We begin by describing our primary model outcomes. We then explain our method for calculating the R 0 of WNV. Finally, we detail how we estimated each parameter in the equation for R 0 using individual sub-models, and how we linked these estimates and propagated uncertainty to calculate R 0. Table 1 describes the components of our overall model and how they fit into our analysis.
First, we focused on spatial and temporal patterns in R 0 at the level of the community; we calculated WNV R 0 for bird communities between and separated spatially by county and temporally by month and year, and then fit a spatio-temporal model to the resulting WNV R 0 estimates which included 11 ecoregions in Texas, human population density, temperature and year as predictor variables. We quantified the importance of each species within each community by calculating the proportional change in R 0 that would be predicted to occur if that species were removed from the community and replaced by the other species in community in proportion to their relative abundance.
We considered species whose removal strongly increases or decreases R 0 as the least or most competent birds for WNV, respectively. We test if more abundant bird species are more physiologically competent for transmitting WNV and if an increase in species richness is predicted to decrease WNV R 0. We calculated R 0 as the expected number of mosquitoes that become infected following the introduction of a single infected mosquito into a population of susceptible birds and otherwise uninfected mosquitoes.
This calculation assumes that all mosquitoes have identical biting preferences and vector competence. We broke R 0 into two transmission steps: mosquito-to-bird transmission, which measures the expected number of each bird of species i that would become infected by a single infected mosquito; and bird-to-mosquito transmission, which calculates the expected number of mosquitoes infected by the infected birds of species i calculated in the mosquito-to-bird transmission step.
Written in this way, the sum of bird-to-mosquito transmission gives the number of new infected mosquitoes resulting from the single infected mosquito, which is the R 0 of WNV. This sum is a measure of vector competence, the total ability of a vector to transmit infection to a susceptible host [ 21 ], a key component of which is the transmission probability per feeding event [ 22 ].
Here we assumed that the mosquito is introduced into the susceptible population of birds on the first day following infection with the WN02 strain of WNV with a dose of 10 5. We predicted mosquito incubation rate of WNV and mosquito survival S Md ; estimates for mosquito survival are taken from a model for mosquito survival fit in [ 6 ] for each Texas bird community using the average temperature in each Texas county by month and year with temperature data obtained from NOAA [ 61 ].
We ignored the effect of mosquito species, which was fitted as a random effect in [ 6 ], due to the absence of data. These simplifications do not affect the relative effect of bird species, but will affect overall R 0 values, and could affect spatio-temporal patterns.
We assumed a constant mosquito biting rate of 0. In Eq. We measured bird titer and survival until day 8, which is one day longer than previous measures of host competence [ 20 , 63 ] and long enough to capture all known detectable measures of titer in birds. While the mosquito-to-bird ratio and mosquito biting rate will in reality be a function of parameters that vary both spatially and temporally such as ecoregion, season and temperature [ 29 ], as well as human population density, we assumed a constant mosquito-to-bird ratio here because of a lack of sufficient data on spatial and seasonal variation in this ratio across Texas and because our primary focus is on estimating R 0 as a function of the bird community.
Because we assumed no interaction between mosquito species and bird species in the probability of infection, and because the remaining parameters are scalars, differences in these parameters will affect the overall magnitude of R 0 estimates but will not affect qualitative patterns in R 0 due to variation among bird communities in space and time.
In Methods, Model components, we further unpack Eqs. The data and models that informed all parameters of both Eqs. The primary difficulty in estimating community competence for a diverse community of birds is that physiological responses to WNV and mosquito biting preferences are unknown for most bird species.
Obtaining these data for every species in a diverse community of birds would be infeasible. The effects of a predictor variable on the response of multiple species can be modeled using the phylogenetic relationships among the species to estimate the correlation among observations.
Classic phylogenetic regression approaches assume a correlated-residual model using phylogenetically independent contrasts PICs , where the residuals evolve as a Brownian motion process [ 64 ]; in other words, residuals are phylogenetically correlated. Here we used a newly implemented method built on the lme4 package in R that incorporates phylogenetic correlations by modeling them as random effects and allows for random slopes i.
Like most previous methods, the evolutionary history for each species is modeled as a sequence of normal independent errors. We estimated missing values for bird responses e. To impute, we first fitted a phylogenetic mixed model to all of the species for which we have data. Then, for each species without data, we first summed the evolutionary change in the response variable that occurred on all branches of the phylogeny leading to the most recent common ancestor between the species with a missing response and the most similarly related species that has data and was included in the mixed model.
To obtain these values we drew random normal multivariate if the mixed model includes multiple correlated species-level random effects samples for each branch, with means equal to the conditional modes of each branch multiplied by the branch length and variances equal to the conditional variances of each branch multiplied by the square of the branch lengths. Then, the evolutionary change that has occurred since the two species diverged is estimated by drawing random normal multivariate normal if the mixed model includes multiple correlated species-level random effects samples with a mean of zero the expected value for each unmeasured species is equal to that of the most closely related measured species because of the assumption of Brownian motion and standard deviation SD equal to the estimated SD of the species-level random effect s multiplied by the evolutionary distance branch length from the most recent common ancestor of the most closely related measured species.
We validated our phylogenetic imputation method in two ways. First, we calculated conditional R 2 using the methods outlined in [ 73 , 74 ] for models with and without a species-level phylogenetic random effect. We estimate conditional R 2 using code from the R package MuMIn [ 75 ], adapted to accommodate the structure of the phylogenetic mixed model objects.
Secondly, we use blocked leave-one-out cross validation [ 76 ] at the level of species for models with and without the species-level phylogenetic random effect to assess the effects of phylogenetic imputation on out-of-sample error. We present additional details and results for each of these forms of validation in Additional file 2 : Text S2; Table S1. For a vignette on the phylogenetic models built on lme4 see [ 69 ]. We modeled bird infection profiles and mortality probabilities using data from experimental infections of 47 bird species collected from 30 publications containing individual infection experiments; most of these data have been presented previously [ 6 ].
For the bird titer, bird survival and bird-to-mosquito transmission models in this paper we grouped data from the two primary WNV strains, NY99 and WN02 in our previous study [ 6 ] we were unable to detect a clear difference between the NY99 and WN02 strains.
To model bird titer profiles we used a log-normal mixed effects model; fixed effects included a Ricker function of day using day and log day as predictors of log-titer; see Additional file 2 : Text S3 or [ 77 ] for more information , infectious dose, bird body size and the interaction between day and bird body size. We used a random intercept and slope over both day and log day , which are constrained by the phylogenetic relationship among the species. We also included random intercepts for citation and infection experiment.
We modeled bird survival using the main effects of titer, day and bird body size as fixed effects; citation, infection experiment and bird species phylogenetically constrained were modeled using random intercepts due to a lack of data we were unable to estimate species-level variation in sensitivity to titer. The bird body size data used in both models were obtained from the searchable digital edition of Dunning [ 49 ]. Body size data were averaged if data for a given species were available for both sexes or multiple subspecies.
The body size for these species was taken as the center of the range. Approximately 0. For these species, the body sizes of all congeners were averaged.
We obtained bird abundances data from the Cornell Laboratory of Ornithology citizen science database eBird [ 52 , 60 ].
We used all complete checklists [ 52 , 79 ] submitted between January and December Complete checklists are defined as a report of all birds number of individuals of all species that are seen on a given outing. Checklists were aggregated spatially at the level of Texas counties for each month between January and December , which resulted in a total of 30, bird communities containing a total of unique species. To match scientific names, which occasionally differed between eBird and the consensus phylogeny, we used an automated lookup procedure to search both the IUCN [ 80 ] and Catalogue of Life [ 81 ] databases.
We present results for the complete Texas eBird dataset, which included all 30, available communities and species in Additional file 2 : Figures S4, S5; Text S5. We subset our data for the main analysis because many of the Texas bird communities were under-sampled e. The bird communities were chosen because they were all sampled with a minimum effort of 80 complete checklists. We chose 80 lists in an attempt to maximize the number of communities for our analysis while minimizing the retention of under-sampled communities.
To optimize the tradeoff between number of communities and data quality, we resampled 5— complete lists from the 46 most sampled communities communities with greater than lists times. We calculated the proportion of species missing in the subsampled communities as well as the root mean squared error RMSE in the relative proportions of all species between the two communities.
Using the rate of change in RMSE and species retention Additional file 2 : Figures S1, S2 , we determined that with fewer than 80 complete lists, the gain in total number of communities was not worth the increased error rate and loss of species representation, while at greater than 80 complete lists the loss in communities was too large for the small decrease in error and species loss.
We scaled raw bird counts by the detectability of each bird species to correct for incomplete sampling of bird communities and to control for variation in the quality of eBird records; alternatively or additionally, eBird lists can be weighted by user skill [ 79 , 82 ]. In all cases the first 60 hits were assessed for relevant information. In total, we took data from 12 sources which contained maximum detection distances for bird species.
However, we failed to find detection distances for waterfowl and shore birds; maximum detection distances roughly intermediate to values between woodland species and seabirds were assigned to 21 waterfowl and shore birds based on detection probabilities in the literature and our knowledge of the natural history of these species personal birding experience [ 83 ].
In order to fill in missing information for detection distance, we used the results of our literature search to fit a phylogenetic mixed effects model. Maximum detection distances for species in the Texas eBird data were estimated using a GLMM with a log-normal error distribution.
Body size was used as a fixed effect and species was included as a phylogenetic random effect. The eBird counts for each species were then adjusted by multiplying counts by the ratio of the maximum detection distance in the community to the detection distance of each species.
Using the square of maximum detection distance to reflect the relative spatial area sampled for each species may also be an appropriate method for adjusting raw eBird counts. Because mosquitoes Culex sp. Experimentally, f i is determined by sampling mosquitoes and determining the species origin of blood recovered from the mosquitoes; bird surveys are used to determine a i [ 20 , 45 , 46 ].
At one extreme, a bird with high infectious potential high titer and low mortality may contribute very little to the spread of WNV if it is avoided by mosquitoes.
For example, American robins Turdus migratorius have been found to infect the largest, or close to the largest, proportion of mosquitoes of any bird species in some bird communities in eastern USA because of their high abundance and mosquito preference [ 20 , 45 , 46 , 85 , 86 ], in spite of their relatively low titer [ 6 ]. In previous studies, when the blood of bird species i was recorded in a mosquito, but bird species i was unobserved in the community, the bird was either assigned a proportion corresponding to the rarest bird measured [ 45 ], or dropped from the analysis [ 20 ].
If bird species i was observed but its blood was not detected in a mosquito, it was assumed that a single mosquito was observed with the blood of bird species i [ 20 , 45 ]. While convenient, the assignment of arbitrary values to missing data leads to biting preferences spanning three orders of magnitude [ 20 , 45 ], which seems biologically implausible. Alternatively, a Bayesian statistical model can be used to estimate mosquito biting preference which is not directly observed , when bird species i or its blood is not observed.
Here we use a multinomial model in Stan [ 87 ], interfaced with R using rstan [ 88 ]. We model bird proportions using data from [ 45 ] and a Dirichlet prior, the conjugate prior to the multinomial distribution [ 89 ]. This prior distribution has a mean equal to one, median less than one and moderate dispersion, which assumes that birds are preferred in proportion to their abundance on average; the majority of bird species are preferred a bit less than proportional to their relative abundance, while a few bird species are preferred much more than proportional to their relative abundance.
Estimates of mosquito blood meals from the Dirichlet-multinomial Stan model were then used to impute biting preference on bird species in the Texas dataset by fitting a GLMM with Poisson-distributed error which includes a species-level phylogenetic random effect to the biting preferences estimated by the Dirichlet-multinomial model.
Biting preference estimates were scaled to a mean of one, and were then used to weight the observed proportions of each bird species. The weighted proportions of each bird species were obtained using:.
We use this as a proof of concept example to show how the imputed physiological responses of birds and mosquito biting preferences can be used to predict larger scale patterns. This model included thin plate splines for the log of human population density, temperature and year. We stress that in the absence of data on mosquito communities on the scale of the bird communities, the R 0 estimates from this model are driven by variation in bird communities and temperature only and cannot be taken at face value as accurate estimates of actual WNV transmission potential.
We first attempted to fit a model using the proportion of each ecoregion in each county, but could not overcome issues of concurvity analogous to co-linearity in a GAM model [ 93 ] in this model. Instead, we fitted a simplified model using a Markov random field to model the effects of ecoregion under the simplified assumption that each county had only a single ecoregion, which we chose as the most abundant ecoregion in each county.
We fitted a random effect of county to control for repeated measures within counties and to account for spatial variation within ecoregion. Ideally, we would also model fine-scale spatial variation using a thin plate spline over latitude and longitude coordinate pairs; however, models that included this predictor suffered greatly from concurvity problems. We used the inverse of the variance in R 0 estimates as weights.
For the 11 major different ecoregions in Texas, population density and county spatial shape data were obtained from [ 94 ]. This model provides estimates of both seasonal and long-term trends in WNV R 0 as the structure of bird communities have changed in the past two decades due to disturbances such as habitat change [ 95 ], habitat destruction [ 96 ], climate change [ 97 ] and the effect of the WNV epidemic itself [ 1 ] as well as spatial estimates of WNV R 0 by county.
Multi-faceted ecological models will underestimate uncertainty e. We focus on results from a model with all uncertainty propagated, but briefly discuss the impacts of ignoring uncertainty on both our quantitative and qualitative conclusions for more detailed results see Additional file 2 : Figure S6.
Table 2 gives a list of the sources of uncertainty and how each source was propagated. We set up our sub-models in the R code see Additional file 1 so that each source of uncertainty can be set individually to be either propagated or ignored, which can be used to obtain a first approximation assuming independence of errors for the relative effects of uncertainty in each sub-model on uncertainty in R 0 and on spatio-temporal patterns in R 0.
We briefly discuss which sources of uncertainty have the largest impact on our conclusions in Additional file 2 : Single sources of uncertainty: Reduced eBird Dataset. WNV transmission is controlled primarily by temperature variation across time and space. Despite these large differences at the larger scale of ecoregions, large uncertainty in the R 0 of individual communities makes it difficult to be certain about the size of the true variation in space and time.
In the least favorable months e. WNV R 0 estimates between months and among Texas counties. Red boxplots show R 0 estimates from a model where each community retained their specific eBird community, but whose temperature was replaced with the average temperature across all of Texas for that month also see Table 3 , Spatially averaged temperature.
Variation in R 0 within months attributable to variation in the bird communities red boxplots is considerably smaller than the variation explained by spatial variation in temperature. For example, in November the mean temperature across Texas is We estimate average mosquito-to-bird transmission per bite over the first 30 days of mosquito infection to be 2. We decomposed the importance of spatial and temporal variation in both temperature and bird community composition by comparing the mean absolute deviation MAD in predictions for R 0 between a full model and models with either the bird community or temperature aggregated across space or time Table 3.
Temperature variation across both space and time is more predictive of R 0 than bird community composition, though ignoring variation in the bird community across space does lead to R 0 estimates that differ from the full model by 0.
Allowing for temporal variation in temperature and bird community composition, the majority of the variation in R 0 within single months is due to spatial variation in temperature; variation in bird community composition is the next most important term Fig.
Ignoring the effects of fluctuations in mosquito populations, WNV R 0 was estimated to vary little across years Fig. Variation due to bird communities among ecoregions after controlling for temperature explained a small fraction of the variation in R 0 among regions Fig.
Spatio-temporal GAM model parameter estimates. Y-axes in panels a — c and the gradient in panel d show the additive effect of centered covariates on R 0. The gradient in panel d shows variation in R 0 among ecoregions explained by variation in bird communities. To evaluate the fit of our focal model relative to the model with latitude and longitude coordinate pairs average estimated concurvities of 0.
Using this method, RMSE for all estimates for our focal model and the model with latitude and longitude coordinate pairs were 0. This suggests that not including the thin plate spline across coordinate pairs results in little loss in terms of predictive power while also minimizing the possibility of over-fitting by reducing concurvity.
Mourning doves Zenaida macroura , recorded in all bird communities accounted for the largest dilution effect median: 1. Only two species were estimated to have a median effect greater than a 1. Keystone species. Bird species whose median estimates for their impact on R 0 when they are removed from each community they occupy are greater than a 1.
With no uncertainty propagated, median WNV R 0 estimates were on average 1. Ignoring uncertainty had a much larger effect on variation among communities: CV in WNV R 0 estimates were on average 1. This increase in magnitude and variation of the R 0 estimates when no uncertainty was propagated is caused by the nonlinear averaging of variation in mosquito-to-bird transmission, mosquito survival, bird-to-mosquito transmission and bird survival.
Species-specific contributions to R 0 also depend on whether uncertainty is propagated. While the most influential bird species northern cardinals and mourning doves were robust to choices about uncertainty propagation, the ranks and identities of some of the top ten most important amplifier and diluter species changed. We present results using the complete eBird data set in Additional file 2 : Figures S4, S5; Text S5, but suggest caution when drawing conclusions from these results because many of the estimates were obtained from poorly sampled bird communities.
Using the complete eBird data resulted in greater variation in estimates for all outcomes: variation in R 0 among communities increased Additional file 2 : Figures S4, S5 , variation explained in the spatio-temporal GAM model decreased, and the estimated impacts of individual bird species on R 0 were more extreme.
Despite our ability to estimate R 0 in individual bird communities, better data, such as mosquito populations on the same scale as the bird communities, are needed to make reliable quantitative estimates of WNV R 0 across space and time. Given the size of our estimated effect of temperature on WNV transmission and the fact that different mosquito species incubate WNV and feed at different rates across temperatures [ 6 , 14 ], variation in mosquito density and species composition among ecoregions and across seasons are likely the most important missing data needed to predict WNV R 0 reliably.
Our WNV R 0 predictions for Texas counties relied on estimates of the mosquito-to-bird ratio and mosquito biting rate based on sparse data from a different geographical region New Haven, CT, USA and are assumed to be spatially and temporally homogeneous. Though these simplifications result in an incomplete mechanistic model for WNV transmission, our model improves on previous models through its extensive use of empirical data, phylogenetic imputation to incorporate all birds within a community and treatment of both temperature-dependent mosquito incubation and survival.
Importantly, all of these models use a tiny fraction of the available data; parameters are often informed by a single study and occasionally neglect uncertainty this is most common in differential equation models, e.
While little data and a simplified life-cycle e. Though we do our best to reduce error and over-confident estimates by using as much empirical data as possible over all aspects of the life-cycle of WNV, because we assume a constant mosquito-to-bird ratio which assumes that the relative ratio of mosquito abundance to bird abundance is constant , mosquito biting rate and mosquito species composition, our estimates for WNV R 0 are likely biased upwards in spring and winter months and underestimate the true variability in WNV R 0 across space and time.
The first limitation arises because we assume a constant mosquito-to-bird ratio across months the value we used is based on data collected in June and July [ 45 ] ; we likely overestimated R 0 in months with low mosquito density and possibly underestimated R 0 in months with large mosquito populations.
This assumption will have the largest influence in spring months when we estimated R 0 transmission to be high because of a favorable temperature. In reality, small mosquito populations in these months probably result in lower WNV transmission. In the coldest winter months e. December through February , our assumption of a constant mosquito-to-bird ratio is unlikely to change our estimate of WNV epidemic potential R 0 greater or less than one in most counties because we already estimate most counties to have a low R 0 because of unfavorable temperatures.
However, in the warmest communities in winter months we estimated R 0 to be greater than one Fig. Even in the absence of any Texas-specific Culex mosquito population data, data on mosquito populations across seasons from anywhere in the mid-west USA could potentially be used to reduce the number of implausible estimates though it may be difficult to find good data on mosquito-to-bird ratios. Some experts suggest that it's acceptable to apply repellent with low concentrations of DEET to infants older than age 2 months.
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