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Effective medium approximation
For a linearized version of Fisher , we first obtain the effect of disorder on invasion velocity using Effective Medium Approximation (EMA). Using the EMA approach, we can replace the spatially irregular diffusion constant with a uniform one in which the effective diffusivity is everywhere equal to (D_{e}) and can replace (D_0(1+xi f(x))) in Eq. (2)44. To obtain (D_{e}), one needs to discretize Eq. (2) using a finite-difference method, which leads to the following equation for the population density39:

$$begin{aligned} frac{partial C_i(t)}{partial t}=sum _{jin {i}}W_{ij}[C_j(t)-C_i(t)]+RC_i(t) ;, end{aligned}$$
(4)

where j belongs to nearest neighbors of i and (W_{ij}=D_{ij}/delta ^2) stands for the density flow rate between units i and j with distance of (delta). Due to spatially irregular diffusion constant we have (W_{ij}=W_0(1+xi f_{ij})). Following44, one has

$$begin{aligned} D_e=bigg ( int _{0}^{infty } frac{g(w) dw}{w} bigg )^{-1} end{aligned}$$
(5)

where g(w) is probability density function for (W_{ij}). Applying this approach to our case leads to

$$begin{aligned} D_{e}=D_0(1-|xi |^2/3) end{aligned}$$
(6)

where (|xi |) stands for dimensionless magnitude of (xi) ((xi ^2) has a physical dimension of length, meter).
Perturbation analysis for invasion front fluctuations
The first step towards the study of dynamics of a propagating front is linearizing (3) by neglecting the (C^2) term in an environment with effect diffusion constant, (D_e), as:

$$begin{aligned} frac{partial C}{partial t}=RC+frac{partial }{partial x} big ((D_e + xi f(x)) frac{partial }{partial x} Cbig ) end{aligned}$$
(7)

This is based on the fact that near the front, the cell density is (C ll 1). In other words, focusing on the dynamics of the front position, automatically grants us the possibility of linearizing (3).
The construction of the solution can be proceeded according to a valuable insight given in the classic paper43, where the particle density is written as follows

$$begin{aligned} C(zeta ,t) approx C_0(zeta +eta (t),t)+ delta C_1(zeta ,t) end{aligned}$$
(8)

where C is written in the comoving frame and (zeta = x-vt). It is assumed that (delta C_1 ll 1) and in the same order as the perturbing function. So that terms containing f(x) and (delta C_1) can be neglected. Furthermore, (C_0) is assumed to satisfy the linearized Eq. (3) with (xi =0), i.e.

$$begin{aligned} frac{partial C_0(zeta ,t)}{partial t}-hat{Gamma } C_0(zeta )=,& {} frac{partial C_0(zeta ,t)}{partial t} nonumber \&-bigg (D_e frac{d^2}{dzeta ^2} + vfrac{d}{dzeta } + Rbigg )C_0(zeta ,t) = 0 end{aligned}$$
(9)

Which has the following solution

$$begin{aligned} C_0(zeta ,t) = frac{1}{sqrt{4pi D_e t}}e^{-frac{1}{2}sqrt{frac{R}{D_e}}zeta }e^{-frac{zeta ^2}{4D_e t}} end{aligned}$$
(10)

The first term in (8) describes the effects of the perturbing function f(x) on the position of the propagating front, while the second term shows the change in the shape of the front. This approach has also been employed and well explained in a recent paper17. As shown in17,43, to determine the effective diffusion coefficient for the fluctuating front, it is sufficient to solve (3) using (8) for (eta (t)). Note also that since we are interested in the dynamics of the system in long times ((t gg frac{1}{R})), (v_e) can be assumed to be equal to (2sqrt{R D_e})45. Plugging (8) in Eq. (9) expressed in comoving coordinates and considering (xi =bar{xi }/D_e) yields

$$begin{aligned} frac{partial delta C_1}{partial t} – hat{Gamma }delta C_1 + dot{eta (t)} C_0(zeta ,t) = bar{xi } bigg (f(zeta ) C’_0(zeta ,t)bigg )’ end{aligned}$$
(11)

Noting that the operator (hat{Gamma }) is not self-adjoint (The adjoint of (hat{Gamma }) is: (hat{Gamma ^dagger }=D_e dfrac{d^2}{dzeta ^2} – v_e dfrac{d}{dzeta } + R)) and following43, we multiply Eq. (11) from the left in the eigenfunction of (hat{Gamma ^dagger }) with 0 eigenvalue (which is (e^{sqrt{frac{R}{D_e}}zeta })) and integrate. Thus,

$$begin{aligned} {,} & int _{-infty }^{infty } e^{sqrt{frac{R}{D_e}}zeta }frac{partial delta C_1(zeta ,t)}{partial t} dzeta + dot{eta (t)}int _{-infty }^{infty }e^{sqrt{frac{R}{D_e}}zeta }C’_0(zeta ,t) dzeta nonumber \&quad = bar{xi } int _{-infty }^{infty } e^{sqrt{frac{R}{D_e}}zeta }bigg (f(zeta ) C’_0(zeta ,t)bigg )’dzeta end{aligned}$$
(12)

Which yields

$$begin{aligned} dot{eta }(t) =bar{xi } dfrac{int _{-infty }^{infty } e^{sqrt{frac{R}{D_e}}zeta }bigg (f(zeta ) C’_0(zeta ,t)bigg )’ dzeta }{int _{-infty }^{infty }e^{sqrt{frac{R}{D_e}}zeta }C’_0(zeta ,t) dzeta } end{aligned}$$
(13)

Which can further be simplified into

$$begin{aligned} dot{eta }(t)=,& {} bar{xi }dfrac{int _{-infty }^{infty } e^{sqrt{frac{R}{D_e}}zeta }f(zeta ) C’_0(zeta ,t) dzeta }{int _{-infty }^{infty }e^{sqrt{frac{R}{D_e}}zeta }C_0(zeta ,t) dzeta } nonumber \=,& {} bar{xi } e^{-frac{R t}{4}}int _{-infty }^{infty } e^{sqrt{frac{R}{D_e}}zeta }f(zeta ) C’_0(zeta ,t) dzeta end{aligned}$$
(14)

Or equivalently,

$$begin{aligned} eta (t) =bar{xi } int _{0}^{t} dtau e^{-frac{Rtau }{4}} int _{-infty }^{infty } dzeta e^{sqrt{frac{R}{D_e}}zeta }f(zeta ) C’_0(zeta ,tau ) end{aligned}$$
(15)

According to17, the effective diffusion would be given by

$$begin{aligned} D_{C} = dfrac{langle eta ^2(t) rangle }{2t} end{aligned}$$
(16)

Or

$$begin{aligned} D_{C}=,& {} frac{bar{xi }^2}{2t}int _{0}^{t}d{T_1}int _{0}^{t}d{T_2}int _{-infty }^{infty } C’_0(zeta ,T_1)C’_0(zeta ,T_2) nonumber \&times e^{-frac{R T_1}{4}}e^{-frac{R T_2}{4}}e^{2sqrt{frac{R}{D_e}}zeta } dzeta end{aligned}$$
(17)

where we have performed an ensemble average over (eta ^2(t)) using the fact that (langle f(x)f(y) rangle = delta (x-y)). A numerical calculation of (16) can be readily computed using any mathematical software. However, valuable insight can still be obtained from (17), if we use dimensionless parameters (tau _i=frac{T_i}{t}) and (sigma =sqrt{frac{R}{D_0}}zeta). In other words

$$begin{aligned} D_{C}=,& {} bar{xi }^2 dfrac{sqrt{R}}{32pi D^{3/2}_0}int _{0}^{1}dtau _1int _{0}^{1}dtau _2int _{-infty }^{infty }dsigma nonumber \&times dfrac{left( 1+dfrac{sigma }{Rttau _1}right) left( 1+dfrac{sigma }{Rttau _2}right) }{sqrt{tau _1tau _2}}e^{-frac{sigma ^2}{4Rttau _1}}e^{-frac{sigma ^2}{4Rttau _2}}e^{sigma }e^{-R tfrac{(tau _1+tau _2)}{4}} end{aligned}$$
(18)

Equation (18) gives the effective diffusion coefficient for the stochastic behavior of the front. For a diffusive behavior, we would expect this effective diffusion coefficient to tend to a constant at large times. At large times, we can approximate the integral as follows,

$$begin{aligned} D_{C} approx bar{xi }^2 dfrac{sqrt{R}}{32pi D^{3/2}_e}int _{0}^{1}dtau _1int _{0}^{1}dtau _2int _{-infty }^{infty }dsigma dfrac{1}{sqrt{tau _1tau _2}}e^{-frac{sigma ^2}{4Rttau _1}} e^{-frac{sigma ^2}{4Rttau _2}}e^{sigma }e^{-R tfrac{(tau _1+tau _2)}{4}} end{aligned}$$
(19)

Luckily, Eq. (19) can be evaluated exactly to yield

$$begin{aligned} D_{C}&approx bar{xi }^2 dfrac{sqrt{R}}{32pi D^{3/2}_e} nonumber \&quad frac{2 pi (2 R t-1) text {Erf}left( frac{sqrt{R t}}{2}right) -2 pi e^{2 R t} text {Erf}left( frac{3 sqrt{R t}}{2}right) +2 pi e^{2 R t} text {Erf}left( sqrt{2} sqrt{R t}right) +8 sqrt{pi } e^{-frac{1}{4} (R t)} sqrt{R t}-4 sqrt{2 pi } sqrt{R t}}{R t} end{aligned}$$
(20)

Where Erf(x) is the error function. As (trightarrow infty) this gives the following simple relation for the diffusion constant for the wave front

$$begin{aligned} D_{C}(trightarrow infty ) = bar{xi }^2 dfrac{sqrt{R}}{8 D^{3/2}_e} end{aligned}$$
(21)

Substituting (xi =bar{xi }/D_e) and (D_e=D_0(1-|xi |^2/3)), we will get the following beautiful equation for the effective diffusion constant of the front at large times:

$$begin{aligned} D_{C}(trightarrow infty ) =dfrac{1}{8} xi ^2 sqrt{R D_0(1- |xi |^2/3)}. end{aligned}$$
(22) More

The early warnings of the outbreak
As early as Dec 31st, 2019, when Wuhan Municipal Health Commission first informed the public about the emerging pneumonia cases21, most of the cities (326 out of 346) exhibited at least some awareness of the emerging SARS-CoV-2 outbreak (Fig. 2b). However, awareness then decreased until Jan 19th, 2020, one day before the Chinese Centre for Disease Control and Prevention confirmed human-to-human transmissions of the novel coronavirus9. Since Jan 20th, 2020, overall awareness increased by a magnitude of at least five, demonstrating significant awareness across all cities (Fig. 2b). Awareness remained low as the epidemic spread, falling close to its lowest point on the starting day of Chunyun (Jan 10th, 2020). Considering cities that showed initial novel coronavirus awareness levels at least 1.5 times that of the search term “common cold”, we found a total of 166 alert cities as early as Dec 31st, 2019 (48 cities at a tighter threshold of (C = 3.0) times, illustrated in Fig. 2a). However, awareness decreased significantly during Chunyun.
Figure 2

Public awareness over time. (a) The frequency distributions of cities that exhibit the first significant signal of awareness over time. The number of cities for which searches for the combined term “Wuhan” and “pneumonia” exceed (user2{ C} = 3) times the search term “common cold” is reported every day. (b) Public awareness on the topic of “pneumonia” over time. All 346 cities exhibit at least some searches of the term “pneumonia” during the initial outbreak period. Of these, 326 cities recorded searches about it as early as Dec 31st, 2019. Cities are divided into two groups according to whether or not they had reported SARS cases in 2003–04. The mean values of awareness magnitude were computed on a daily basis for two groups of cities respectively. Accordingly, a paired t-test was performed on those two time-series, and we found the cities that had reported SARS cases had greater of awareness (t-statistic: 3.56; degrees of freedom: 23; p  More

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Fish and vaccines
Two pedigree populations of Atlantic salmon (Salmo salar) called Fanad and Lochy were used in this study18 (Table 1). The two populations were managed separately and had different origins. Fish were provided in 2016 by the salmon fish farming company Salmones Camanchaca and pit tagged in April 2016 at an average weight of 26.4 ± 3.9 g and 30.2 ± 4.2 g, for populations Fanad and Lochy, respectively. During the freshwater growth period, salmon were immunized twice using commercial vaccines, following the strict Salmones Camanchaca protocols. First, fish were vaccinated by intraperitoneal (IP) injection with a pentavalent vaccine against P. salmonis, Vibrio ordalii, A. salmonicida, IPNV (infectious pancreatic necrosis virus) and ISAV (infectious salmon anemia virus). Second, fish were immunized by IP injection against P. salmonis using a monovalent live attenuated vaccine at the same time as the first vaccination. Since 2016, this double vaccination strategy has been a common practice in the Chilean salmon industry17. Fish were transferred as smolts to the Aquadvice experimental station in Puerto Montt, Chile. Unvaccinated fish were injected with PBS (phosphate-buffered saline) and used as control (Table 1). Prior to transferring the fish, a health check by RT-PCR was performed to verify that the fish were free of viral (ISAV and IPNV) and bacterial pathogens (Vibrio sp., Flavobacterium sp., P. salmonis, and Renibacterium salmoninarum). At the experimental station, all fish underwent a 15 days acclimatization period in seawater (salinity of 32% and a temperature of 15 ± 1 °C). Fish were fed daily ad libitum with a commercial diet.
Calculation of Piscirickettsia salmonis LD50
The median lethal dose (LD50) of P. salmonis (EM-90 type) was determined as previously described18. Briefly, animals from both populations were distributed in eight tanks of 350 L (n = 60 fish per tank). The LD50 was calculated in fish infected by IP injection with 200 μL of a P. salmonis suspension. Three dilutions were assessed from stock with concentrations of 1 × 106.63 TCID/mL (TCID = median tissue culture infective dose): 1 × 10–3 TCID/mL, 1 × 10–4 TCID/mL, and 1 × 10–5 TCID/mL. Controls were injected with 200 μL of PBS. Fish were monitored daily for 30 days, and mortalities were recorded. The presence of bacteria was assessed by qRT-PCR. In both infection scenarios, a single infection and coinfection, the highest dose of P. salmonis was used (1 × 10–3 TCID/mL) as a conservative measure because the fish grow about 100 g between LD50 and the main challenge (50 days).
Infection design, trait of resistance and protection added by vaccine
Fish were treated with two different types of infection, a single infection with P. salmonis (PS) or coinfection with both C. rogercresseyi and P. salmonis (CAL + PS) as previously described18. In short, infections against P. salmonis occurred at 822 ATU (accumulated thermal units) within the immunization period described by the vaccine manufacturer. Vaccinated and unvaccinated fish from populations Fanad and Lochy were equally distributed in four tanks of 6 m3, with two replicates per type of infection. For the single infection with P. salmonis, fish were IP injected. For the coinfection, fish were exposed first to sea lice and then to P. salmonis. A coinfection procedure was established based on our previous experience with this study model45,55. Infections with sea lice were performed by adding 60 copepodites per fish to each tank of coinfection. Copepodites were collected from egg-bearing females reared in the laboratory and confirmed as “pathogen-free” (P. salmonis, R. salmoninarum, IPNV, and ISAV) by RT-PCR diagnostic. After the addition of parasites, water flow was stopped for a period of 8 h, and tanks were covered to decrease light intensity, which favors a successful settlement of sea lice on fish55. A placebo procedure was applied to single infection tanks, keeping them in darkness and controlling the volume of water, temperature, oxygen levels, and fish density equivalent to those that were measured in coinfected tanks18. The secondary infection was performed with P. salmonis after seven days of sea lice infestation, and the establishment of the parasites was confirmed and quantified on all fish. Therefore, our experimental design had two types of treatments: (1) single infection (PS) or coinfection (CAL + PS); and (2) vaccinated or unvaccinated fish. Vaccinated and unvaccinated fish with a single infection were distributed in tanks 1 and 2, and vaccinated and unvaccinated fish with a coinfection were distributed in tanks 3 and 4. Further, fish were fasted for one day prior to each procedure to minimize the detrimental effects of stress on water quality parameters. Finally, fish were sedated with AQUI-S (50% Isoeugenol, 17 mL/100 L water) to reduce stress during handling. Fish were monitored daily for 30 days, and resistance to P. salmonis was measured individually as days to death. Protection added by vaccination was calculated as the difference of resistance between vaccinated fish and their unvaccinated full-sibs and represented under a single Genetic and Environment model (GxE, G = full-sib family; E = Vaccination treatment).
Comparison of moribund and survivor fish
Bacterial load, growth, and macroscopic lesions were evaluated in survivors and moribund fish. Moribund fish were obtained as dying fish when 50% of mortality was reached in both a single infection and coinfection treatments. Moribund fish were recognized and collected by three behavioral traits: lethargy, no response to stimuli, and slow swimming close to the tank wall. Resistance to P. salmonis was measured by days to death and mortality (alive versus dead) and monitored for 30 days15,45, survivors fish comprised those that lived at the end of experiment15. Forty fish were collected from each group of moribund and survivors, and from each treatment (PS and CAL + PS) and comparisons were performed between unvaccinated and vaccinated fish, twenty fish each group. However, due to the low number of unvaccinated survivors fish coinfected with P. salmonis and sea lice, it was not possible to compare with the vaccinated survivors fish.
Specific growth rate (SGR)
SGR was evaluated for moribund and survivors fish. The specific growth rate was calculated previous to infection, and post-infection as SGR = ((lnw2 − lnw1)*t−1)*100, where w2 corresponds to final weight, w1 to the initial weight, and t corresponds to the number of days between infection and death of the fish or the end of the trial if they survived56.
Piscirickettsia salmonis load
Piscirickettsia salmonis load was evaluated for moribund and survivors fish. P. salmonis load was estimated based on the amount of specific ribosomal RNA from the bacteria in the head kidneys of the infected fish, as measured by qRT-PCR. Dead fish were not used to evaluate bacterial load. Threshold cycle (CT) values from bacterial RNA was used as an indication of the bacterial load as previously described18. Head kidney samples were extracted from 20 moribund and survivors fish per group and preserved in RNAlater at − 80 °C until RNA extraction. RNA was extracted from tissue samples with the TRIzol reagent (Thermo Fisher Scientific, MA, USA) following the instructions provided by the manufacturer. DNA was removed through an additional step using a DNase incubation for 60 min at 37 °C. The quality of the RNA extraction was checked by visualizing the 28S and 18S rRNA bands resolved in 1% of agarose gels stained with SYBR Safe DNA gel stain (Invitrogen, CA, USA), and the total concentration of the RNA was measured spectrophotometrically in a MaestroNano device (MAESTROGEN, Hsinchu, Taiwan). One hundred nanograms of purified total RNA was used for the qRT-PCR reactions. The qRT-PCR reaction was prepared using the Brilliant III SYBR master mix (Agilent Technologies, CA, USA) by adding the template RNA, probes, and primers as described previously57. qRT-PCR was performed in the Eco Real-Time PCR system (Illumina, CA, USA), whose results were expressed in terms of CT. All samples were tested in triplicates and were calibrated to a plate standard that contained a combination of samples from all groups tested. Primers used for 23S gene of S. salar were forward primer TCTGGGAAGTGTGGCGATAGA and reverse primer TCCCGACCTACTCTTGTTTCATC.
Necropsy analysis
Macroscopic lesions from 20 fish per treatment were analyzed on moribund and survivors fish13; almost all survivors sampled fish were vaccinated, except one unvaccinated fish that survived to P. salmonis infection (data not shown). Fresh samples were analyzed by two veterinarians who were blinded to the treatments. Macroscopic lesions evaluated in the tissues were peeling or undergoing desquamation, congestion, and ecchymosis in the skin, paleness, and melanomacrophages in the gills, white hepatic nodules, hepatomegaly, spleen paleness, and splenomegaly. Macroscopic lesions were indicated as present or absent.
Statistical analysis
Significance levels of resistant to P. salmonis were obtained using a two-way ANOVA followed by a Tukey post-hoc test and unpaired t-test. The effects of populations and sex of fish on SGR and P. salmonis load were analyzed using a non-parametric Kruskal–Wallis test followed by a Dunn post-hoc test. Additionally, differences in the clinical signs of the P. salmonis infection between different treatments were analyzed using a non-parametric Chi-square proportion. All statistical analyses were performed using R Core Team (RStudio, Vienna, Austria). Graphs were designed with GraphPad Prism 8.0 software (GraphPad Software, CA, USA).
Quantitative genetic analysis
Each population in this study has a different genetic origin and has been managed as closed populations during the domestication process. Thus, (co) variance components of days to death were estimated independently for each population from the data of its genealogy (Table 1) using VCE 6.0 software by Groeneveld et al.58.
Heritability of days to death was estimated using the following univariate animal model:

$$y = upsilon 1 , + X_{1} t + X_{2} i + X_{3} v + Za + e,,,,{text{Model}},1$$

where y is the vector of the trait days to death, μ is the overall mean effect, t is the fixed effect of tank; i is the fixed effect of type of infection; v is the fixed effect of group of vaccination; a is the random effects vector of animal effects, with a ~ N(0, σa2A); and e is the random vector of errors, with e ~ N(0, σe2Ie). X1, X2, X3, and Z are incidence matrices, and A is the numerator relationship matrix obtained from pedigree information. The magnitude of estimated heritability was established following the classification of Cardellino and Rovira59: low (0.05–0.15), medium (0.20–0.40), and high (0.45–0.60) and very high ( > 0.65).
Genotype–environment interactions (GxE) were estimated by means of genetic correlations between the trait days to death measured in one environment (i.e., unvaccinated and single infection with P. salmonis) and the same trait measured in the other environment (i.e., vaccinated and coinfection).
Genetic correlations were estimated using the following bivariate animal model:

$$y_{1} ,;y_{2} = X_{1} d + X_{2} t(d) + X_{3} i(d) + X_{4} v(d) + Za(d) + e,,,,{text{Model}},1$$

where, y1 and y2 are the data vectors for the traits of interest (days to death in vaccinated and unvaccinated fish); d is the fixed vector of trait effects; t(d), i(d), v(d), are the fixed effects of tank, type of infection and group of vaccination effects within trait, respectively; a(d) is the random vector of animal effects within trait, with a(d) ~ N(0, A ⊗ G); and e is the random vector of errors, with e ~ N(0, I ⊗ R). The matrix G is a 2 × 2 variance–covariance matrix between traits defined by a genetic additive correlation term, rg, and a genetic variance (σgj2) for each trait. The matrix R is an unstructured 2 × 2 residual variance–covariance matrix with a different variance for each trait (σej2), and a covariance between traits (σeij). All other terms were previously defined. Correlations were classified as low (0–0.39), medium (0.40–0.59), high (0.60–0.79), and very high (0.80–1), regardless whether it was positive or negative. Significance testing of the estimates of heritability and genetic correlation were approximate as suggest by Åkesson et al.60. Thus, any genetic parameter value was considered significantly different from zero with P  More

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