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To enable successful spawning of Atlantic eels in remote offshore areas of the ocean, three conditions need to be met. This requires, first, appropriate navigation abilities and cues leading to the remote spawning area; second, a meeting point; and third, an adequate timing.
Orientation and navigation cues towards the spawning area
Eels are thought to imprint a magnetic map on their first transoceanic migration from the spawning areas to the coasts21. Moreover, silver eels are known to be sensitive to magnetic cues22 that are likely involved in navigation towards the Sargasso sea with a very high spatial accuracy23. Under this hypothesis, silver eels are expected to choose the fastest or shortest route to join the Sargasso Sea. Indeed, recent studies showed that European silver eel swam south-westward24, 25 while American silver eels swam south-eastward26. Surprisingly, none of the tagged eels reached the spawning areas within the Sargasso Sea26. One single American eel reached the north west boundaries of the North Atlantic Convergence zone at  > 2,000 km from the center of the Sargasso Sea26, while a few European eels where detected at the North East of the Azores at c.a. 3,000 km from the Sargasso Sea24, 25. This was interpreted as a consequence of the tagging rather than a biological fact.
Interestingly, all European eels, whatever their release points (Baltic Sea, Ireland, the Bay of Biscay, Mediterranean) converged towards the Azores, which is not the shortest way back to the Sargasso Sea24. So what could be the advantage for silver eels for choosing a longer route? The most parsimonious hypothesis is that the Azores serve as a meeting point located along the Mid-Atlantic Ridge. Once they reach this point they turn southwest, following the Mid-Atlantic Ridge. This could be made possible by the striking vertical diel migration behavior that takes eels from epipelagic layers (150–300 m) during the night to mesopelagic and bathypelagic depths during the daytime (down to 1,200 m)24,25,26. This behavior could enable silver eels to detect and follow the Mid Atlantic Ridge and associated seamounts that culminate at 2,000 m to 3,500 m above the seafloor that lies at  > 4,000 m depths. Moreover, it is likely that eels detect chemical variations of the seawater using their high olfactory abilities enabling them to detect specific odors or plumes from subducted or convected deep layer waters27, 28. Indeed, the volcanic activity and deep currents disturbed by the sea level rise around the ridge likely modify the chemical composition and related odor of the water thus providing signposts28.
Following this north south Y axis, silver eels may finally reach favorable thermic conditions of 22 to 24 °C to spawn, which are located between two parallel east–west thermal fronts that occur in the Sargasso Sea at about 24°N and 28°N (X-axis)8, 29. Worth mentioning, small leptocephali of both Atlantic eel species have been collected over a wide longitudinal range (75–50°W) between these two fronts8. Although the collected area of American and European eel larvae partly overlapped in Sargasso Sea, the southern-most collection of European eel larvae was about 100–200 km north compared to American eel larvae8, apart from thermal fronts that were suggested earlier as an X-axis, European eels may follow a different hint. One of the major water masses in the Sargasso Sea is the North Atlantic Subtropical Mode Water, which has unique vertical temperature distribution, in which the temperature is nearly uniform in the Mode Water layer, especially in winter and early spring30, 31. Its southern boundary is around 22–26°N; therefore, the mode water’s boundary could also potentially serve as a destination hint (X-axis) for European eels.
Meeting point to mitigate lack of migration timing
Once eels have reached favorable habitat conditions to spawn, they have to find their mates to breed. Random mating in the huge Sargasso Sea (c.a. 3 million km2) is highly unlikely. Indeed, male and female silver eels do not have a synchronized migration. Males start their migration from August to September, whereas females migrate between November and December24. Telemetry data demonstrated that migrating silver eels disperse after they are released24. Migration speed is highly variable according to size24, 32, because males that are approximately 45 cm long on average have much lower swimming speeds than female eels, which have bodies up to twice the size as males. This suggests that, unlike tuna or mackerel, eels do not form schools, and even if they start their spawning migration in a school from continental rivers, they eventually scatter and arrive in the Sargasso Sea one by one. These arguments strongly suggest that synchronized migration and schooling do not likely occur, meaning that successful mating and spawning depends on the existence of clear physical, chemical, geological, or biological signals that eels can use to locate a meeting point in the ocean. However, such east–west and north–south hints (X and Y axis) or any kind of gradient do not exist in the large Sargasso Sea.
Egg distributions of Japanese eel within the spawning area indicated that spawning occurred just south of the crossing point where north–south seamount chain and east–west salinity front between two water masses with different salinities—caused by evaporation in the north and tropical rainfall in the south13, 16. It has been speculated that eels can locate the spawning site using a combination of the seamount chain (Y-axis) and salinity front (X-axis) as a signpost for forming spawning aggregations in the ocean.
To ensure successful external fertilization of eggs, eels must meet their mates in the ocean, meaning that time and space must precisely coincide for successful mating. If the same strategy can be adapted to Atlantic eels, waters near the Mid-Atlantic Ridge could be chosen as a spawning site because of unusual topographical features, geomagnetic anomalies33 or variation of chemical compositions that could serve as an olfactory cue for eels. Indeed, active hydrothermal vents have been observed along the Mid-Atlantic Ridge across the entire Atlantic34, and the release of chemical elements from hydrothermal vents may serve as a cue for locating a spawning site. This kind of signpost remains very large, and therefore it is likely that pheromones might be released by silver eels to favor the final meeting of the partners.
Simulating departure from the Mid Atlantic Ridge and from the Sargasso Sea
Using the same principle as Japanese eel, volcanically active parts of the Mid-Atlantic Ridge could be one of the spawning sites for Atlantic eels due to unusual topographical features, geomagnetic anomalies, or differing water chemical composition21. The 22 °C and 24 °C thermal fronts between which Atlantic eel larvae have been frequently observed8 are used to extend farther east, interacting with the Mid-Atlantic Ridge at around 27 and 20°N, respectively. To the south of these thermal fronts exists a discernible salinity front around the northern limit of the North Equatorial Current (NEC) in the Atlantic at 15–18°N. Thus, we modeled the transport of virtual leptocephali larvae from the area chosen to be 15–29°N and 43–48°W which included intersections of the Mid-Atlantic Ridge by one salinity front and two thermal fronts (Fig. 1 top).
We then released v-larvae near the Mid-Atlantic Ridge from 15 to 29°N. We classified v-larvae by their initial positions as north of the 22 °C isotherm (yellow), between the 22 and 23 °C isotherm (blue), between the 23 and 24 °C isotherm (green), south of the 24 °C isotherm (red), and the north of NEC with a salinity front at around 18–19°N (cyan) (Figs. 1, 2). Passive swimming v-larvae were widely dispersed to the west and east of the release area after 720 days of migration (Fig. 2). V-larvae departing from north of 24°N (yellow and blue dots) finally arrived at the Azores front and North Atlantic drift, with easternmost positions near 15°W, showing similar distribution to observed European eel larvae. In contrast, v-larvae departing from south of 24°N (green, red, and cyan dots) could make it to the Caribbean Sea and the Gulf of Mexico, and some v-larvae entrained in the Loop Current and Gulf Stream, arriving at the east coast of North America, that is similar to the observed American eel larvae distribution. The percentage of v-larvae reaching 25°W after 720 days decreased from north to south: 0.71% in the northernmost area (yellow in Fig. 2), 0.13% (blue), and 0% (green, red, and cyan). Arrival at the Caribbean Sea and Gulf of Mexico increased from north to south: 0.13% (yellow in Fig. 2), 0.77% (blue), 4.64% (green), 19.3% (red), and 38.9% in cyan.
Figure 2

Distribution of passive swimming v-larvae departing from near the Mid-Atlantic Ridge. Colors correspond to release areas (north of the 22 °C isotherm (yellow), between the 22 and 23 °C isotherm (blue), between the 23 and 24 °C isotherm (green), south of the 24 °C isotherm (red), and the north of NEC with a salinity front at around 18–19°N (cyan)) as indicated in the top panel. The simulation period was 1993–2000 and included both positive (1993–1994, 1999–2000) and negative (1995–1996, 1997–1998) North Atlantic Oscillation events, and the results are based on an eight-year composite.

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By comparison, v-larvae released in the Sargasso Sea were widely distributed throughout the northwestern Atlantic Ocean, including the Caribbean Sea and Gulf of Mexico (Fig. 3). A total of 0.14% of the v-larvae released from the suggested European eel spawning area in the Sargasso Sea reached 25°W after 720 days (Fig. 3 right), whereas 0.27% of those released in the American eel spawning area reached 25°W (Fig. 3 left). Arrival at the Caribbean Sea and Gulf of Mexico was 6.56% and 11.9% of those released from European and American eel spawning areas, respectively.
Figure 3

Distribution of v-larvae released in the Sargasso Sea for American (left) and European eels (right).

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Although the distribution patterns were similar to v-larvae departing from the newly proposed spawning area (Figs. 3, 5), differences were detected. A significant fraction of v-larvae representing European eels released in the Sargasso Sea dispersed to the Caribbean Sea and Gulf of Mexico (Fig. 3, right). As the v-larvae departing from the northern proposed area did not enter these areas (Fig. 2), and only American eel larvae but not European eel larvae have been collected in the Caribbean Sea and Gulf of Mexico. In addition, some of v-larvae representing American eel departing from Sargasso Sea were transported far northeast by Gulf Stream and North Atlantic Drift to east of 40°W, where American eel larvae were not observed8. In contrast, v-larvae departing from southern proposed area showed closer distribution to observations of American eel leptocephali, while v-larvae departing from central to northern sub areas of the Mid-Atlantic Ridge presented similar distributions to observations of European eel leptocephali. Therefore, it could be suggested that both European and American eels may indeed spawn in the newly proposed area near the Mid-Atlantic Ridge.
Interestingly, distributions of v-larvae departing from the American eel spawning area or from the European eel are very similar (Fig. 3) suggesting that swimming and orientations are likely. If v-larvae could swim at 1 body length per second (BL/s) northeastward, arrival rate at 25°W would increase substantially, especially for those departing from the northern area (Fig. 4, left). On the other hand, v-larvae swimming at the same speed of 1BL/s but heading northwestward would not reach 25°W (Fig. 4, right), instead, distribution of v-larvae would be concentrated at northwestern Atlantic Ocean. The simulations with swimming ability indeed also revealed similar distribution as observations. V-larvae departure from northern area would swim towards eastern north Atlantic (yellow and blue, Fig. 4 left), whereas those departing from southern area (cyan and red, Fig. 4 right) would move towards western north Atlantic and some of them may bypass Caribbean Sea and Gulf of Mexico.
Figure 4

Same as Fig. 2, but for northwestward swimming (left), and northeastward swimming (right) at swimming speed of 1 BL/s.

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Estimating spawning location of small leptocephali caught in historical surveys
Historical surveys have spent great effort to search the eggs and spawning adult eels in the past century. However, the larval surveys to date have not explicitly considered the possibility of alternative spawning areas or an extension eastward. This introduced an evident gap, both geographically and temporally, in larval surveys. Indeed, our numerical simulation showed that a different departure point (spawning area), located above the Mid-Atlantic Ridge, resulted in a distribution of leptocephali larvae similar to historical observations in the Atlantic Ocean. Hence, these results strongly suggest that oceanographic surveys should be organized outside the Sargasso Sea, in the vicinity of the Mid Atlantic Ridge.
We applied passive backward particle tracking to trace the origin of those observed Atlantic eel larvae. We released v-larvae in the Sargasso Sea where ≤ 10.9 mm Atlantic eel larvae have been collected8. The distribution of potential v-larvae origins 30 days prior to on-site collections was not far from where eel larvae have been collected (Fig. 5), as ocean currents were rather weak and lacked a unified direction. The results suggest a few possibilities, such as eggs may occur nearby the area where eel larvae were collected although they have not been collected; eel larvae indeed were observed in rather a wide region, suggesting eel larvae (or eggs) may also occur in areas located outside the hot-spot survey zone of the Sargasso Sea. Learning from the experience of Japanese eel surveys would allow exploring the hypothesis of alternative spawning locations.
Figure 5

Distributions of passive backward tracking v-larvae 30 days prior to collection for (a) American eels, and (b) European eels. Black crosses showed the released locations that followed the positions where eel larvae were collected8.

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Disagreement from microchemistry aspect
Ocean currents in the Sargasso Sea are generally weak, with average speeds less than 5 cm/s in the top 200 m (Fig. 1 bottom). Eddy activity is inactive in the Sargasso Sea and eddy nonlinearity is relatively low compared to those formed near the Gulf Stream or Azores Current35, indicating less trapping and transporting by westward propagating eddies for marine organisms. Additionally, Japanese eels are spawned in the faster (10–20 cm/s) NEC in the Pacific (Fig. 6), thus, it can be said that the Sargasso Sea, which is the presumed Atlantic eel spawning area, is relatively quiet and has less transporting ability because of a subtropical gyre convergence zone. This convergence zone is unfavorable for the transport of eel larvae to continental rivers.
Figure 6

Bathymetry (shading) and mean ocean circulation (vectors) in the western Pacific. Fast and slow currents with criteria of 0.15 m/s are indicated by magenta and white vectors, respectively. The yellow circle marks the spawning area of Japanese eels. See the analogy of ocean current systems in both the Atlantic (Fig. 1) and Pacific (Fig. 6), i.e., the relationship between possible eel spawning locations and currents in the western subtropical gyre such as the North Equatorial Current and western boundary currents (Gulf Stream or Kuroshio).

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Interestingly, concentrations of Mn, a trace element signature, in the central part of otoliths of glass eels caught in western European estuaries were significantly greater than those of leptocephali collected in the Sargasso Sea20. Mn is a geochemical fingerprint of volcanic activity mainly found along the Mid-Atlantic Ridge34.The numerical experiment had shown that the buoyant hydrothermal plume could transport dissolved elements vertically by 1000–1500 m, and could also spread thousands of kilometers by horizontal advection. This suggests that glass eels caught in European estuaries spent their early life in the plume of a volcanic activity zone whereas leptocephali born in the Sargasso Sea may not successfully migrate to Europe due to being trapped in the convergence zone. This supports the existence of multiple spawning areas or batches suggested by Baltazar-Soares36, without affecting the well-established panmixia36,37,38,39 given that larvae seeded from any location are dispersed along with large recovering areas (Figs. 2, 4).
General discussion
The present study explores the existence of another spawning area near the Mid-Atlantic Ridge at the east of the Sargasso Sea, that has been assumed to be the sole spawning area for almost 100 years since Schmidt’s research. This scenario relies on the combination of ecological and environmental inferences, comparative biology, the need of biotracers to pilot the catadromous fish and modelling.
The distribution of v-larvae departing from the newly proposed spawning area near the Mid-Atlantic Ridge showed possibilities of successful migration of both species to their respective geographic continental distributions. The v-larvae released in the northern region of the newly proposed spawning area showed distributions similar to those of collected European eel larvae, whereas those that departed from the southern region, within the salinity front, had distributions closer to the those of American eel larvae8. These fits between our model and observations of larval distribution are even stronger when assigning orientation and swimming skills to v-larvae. We therefore assume that swimming and orientation behaviors likely occur supporting previous findings and hypothesis21, 40.
Salinity fronts have been suggested to be related to Japanese eel spawning41 in the Pacific Ocean. Indeed, approximately 600 eggs have been collected over five research cruises at the intersection of the salinity front and the West Mariana Ridge. Similarly, a salinity front has also been observed in the Atlantic Ocean between the Sargasso Sea and NEC at around 15–18°N. In the new spawning area tested in this study, a strong salinity front with a rapid increased from 36.3 PSU at 15°N to 36.9 PSU at 19°N down to depths around 200 m was observed, and the front extended below 200 m south of 18°N. The salinity front could potentially provide a landmark for silver eels during breeding migration. In this study, v-larvae released near the salinity front showed quick dispersion westward, entering the Caribbean Sea and the Gulf of Mexico with some going to the Gulf Stream. This pattern is similar to that of Japanese eels in the Pacific Ocean that established their migration loop in the southwest corner of the subtropical gyre using the NEC and the Kuroshio42 (Fig. 6).
For the migration of adult eels, routes proposed by Righton et al.24 from a pop-up tag study showed that silver European eels seemed to converge toward the Azores regardless of origin (Baltic, North Sea, Celtic Sea, Bay of Biscay, Mediterranean). This does not fit with the Sargasso Sea hypothesis as the most direct routes from northern Europe and the Mediterranean to the Sargasso Sea do not encompass the Azores. Our hypothesis is that the Azores acts as a landmark for silver eels swimming southwest.
On their spawning migration, silver eels need to find the most efficient way to reach the spawning area using the safest and less energy costly route. It could be suggested that silver eels simply backtrack the migration route they used as leptocephali. This would imply that eels imprint their larval route, and that silver eels would have to swim against the strongest currents of the North Atlantic Ocean as the Gulf Stream, the Azores Currents and the North Atlantic drift (ie Miller and Tsukamoto43). This strategy would probably cost too much energy. Alternatively, by converging towards the Azores, as suggested by Righton et al.24, Silver eels avoid the strongest marine currents thus saving energy expenditures, which is a more likely evolutionary scenario. However, this would involve the existence of a genetically imprinted geomagnetic map that would enable eels to navigate towards the Azores whatever their departure point. Although possible, this assumption remains speculative as to date, science has not addressed how DNA encodes for such a behavior.
Because of their diel vertical migration ranging from ~ 800 m during the day to 300 m at night24, 25, 44, these eels could detect the topography and specific odors of the ridge they follow until they reach to favorable thermal fronts. Strong magnetic abnormalities occur along the Mid-Atlantic Ridge from the Azores to the junction with the Kane fracture zone (23.5 N; 46.4 W) and then make a bend westward along the Krane fracture33. For the American eel, an individual released from the Gulf of St. Lawrence near the northernmost distributional range of American eel leptocephali showed a long-distance migration to the northern Sargasso Sea26. We need to further observe the route in the southern Sargasso Sea. Additionally, the release of silver eels with pop-up tags from the Caribbean Sea near the southernmost distributional range and nearest areas to both the Sargasso Sea and Mid-Atlantic Ridge is the next step to confirm the success of adults migrating to their spawning area.
The collection of tiny larvae, known as preleptocephali, has been reported for both species in the Sargasso Sea. Preleptocephali are newly hatched larvae less than 6 mm long, and are genetically identified to be American eel, European eel, or other marine eel species. Molecular techniques are indispensable because morphological species identification does not work for undeveloped eggs and preleptocephali. Preleptocephali collected in the Sargasso Sea appear to be approximately one week old after hatching, which seems a too short duration for transportation of eggs and preleptocephali by currents from the newly proposed spawning area to the collection area in the Sargasso Sea. Therefore, it is indeed a fact that eel spawning occurs in the Sargasso Sea. Although eggs and spawning-condition adults have not been collected there, this lack of collection does not mean absence. There has also been no collection of eggs and adults or even preleptocephali outside Sargasso Sea. These apparent “false negatives” may result from insufficient sampling efforts in the Sargasso Sea and Mid-Atlantic Ridge areas as shown by Westerberg et al. 201845. It is also noteworthy that sampling efforts were not necessarily conducted with appropriate timing, place, and sampling methods, for example, with attention to peak spawning season, lunar phase, sampling grid mesh size of sampling grid, etc.
Based on molecular phylogenetic analyses of all anguillid eels, Atlantic eel ancestors were speculated to have invaded the North Atlantic from the Indo-Pacific through the ancient Tethys Sea before the Isthmus of Suez closed 30 million years ago46. They established their small migration loop around the coasts of the North Atlantic. They had a spawning area near the Mid-Atlantic Ridge in the narrow ancient North Atlantic that had not yet well expanded, and larvae were transported to Europe and North America randomly. Based on the expansion of the Atlantic Ocean floor, it is likely that the Atlantic eel split into two distinct species, American and European eels, due to the separation of their spawning areas, migration routes, and recruitment places42. The segregation of the two spawning areas probably is still the current situation considering the limited hybridization between both species and the introgression from American eels to European eels47. Moreover, the introgression force declines from northern to southern Europe, suggesting that spawning may have taken place in the central part of the newly proposed hatching zone near the Mid-Atlantic Ridge. For effective conservation of these endangered species, we must understand Atlantic eel reproductive ecology, including their respective present-day spawning areas and the evolutionary processes of both eel species. The first step in this process is to organize research cruises to enlarge the domain of survey and to validate a newly proposed Mid-Atlantic ridge hypothesis. More

Our overall approach is as follows: (1) develop a mathematical model (an SEIR-type compartmental model)18,19 that incorporates social-distancing data, case identification via testing, isolation of detected cases and contact tracing; (2) assess the model’s predictive performance by training (calibrating) it to reported cases and mortality data from 19 March to 30 April 2020 and validating its predictions against data from 1 May to 20 June 2020; and (3) use the model, trained on data to 22 July 2020, to predict future incidence and mortality. The final stage of our approach predicts future events under a set of scenarios that include increased case detection through expansion of testing rate, contact tracing and relaxation or increase of measures to promote social distancing. All model fitting is performed in a Bayesian framework to incorporate available prior information and address multivariate uncertainty in model parameters.
Model formulation
We modified the standard SEIR model to address testing and contact tracing, as well as asymptomatic individuals. A fraction fA of those exposed (E) to enter the asymptomatic A class (divided into AU for untested and AC for contact traced) instead of the infected I class, which in our model formulation also includes infectious presymptomatic individuals. With respect to testing, separate compartments were added for untested, ‘freely roaming’ infected individuals (IU), tested/isolated cases (IT) and fatalities (FT). Following recovery, untested infected individuals (IU) and all asymptomatic individuals move to the untested recovered compartment, IU, and tested infected individuals move to the tested recovered compartment, IT. In balancing considerations of model fidelity and parameter identifiability, we made the reasonably conservative assumptions that all tested cases are effectively isolated (through self-quarantine or hospitalization) and thus unavailable for transmission, and that all COVID-related deaths are identified/tested.
With respect to contact tracing, the additional compartment SC represents unexposed contacts who undergo a period of isolation during which they are not susceptible before returning to S, while EC, AC and IC represent contacts who were exposed. Again, the reasonably conservative assumption was made that all exposed contacts undergo testing, with an accelerated testing rate compared to the general population. We assume a closed population of constant size, N, for each state.
The ordinary differential equations governing our model are as follows:

$$begin{array}{l}frac{{mathrm{d}S}}{{mathrm{d}t}} = – S times c times left[ {beta + (1 – beta ) times f_{mathrm{C}}} right] times (I_{mathrm{U}} + A_{mathrm{U}})/N + S_{mathrm{C}} times gamma \ frac{{mathrm{d}S_{mathrm{C}}}}{{mathrm{d}t}} = – S_{mathrm{C}} times gamma + S times c times (1 – beta ) times f_{mathrm{C}} times (I_{mathrm{U}} + A_{mathrm{U}})/N\ frac{{mathrm{d}E}}{{mathrm{d}t}} = – E times kappa + S times c times beta times (1 – f_{mathrm{C}}) times (I_{mathrm{U}} + A_{mathrm{U}})/N\ frac{{mathrm{d}E_{mathrm{C}}}}{{mathrm{d}t}} = – E_{mathrm{C}} times kappa + S times c times beta times f_{mathrm{C}} times (I_{mathrm{U}} + A_{mathrm{U}})/N\ frac{{mathrm{d}I_{mathrm{U}}}}{{mathrm{d}t}} = – I_{mathrm{U}} times (lambda + rho ) + E times kappa times (1 – f_{mathrm{A}})\ frac{{mathrm{d}A_{mathrm{U}}}}{{mathrm{d}t}} = – A_{mathrm{U}} times rho + E times kappa times f_{mathrm{A}}\ frac{{mathrm{d}I_{mathrm{C}}}}{{mathrm{d}t}} = – I_{mathrm{C}} times (lambda _{mathrm{C}} + rho _{mathrm{C}}) + E_{mathrm{C}} times kappa times (1 – f_{mathrm{A}})\ frac{{mathrm{d}A_{mathrm{C}}}}{{mathrm{d}t}} = – A_{mathrm{C}} times rho _{mathrm{C}} + E_{mathrm{C}} times kappa times f_{mathrm{A}}\ frac{{mathrm{d}R_{mathrm{U}}}}{{mathrm{d}t}} = (I_{mathrm{U}} + A_{mathrm{U}} + A_{mathrm{C}}) times rho + I_{mathrm{C}} times rho _{mathrm{C}}\ frac{{mathrm{d}I_{mathrm{T}}}}{{mathrm{d}t}} = – I_{mathrm{T}} times (rho + delta ) + I_{mathrm{U}} times lambda + I_{mathrm{C}} times lambda _{mathrm{C}}\ frac{{mathrm{d}R_{mathrm{T}}}}{{mathrm{d}t}} = I_{mathrm{T}} times rho \ frac{{mathrm{d}F_{mathrm{T}}}}{{mathrm{d}t}} = I_{mathrm{T}} times delta end{array}$$

where c is the contact rate between individuals, β is the transmission probability per infected contact, fC is the fraction of contacts identified through contact tracing, 1/γ is the duration of self-isolation after contact tracing, 1/κ is the latent period, fA is the fraction of exposed who are asymptomatic, λ is the testing rate, δ is the fatality rate, ρ is the recovery rate and λC and ρC are the testing and recovery rates, respectively, of contact-traced individuals. The testing rates λ and λC, the fatality rate δ and the recovery rate of traced contacts ρC are each composites of several underlying parameters. The testing rate defined as

$$lambda (t) = F_{{mathrm{test}},0} times left[ {1 – frac{1}{{1 + mathrm{e}^{(t – T50_T)/tau _T}}}} right] times {mathrm{Sens}_{rm{test}}} times k_{{mathrm{test}}},$$

where Ftest,0 is the current testing coverage (fraction of infected individuals tested), Senstest is the test sensitivity (true positive rate) and ktest is the rate of testing for those tested, with a typical time-to-test equal to 1/ktest. The time-dependence term models the ramping up of testing using a logistic function with a growth rate of 1/τT d−1, where T50T is the time where 50% of the current testing rate is achieved. Similarly, for testing of traced contacts, the same definition is used with the assumption that all identified contacts are tested, Ftest,0 = 1 and at a faster assumed testing rate, kC,test:

$$lambda _{mathrm{C}}(t) = left[ {1 – frac{1}{{1 + mathrm{e}^{(t – T50_T)/tau _T}}}} right] times {mathrm{Sens}_{rm{test}}} times k_{{mathrm{C,test}}},$$

Because all contacts are assumed to be tested, the rate ρC at which they enter the ‘recovered’ compartment, RU is simply the rate of false negative test results:

$$rho _{mathrm{C}}(t) = left[ {1 – frac{1}{{1 + mathrm{e}^{(t – T50_T)/tau _T}}}} right] times (1 – {mathrm{Sens}_{rm{test}}}) times k_{{mathrm{test}}}$$

The fatality rate is adjusted to maintain consistency with the assumption that all COVID-19 deaths are identified, assuming constant IFR. Specifically, we first calculated the fraction of infected that is tested and positive:

$$f_{{mathrm{pos}}}(t) = f_{mathrm{C}}frac{{lambda _{mathrm{C}}(t)}}{{lambda _{mathrm{C}}(t) + rho _{mathrm{C}}(t)}} + (1 – f_{mathrm{C}})frac{{lambda (t)}}{{lambda (t) + rho }}.$$

Then the case fatality rate CFR(t) = IFR/fpos(t). Because CFR = δ/(δ + ρ), this implies

$$delta (t) = rho frac{{{mathrm{CFR}}(t)}}{{1 – {mathrm{CFR}}(t)}} = rho frac{{{mathrm{IFR}}}}{{f_{{mathrm{pos}}}(t) – {mathrm{IFR}}}}.$$

The model is ‘seeded’ Ninitial cases on 29 February 2020. Because in the early stages of the outbreak there may be multiple ‘imported’ cases, we fit to data only from 19 March 2020 onwards, 1 week after the US travel ban was put in place31.
Our model is fit to daily case yc and death yd data (cumulative data are not used for fitting because of autocorrelation). To adequately fit the case and mortality data, we accounted for two lag times. First, a lag is assumed between leaving the IU compartment and public reporting of a positive test result, accounting for the time it takes to seek a test, obtain testing and have the result reported. No lag is assumed for tests from contact tracing. Second, a lag time is assumed between entering the fatally ill compartment FT and publicly reported deaths. Additionally, we use a negative binomial likelihood to account for the substantial day-to-day over-dispersion in reporting results. The corresponding equations are as follows:

$$begin{array}{l}y_{{mathrm{obs}},[c,d]}(t) approx {mathrm{NegBin}}[alpha _{[c,d]},p_{[c,d]}(t)]\ p_{[c,d]}(t) = frac{{y_{{mathrm{pred}},[c,d]}(t)}}{{alpha _{[c,d]} + y_{{mathrm{pred}},[c,d]}(t)}}\ y_{{mathrm{pred}},c}(t) = I_{mathrm{U}}(t – tau _{{mathrm{case}}}) times lambda (t) + I_{mathrm{C}}(t) times lambda _{mathrm{C}}(t)\ y_{{mathrm{pred}},d}(t) = I_{mathrm{T}}(t – tau _{{mathrm{death}}}) times delta (t)end{array}$$

In this parameterization, because the dispersion parameter α → ∞, the likelihood becomes a Poisson distribution with expected value ypred,[c,d], whereas for small values of α there is substantial interindividual variability. Case and death data were sourced from The COVID Tracking Project32.
Finally, we derived the time-dependent reproduction number, R(t) and the effective reproduction number, Reff(t) of this model, given by

$$R(t) = c times beta times (1 – f_{mathrm{C}})left( {frac{{1 – f_{mathrm{A}}}}{{lambda + rho }} + frac{{f_{mathrm{A}}}}{rho }} right)$$

and

$$R_{{mathrm{eff}}}(t) = R(t) times frac{{{{S}}(t)}}{N}$$

Reff(t) is the average number of secondary infection cases generated by a single infectious individual during their infectious period in partially susceptible population at time t. It is equal to the product of the transmission risk per contact of an infectious individual with their untraced contacts, c × β × (1 − fC), times their average duration of infection, (left( {frac{{1 – f_{mathrm{A}}}}{{lambda + rho }} + frac{{f_{mathrm{A}}}}{rho }} right)), and the portion of contacts that are susceptible, (frac{{{{S}}(t)}}{N}). This accounts for the relative contribution of asymptomatic, (c times beta times left( {1 – f_{mathrm{C}}} right)left( {frac{{f_{mathrm{A}}}}{rho }} right) times frac{{{{S}}(t)}}{N}) and symptomatic infection, (c times beta times (1 – f_{mathrm{C}})left( {frac{{1 – f_{mathrm{A}}}}{{lambda + rho }}} right) times frac{{{{S}}(t)}}{N}). Using posterior samples for all 50 states and the District of Columbia, we conducted an analysis of variance using a linear model to characterize the contributions to the combined interstate and intrastate variation in Reff. Specifically, we used a linear model for Reff with the model parameters R0, η, θmin, rmax, fC, fA, λ and ρ as predictors, and evaluated the percentage of variance in Reff contributed by each parameter.
Incorporating social distancing, enhanced hygiene practices and reopening
The impact of social distancing, hygiene practices and reopening was modelled through a time dependence in the contact rate, c and the transmission probability per infected contact, β:

$$begin{array}{l}c(t) = c_0 times left[ {theta (t) + (1 – theta _{mathrm{min}}) times r(t)} right]\ beta (t) = beta _0 times theta (t)^eta end{array}$$

The θ(t) function parameterized social distancing during the progression to shelter-in-place, and is modelled as a Weibull function:

$$theta (t) = theta _{{mathrm{min}}} + (1 – theta _{{mathrm{min}}}){mathrm{e}}^{ – (t/tau _theta )^{n_theta }},$$

which starts as unity and decreases to θmin, with τθ being the Weibull scale parameter and nθ the Weibull shape parameter (Fig. 1).
The r(t) function parameterized relative increase in contacts due to reopening after shelter-in-place, with r = 1 corresponding to a return to baseline c = c0.

$$begin{array}{l}r(t) = r_{{mathrm{max}}}frac{{t – tau _theta – tau _s}}{{tau _r}}left[ {u(t – t_r) – u(t – t_{r{mathrm{max}}})} right] + u(t – t_{r{mathrm{max}}})\ u(t) = {mathrm{Heaviside}}(t) approx 1 – frac{1}{{1 + {mathrm{e}}^{4t}}}\ t_r = tau _theta + tau _s\ t_{r{mathrm{max}}} = tau _theta + tau _s + tau _rend{array}$$

The term r(t) is 0 before tr, linear between tr and trmax and constant at a value of rmax after that, and made continuous by approximating the Heaviside function by a logistic function. The reopening time is defined as τs days after τθ, and the maximum relative increase in contacts rmax happens τr days after that.
We selected the functional form above for c(t) because it was found to be able to represent a wide variety of social-distancing data, including mobile phone mobility data from Unacast33 and Google34 as well as restaurant booking data from OpenTable35. We used these different mobility sources to derive state-specific prior distributions because different social-distancing datasets had different values for θmin, τθ, nθ, τs, rmax and τr (Supplementary Fig. 1).
With respect to the reduction in transmission probability β, we assumed that during the shelter-in-place phase, hygiene-based mitigation paralleled this decline with an effectiveness power η, and that this mitigation continued through reopening.
Finally, we define an overall reopening parameter Δ that measures the rebound in disease transmission, c × β relative to its minimum, defined to be 0 during shelter-in-place (that is, R(t) is at a minimum) and 1 when all restrictions are removed (when R(t) = R0), which can be derived as:

$${Delta}(t) = frac{{c times beta /(c_0 times beta _0) – theta _{{mathrm{min}}}^{1 + eta }}}{{1 – theta _{{mathrm{min}}}^{1 + eta }}}.$$

Our model is illustrated in Fig. 1, with parameters and prior distributions listed in Table 1.
Scenario evaluation
We used the model to make several inferences about the current and future course of the pandemic in each state. First, we consider the effective reproduction number. Two time points of particular interest are the time of minimum Reff, reflecting the degree to which shelter-in-place and other interventions were effective in reducing transmission, and the final time of the simulation, 22 July 2020, reflecting the extent to which reopening has increased Reff. Additional parameters of interest are the current levels of reopening Δ(t), testing λ and contact tracing fC.
We then conducted scenario-based prospective predictions using our model’s parameters as estimated to 22 July 2020. We then asked the following questions:
(1)
Assuming current levels of reopening, what increases in general testing λ and/or contact tracing fC would be necessary to bring Reff  More

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Ocean temperature
The independent bottom water temperature estimates all showed variable but increasing autumn temperatures from 1996 to 2010, followed by a decline through to 2018 in the mid-depth offshore stations and the survey, but less of a decline in the inshore or deep offshore stations (Fig. 1, 2a). Both deep offshore stations warmed by ~ 0.2 °C over the time series. The ~ 1 °C temperature range evident in the southwestern mid-depth offshore FX8 autumn hydrographic series was larger than that of the less variable northern mid-depth offshore SI7 hydrographic station, but neither of the mid-depth offshore stations showed any net warming over the 22-year period. The shallow inshore stations both warmed considerably (1.5–2 °C) over the time series. Although no net increase was apparent in either of the mid-depth offshore hydrographic station autumn time series, 0.25–0.40 °C increases were observed in both the winter and spring measurement time series at both mid-depth hydrographic stations.
Figure 1

Map of 245 autumn groundfish survey core stations (black dots) sampled around Iceland every October between 1996 and 2018 (except 2011). Bottom-temperature profiles were collected independently at hydrographic stations FX3, FX8 and FX9 (red triangles) and SI1, SI7 and SI8 (blue triangles). Depth contours are 200, 500 and 1000 m.

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The mean environmental temperature from the autumn survey (Tempe) lay midway between that of the two mid-depth offshore hydrographic stations, over a similar ~ 1 °C temperature range. Based on a fitted regression to the environmental temperature time series, there was a net warming of 0.33 °C across the survey area over the 22-year time period, although the increase was certainly not linear (Fig. 2c). There were large differences in both the temperature range and temperature-at-depth among some of the regions (Fig. 2b). Bottom water temperatures at 200 m depth were 3–4 °C warmer in the south (SW = 7.3 °C; SE = 8.2 °C) than in the north (NW = 5.4 °C; NE = 4.5 °C), and 4–5 °C warmer at a depth of 500 m (NW = 1.4 °C; NE = −0.5 °C; SW = 6.6 °C; SE = 3.9 °C). Although most stations deeper than 500 m in the NE were  3 °C.
Figure 2

Bottom-temperatures from hydrographic stations (A: left panel) and from the autumn groundfish survey (B: top right and C: bottom right panel). (A) Near-bottom temperatures at hydrographic stations FX3 and SI1 (~ 70 m), FX8 and SI7 (~ 400 m), and FX9 and SI8 (~ 1000 m) between 1996 and 2018 are shown for winter (January–March), spring (May–June) and autumn (October–December). A geometric smooth has been fitted to the annual means. Note different scales on the y-axes. (B) Mean depth-temperature profiles for each region based on bottom temperatures from the autumn survey. Fitted lines are loess regressions. (C) The mean annual environmental temperature time series estimated from a GLM of the autumn survey bottom temperature measurements (see text for details). The linear regression fitted to the annual means is intended only to show the overall rate of warming through the time series.

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Survey catches
A total of 5390 tows were fished over the 22-year time series. Station depths ranged from 26 to 1203 m (mean of 377 m) while bottom temperatures ranged between − 2 and 11 °C (mean of 4.85 °C). Of the 7,246,474 fish representing 200 species that were caught, 82 species were caught during at least 19 of the survey years, and thus are the primary subject of this analysis (Suppl. Table 1). The mean standardized annual abundance of these 82 species ranged between 1 and 27,513 fish, with an overall mean across species of 4096; 70% of the species had a mean standardized annual abundance of less than 100.
There was no clear identifier of species that were rare, and thus caught sporadically, versus those that were newly immigrated to the survey area in response to the warming environment. Nor was it possible to exclude the possibility that certain rare species were incorrectly identified in some survey years. However, there were 6 species that were never caught prior to 1998, yet appeared in increasing numbers in subsequent years (Table 1). All of these species were caught in at least 8 survey years, and as many as 19. However, three of the species were deepwater ( > 800 m), and thus unlikely to have been subject to temperature increases of > 0.2 °C. Of the remaining three species, Atlantic mackerel (Scomber scombrus), blackbelly rosefish (Helicolenus dactylopterus) and hollowsnout grenadier (Coelorinchus caelorhincus) were all warmwater (TB  > 2) and stenothermal (Steno  More