State-wide clustering analysis of epidemiological time series

To understand which counties in Washington state experienced similar epidemic trajectories, and whether these similarities were related to spatial proximity, we performed a clustering analysis of the rank correlation (quantified using Spearman’s ρ) among county-level time series of weekly confirmed cases per 10⁵ (Fig 2; see S1 and S2 Figs for the underlying time series data). Three sparsely populated counties, Garfield, Skamania and Wahkiakum, with populations under 5,000 were excluded due to insufficient data. We found dynamics of COVID-19 were positively correlated between all pairs of counties, with correlation ranging from ρ = 0.27 (between Yakima and Whitman) to ρ = 0.98 (between Clark and Spokane) (Fig 2A). To explore groupings in the pair-wise correlation between counties, we performed a clustering analysis using hierarchical agglomerative clustering with a maximum linkage criterion (Fig 2B). We found specific closely linked groupings of neighboring counties, including Pierce, Kitsap and Mason, Franklin and Benton, and King and Snohomish. However, other closely linked counties were geographically far apart, in particular Clark, Spokane and Thurston. Based on the silhouette score (a measure of clustering performance [34], see S3 Fig), we detected five clusters (Fig 2B). Mapping these clusters elucidates clear geographic patterns (Fig 2C).

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Fig 2. Analysis of temporal correlation in pre-vaccination incidence time series in WA state.

A) Correlation matrix of Spearman’s ρ for each pairing of WA counties. Three sparsely populated counties (Garfield, Skamania and Wahkiakum) were excluded from the analysis due to insufficient data. B) Results of agglomerative clustering performed on the correlation matrix. Based on the silhouette score, we identify five distinct epidemiological clusters. C) Map of identified clusters. We observe a clear spatial structure to the clustering: i) a cluster predominantly of counties from Seattle northwards bordering the Puget Sound (yellow), ii) a cluster including the Yakima valley and Tri-cities area (orange), iii) two rural clusters in eastern Washington (teal and blue) and iv) a state-wide cluster (pink). Map makes use of TIGER/Line Shapefiles from the U.S. Census Bureau (https://www.census.gov/geographies/mapping-files.html).

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Three of the five clusters were contiguous: the first (yellow) centered around the Puget Sound region includes Seattle (located in King), the second (orange) includes the Yakima valley and its confluence with the Columbia river in the Tri-cities area (located on the border of Franklin and Benton) and the third (teal) in the rural north central region. Of the two remaining clusters, one (blue) is mostly contiguous apart from Ferry, which is the the most distantly linked county in the data set. The final cluster (pink) is the largest, comprising a number of urban and rural areas across the entire state (including the previously mentioned Pierce, Kitsap, Clark, Spokane and Thurston counties).

We repeated the analysis on weekly hospitalizations data (S4 Fig). The data were substantially noisier, due to the smaller numbers of infections that were hospitalized, which is reflected in lower entries in the correlation matrix. Overall, the clusters were less well defined, as quantified using the silhouette score (S5 Fig). As might be expected, the impact of increased noise is most pronounced on the clustering of smaller counties. Apart from Thurston, the top ten most populous counties all remain in the same clusters (S4 Fig).

Statistical inference using spatio-temporal data from the Puget Sound region

To investigate further the factors driving county-level differences in epidemic trajectories, including the role of human mobility patterns and non-pharmaceutical interventions, we used a stochastic spatial transmission model to perform statistical inference on county-level weekly hospitalizations data. For the analysis, we selected five counties, belonging to two clusters, that form the core of the Puget Sound metropolitan area: King, Pierce, Snohomish, Thurston and Kitsap. Together these counties account for over 70% of the total state population. We focused on hospitalization data, rather than confirmed cases, to alleviate issues with time-varying testing rates (see S6 Fig). To allow us to focus solely on questions around the impact of movement patterns and local non-pharmaceutical interventions on the spatial spread of SARS-CoV-2 in Washington state, we restricted our analysis to the dynamics of SARS-CoV-2 up to February 2021, prior to both i) the introduction of mass vaccination and ii) the introduction of VOCs in 2021 (with phylogenetic evidence suggesting that in 2020 co-circulating strains in Washington had comparable reproductive numbers [35]). We further filtered the data to include only hospitalized cases aged under 60, due to sustained transients in older age groups (possibly due to their increased ability to socially distance compared to school- and work-aged individuals, see S6 Fig). To model the impact of county-specific non-pharmaceutical interventions, for each county we introduced a resident reproductive number, , defined as the expected number of cases caused by a non-commuting individual who remains in county i with a fully susceptible population. This function was parameterised for each county via a set of basis function coefficients, which were estimated from incidence data (see Methods). In addition, we also obtained maximum likelihood estimates for four region-wide parameters: the baseline reproductive number (R₀), the probability an infection was aged under 60 and hospitalized (ρ), the average number of weekly cases imported to the region (η) and the baseline pre-social distancing fraction of total contacts a commuting individual would have in their commuting destination (ϵ). Maximising the log-likelihood was accomplished in two steps (see Methods). First, for each point on a grid of region-wide parameter values we used iterated filtering to maximise the log-likelihood over the space of basis function parameters (S7 Fig). This likelihood surface shows a trade-off in model performance between R₀, ρ and η, which we attribute to all three parameters affecting the apparent size of a growing epidemic. The fraction of contacts in the commuting destination, ϵ, does not trade off with any of the other region-wide parameters. Maximum likelihood estimates for all parameters were subsequently obtained by maximising the likelihood over the grid, with profile likelihoods and 95% confidence intervals (estimated using Wilks’ theorem [36]) for each region-wide parameter shown in Fig 3A–3D.

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Fig 3. Profile likelihoods and reconstructed connectivity matrix from the inference analysis.

A–D) Profile likelihoods are shown for the four estimated global parameters: the baseline reproductive number (A), the probability an infection is under 60 and hospitalized (B), the weekly rate infections are imported into the population assuming full susceptibility (C) and the pre-pandemic fraction of contacts that commuting individuals have in their commuting destination (D). The dashed horizontal line corresponds to 2 log-likelihoods below the maximum. E) Reconstructed connectivity matrix (see Methods) incorporating census commuting data and the maximum likelihood estimate of the commuting parameter ϵ (see panel D).

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A sensitive question in this inference is the start date of the epidemic. In the analysis above, we assumed a starting date of January 12 (the date of the first reported case in Washington state). However, a phylogenetic study by Worobey et al. [7] suggested the outbreak was initiated during the week beginning January 26 (the median of their posterior distribution), we therefore examined the sensitivity of our statistical inference to initializing on this date. We found the model performance (considering only the data post-January 26) was 7.7 log-likelihoods lower compared with an outbreak starting in the week of January 12. In addition, with the later start date, the maximum likelihood estimate for η suggests an average of 8 introductions per week, with η = 0 lying outside the 95% confidence interval, making it extremely improbable that all infections in the outbreak were descended from a single introduction (as suggested by the results of Worobey et al. [7]). This is in contrast with our inference using the earlier January 12 start date, where the MLE was η = 0 (Fig 3C).

Using the demographic data, commuting data and our maximum likelihood estimate for ϵ, we reconstructed the connectivity matrix: the proportion of daily contacts an individual resident in the origin county has with individuals in the destination county. We found that the reconstructed connectivity matrix is the same regardless of whether a January 12 or January 26 start date was used (Fig 3E). At the beginning of the pandemic (before the impact of local variation in ), this also gives the proportion of transmission in each destination county by an infectious individual resident in the origin.

As explained above, is the reproductive number of individuals who do not travel outside of their county of residency. The reproductive number for commuters is a weighted sum of in their origin and commuting destination. Time-variation in alters the rate of transmission between individuals located in a county–either due to residency or commuting (Fig 4). We find that non-pharmaceutical interventions sharply reduced in all five counties, with a particularly sharp drop in Thurston (Fig 4I) and Kitsap (Fig 4J). Crucially, while non-pharmaceutical interventions drove below 1, they were of insufficient extent and duration to eliminate SARS-CoV-2 in the Puget Sound region. Subsequent waves in the summer and winter were driven by time-varying trends in .

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Fig 4. Time series of weekly COVID-19 hospitalizations and maximum likelihood estimates of the resident reproductive number for the five counties studied in inference analysis.

A–E) Weekly hospitalized cases under the age of 60 for King (A), Pierce (B), Snohomish (C), Thurston (D) and Kitsap (E). Inference was performed on hospitalizations data rather than confirmed cases due to temporal variation in testing access (see main text). F–J) Reconstructed maximum-likelihood estimates of the resident reproductive number for each county (see Methods for details). Time-varying trends reflect the combined impact of all non-pharmaceutical interventions (including social distancing and mask wearing). Non-pharmaceutical interventions were assumed to start in the week beginning March 1 2020, i.e. immediately after the first discovery of sustained SARS-CoV-2 transmission in the state.

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Impact of stochasticity on the epidemic trajectory in the Puget Sound region

Our maximum likelihood estimate (MLE) for the commuting parameter was ϵ = 1, i.e. the strongest connectivity possible (see Fig 3D). To understand why the model favoured high spatial connectivity, we systematically varied ϵ and investigated the log-likelihood of each week’s incidence conditioned on the preceding data, i.e. ln P(x_t|{x_s}_s<t), which we refer to as the conditional log-likelihood (CLL; Fig 5A). The CLL paints a picture of how well the model explains each data point and helps identify those that contribute most to the overall log-likelihood, which is the sum of the CLL across all weeks. We find that the CLL follows a similar trajectory regardless of the value of ϵ, and is generally lowest when the epidemic is largest, an expected consequence of both demographic stochasticity [37] and observation error (see Eq 10). More insight can be gained into the dependence of the CLL on ϵ by calculating the relative CLL, found by subtracting the CLL corresponding to the MLE ϵ = 1 (Fig 5B). We see that lower values of ϵ (darker colors) are penalised the most early in the epidemic (pre-April 2020), and have the lowest relative CLL. During April and May, there is a suggestion that smaller values of ϵ have a larger CLL than the MLE (relative CLL > 0). To quantify these observations further, we measure the association between the CLL and ϵ for each week using Kendall’s τ (Fig 5C). Pre-March 2020 lockdown, we see the largest values of τ, implying the strongest positive association between ϵ and CLL. After the start of lockdown, Kendall’s τ drops. Indeed, during late April and May, we see a period where τ < 0, implying a negative association between ϵ and CLL. From June onwards τ stabilises, fluctuating about a smaller positive value than pre-lockdown, indicating a weaker positive association. Overall, we see that the main reason the model parameterized with higher values of ϵ performs better is due to an increased ability to fit the data pre-April 2020.

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Fig 5. Investigation of the dependence of model performance on the commuting parameter, ϵ.

A) Conditional log-likelihood for each week of data. The model parameters for each line maximise the overall log-likelihood with ϵ constrained to the value indicated in the legend. Consequently, summing over all weeks for a particular value of ϵ gives the corresponding value of the profile log-likelihood shown in Fig 3D. B) Conditional log-likelihood relative to the maximum (ϵ = 1). C) Kendall’s τ coefficient for each week, quantifying the association between conditional log-likelihood and ϵ. A more positive (negative) τ indicates a stronger positive (negative) association between ϵ and the conditional log-likelihood for that week’s data. D) Log-probability of the final week’s data (week beginning January 31 2021) conditioned on all data up to and including the week indicated (Q_t). Results show a large jump in Q_t over the course of the first month, indicating the importance of atypical spread in disseminating the virus across the region.

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In addition to investigating the CLL, we also investigated the importance of each weekly hospitalisations data point in determining the subsequent course of the regional epidemic (Fig 5D). This was quantified via the log-probability of the final week’s hospitalizations data (week beginning January 31 2021), x_T, conditioned on all data up-to-and-including week t, Q_t = ln P(x_T|{x_s}_s≤t). An increase (decrease) in Q_t in week t implies that the new observation x_t increases (decreases) the probability of observing x_T at the later time T.

We see the largest rise in Q_t during March and April, implying that the impact of early stochastic events persisted over the entire period of study. Subsequently, there is little increase in Q_t between April and October 2020, implying that individual stochastic fluctuations which occurred during this second period had limited lasting impact. Finally, from around October onward we see an accelerating rise in Q_t. This observation is to be expected, and is analogous to the time series autocorrelation rising as the lag between observations shrinks [37]. Overall we find Q_t is positively associated with ϵ, as expected given our results for the CLL (Fig 5A–5C).

To investigate the dynamical impacts of these early stochastic events, we simulated 50,000 realizations of our fitted transmission model (Fig 6). We find that if the simulations are initialized in the week beginning January 12 2020 (when the first case of COVID-19 was detected in Washington state), the mean trajectory (calculated by averaging over all realizations) predicts lower hospitalizations than observed throughout the epidemic (Fig 6A). The most similar realization (as quantified via sum of squared errors) tracks the data closely, however falls in the tail of the distribution, suggesting it is an atypical trajectory. If, instead, we simulate from the model using the state of the most similar trajectory in the week of February 9 (i.e. 4 weeks after the first confirmed case) as the initial condition, we find that the sample mean overlaps the data, and the variance among simulated realizations is much smaller (Fig 6B). This substantiates our earlier observations regarding the pronounced early increase in Q_t (Fig 5D). Our results suggest that rapid, atypically intensive transmission within the first couple of weeks after introduction is necessary to generate the epidemic trajectories observed in King and Snohomish counties. Subsequently, the epidemic is sufficiently large that it followed a roughly deterministic trajectory determined by the combined effects of non-pharmaceutical interventions and susceptible depletion.

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Fig 6. Comparison of simulated trajectories using the maximum likelihood parameter estimates with observed data.

A) Simulated trajectories for King county initialized on January 12 2020. Blue shading corresponds to the probability mass estimated from 50,000 sampled trajectories, with the sample mean shown in red. The orange curve is the observed hospitalizations data used for parameter inference, and lies in the tail of the probability distribution—suggesting an atypical trajectory. Also shown is the simulated trajectory most similar to the observed data (grey curve) as calculated via the sum of errors. B) Simulated trajectories for King county initialized on February 9 2020 using the state variables of the most similar sample (see panel A). Our results suggest that, once the initial atypical stochastic spread had occurred, the subsequent epidemic trajectory was largely a deterministic response to non-pharmaceutical interventions and susceptible depletion.

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Model validation

Finally, from our simulated trajectories at the MLE initialised on February 9 2020 (Fig 6B), we reconstructed the cumulative prevalence of SARS-CoV-2 infection in Washington state through time (Fig 7). We compared our reconstruction with those from four published studies [38–41]. We also compared our simulations with estimates of SARS-CoV-2 antibody seroprevalence released by the CDC [42, 43]. We find our simulations are broadly consistent with two studies (Davis et al. [38] and Wu et al. [41]), and are smaller than the estimates of Pei et al. [39] and Reese et al. [40] by a factor about 1.5 and 3 respectively.

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Fig 7. Comparison of reconstructed prevalence with estimates from the literature.

Using our simulated trajectories initialised on February 9 2020 (see Fig 6B), we estimated the cumulative percentage of the population exposed to SARS-CoV-2 in the Puget Sound region (blue). We compared our estimate to a) time series data of confirmed cases in the Puget sound region (black) b) estimates from four studies (orange, green, red, brown) [38–41] and c) estimates of the seroprevalence in Washington state from serological data (purple) [42, 43]. Caution must be taken in reading these results as the studies used different statistical approaches and spatial aggregations. Davis et al., Pei et al. and Wu et al. all used Bayesian approaches, results shown are the median and the 90% (Davis et al.) and 95% (Pei et al. and Wu et al.) credible interval. The results shown from Reese et al. and the seroprevalence study are the mean and 95% confidence intervals. Our results show the mean and 95% percentile of simulations, accounting for stochastic variability but not parametric uncertainty (see Fig 3). Furthermore, the results from Pei et al. are for the Western Washington region (which includes the Puget sound region), the results from Davis et al., Wu et al., and the seroprevalence study are for the entire state, and the results of Reese et al. are for the combined states of Alaska, Idaho, Oregon and Washington.

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Data sources

Throughout the pandemic, the Washington state Department of Public Health has released weekly COVID-19 data, consisting of confirmed cases, new hospitalizations and deaths. Data are disaggregated on the county-level and by age [57]. As our study focused on understanding spatial patterns in SARS-CoV-2 transmission, we restricted our analysis to data from the period January 12 2020 (the week of the first confirmed case) to January 30 2021. This end date was chosen to be prior to the introduction of novel VOC and mass vaccination, both of which have a confounding spatially heterogeneous effect on viral transmission.

For our likelihood-based inference we used data on the weekly number of new COVID-19 hospitalizations, in order to avoid confounding time-varying trends in confirmed case reports and mortality data (due to the limited initial testing capacity and improvements in treatment respectively). In addition, we excluded hospitalizations in individuals over 60 from the data due to early outbreaks in residential care homes [6].

Demographic data on the population size of each Washington county were sourced from the US census [58]. Mobility data recording the number of daily commuters between each county in Washington state were also sourced from the US census [59]. State-level seroprevalence estimates of cumulative SARS-CoV-2 infections were periodically made publicly available by the CDC [42, 43].

Clustering analysis

To understand state-wide patterns in SARS-CoV-2 transmission, we performed a clustering analysis on time series data [60]. We used county-level confirmed case data, excluding Garfield, Skamania and Wahkiakum counties (each with population <5000) from the analysis due to insufficient data. In the clustering analysis, we used 1 − ρ as our distance matrix, where ρ is the matrix of pairwise values of Spearman’s ρ calculated from the county-level confirmed case time series. We used hierarchical agglomerative clustering with a maximum linkage criterion to identify clusters of counties [61]. The number of clusters present in the data was determined by maximising the silhouette score [34]. We repeated our analysis using hospitalization data in place of case counts.

Transmission model

Statistical inference was performed using a stochastic SEIR-based transmission model that modelled transmission of SARS-CoV-2 in a population of n_c counties. As each county had distinct transmission parameters (see below), the curse of dimensionality limited the computational feasibility of our statistical inference to n_c = 5. We chose to focus on the five counties which form the core of the Puget Sound region: King, Pierce, Snohomish, Thurston and Kitsap. We denote the number of new infections in week [t, t + 1) among residents of county i by I_i,t+1. Assuming a 7-day serial interval (consistent with multiple estimates for the original 2020 variant [2, 62, 63]), I_i,t+1 can be modelled using a Poisson distribution, with expectation (1) where λ_i,t is the force of infection and S_i,t is the number of susceptible individuals resident in i at time t. We update S_i,t according to the update rule (2)

Due to a lack of evidence to the contrary, we assumed the entire population had no prior immunity against SARS-CoV-2, i.e. that S_i,0 = N_i where N_i is the population of county i.

The force of infection λ_i,t depends on the contact rates of susceptible individuals resident in location i with infectious individuals, and is given by (3) where η is the average weekly number of SARS-CoV-2 infections imported into the region and ξ_t is a zero-mean gamma noise process with standard deviation σ_SE = 0.126 that accounts for environmental stochasticity in the transmission process [64]. The parameter is the resident reproductive number for location i, which we detail below. The tensor element β_i,j,t is the transmission rate from infectious individuals in j to susceptible individuals in i at time t, (4) where the weight is determined by the connectivity matrix, whose elements K_i,j correspond to the probability that an individual resident in j is present in i. The denominator corresponds to the mobility-adjusted population size of location k. The sum over k in Eq 4 accounts for the small but non-zero probability that an individual resident in i might encounter an individual resident in j in a third county, k. The connectivity matrix was parameterised using commuting data, and has the structure (5) where T_i,j is the number of daily commuters from j to i in the commuting data and the commuting parameter ϵ, which can take values in [0, 1], is the proportion of contacts that a commuter has in their commuting destination. The diagonal elements of the contact matrix account for both the contacts of non-commuting individuals and commuters in their residence location (i.e. while not at work).

The resident reproductive number for location i, , is the expected number of secondary infections caused in a fully susceptible population by a non-commuting individual who lives in i. This parameter varies through time due to the effects of non-pharmaceutical infections, such as social distancing and mask wearing, and is county-specific to allow for differences in measure implementation and adherence. We model using a set of n_α basis functions, , via the relationship (6) The functions c_μ(t) are in turn defined in terms of a basis of n_α + 1 cubic B-splines, , which are constructed over the period [t_sd, t_end] where t_sd and t_end are the start time of social distancing and the end time of the time series respectively (see [65]). Explicitly, we define (7) This definition ensures that: i) for t < t_sd and ii) is a continuous function of time regardless of α_i,μ and b_μ(t) (as can be seen by sending t → t_sd from above in Eq 7). We fix the start of social distancing as t_sd = 7, corresponding to the week beginning March 1 2020 (the first week after the discovery of community transmission) and, based on preliminary investigation, used a basis of n_α = 8 functions. Commuters experience a reduction in contact rates through social distancing in both their resident and commuting location (see Eq 4).

In what follows, we use hospitalizations to exclusively refer to those under 60, i.e. the subset we are fitting to. We assumed that the number of weekly infections in each location that are under 60 and require hospitalization, H_i,t, followed an overdispersed normal distribution [64], with probability density function specified in terms of the mean and variance. Due to the typical delay between infection and hospitalization, infections were lagged by one week in our hospitalizations model. The mean and variance in the number of weekly hospitalizations in location i at time t are given by, (8) (9) with ρ the probability a case in under 60 and hospitalized and ψ the overdispersion parameter. Together, Eqs 1–9 define a discrete time Markov process with state at time t.

Statistical inference

Statistical inference was performed on T = 56 weeks of hospitalizations data from n_c = 5 counties, . The likelihood function for the observed data given the model parameters Θ, denoted ℓ(Θ), is found by marginalizing the probability of a trajectory over the unobserved state variables, (10) where is the probability density function of the normal distribution specified by Eqs 8 and 9. If the populations are disconnected (ϵ = 0) then the transmission processes in each location are mutually independent and the likelihood function factorizes, , with the likelihood in location i given by (11) The structure of our model makes Eqs 10 and 11 analytically intractable, and solving either of them requires using Monte Carlo methods, i.e. repeatedly simulating from the transmission model Eqs 1–7. To perform this sampling efficiently we used particle filtering, see [33, 66].

Due to a lack of compelling evidence for pre-2020 immunity to SARS-CoV-2 infection, we assumed the population was initially fully susceptible with one infectious individual in Snohomish county in the week of January 12 (the week and location of the first identified case in Washington state). This infection was not necessarily the ancestor of all subsequent infections as our model allowed for introductions in the force of infection (see Eq 3).

Statistical inference to find maximum likelihood estimates for unknown model parameters was performed using data from five counties in the Puget Sound region: King, Pierce, Snohomish, Thurston and Kitsap. In total, our model had 44 unknown parameters: 4 region-wide parameters and 8 × 5 county-specific basis function coefficients. Given the high-dimensionality of the search space, we used a hybrid scheme to maximise the likelihood.

Maximisation over the four region-wide parameters was performed via a grid search. At each grid point, the region-wide parameters were set to their grid point values and the likelihood was maximised over the remaining 40 basis function parameters. To ensure convergence we replicated the estimation 27 times with different initial parameter guesses. Initial guesses for each basis function coefficient α_i,μ within each replicate were generated using latin-hypercube sampling over the interval [−3, 3] (substantially larger than any final estimated values, based on preliminary studies). Likelihood maximisation was then achieved in two stages, both using the iterated filtering algorithm implemented in the mif2 function of the R package “pomp” [32, 33]:

1. To refine our initial guesses for , iterated filtering was run for each county separately, assuming no spatial connectivity (i.e. ϵ = 0, see Eq 11). For this first run we used the following iterated filtering hyperparameters: 2000 particles, 50 iterations, a cooling fraction of 0.8 and a random walk standard deviation of σ_RW = 0.05.
2. The estimates from the first run were used to initialize for a second iterated filtering search with the full connected model (i.e. with ϵ at its grid point value, Eq 10), which used the same hyperparameters apart from a smaller random walk σ_RW = 0.01. This second search returned estimates of the basis function coefficients which maximise the likelihood at each grid point.

Using these estimated basis function coefficients, we then calculated the maximimum likelihood at each grid point via particle filtering, again using 2000 particles. Finally, the MLE for all model parameters we found by maximising the resulting likelihood surface over the entire grid. Confidence intervals on the global parameters were constructed in a similar way: For each value of the parameter of interest, the profile likelihood was calculated by maximising the likelihood surface over all the other grid dimensions. Statistical significance was then quantified using a likelihood-ratio test and the application of Wilks’ theorem, with results shown at the 95% confidence level.