaccard similarity index for companion plant species composition among three habitats.Full size table More

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Data descriptionIn Indian cities, cases of falciparum malaria rise after the monsoon rains and peak in October–November. To address the question of whether humidity influences the seasonality and inter-annual variability of urban malaria we focus on 2 cities, Ahmedabad and Surat, with over 3 million people in the semi-arid state of Gujarat, India. These cities exhibit a rising population where sustained, extensive, and consistent surveillance programs have been conducted for over two decades. Despite their close proximity, these cities also exhibit distinct environments. While Ahmedabad is semi-arid, Surat is coastal with a maritime influence on its climate and is prone to flooding from the Tapi river.The malaria data consists of monthly cases collected from 1997 to 2014 by the respective Municipal Corporations of the cities of Ahmedabad and Surat (Fig. 1A, B). The epidemiological data result from two kinds of surveillance: (a) the collection of blood slides from fever patients by house-to-house visits by a health worker and examination of these slides for positive malaria parasites at the Primary/Community Health Center (active surveillance); (b) examination of blood slides from fever patients reporting directly to the Primary/Community Health Center (passive surveillance). Both types of data are pooled into a temporal record for each city. We used climate data of monthly RH, rainfall, and temperature for the same 18 years recorded at a local weather station within each city, supplied by the Indian Meteorological Department in Pune (India) and verified in the GHCN network of climate data (https://www.ncdc.noaa.gov/ghcn-daily-description). Since station data sometimes exhibit biases and can fail to represent the climate of the whole area of interest, here the whole city, we used gridded climate products (https://www.chc.ucsb.edu/data/chirps for precipitation and https://modis.gsfc.nasa.gov/data/dataprod/mod11.php for temperature) and constructed an average of grid cells to verify if climate covariates from the station data coincide with the satellite-based products (Supplementary Fig. 14). Time series for total population size were obtained through estimates by the respective municipal corporation.Data analysesThe temporal lagged correlation between monthly malaria cases and monthly meteorological factors from 1997 to 2014 was explored first for the two cities, by defining an interannual association based on maximum lagged correlations between the mean of the cases in the peak months (Aug–Nov) and the climate covariates. For humidity, we defined a three months window preceding the case epidemic season. This period was determined to fall between April and July for Surat and from May to July for Ahmedabad (Supplementary Table 2). The windows defined for the other covariates are shown in Supplementary Fig. 2.In addition, the temporal and possibly transient association of variability at different periods between the times series for malaria and humidity was also examined using wavelet coherence analysis3,42. In contrast to the Fourier spectral approaches, wavelet analyses are well suited for the study of signals whose frequency composition changes in time. The wavelet spectrum specifically provides a time-frequency decomposition of the total variance that is local in time42. The wavelet coherence analysis indicates the co-occurrence of a particular frequency at a given time in the number of cases and in the climate covariate.The wavelet cross-spectrum is given by ({W}_{x,y}(f,tau )={W}_{x,y}(f,tau ){W}_{x,y}^{* }(f,tau )) where x and y represent the two-time series, f is the scale parameter and (tau), the time parameter, with * denoting the complex conjugate. As in the Fourier spectral approaches, the wavelet coherence is defined as the cross-spectrum normalized by the spectrum of each signal$${R}_{x,y}(f,tau )=frac{|langle {W}_{x,y}(f,tau )rangle |}{{|langle {W}_{x,x}(f,tau )rangle |}^{1/2}{|langle {W}_{y,y}(f,tau )rangle |}^{1/2}}$$
(1)
where (langle rangle) denotes a smoothing operator in both time and scale. Using this definition, ({R}_{x,y}(f,tau )) is bounded by (0 , < , {R}_{x,y}(f,tau ), < , 1). The smoothing is performed, as in Fourier spectral approaches, by a convolution with a constant length window function both in the time and frequency directions42. We have chosen to use a procedure based on resampling the observed data with a Markov process scheme that preserves only the short temporal correlations. Our aim is to test whether the wavelet-based quantities (the coherence) observed at a particular position on the time-scale plane are not due to a random process with the same Markov transitions (time order) as the original time series42. In our wavelet coherence spectrum, the white lines indicate the α = 5% significant level computed on the basis of 1000 bootstrapped series, and the shaded area, known as the cone of influence, indicates the influence of edge effects.Transmission modelWith a stochastic transmission model (Supplementary Fig. 5), we test the hypothesis that humidity is important in driving the temporal dynamics of malaria. The model subdivides the total population P into two classes of infectious and susceptible individuals respectively, to allow for heterogeneity in the degree of clinical symptoms and protection conferred by the previous infection. Specifically, the number of individuals in those classes is denoted by S1 for those susceptible to infection, E, for those exposed to infection, I1, for those infected, symptomatic and infectious, I2, for those that are infected but are asymptomatic and still infectious, and S2, for those recovering from initial infection with partial protection. In the equation for S1, the flow of newborns combined with the death rate of each class results in population numbers equal to those observed for the overall demographic growth of the city. The system of stochastic differential equations is given by the following equations:$$d{S}_{1}/{dt}=left(delta P+{dP}/{dt}right)+{mu }_{{S}_{2}{S}_{1}}{S}_{2}-{mu }_{{SE}}(t){S}_{1}-delta {S}_{1,}$$ (2) $${dE}/{dt}={mu }_{{SE}}(t){S}_{1}-{mu }_{E{I}_{1}}E-delta E,$$ (3) $$d{I}_{1}/{dt}={mu }_{E{I}_{1}}E+{mu }_{{I}_{1}{S}_{2}}{I}_{1}-delta {I}_{1},$$ (4) $$d{S}_{2}/{dt}={mu }_{{I}_{1}{S}_{2}}{I}_{1}+{mu }_{{I}_{1}{S}_{2}}{I}_{2}-{mu }_{{S}_{2}{S}_{1}}{S}_{2}-{mu }_{{SE}}(t){S}_{2}-delta {S}_{2},$$ (5) $$d{I}_{2}/{dt}={mu }_{{SE}}(t){S}_{2}+{mu }_{{I}_{2}{S}_{2}}{I}_{2}-delta {I}_{2},$$ (6) We rely on a model that represents vector dynamics implicitly by implementing a Gamma-distributed time delay with mean in the force of infection (the rate of transmission per susceptible individual)9,16,43. This distributed lag is meant to account for the developmental delay of P. falciparum parasites within surviving mosquitoes. For this purpose, we follow the phenomenological representation of transmission via a mosquito vector introduced in refs. 16,43,44, which includes a distributed delay in the transmission from infected to susceptible humans. That is, the force of infection generated by the number of infections at any given time is not experienced at that same time by susceptible individuals, as would be the case in a directly transmitted disease. Under vector transmission, susceptible individuals experience it with a delay, which we consider Gamma distributed, to avoid the unrealistic assumption of a perfectly fixed delay, and to use a positive distribution with a flexible shape and a well-defined mode.Specifically, the development of the parasite within the mosquito introduces a distributed delay in the “latent” force of infection λ (s) resulting in the realized rate of infection of susceptible individuals$${mu }_{{SE}}left(tright)={int }_{{{{{{rm{infty }}}}}}}^{t}gamma (t-s)lambda (s){ds},$$ (7) where the delay probability function follows a gamma distribution. In this expression, λ(s) corresponds to the “latent” force of infection$$lambda (t)=left(frac{{I}_{1}+{I}_{2}}{Pleft(tright)}right)beta (t)$$ (8) where parameter β denotes the transmission rate. The transmission rate is specified to include the effects of seasonality, (interannual) climate variability, and environmental noise with the following expression$$beta left(tright)={{exp }}left[{sum }_{k=1}^{6}{b}_{k}{S}_{k}+{b}_{{RH}}{S}_{4}Cright]left[frac{dGamma }{{dt}}right]$$ (9) where seasonality is represented nonparametrically as the sum of six terms with a basis of periodic b-splines (t) (k = 1…, 6), and the coefficients (({b}_{k})) are parameters to be fitted determining the temporal (seasonal) shape. The b-splines are shown in Supplementary Fig. 6. The first term in Eq. (9) (the exponential of the weighted sum of these six splines) provides the basic, seasonal shape of the transmission rate (Supplementary Fig. 12). We superimpose this seasonality variability in the transmission rate across years through explicit consideration of a specific covariate (temperature, rainfall or humidity, depending on the model). We explain first how the covariate C is defined and second, how its effect is introduced in Eq. (9). C represents respectively in the different models, the mean of monthly humidity, the mean of monthly temperature, and the accumulated monthly rainfall, for a defined temporal window. That is, the covariate is defined here to represent yearly effects in a given window of time that is critical for the way a specific climate factor affects transmission. This window was chosen as the one with the highest correlation to the total cases aggregated for the epidemic season. We examined windows of all possible sizes within the previous six months which precede the epidemic season, as climate factors influence the abundance of the vector and the fraction of vectors infected, and these effects on the vector are manifested in the human cases with a delay. The resulting windows chosen to calculate C are shown in Supplementary Fig. 4. The effect of the covariate on the transmission rate was then localized in time in Eq. (9), by multiplying C to spline S4, which corresponds to the time of the year preceding the epidemic season (and including the window during which C was obtained) (Supplementary Fig. 6). Parameter ({b}_{{{{{rm{RH}}}}}}) then quantifies the strength of the climate effect by modulating the seasonal component of the transmission rate corresponding to this time of the year. Finally, environmental noise is introduced in the transmission rate with a Gamma distribution Γ to represent additional fluctuations absent in the climate covariate (details are provided in ref. 46).In practice and for ease of implementation (including parameter inference), we transform the integral in Eq. (7) into a Markovian chain of differential equations going from the equation for ({lambda }_{1}) to that for ({lambda }_{j}) (Eqs. 10–11) following16,44:$${dlambda }_{1}/{dt}=(lambda -{k}_{1})k{tau }^{-1}$$ (10) $${dlambda }_{j}/{dt}=left({k}_{j-1}-{k}_{j}right)k{tau }^{-1},{for; j}=2$$ (11) Parameter estimationWe estimated parameters with an iterated filtering approach to maximize the likelihood for partially observed, nonlinear and stochastic dynamical models. Specifically, the estimation of parameters and initial conditions for all state variables was carried out with the iterated filtering algorithm known as MIF, for maximum likelihood iterated filtering, implemented in the R package “pomp” (partially observed Markov processes44,45,46. This “plug-and-play” method45,47 is simulation-based, meaning that parameter search relies on a large number of stochastic simulations from initial conditions to the end of the time series.For details on the method, see46 and for other applications to malaria and climate forcing, see refs. 9,16,43,48. This algorithm allows for consideration of both measurement and process noise, in addition to hidden variables, which are a typical limitation of surveillance records providing a single observed variable for the incidence. It consists of two loops, with the external loop essentially iterating an internal, “filtering” loop, and in so doing generating a new, improved estimate of the parameter values at each iteration. The filtering loop implements a selection process for a large number of “particles” over time. For each time step, a particle can be seen as a simulation characterized by its own set of parameter values. Particles can survive or die as the result of a resampling process, with probabilities determined by their likelihood given the data. From this selection process over the whole extent of the data, a new estimate of the parameters is generated, and from this estimate, a cloud of new particles is reinitialized using a given noise intensity adjusted by a cooling factor. The initial search in parameter space was performed with a grid of 10,000 random parameter combinations, and the output of this search was used as the initial conditions of a more local search46,49.The fitting algorithm provides an estimate of the likelihood itself. On the basis of the likelihood, we can then implement model comparisons (i.e., model selection) on the basis of the likelihood ratio test and DIC (Table 1). We further compared the ability of the different models to explain the temporal patterns of the data with different comparisons of the observed cases and the predicted ones via model simulation. Namely, we simulated 1000 runs from the respective stochastic MLE models from the estimated initial conditions. We obtained the median of monthly cases from these simulations as well as the uncertainty as to the 10–90% quantiles of the monthly cases. We considered visually whether this interval includes the observation and how close the median simulated cases are to the observed cases. We also considered whether the interannual cycles (in particular, their highs and lows) in the data and the simulations are in phase. We further compared the simulated predictions and observations by aggregating cases for the epidemic season. In a scatter plot of predictions against observations, we can assess how close the points fall to the diagonal, and whether the uncertainty of predictions contains the diagonal (where predictions equal observations). We more formally implemented this comparison with a criterion for evaluating stochastic predictions known as the CRPS, which is a commonly used measure of performance for probabilistic prediction of a scalar observation. It is a quadratic measure of the difference between the prediction cumulative distribution function (CDF) and the empirical CDF of the observation.Permutation testWe used a permutation procedure to test whether the association between humidity and malaria transmission might be confounded by season. In this procedure, we selected the humidity data for each of the 12-months and randomized these humidity data across years, rerunning the analysis with the randomized explanatory variables. Then, we correlated the predicted cases in a year with the humidity in the random window selected. We conducted 10,000 permutations, and sampling was done with replacement. For each permutation, we then calculated how well the humidity correlated with the time series of malaria. If the correlation between humidity and malaria incidence in the actual time series was significantly stronger than the correlations we observed in the randomized samples, we concluded that confounding by season was an unlikely explanation for this correlation.Out-of-fit predictionTo examine the ability of the process-based model to predict malaria incidence, we compared the total number of malaria cases observed for each city to those predicted by model simulations in a window of time not used to estimate the parameters. That is, monthly cases from January 1997 to only December 2008 were used as a training set for parameter estimation. We chose this length of the data set, to place ourselves in the position of having about two characteristic multiannual cycles (of 4–5 years) of the reported cases inform inference, while still leaving a sufficient number of seasons to test prediction on at least one such full cycle. The resulting MLE model relies on estimated state variables at the end of the training period as the initial conditions for predicting the first “out-of-fit” year. The estimated initial states are then obtained for January of each predicted year (between 2009 and 2014) by extending sequential filtering and assimilating the new data for the past 12 months. That is, because the inference method provides filtered values of the hidden variables, we can use these estimates and their distribution at a given time as initial conditions from which to simulate the following year. Parameter estimates are also continuously updated with the addition of one more year of data. Predictions are obtained by simulating the model forward over the next 12 months. To consider the uncertainty arising from both dynamic and measurement noise, the distribution of predicted observed cases is obtained for each month from 1000 simulations with initial conditions resampled from their estimated values. Departures between the yearly projections and the out-of-fit data can be used to evaluate the impact of humidity variability on the predictability of the upcoming season.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

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