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We have demonstrated that Mexican tetra fish can locate their depth with high fidelity by using hydrostatic pressure alone. Crucially, the fish can use hydrostatic pressure not only as a gradient, giving information about upward and downward movement but also as a distance-based cue that can allow precise localisation of their vertical position. This newly identified sensory capability indicates how fish can achieve the complex task of navigating through three-dimensional environments.The basis of navigation in all animals, hinges on the individual knowing the spatial relationship between their current location and an intended destination. Although all animals inhabit a three-dimensional world, many, including humans, are constrained to travelling over surfaces with three degrees of freedom: two translational and one rotational14. The addition of the vertical dimension enlarges the size of the navigable space from a two-dimensional plane to a three-dimensional volume2, leading to a multiplicative increase in the complexity of a navigational task14,15,16. Reliable information on vertical position would therefore be a significant benefit for three-dimensional navigation.Although it is likely that in the wild fish rely on multiple cues to navigate, a sense of pressure would be particularly useful when other cues are unavailable or unreliable, for example, in turbid waters where visual landmarks are absent or obscured, and in turbulent waters where olfactory plumes cannot provide fine-scale information. The stability and ubiquity of hydrostatic pressure in aquatic environments allow fish access to a reliable navigational cue and could explain why two separate experiments, each testing a different species, found that fish perceived vertical information as the more reliable cue when horizontal and vertical information conflicted.The physiological mechanism underlying depth perception in fish is yet to be identified, although the swim-bladder has been implicated. In this putative mechanism, absolute depth is estimated during fast, steady vertical displacements by combining a measurement of vertical speed with a measurement of the fractional rate of change of swim-bladder volume. If this is the mechanism that these and other bony fish are using to sense their depth, there are likely to be important ecological and welfare consequences for fish that suffer barotrauma from angling or transit through hydroelectric power facilities, where the damage caused from exposure to rapid changes in barometric pressure may cause swim bladder ruptures17. Therefore, governments need to be aware of key migratory paths that fish use to move between feeding and breeding sites to enable them to protect important species. Similarly, fish that contract parasitic infections of the swim bladder are likely to find their vertical navigation is severely compromised. While there are currently no studies on the pressure sensing in fish with parasitic infections of the swim bladder, previous research has reported that infected Koi carp (Cyprinus carpio) are less able to achieve and sustain neutral buoyancy and demonstrate abnormal swimming behaviour18. Similarly, silver eels (Anguilla Anguilla) infected with a swim bladder nematode experienced a loss of buoyancy resulting in them expending more energy while swimming, impeding their migration19.While the swim bladder appears to be a good candidate organ for sensing hydrostatic pressure in bony fish, many cartilaginous or deep-sea species do not possess a gas phase, despite still being able to navigate vertically. Previous research has suggested that instead of relying on fractional changes in swim bladder volume, these species may rely on the sensory afferents of their lateral line system; with evidence that swimming crabs (Callinectes ornatus), mud crabs (Panopeus herbstii) and dogfish (Scyliorhinus canicula) sense pressure changes via the bending of hair cells oriented to sense either vertical or horizontal displacements20. The ability of fish to use hydrostatic pressure to accurately locate a point in the vertical dimension may be important for fisheries management. Known points of interest, for example, food sites, refugia, heavily predated areas, present in the vertical column could be learned and remembered by fish, with them either returning to or avoiding these areas as necessary. Further field studies on individual fish and shoals of fish using hydrostatic pressure in this context are needed to identify how this cue is used both in the wild and in farmed fisheries.Our findings reveal novel sensory information that A. mexicanus, and possibly other fish species, use to gain detailed navigational information over short distances in the vertical dimension. Extrapolating from this, we argue that it is likely that fish could use pressure to navigate over larger distances as the pressure magnitudes will increase as the vertical distance increases. Together, this study reveals a new sensory capacity that has great adaptive value in the fish’s volumetric world. More

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Study site and demographic dataThe study was carried out in Volcanoes National Park, the Rwandan part of the Virunga massif, which is further shared with Uganda and the Democratic Republic of the Congo. We focused on habituated mountain gorilla groups monitored by the Dian Fossey Gorilla Fund’s Karisoke Research Center, often referred to as the Karisoke subpopulation. Since 1967, groups in this subpopulation have been followed on a near daily basis. Through the mid-2000s, the Karisoke groups generally numbered three but over the last decade, group fission events and new group formations resulted in an average of ten groups in the region (see42,43). During daily observations, detailed demographic data are recorded, such as group composition, birthdate and death date, group transfers (for further details see Strier et al.50). The data used for this study covers demographic data from 1967 to 2018 and includes 396 recognized individuals.Epidemiological dataWe obtained published data on four variables that control the disease dynamics of COVID-19 in humans, namely (a) the basic reproductive number (R0)34,35, (b) the infection fatality rate (IFR) based on estimates from China and Italy24,25,36,37, (c) the probability of developing immunity and (d) the duration of immunity37,38,39,41.Stochastic projection modelWe used the stochastic projection model proposed by Colchero et al.51, that models population dynamics for both sexes on fully age-dependent demographic rates. The model incorporates the yearly variance–covariance between demographic rates, while it accounts for infanticide as a function of the number of silverbacks (mature males > 12 years old) in the population51. Because of this relationship between infanticide and number of silverbacks, this source of mortality changes in time and cannot be assumed to be part of the infant mortality rate. To explore the extinction probability for the Karisoke subpopulation as a function of different diseases, we gathered information from the model on the proportion of individuals that died for each disease and the frequency of outbreaks (i.e., how often outbreaks occurred).Demographic-epidemiological projection model for COVID-19We constructed a predictive population model that combines the species’ baseline demographic rates with a model based on the susceptible-infected-recovered-susceptible (SIRS) framework. As the baseline demographic rates, we used the age-specific mortality and fecundity estimated by Colchero et al.51 for mountain gorillas (Karisoke subpopulation). We defined four epidemiological stages, namely (a) susceptible, (b) infected, (c) immune and (d) dead, each of which we further divided into a fully age-specific structure (Fig. 1). Based on recent research on COVID-19 on humans, we assumed that the dynamics of the model allowed for the recovered individuals to be divided into either susceptible or immune37,38,39,41. Furthermore, we incorporated the potential age-specific infection fatality rate (IFR) based on current estimates from medical and epidemiological research24,25,36,37, adjusted to the lifespan of the gorillas by means of the logistic function$$qleft(xright)=frac{{q}_{M}}{1+{text{exp}}left[-0.2left(x-25right)right]},$$
(1)
where qM is the maximum infected mortality probability. Similarly, we modeled the probability of developing immunity as a function of the strength of the disease, which, based on recent research, we measured as mirroring Eq. (1) as$$mleft(xright)=frac{{M}_{I}}{1+{text{exp}}left[-0.2left(x-25right)right]},$$
(2)
where MI is the maximum immunity probability (Fig. 2B).To explore the potential impact of COVID-19 on the growth rate of the Karisoke mountain gorilla subpopulation, we varied four of the critical epidemiological variables, namely (a) the basic reproductive number, R0, from 0.5 to 6 (which helps to simulate factors such as increased group density, which may increase the likelihood of transmission), (b) the maximum infected mortality probability, qM = (0.3, 0.6) (Fig. 2A), (c) the immunity duration, TI to 1, 3, 6, and 12 months, and (d) the maximum immunity probability, MI, from 0.2 to 0.8 (Fig. 2B). As time units we used year fractions in half months (i.e., t1 − t0 = 0.5/12), which allowed us to simplify the model, based on current information on the average time of serial interval and incubation period in humans21. This implementation assumes that susceptible individuals could become infected at the beginning of the time interval, while infected individuals in time interval t would either recover (immune or susceptible) or die in t + 1.The deterministic structure of the model implies that the number of individuals in each sex, age and epidemiological stage was given by the possible contribution from the other stages 1/2 month before. This is, the number of susceptible individuals of age x at time t is given by the difference equation$$begin{aligned} n_{s,x,t} & = p_{x – 1} left{ {n_{s,x – 1,t – 1} + n_{i,x – 1,t – 1} left[ {1 – qleft( {x – 1} right)} right]left[ {1 – mleft( {x – 1} right)} right]} right} \ & quad + n_{{m,x – T_{i} ,t – T_{i} }} prodlimits_{{j = x – T_{i} :j > 0}}^{x – 1} {p_{j} – n_{i,x,t} } , \ end{aligned}$$where the ns,x,t is the number of susceptible individuals of age x at time t, and subscripts i and m refer to infected and immune individuals, respectively. For simplicity of notation, we do not include a subscript for sex, although the model does distinguish between sexes. The probability px is the age-specific survival probability. Functions q(x) and m(x) are as in Eqs. (1) and (2). Similarly, the number of immune individuals at time t and age x are$${n}_{m,x,t}={n}_{i,x-1,t-1}left[1-qleft(x-1right)right]mleft(xright)+sum_{{j:0le jle {T}_{i}wedge x-j >0}}{p}_{x-j}{n}_{i,x-j,t-j}.$$We incorporated this mechanistic structure into a stochastic model, where all contributions from time t to t + 1 were drawn from binomial or Poisson distributions. For instance, the total new number of infected individuals, Ni,t, was obtained as a random draw from a Poisson distribution with expected value$$Eleft[{N}_{i,t}right]={text{min}}left[{{R}_{0}N}_{i,t-1},{N}_{t}right],$$where Nt is the total number of individuals in the study subpopulation. We then distributed randomly these individuals into different available ages and sex corresponding to the term ni,x,t, in the susceptible equation above. The number of newborns, Bx,t, at each age for which there were available females at time t was drawn from a binomial distribution with expected value$$Eleft[{B}_{x,t}right]=left({n}_{s,x,t}+{n}_{m,x,t}right){f}_{x}$$where fx is the age-specific average female fecundity rate and ns,x,t and nm,x,t refers to the number of susceptible and immune females, respectively, of age x at time t. The sex of each newborn was then determined by means of a Bernoulli draw with probability given by the proportion of males in the population. Thus, if the draw produced 1 for that individual, it became a male, and if 0 a female.For each scenario, we ran stochastic simulations for 2000 iterations for 10 years and recorded the average number of individuals at each age–sex and epidemiological state at every month. We then ran long-term stochastic simulations for four scenarios with R0 = 3 and maximum immunity probability MI = 0.2. For these, we recorded also the number of subpopulations that went extinct at each month. More

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Global statistical analysisOur first attempt to identify plausible effects of meteorological covariates on COVID-19 spread applied a comparative regression analysis. To this end, we focused on the exponential onset of the disease, as it is the epidemic phase that allows for a better comparison between countries or regions, without the confounding effect of intervention policies. We first determined, for each of the spatial units (either countries or NUTS (nomenclature of territorial units for statistics) 2 regions), the day in which 20 or more cumulative cases were officially reported. We then fitted the first-order polynomial function f(t) = x0 + rt for the next 20 days of log-transformed data, where t represents time (in days) and ({{x}_0}) is the value at initial condition t = 0. The r parameter can be understood as the exponential growth rate, and is then used to estimate the basic reproduction number (R0) using the estimated serial interval T for COVID-19 of 4.7 days53, such that R0 = 1 + rT (ref. 54). (We note that we are interested here in the relationship between the reproductive number and not in the actual inference of R0.) Once R0 was obtained for all our spatial units, we filtered our meteorological data to match the same fitting period (with a 10-day negative delay to account for an incubation and reporting lapse) for every spatial unit. To compute a single average of the meteorological variables per regional unit, we computed a weighted average on the basis of the population contribution of each grid cell to the total population of the region. We did so to have an aggregated value that would better represent the impact of these factors on the population transmission of COVID-19, as the same variation in weather in a high-density urban area is more likely to contribute to a change in population-level transmission than that of an unpopulated rural area. We then averaged the daily values of temperature and AH for each country and computed univariate linear models for each of these variables as predictors of R0. Given the somewhat arbitrary criteria to select the dates to estimate the R0 in each country, a sensitivity analysis was run to test the robustness of the regressions to changes in the related parameters. We tested 70 different combinations of two parameters: the total number of days used for the fit (18–27) and the threshold of cumulative COVID-19 cases used to select the initial day of the fit (15–45). We also calculated the weather averages by shifting the selected dates accordingly. Then, a linear model for each of the estimates was fitted for both T and AH. A summary of the distribution of parameter estimates (the regression slope coefficients and the R2 of the models) is shown in Extended Data Fig. 3.Bivariate time-series analysis with scale-dependent correlationsTo examine associations between cases and climate factors in more detail, SDC was performed on the daily time series of both COVID-19 incidence and a given meteorological variable. SDC is an optimal method for identifying dynamical couplings in short and noisy time series20,21. In general, Spearman correlations between incidence and a meteorological time series assess whether there is a monotonic relation between the variables. SDC analysis was specifically developed to study transitory associations that are local in time at a specified temporal scale corresponding to the size of the time intervals considered (s). The two-way implementation (TW-SDC) is a bivariate method that computes non-parametric Spearman rank correlations between two time series, for different pairs of time intervals along these series. Different window sizes (s) can be used to examine increasingly finer temporal resolution. The results are sensitive to the value of this window size, s, with expected significant and highest correlation values at the scale of the transient coupling between variables. Correlation values decrease in magnitude as window size increases, and averages are computed over too long a time interval. Values can also decrease and become non-significant for small windows when correlations are spurious. Here, the method was applied for windows of different length (from s = 75 to 14 days) and, despite a weekly cycle showing up in some cases for small s, results removing this cycle were robust. We therefore did not remove this cycle.The results are typically displayed in a figure with the following subplots: (1) the two time series, to the left and top of the matrix of correlation values, respectively; (2) the matrix or grid of correlation values itself in the center, with significant correlations colored in blue when positive and in red when negative, with rows and columns corresponding to the temporal localization of the moving window along the time series on the left and top, respectively; (3) a time series at the bottom, below this grid, with the highest significant correlations for a given time (vertically, and therefore for the variable that acts as the driver, here the meteorological time series). To read the results, one starts at the diagonal and moves vertically down from it to identify a given lag for which significant correlations are found (the closest to the main diagonal). In some of the SDC figures, the time intervals with high local correlations are highlighted with boxes. These intervals alternate with other ones (left blank) for which no significant correlation is found. All colored areas correspond to significance levels of at least P  fs/fr, where fs is the sampling rate and fr the minimum frequency. Another strategy is that M be large enough that the M-lagged vector incorporates the temporal scale of the time series that is of interest. The larger the M, the more detailed the resulting decomposition of the signal. In particular, the most detailed decomposition is achieved when the embedding dimension is approximately equal to half of the total signal length. A compromise must be reached, however, as a large M implies increased computation, and too large a value may produce mixing of components. SSA is especially well suited for separating components corresponding to different frequencies in nonlinear systems. Here, we applied it to remove the weekly cycle.MSDC analysisMSDC provides a scan of the SDC analyses over a range of different scales (here, S from 5 to 100 days at 5-day intervals), by selecting the maximum correlation values (positive or negative) closer to the diagonal. The goal is to consider the evolution of transient correlations at all scales pooled together in a single analysis. The MSDC plot displays time on the x axis and scale (S) on the y axis, and positive and negative correlations either jointly or separately. The rationale behind MSDC is that correlations at very small scales can occur by chance because of coincident similar patterns, but that as one moves up to larger scales (by increasing S), the correlation patterns that are spurious tend to vanish, whereas those reflecting mechanistic links increase in strength. This increase in correlation values should occur up to the real scale of interaction, decreasing afterwards. By ‘real’, we mean here the temporal scale covering the extent of the interaction between the driver and the response process (in this case, the response of disease transmission to a given climate factor). Thus, continuity of the same sign correlations together with transitions to larger values are indicative of causal effects, whereas the rapid vanishing of small-scale significant correlations signals spurious ones.Process-based modelDescriptionThe dynamical model is a discrete stochastic model that incorporates seven different compartments: S, E, I, C, Q, R and D. The model structure is illustrated in Fig. 4. The transition probabilities of the stochastic model are based on the corresponding rates of the transitions between classes in the deterministic (mean-field) model (specified in Fig. 4b). These probabilities are defined as follows. P(e) = (1.0 − exp(−β dt)) is the probability of infection exposure of the susceptible class, where β = (1/N)(βII + βQQ) is the infection rate (of the deterministic model). P(i) = (1.0 − exp(−γ dt)) is the probability that an new exposed individual becomes infectious, where γ denotes the incubation rate. P(r) = (1.0 − exp(−Λ dt)) is the recovery probability, where λ0(1 − exp(λ1t)) is the (deterministic) recovery rate. P(p) = (1.0 − exp(−α dt)) is the protection probability, where α = α0exp(α1t). P(d) = (1.0 − exp(−K dt)) is the mortality probability, with K = k0exp(k1t). P(re) = (1.0 − exp(−τ dt)) is the release probability from confinement, where τ = τ0exp(τ1t). Finally, P(q) = (1.0 − exp(−δ  dt)) is the detection probability, where δ is the quarantine rate (for example, at which infected individuals are isolated from the rest of the population).In the model, both infected non-detected and infected detected individuals can infect susceptible ones. In the model incorporating temperature in the transmission rate, the respective values of βI and βQ are calculated as follows:$${beta }_{I}(t)={beta }_{I},T_{mathrm{inv}}(t);quad {beta }_{Q}(t)={beta }_{Q},T_{mathrm{inv}}(t)$$where (T_{mathrm{inv}}=fleft(frac{1-T(t)}{bar{T}}right)), with (bar{T}) corresponding to the overall mean of the temperature time series and f(·) to a Savitzky–Golay filter, used to smooth the temperature series with a window size of 50 data points and a polynomial order of 3. When the infection rate is constant, we simply omit the temperature term. For further comparison, in a third model, β is specified with a sinusoidal function of period equal to 12 months and an estimated phase.The number of individuals transitioning from compartment i to j at time t are determined by means of binomial distributions P(Xi,P(y)), where Xi corresponds to one of the compartments S, E, I, Q, R, D, C, and P(y) to the respective transition probability defined above. Thus,

e(t) = P(S(t), P(e)), new exposed individuals at time t

p(t) = P(S(t), P(p)), protected individuals at time t

i(t) = P(E(t), P(i)), new infected not detected individuals at time t

q(t) = P(I(t), P(q)), new infected and detected individuals at time t

r(t) = P(Q(t), P(r)), total recovered individuals at time t

d(t) = P(Q(t), P(d)), total dead individuals at time t

re(t) = P(C(t), P(re)), individuals released from confinement at time t

Then, the final dynamics are given by the following equations:$$S(t)=S(t-{rm{d}}t)-e(t)-p(t)+re(t)$$$$E(t)=E(t-{rm{d}}t)+e(t)-i(t)$$$$I(t)=I(t-{rm{d}}t)+i(t)-q(t)$$$$Q(t)=Q(t-{rm{d}}t)+q(t)-r(t)-d(t)$$$$R(t)=R(t-{rm{d}}t)+r(t)$$$$D(t)=D(t-{rm{d}}t)+d(t)$$$$C(t)=C(t-{rm{d}}t)+p(t)-re(t)$$CalibrationThe model was implemented using Python and calibrated by means of the least squares algorithm of the scipy library. The error function minimized with this algorithm was obtained from the normalized residuals on the basis of total cases (Q + R + D) and deaths (D).To search parameter space, we ran 100 calibrations starting from different initial choices of parameter combinations. The tolerance for termination in the change of the cost function was set to 1 × 10−10. Tolerance for termination by the norm of the gradient was also set to 1 × 10−10, and the tolerance for termination by the change of the independent variables was set to 1 × 10−10. The solver was the lsmr method (which is suitable for problems with sparse and large Jacobian matrices) with a differential step of 1 × 10−5. With this configuration, each fitting run usually converged after ~500 iterations.ValidationTo compare the model including an effect of T in the transmission rate to those without it, we calculated the chi-square, Akaike information criterion (AIC) and Bayesian information criterion (BIC) indices for the residuals obtained from the optimization process. The resulting values are shown in Supplementary Table 1.Our choice of T to modulate the infection rate (β) instead of AH underlies the fact that the temporal dynamics of both factors roughly follow the same shape, with the advantage that T shows less oscillatory behavior than AH. This fact adds stability to the model when the inverse relationship is used in the calculation of β (Supplementary Information). This selection is further reinforced by the results from the SDC analyses, which yielded larger correlations for temperature, even when penalizing for the larger autocorrelation structure.Our choice to modulate β using T instead of AH follows from the fact that the temporal dynamics of both climate variables present roughly the same shape, with the advantage that T exhibits weaker oscillations. This less fluctuating pattern provides stability to the model fitting when the inverse relationship is used in the calculation of β (Supplementary Information). Additionally, the transient correlations obtained with SDC yielded higher values for T than for AH (even when accounting for concurrent levels of autoregression in the two variables). More

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