Philippa Kaur

Candidate bird deterrentsWe identified four candidate bird deterrents that were suitable for deployment within a SN-toxic baiting program (Fig. 2). Specifically, we searched published studies and vendor websites to identify candidate bird deterrents that had a proven record of deterring birds, or features that we expected would deter all birds after a deployment of SN-toxic bait while not deterring wild pigs. These features included: (1) not deterring wild pigs (i.e., user programmable operating hours for after wild pigs visits or being bird-specific), (2) aversive to birds (i.e., erratic movements or irritating to birds), and (3) remotely operated (i.e., battery operated or effects lasting ~ 12 h if user applied).Figure 2Examples of potential bird deterrents tested in in north-central Colorado, USA during April–May 2020, including (A) control, no deterrent, (B) 7.5% concentration of methyl anthranilate, (C) a metal grate, (D), an inflatable scarecrow, and (E) a scare dancer. Photos property of USDA.Full size imageWe selected two frightening devices that offered visual and auditory stimuli, were battery-powered, and programmable to have a user-specified start time. The first frightening device was a 1.8 m inflatable scare dancer (Snake 6 ft Cordless Inflatable Scarecrow, AirCrow LLC, Lake Charles, LA, USA). The scare dancer was a yellow nylon tube shaped like a snake and inflated by a small fan and control unit powered by a 12 V battery connected to a programmable control panel. If using the scare dancer for SN-toxic bait deployment, our strategy would be to program the device to operate continuously starting 1 h before first light the morning after toxic bait was deployed. Our expectation would be that wild pigs would have already visited bait sites and consumed SN-toxic bait prior to scare dancer activation. Once activated, the scare dancer would deter non-targets away from any spilled SN-toxic bait during the morning after toxic baiting until operators arrived to clean the site.The second frightening device was an inflatable scarecrow called the Scarey Man Birdscarer (Clarratts Ltc, United Kingdom). This device was also powered by a small fan using a 12 V battery, activated by a timer, and inflated for 25 s every 18 min accompanied by an audible 112 db siren. The timing of the inflation could not be altered. The blaze-orange inflatable scarecrow bobbed up and down as it inflated and deflated, and emitted a siren wail. Our strategy with the inflatable scarecrow, following SN-toxic bait deployment, would be the same as the scare dancer, except the inflatable scarecrow could not be programmed to operate continuously.For the physical barrier treatment, we constructed a metal grate using a 2.4 m × 1.2 m sheet of #13-gauge steel diamond-shaped expanded metal. The maximum openings of the expanded metal were 1.0 cm and were raised (i.e., tapered upwards) to facilitate bait falling through the grate. We constructed the grate to sit 9.0 cm above ground using a frame of standard construction lumber. We also tapered the top of the wooden frame to reduce surface area and facilitate bait falling through the grate. If using the grate for SN-toxic bait deployment, our strategy would be to put the bait station on top of the grate. Our expectation would be that wild pigs would stand on the grate to access the bait station, and spilled particles would fall under the grate and be inaccessible to non-target animals.The chemical repellent treatment we tested was Avian Migrate™ Goose and Bird Repellent (Avian Enterprises, Jupiter, FL, USA) which contained 14.5% methyl anthranilate. Avian Migrate required dilution with water for all applications. We followed the label instructions for spot repelling, and used the strongest dilution recommended at 50:50 Avian Migrate and water, resulting in 7.5% methyl anthranilate. We used a hand-pump-pressurized garden sprayer to apply 500 ml of the mixture to a 3 × 3 m area which resulted in an even and thorough coating of the area. Aversion to methyl anthranilate may be a learned behavior as an irritant for birds36, therefore would need to be applied daily for 1–2 days prior to SN-toxic bating. If using the repellent for SN-toxic bait deployment, our strategy would be to spray the ground immediately surrounding bait stations for 2 nights prior to deploying toxic bait, and the night of toxic baiting. Our expectation would be that by the 3rd night of application non-target birds would be repelled from consuming particles of spilled bait that fell on the treated ground; after which, we could safely deploy SN-toxic bait.Field study on deterrent effectiveness for birdsWe initially selected and pre-baited ~ 60 sites in north-central CO using 5 kg of bird seed (Deluxe Blend Bird Seed, Wild Birds Unlimited, Fort Collins, CO, USA). Sites were selected in diverse land covers that were likely to hold small passerine birds, such as thickets, wind rows, near water sources, or along shelter belts; and based on distance to nearby sites (i.e., goal of  > 500 m to nearest site). We cleared sites of tall grass and debris to ease discovery and access to the bird seed by smaller birds. We visited sites every 2–3 days to replenish and maintain ~ 2 kg of bait at the sites. We pre-baited sites for ~ 4 weeks to ensure birds were well-acclimated to visiting sites daily.We monitored visitation to sites using remote cameras (RECONYX PC900, RECONYX Inc, Holmen, WI, USA) mounted on T-post approximately 5 m from the bait pile, 1.5 m above ground, and angled down at 70° to provide a consistent field of view at each site. Cameras were programmed to record time-lapse imagery every 2 min (i.e., 720 images/day) which was used to calculate indices of species visitation. We used the Colorado Parks and Wildlife Photo Database to process all time-lapse imagery (Ivan and Newkirk 2016). For each image, a single observer recorded presence and count of each unique species present. We selected the best 20 sites (Fig. 1) based on the greatest rates of bird visitation, greatest diversity of bird species visiting, and lowest presence of other species that consumed large quantities of the bird seed (e.g., raccoons, deer, skunks).For the trial, we randomly assigned a deterrent treatment (i.e., inflatable scarecrow, metal grate, methyl anthranilate) or control (i.e., no deterrent method) to five sites each. We re-used the control sites to test the scare dancer after testing the initial four treatments, because the scare dancers were received later than first three treatments. We visited bait sites daily and weighed the amount of bird seed remaining to calculate the amount consumed with digital scales (MeasureTek GGS_42964, MeasureTek Scale Co, Ltd, Vancouver, BC, Canada). We replenished each site to ensure ~ 2 kg of fresh bird seed was available each day.The trials were seven consecutive days (Table 1). We focused on species visitation from 1 h before first light (~ 0500 h) to midday (1200 h) each day, because this time period represented the critical hours in which hazards occurred at toxic bait sites22,24. We visited the bait sites between 1200 and 1400 h each day to replenish bait and prepare sites for the following day. The 7-day trial consisted of:

Days 1–2 = Pre-baiting days. No deterrent deployed.

Day 3 = Acclimation day. We deployed the deterrent devices but did not activate. Scare dancers were installed on a t-post 1.5 m above the bait sites. Inflatable scarecrows were placed on the ground 3 m away from the bait sites. Metal grates were deployed 3 m away from the bait sites. Methyl anthranilate was sprayed for first time in the 3 × 3 m area surrounding bait sites to initiate the learned repellency.

Day 4 = Pre-treatment day. This was the day we collected pre-treatment data (i.e., consumption and remote camera data) for comparison with treatment and post-treatment below. All deterrent devices remained inactive as described for acclimation day. The methyl anthranilate was sprayed in the same manner as before for the second time.

Day 5 = Treatment day. Both frightening devices were activated at 1 h prior to first light. The metal grate was installed over the bird seed. Methyl anthranilate was sprayed in the same manner as before for the third and final time.

Day 6 = Post-treatment day. All deterrent devices were inactivated but left in place similar to the pre-treatment day. The metal grate was moved 3 m away from the bait site. No methyl anthranilate was sprayed.

Day 7 = Removal day. We removed all our cameras and deterrent devices and ceased re-baiting at all sites.

Table 1 Strategies used to evaluate effectiveness of bird deterrents during a 7-day trial in north-central Colorado, USA during April–May 2020.Full size tableFor each site, we calculated an index of the number of passerine birds observed in each two-min time-lapse image (rate = average number of birds/two mins) during morning hours (i.e., 0500–1200) for the morning of pre-treatment, treatment, and post-treatment. We compared indices among each of the 3 days and five treatments using negative binomial mixed models and log-links with package glmmTMB37 in Program R v3.6.338. We used offsets of the number of hours monitored and site ID as a random effect to account for repeated (i.e., daily) measures taken at each site. We did not analyze for other species (i.e., predatory birds and mammals) because visitations were rare. For all analyses we calculated and examined the 95% confidence intervals (CIs) surrounding the regression coefficients (β) for non-overlap of zero to indicate statistical and biological differences.Effects of deterrents on captive wild pigsWe evaluated whether the deterrents influenced feeding behaviors of captive wild pigs. Specifically, we evaluated how wild pigs responded to the metal grate and methyl anthranilate, because these deterrent strategies would need to be in place as wild pigs visited bait sites, and we wanted to ensure wild pigs would not be deterred from feeding. Contrarily, neither of the deterrent devices should be encountered by wild pigs because these devices would be operated on a timer and set to activate after wild pigs visited toxic baiting sites. Therefore, we did not evaluate those treatments with captive wild pigs.For testing methyl anthranilate, we randomly selected and placed three captive wild pigs from the larger holding pen (i.e., two males and one female) into three 0.02 ha pens, respectively. We replicated this design twice, for a total of six pens (n = 18 wild pigs) tested. The wild pigs in each pen were acclimated for one night to the new pens and to feeding from two identical feed troughs (1.8 × 0.3 × 0.1 m) placed 3.2 m apart. Each night we fed ~ 10 kg of whole kernel corn in each trough and weighed any remaining corn the following morning. A 2-choice feeding test was conducted on nights two, three, and four, where we applied methyl anthranilate to a 3 × 3 m area surrounding one of the troughs using the same mixture as described above in CO. For the other trough, we did not apply methyl anthranilate to the surrounding soil. We applied the methyl anthranilate and whole kernel corn each evening of the 3-day treatment period.For testing the metal grate, we randomly selected and placed four captive wild pigs from the larger holding pen into two 0.2 ha pens, respectively. We replicated this design twice, for a total of four pens (n = 16 wild pigs) tested. A single feed trough (1.8 × 0.3 × 0.1 m) was placed in each pen. We placed the metal grate under the trough in one pen where it remained for the three nights of study. Two kg of pelleted sow ration were fed in each pen on night 1. On night two, ~ 10 kg of a placebo SN-toxic bait (i.e., HOGGONE without SN) and 1 kg of pelleted sow ration were fed in each pen. On night three we offered just 10 kg of placebo bait to evaluate whether spilled particles of the peanut paste-based bait16 would stick to the metal grate. We ceased testing the metal grate after the second replicate because we observed that wild pigs spilled small particles of the placebo bait which stuck to the top of the metal grate in the first replicate, followed by 100% aversion by wild pigs to the metal grate in the second replicate, rendering the metal grate a non-viable option for operational use.For the methyl anthranilate, we compared proportions of whole-kernel corn consumed in the 2-choice test using a linear model in Program R. We evaluated the interaction of treatment × night to determine if the application of methyl anthranilate influenced the amount of corn wild pigs consumed over time. We also tested the reduced model without the interaction to best interpret the unconditional main effects39. We did not analyze data from the metal grate treatment because the evaluation was stopped early, and the results were clear.Field evaluation of deterrent with toxic baitFor the final phase of this study, we evaluated the most effective deterrent identified in the first phase of the study (i.e., scare dancer deterrent device, see results) and implemented this deterrent device into a SN-toxic toxic baiting program for wild pigs in north-central TX. We followed methodologies established in previous studies (Table 2) to initiate a SN-baiting program24,40,41,42. Specifically, we initially deployed ~ 30 bait sites by placing ~ 11 kg of whole-kernel corn on the ground at locations with recent sign of wild pigs (e.g., fresh tracks, feces, wallowing, rooting). We installed one remote camera on a t-post 5 m away from each bait site, 1.5 m above ground, and angled down at 70°. We programmed cameras to capture time-lapse images every 5 min (i.e., 288 images/day). We revisited bait sites every day for 5 days to refresh bait (i.e., maintain 11 kg of corn) and view camera images for wild pigs. After day 5, we selected the 10 best sites (Fig. 1) using the highest ranked sites from this ranking system: (1) consistent wild pig visitation (i.e., ≥ 2 days in a row), (2) consistent visitation by a family group of wild pigs (i.e., ≥ 1 female with multiple juveniles or piglets), (3) consistent visitation by multiple family groups (4) consistent visitation of independent family groups not visiting other sites42. We also made sure to select bait sites that were  > 500 m apart to maintain independence among the groups of pigs visiting each site41,43.Table 2 Baiting strategy to locate, pre-bait, and train wild pigs to use bait stations and consume SN-toxic bait used in north-central Texas, USA during July 2020.Full size tableWe deployed wild pig-specific bait stations20 with ~ 13 kg of magnetic resistance on the lids21 at the 10 final sites and initiated a series of conditioning phases to acclimate wild pigs to open and consume bait from inside the bait stations (Table 2). We deployed two bait stations at sites with ≥ 10 wild pigs to ensure all wild pigs had sufficient access to bait. We deployed bait stations 10–30 m away from initial pre-baiting sites (where we originally placed corn on the ground) to reduce visitation by non-target animals that may be attracted to residual particles of corn. Where cattle were present, we also constructed 3-strand barbed-wire fences around the site to exclude them from accessing SN-toxic bait.We randomly selected five sites to deploy the deterrent devices, and five sites as controls (no deterrent devices). Three days prior to deploying SN-toxic bait, we deployed the deterrent devices but left them inactive to condition wild pigs to the presence of the devices. We mounted the deterrent devices on T-posts approximately 1.8 m above ground directly over each bait station with the battery box secured at the base of the T-post (Fig. 3). When we deployed SN-toxic bait, we programmed the deterrent devices to activate at 0520 h the next morning (i.e., 1 h before first-light). We waited until 0900–1200 h the next morning before visiting bait sites to allow ample testing time of the deterrent devices to deter birds, and to simulate realistic use in an operational setting. When we arrived at the bait site, we deactivated the deterrent devices and cleaned the surrounding area of any remaining spilled bait. We collected and weighed all spilled bait we could locate and turned over the soil surrounding the bait station to bury any small particles of spilled bait we could not collect.Figure 3Example of activated deterrent devices (scare dancers) mounted above bait stations containing a sodium nitrite toxic bait in north-central Texas, USA during July 2020. Photo property of USDA.Full size imageWe conducted systematic carcass searches along transects following the SN-toxic bait deployment. Specifically, we searched 400 m × 400 m transect grids centered on the bait sites every 50 m, walking transects oriented North/South the first day and East/West the second day. We generated the transects in ArcGIS (v10.8.1, Environmental Systems Research Institute, Redlands, CA, USA), and uploaded them to handheld devices (i.e., mobile phones or tablets) using ArcGIS Explorer (v20.0.1) to navigate along the transects. Additionally, we searched a smaller 50 m × 50 m transect grid centered on the bait sites every 5 m for three consecutive days, again switching between North/South, East/West, and North/South orientation each day, respectively. Transect spacing and distances were based on locations of carcasses found in a previous study with SN-toxic bait24. We searched transects for multiple days to ensure any carcasses were located and to determine if any animals succumbed to consuming spilled SN-toxic bait that may have been missed during our clean-up process days after deployment.We recorded sex, age based on tooth eruption44, weight, location, and evidence of SN-toxic bait consumption of any dead wild pigs that we located. Bait consumption was determined by observing bait in the mouth or stomach, or based on the percentage of methemoglobin in the blood by comparing the red-color-value of a drop of blood on a white laminated card to a standard curve45. For any non-target animals found dead, we recorded species, location, and evidence of SN-toxic bait consumption (as described above).We processed all time-lapse imagery from each bait station using the Colorado Parks and Wildlife Photo Database46. For each image, a single observer recorded the count of each species present. We did not include cattle because they were excluded from bait sites. We used two indices from the images for comparing the rates of visitation by different species. First, we used an index of the count of non-target animals/image during the hours that the deterrent devices were operating (0520–1200 h). We compared this index among the days of pre-, during, and post-activation periods of the deterrent devices to assess if the devices influenced the rate of visitation using linear models in program R. We analyzed sites with and without the deterrent devices separately to assess the effects of each treatment throughout the days independently.For the second index, we estimated rates of the number of wild pigs, non-target mammals, and non-target birds, respectively, observed per hour that visited bait sites. We followed methodology established by22, and used negative binomial generalized mixed models with package glmmTMB37 to compare rates of visitation between periods of pre- and post-SN-toxic bait deployment to assess changes relative to toxic baiting. We considered the change in rates of visitation to be attributed to lethality from SN-toxic bait for the populations of animals visiting the bait sites. We expect this methodology met the assumption that detection of animals remained consistent47 at bait sites because pre- and post-toxic periods were only separated by a single 24-h period when the toxic bait was deployed, and we refreshed the bait daily. We also compared the indices between treatments (with vs without deterrents) and the interaction of period × treatment. The models examined for each group of species were: rate of hourly visitation ~ period + treatment + period × treatment. We also used Site ID as random effects to account for repeated measures taken at each bait site.For the transect analysis, we calculated descriptive summaries of sexes, ages, and distances from carcass to nearest bait station for wild pigs that succumbed to the SN-toxic bait. We also summarized any non-target deaths and distances from the nearest bait site. All research methods for all phases of this study were approved under the USDA National Wildlife Research Center, Institutional Animal Care and Use Committee (protocol QA-3068), and performed and reported in accordance with ARRIVE guidelines and US EPA regulations. More

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Infection-age structured dynamicsFor the description of the dynamics, we follow the customary infection-age structured approach (for details see for instance Refs.4,10,11,12). Explicitly, we consider the infection-age structured dynamics of the number of individuals ({u}_{I}left(t,tau right)) at time (t) who were infected at time (t-tau) given by$$begin{array}{c}frac{partial }{partial t}{u}_{I}left(t,tau right)+frac{partial }{partial tau }{u}_{I}left(t,tau right)=0end{array}$$
(7)
with boundary condition$$begin{array}{c}{u}_{I}left(t,0right)=jleft(tright).end{array}$$
(8)
Here, (tau) is the time elapsed after infection, referred to as infection age, and (jleft(tright)={int }_{0}^{infty }{k}_{I}(t,tau ){u}_{I}left(t,tau right)dtau) is the incidence, with ({k}_{I}(t,tau )) being the rate of secondary transmissions per single primary case.The solution is obtained through the method of characteristics32 as$$begin{array}{c}{u}_{I}left(t,tau right)=jleft(t-tau right)end{array}$$
(9)
for (tge tau) and ({u}_{I}left(t,tau right)=0) for (t1 for countries and for US locations.The daily death counts (Delta {n}_{W}left(tright)={n}_{W}left(tright)-{n}_{W}left(t-1right)) are considered to contain reporting artifacts if they are negative or if they are unrealistically large. This last condition is defined explicitly as larger than 4 times its previous 14-day average value plus 10 deaths, (Delta {n}_{W}left(tright) >10+4times frac{1}{14}left({n}_{W}left(tright)-{n}_{W}left(t-14right)right)), from a non-sparse reporting schedule with at least 2 consecutive non-zero values before and after the time (t), (Delta {n}_{W}left(tright)ne frac{1}{5}left({n}_{W}left(t+2right)-{n}_{W}left(t-3right)right)).Reporting artifacts identified at time (t) are considered to be the result of previous miscounting. The excess or lack of deaths are imputed proportionally to previous death counts. Explicitly, death counts are updated as$$begin{array}{c}{n}_{W}left(t-1-iright)leftarrow {n}_{W}left(t-1-iright)frac{{n}_{W}{left(t-1right)}_{estimated}}{{n}_{W}left(t-1right)}end{array}$$
(32)
with ({n}_{W}{left(t-1right)}_{estimated}={n}_{W}left(tright)-frac{1}{7}left({n}_{W}left(t-1right)-{n}_{W}left(t-8right)right)) for all (ige 0). In this way, (Delta {n}_{W}left(tright)) is assigned its previous seven-day average value.The expected daily deaths, (Delta {n}_{D}(t)), are obtained through a density estimation multiscale functional, ({f}_{de}), as (Delta {n}_{D}(t)={f}_{de}left(Delta {n}_{W}left(tright)right)), which leads to the estimation of the expected cumulative deaths at time (t) as ({n}_{D}left(tright)={n}_{W}left({t}_{0}right)+{sum }_{s={t}_{0}+1}^{t}Delta {n}_{D}(s)). Specifically,$$begin{array}{c}{f}_{de}left(Delta {n}_{W}left(tright)right)=left(1-{r}_{1}right)d{d}_{0}+{r}_{1}left(left(1-{r}_{2}right)d{d}_{1}+{r}_{2}d{d}_{2}right)end{array}$$
(33)
with$$begin{array}{c}{r}_{1} = {e}^{-0.3d{d}_{1}},end{array}$$
(34)
$$begin{array}{c}{r}_{2} = {e}^{-3d{d}_{2}},end{array}$$
(35)
$$begin{array}{c}d{d}_{0}={ma}_{14}left({ma}_{14}left(Delta {n}_{W}left(tright)right)right),end{array}$$
(36)
$$begin{array}{c}d{d}_{1}={rg}_{12}left({ma}_{14}left(Delta {n}_{W}left(tright)right)right),end{array}$$
(37)
$$begin{array}{c}d{d}_{2}={rg}_{48}left({ma}_{14}left(Delta {n}_{W}left(tright)right)right),end{array}$$
(38)
where ({ma}_{14}left(cdot right)) is a centered moving average with window size of 14 days and ({rg}_{sigma }left(cdot right)) is a centered rolling average through a Gaussian window with standard deviation (sigma). The specific value of the window size has been chosen to mitigate weekly reporting effects. The values of the standard deviations of the Gaussian windows have been selected to achieve a smooth representation of the expected death estimation for each country as shown in the bottom panels of Supplementary Fig. S1.Reporting delaysWe consider an average delay of two days between reporting a death and its occurrence. This value is obtained by comparing the daily death counts reported for Spain1 and their actual values33 from February 15 to March 31, 2020. The values of the root-mean-squared deviation between reported and actual deaths shifted by 0, 1, 2, 3, and 4 days are 77.9, 58.4, 38.5, 58.7, and 88.6 deaths respectively.Infection fatality rate ((IFR))The infection fatality rate is computed assuming homogeneous attack rate as$$begin{array}{c}IFR=frac{1}{{sum }_{a}{g}_{a}}{sum }_{a}{IFR}_{a}{g}_{a} ,end{array}$$
(39)
where ({mathrm{IFR}}_{a}) is the previously estimated (IFR) for the age group (a)5 and ({g}_{a}) is the population in the age group (a) as reported by the United Nations for countries18 and the US Census for states19.Clinical parametersWe obtained the values of the average ({tau }_{G}) and standard deviation ({sigma }_{G}) of the generation time from Ref.13, of the averages of the incubation ({tau }_{I}) and symptom onset-to-death ({tau }_{OD}) times from Refs.5,14, and of the average number of days (Delta {t}_{TP}) of positive testing by an infected individual from Refs.15,17. The average time at which an individual tested positive after infection ({tau }_{TP}) was computed as ({tau }_{TP}={tau }_{I}-2+Delta {t}_{TP}/2), where we have assumed that on average an individual started to test positive 2 days before symptom onset. The average seroconversion time after infection ({tau }_{SP}) was estimated as ({tau }_{I}) plus the 7 days of 50% seroconversion after symptom onset reported in Ref.16.Dynamical constraints implementation with discrete timeWe implemented the dynamical constraints to compute the infectious and infected population as outlined in the main text and as detailed in the previous section of this document, using days as time units. Time delays were rounded to days to assign daily values.The first derivative of the cumulative number of deaths is computed as$$begin{array}{c}frac{d{n}_{D}left(tright)}{dt}=Delta {n}_{D}left(tright),end{array}$$
(40)
with (Delta {n}_{D}left(tright)={n}_{D}left(tright)-{n}_{D}(t-1)).The growth rate was computed explicitly from the discrete time series as the centered 7-day difference$$begin{array}{c}{k}_{G}left(tright)=frac{1}{7}left({mathrm{ln}}left(Delta {n}_{D}left(t+4right)+Delta {n}_{D}left(t+3right)right)-{mathrm{ln}}left(Delta {n}_{D}left(t-3right)+Delta {n}_{D}left(t-4right)right)right).end{array}$$
(41)
The 7-day value was chosen to mitigate reporting artifacts.Confidence and credibility intervalsConfidence intervals associated with death counts were computed using bootstrapping with 10,000 realizations34. These confidence intervals were combined with the credibility intervals of the (IFR) in infectious and infected populations assuming independence and additivity on a logarithmic scale.Fold accuracyThe fold accuracy, ({F}_{A}), is explicitly computed as$$begin{array}{c}{mathrm{log}}{F}_{A}=frac{1}{N}{sum }_{i=1}^{N}left|{mathrm{log}}{x}_{i}^{obs}-{mathrm{log}}{x}_{i}^{est}right|,end{array}$$
(42)
where (left|cdot right|) is the absolute value function, ({x}_{i}^{obs}) is the ({i}^{th}) observation, ({x}_{i}^{est}) is its corresponding estimation, and (N) is the total number of observations.Inference and extrapolationBecause of the delay between infections and deaths, inference for the values of the growth rate and infectious populations ends on December 30, 2020 and for the values of the infected populations ends on December 26, 2020. Extrapolation to the current time (January 21, 2021) is carried out assuming the last growth rate computed.Reproduction numberThe quantities ({R}_{t}) and ({k}_{G}left(tright)) are related to each other through the Euler–Lotka equation, ({R}_{t}^{-1}={int }_{0}^{infty }{f}_{GT}left(tau right){e}^{-{k}_{G}left(tright)tau }dtau ,) which considers (jleft(t-tau right)simeq {e}^{-{k}_{G}left(tright)tau }jleft(tright)) in the renewal equation (jleft(tright)={int }_{0}^{infty }{k}_{I}left(t,tau right)jleft(t-tau right)dtau). Generation times can generally be described through a gamma distribution ({f}_{GT}left(tau right)=frac{{beta }^{alpha }}{Gamma left(alpha right)}{tau }^{alpha -1}{e}^{-beta tau }) with (alpha ={tau }_{G}^{2}/{sigma }_{G}^{2}) and (beta ={tau }_{G}/{sigma }_{G}^{2}), which leads to ({R}_{t}={left(1+{k}_{G}(t)/beta right)}^{alpha }) for ({k}_{G}(t) >-beta) and ({R}_{t}=0) for ({k}_{G}left(tright)le -beta). In the case of the exponentially distributed limit ((alpha simeq 1)) or small values of ({k}_{G}(t)/beta), it simplifies to ({R}_{t}=1+{k}_{G}left(tright){tau }_{G}) for ({k}_{G}left(tright) >-1/{tau }_{G}) and ({R}_{t}=0) for ({k}_{G}left(tright)le -1/{tau }_{G}). Global prevalence data were obtained from multiple data sources35,36,37,38,39,40,41,42, as described in Supplementary Table S1. More

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In this study, the HFLUX model was coupled with the SCEM-UA algorithm for analyzing the uncertainties of the model inputs. The specific procedures started with selecting the inputs of the HFLUX model. With the linked HFLUX and SCEM-UA model and implementation of an iteration scheme, the uncertainty of each of the selected inputs was obtained based on the ranges (minimum and maximum values) of the input data/parameters and the Latin hypercube sampling. The simulations were then compared against the observed data to evaluate the performance of the SCEM-UA algorithm. These steps are depicted in Fig. 1.Figure 1Flowchart for the uncertainty analysis.Full size imageRiver water temperatures simulated by the HFLUX modelRiver water temperature affects the water quality and the ecosystem health, and hence control of river water temperature is important to mitigation of its adverse effects1. The HFLUX model was used to simulate the streamflow temperatures at different locations and times. The model is highly flexible in terms of choosing the solution methods for solving the governing equations and selecting the energy budget terms such as shortwave solar radiation, latent heat flux, and sensible heat transfer flux. The model input data include the initial spatial and temporal temperature conditions, stream geometry data, discharge data, and meteorological data8. The water balance and energy balance equations are respectively given by8:$$frac{partial A}{{partial t}} + frac{partial Q}{{partial x}} = mathop qnolimits_{L}$$
(1)
$$frac{{partial left( {Amathop Tnolimits_{w} } right)}}{partial t} + frac{{partial left( {Qmathop Tnolimits_{w} } right)}}{partial x} = mathop qnolimits_{L} mathop Tnolimits_{L} + R$$
(2)
$$R = frac{{Bmathop varphi nolimits_{total} }}{{mathop rho nolimits_{w} mathop Cnolimits_{w} }}$$
(3)
where A is the cross section area of the stream (m2), x is the distance along the stream (m), t is the time (s), Q is the discharge of the stream (m3/s), qL is the lateral inflow per unit stream length (m2/s), Tw is the stream temperature ((^circ C)), TL is the temperature of the lateral inflow ((^circ C)), R is the energy flux (source or sink) per unit stream length ((^circ C) m2/s), B is the width of the stream (m), (mathop varphi nolimits_{total}) is the total energy flux to the stream per surface area (W/m2), (mathop rho nolimits_{w}) is the density of water (kg/m3), and (mathop Cnolimits_{w}) is the specific heat of water (J/kg (^circ C)). Equation (3) is based on a thermal datum of 0 (^circ C) and the impact on the absolute value of the advective heat flux term. In Eq. (2), if qL is negative, the first term on the right-hand side of the equation becomes a loss of qLTw. Also, dispersive heat transport that is omitted in Eq. 2 is negligible when the longitudinal change in water temperature is small in comparison to the temporal changes8.SCEM-UA algorithmThe SCEM-UA algorithm provides posterior distribution functions for the model parameters and input data by generating an initial sample from the parameter space. First, the indicators of n, q, and s that are respectively dimension (the number of investigate inputs), number of complexes (the population to be divided), and population (the number of sample points) are determined for the algorithm. Then, the algorithm searches the sampling points in the feasible space and sorts the points according to the density. The algorithm determines the sequence and complexes based on those points. The sequence is the first q points of the population and complexes are a collection of m points from the population. Note that m = s/q. In the next step, the points of each complex are sorted based on the density, which can be mathematically expressed as20:$$left{ {begin{array}{*{20}c} {mathop alpha nolimits^{k} le T,,,,,,,,,mathop theta nolimits^{t + 1} = Nleft( {mathop theta nolimits^{t} ,,mathop Cnolimits_{n}^{2} mathop Sigma nolimits^{k} } right)} \ {mathop alpha nolimits^{k} > T,,,,,,,,mathop theta nolimits^{t + 1} = Nleft( {mathop mu nolimits^{k} ,,mathop Cnolimits_{n}^{2} mathop Sigma nolimits^{k} } right)} \ end{array} } right.$$
(4)
where k = 1,2,…,q, α is the ratio of the mean posterior density of the m points of complexes to the mean posterior density of the last m generated points of sequences, (theta) is the points of complexes, ({c}_{n}=frac{2.4}{sqrt{n}}) , (T={10}^{6}), (mu) is the mean, and ∑ denotes the covariance. To investigate the new points created by the algorithm, the points of complexes are replaced by20:$$left{ {begin{array}{*{20}l} {Omega ge Zquad replace,best,member,of,mathop Cnolimits^{k} ,with,mathop theta nolimits^{t + 1} } \ {Omega < Zquad mathop theta nolimits^{t + 1} = mathop theta nolimits^{t} ,,,,,,,,,,,,,,,,,,,,,} \ end{array} } right.$$ (5) where (mathop Cnolimits^{k}) is the Kth complex, Z is drawn from the uniform distribution in the range of 0–1, and Ω is calculated by20:$$Omega = frac{{Pleft( {left. {mathop theta nolimits^{t + 1} } right|y} right)}}{{Pleft( {left. {mathop theta nolimits^{t} } right|y} right)}}$$ (6) where (Pleft( {left. {mathop theta nolimits^{t + 1} } right|y} right)) and (Pleft( {left. {mathop theta nolimits^{t} } right|y} right)) are the posterior probability distributions for (mathop theta nolimits^{t + 1}) and (mathop theta nolimits^{t}), respectively. Then, the algorithm examines the following condition for each complex. If it is rejected, the algorithm replaces the worst member ({c}^{k})(the point with the lowest density) with ({theta }^{t+1}) 20.$$mathop Gamma nolimits^{k} le T,,and,,Pleft( {{{mathop theta nolimits^{t + 1} } mathord{left/ {vphantom {{mathop theta nolimits^{t + 1} } y}} right. kern-nulldelimiterspace} y}} right) < ,Pleft( {{{mathop Cnolimits_{m}^{k} } mathord{left/ {vphantom {{mathop Cnolimits_{m}^{k} } y}} right. kern-nulldelimiterspace} y}} right)$$ (7) where ({Gamma }^{k}) is the ratio of the posterior density of the best (the point with the highest density) to the posterior density of the worst member of ({c}^{k}). The last step is to examine (beta) and L. Note that (beta) = 1 and L = m/10. If (beta < L), (beta = beta + 1) and the algorithm returns to sort complex points. Otherwise, the algorithm examines the Gelman and Rubin convergence6, and eventually provides the posterior distribution functions20. The value of the Gelman and Rubin convergence should be less than 1.2. The Gelman and Rubin convergence is examined by:$$R = sqrt {frac{g - 1}{g} + frac{q + 1}{{q.g}}frac{B}{W}}$$ (8) where g is the number of iterations within each sequence, B is the variance between the q sequence means, and W is the average of the q within-sequence variances for the parameter under consideration20.Study AREAMeadowbrook Creek was selected to test the methods proposed in this study8. The creek flows through the City of Syracuse in New York. Thus, this catchment consists of high residential and industrial land covers, which contribute runoff to the main channel. The creek is about 4 km long. A portion of this creek (475 m long) was selected for the modeling for a period of June 13–19, 2012 in this study. The upstream boundary condition in the HFLUX model was set based on the water temperature of the creek observed at the upstream station8. The uncertainty of the model inputs was examined at three selected points as shown in Fig. 2. Note that the input values at these three points had greater relative changes than the changes at other locations, which provided the possibility to improve the evaluation of the algorithm performance. In addition, these three locations had the same sampling of the selected input data. During the simulation period, the streamflow velocity varied within a range of 0.06–0.63 (m/s). The daily temperature changed between 8.9 and 28.2 °C. The relative humidity, used to calculate the total energy flux to the stream per surface area, changed from 36 to 93%. The creek bed mainly consisted of clay, cobbles, sand, and gravel materials. The basic statistics of the data/variables used in the HFLUX model are presented in Table 1. Figure 2 shows the study area, the creek, and the three selected points for analysis.Figure 2Study area and the locations of three evaluation sections (the gray enlarged map shows the State of New York), the map in this Figure is created by Google Earth 7.0.2.8415 (https://google.com/earth/versions).Full size imageTable 1 Basic statistics of the data/variables used in the HFLUX model.Full size tableEthical approvalAll authors accept all ethical approvals.Consent to participateAll authors consent to participate.Consent to publishAll authors consent to publish. More

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