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Host and gut content samplingA total of 94 individuals (42 A. speciosus, 9 A. argenteus, and 43 M. rufocanus) were captured from four sites within the Kamikawa Chubu national forest in the central area on the island of Hokkaido, Japan (Supplementary Table S1), and a total of 280 gut content (from the small intestine, cecum, and colon) and fecal matter (from the rectum) samples were collected for microbiome analysis (Supplementary Table S2). Based on 16S rRNA amplicon sequencing using Illumina Miseq, a total of 12,286,171 paired-end reads were obtained after quality filtering and chimeric sequence removal. There was an average of 43,879 reads per sample, although it varied among species and gut region (Supplementary Table S3).Within host species/among gut region gut microbiota alpha diversityAlpha diversity of the gut microbiota in the small intestine was significantly lower than the rectum, colon, and cecum in all three host species based on Shannon diversity, Faith’s PD, evenness, and number of ASVs as expected (GLME: all p  0.05; Fig. 1, Supplementary Fig. S2, Supplementary Tables S4–S7). Males had significantly higher alpha diversity within all gut regions of A. speciosus while female A. argenteus had significantly higher alpha diversity as compared to males (GLME, all p  0.05; Supplementary Tables S4–S7) while age had no effect in any gut region of any rodent species (GLME: all p  > 0.05; Supplementary Tables S4–S7).Figure 1Alpha diversity within each gut region of each species based on (a) Shannon diversity and (b) Faith’s PD. Dashed lines separate host species.Full size imageAmong host species alpha diversityMyodes rufocanus had significantly higher alpha diversity in all four gut regions as compared to both A. speciosus and A. argenteus based on all four diversity measurements (GLME: all p  More

Isolation and characterization of bean dreg-degrading strainsA 1362-bp amplification fragment of 16S rDNA was obtained by PCR (GenBank accession number MW406939). This sequence was compared with others in the GenBank database, aligning the 16S rDNA sequences with several Geobacillus sp. strains and constructed a phylogenetic tree (Fig. 2a). The phylogenetic tree clearly showed that strain GT-02 belongs to the G.toebii branch and was similar to G.toebii R-32652, G.toebii NBRC 107807, and G.toebii SK-1 with 99.78%, 99.63% and 99.05% similarities, respectively. According to the study described previously, G.toebii was a gram-positive, aerobic rod and motile bacterial26. G.toebii could produce acid from inositol and gas from nitrate. G.toebii could hydrolysis casein and utilize n-alkanes as carbon source27.Figure 2(a) Phylogenetic tree based on 16S rDNA gene sequences from related species of the genus Geobacillus constructed using the neighbour-joining method with 1000 bootstrap replicates. Branch length is indicated at each node. (b) The growth curve of strain GT-02 with temperature. (c) The growth curve of strain GT-02 with pH.Full size imageThe growth characteristics of strain GT-02, such as temperature and pH values, were investigated. The bacterial strain could grow within a range of 40–75 °C and pH 6.50–9.50, and the optimum temperature and pH were 65 °C and 7.50, respectively (Fig. 2b,c). Compared to other G.toebii strains, the maximum growth temperature and pH of strains R-32652 and SK-1 were 70 °C and 9.0026,28, respectively. These results showed that strain GT-02 was more resistant to high temperature and alkalinity. Fermentation temperature above 70 °C could effectively inactivate harmful microorganisms in organic solid waste12. Therefore, the fermentation temperature was set at 70 °C in this study.Changes in the composition of bean dregs during fermentationChanges in GI, TOC and TN of bean dregs during fermentationThe GI is traditionally used to evaluate the phytotoxicity and maturity of organic fertilizer12. As shown in Fig. 3a, both groups of experiments reached the standard of maturity (GI ≥ 85.00%). Therefore, the fermentation was terminated in five days. In the initial stage of fermentation, the GI of CK dropped to 51.85% on day 2, and the GI of T1 dropped to 41.98% on day 1. Phytotoxicity, which is usually caused by various heavy metals and low-molecular-weight substances, such as NH3 and organic acids, can reduce seed germination and inhibit root development29. During fermentation, bean dregs might produce NH3, organic acids and other substances, which could trigger a decrease in the GI. The GI of T1 showed a clear decrease, which was likely due to the production of toxic organic acids and might also explain the decrease in pH observed in T1 (Fig. 3d). Due to the degradation of organic acids, the GI of T1 increased to 95.06% on the third day and continued to increase to more than 100.00%, whereas in CK, the GI only reached 86.42% at the end of the fermentation. These results revealed that the maturity of T1 on day 3 was markedly higher than that of CK on day 5 and thus suggest that G.toebii can significantly enhance the fermentation efficiency by accelerating the maturation process and thus reducing the thermophilic fermentation period from 5 to 3 days.Figure 3Profiles of GI (a), TOC (b), TN (c), pH (d) and EC (e) during the fermentation process of CK and T1. The data represent the means ± standard deviations from three measurements.Full size imageTOC is usually used as an energy source by microorganisms30. The TOC loss in both CK and T1 increased during fermentation (Fig. 3b). The reduction of TOC was mainly caused by the production of carbon dioxide from bacterial respiration. The rate of TOC loss in T1 was higher than that in CK. At the end of the fermentation, the TOC loss of T1 was 11.78% higher than that in CK. Because of the addition of G.toebii, bacterial metabolism in T1 was more active, and organic degradation was faster.The TN loss in both CK and T1 also showed an upward trend (Fig. 3c). The loss of TN was mainly caused by the volatilization of ammonia nitrogen31. The rate of TN loss in T1 increased more than that of CK group. After fermentation (day 5), the TN loss in T1 was 6.83% higher than that of CK. The mineralization in T1 was more active and thus ammonia nitrogen was more, which was easy to cause volatilization. However, the bean dregs in CK were mature on the 5th day, while those in T1 were on the 3rd day. At this time, the TN loss of mature bean dregs in T1 was 5.66% lower than that in CK, which indicated that the bean dregs lost less nitrogen source when they reached the standard of maturity after the addition of G.toebii.Changes in pH and EC of bean dregs during fermentationThe variation in pH observed during fermentation is due to the interaction between inorganic nitrogen and organic acids produced by the decomposition of organic matter32. As shown in Fig. 3d, the pH of CK gradually increased to 8.72 at the end of the fermentation. The ammonification process and the release of free NH3 during organic matter (OM) degradation lead to increases in pH33. The pH of T1 decreased to 5.73 on day 1, which was due to the formation of more organic acids than CK, and then increased to 8.76 on day 2, which was due to acid consumption and ammonia formation. Figure 2c showed that GT-02 could hardly grow when the pH was lower than 6.00, but the heterogeneity of solid fermentation provided a possible living environment for the growth of GT-02. Subsequently, the pH of T1 slowly decreased to 8.10 due to ammonia volatilization or ammonia conversion. These study findings showed that the pH value of the fermentation process was significantly affected by the addition of GT-02. G.toebii can produce abundant high-temperature enzymes, such as amylase, protease, cellulase, xylanase, and mannanase17, which explains why the ammonification process was faster in T1 than in CK and thus the higher pH was found in T1.The EC, which is a measure of the total ion concentration, describes changes in the levels of organic and inorganic ions such as SO42−, Na+, NH4+, K+, Cl−, and NO3− during the fermentation process34. As shown in Fig. 3e, the EC of the two groups increased significantly during fermentation process (P  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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