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Data and data processingThe modeling tools described in the following sections are applied to the Italian COVID-19 epidemic at the scale of second-level administrative divisions, i.e., provinces and metropolitan cities (as of 2020, 107 spatial units). Official data about resident population at the provincial level are produced yearly by the Italian National Institute of Statistics (Istituto Nazionale di Statistica, ISTAT; data available at http://dati.istat.it/Index.aspx?QueryId=18460). The January 2019 update has been used to inform the spatial distribution of the population.The data to quantify nation-wide human mobility prior to the pandemic come from ISTAT (specifically, from the 2011 national census; data available online at https://www.istat.it/it/archivio/139381). Mobility fluxes, mostly reflecting commuting patterns related to work and study purposes, are provided at the scale of third-level administrative units (municipalities)53,54. These fluxes were upscaled to the provincial level following the administrative divisions of 2019, and used to evaluate the fraction pi of mobile people and the fraction qij of mobile people between i and all other administrative units j (see Supplementary Material in Gatto et al.7).Airport traffic data for year 2019, used to inform the simulation shown in Fig. 4c, d, are from the Italian Airports Association (Assaeroporti; data available at http://assaeroporti.com/statistiche_201912/). Note that airports have been assigned to the main Metropolitan Area they serve, rather than to the province where they are geographically located (e.g., Malpensa Airport has been assigned to the Metropolitan City of Milano, rather than to the neighboring Varese province, where it actually lies).Model parameters are taken from a paper by Bertuzzo et al.14, where they were inferred in a Bayesian framework on the basis of the official epidemiological bulletins released daily by Dipartimento della Protezione Civile55 (data available online at https://github.com/pcm-dpc/COVID-19) and the bulletins of Epicentro, at ISS51,56. The parameters estimated for the initial phase of the Italian COVID-19 epidemic14, during which SARS-CoV-2 was spreading unnoticed in the population, reflect a situation of unperturbed social mixing and human mobility, absent any effort devoted to disease control. This parameterization, in which all parameters (including the transmission rates) are spatially homogeneous, is reported in Table 2 and has been used to produce all the results presented in the main text, except for those of Fig. 6. In this case, to account for the containment measures put in place by the Italian authorities and their effects on transmission rates and mobility patterns during the first months of the pandemic, a time-varying parameterization14 for the period February 24 to May 1, 2020 has been used. In this parameterization, the transmission rates were allowed to take different values over different time windows, corresponding to the timing of the implementation of the main nation-wide restrictions, or lifting thereof. Specifically, the effect of the containment measures was parameterized by assuming that the transmission parameters had a sharp decrease after the containment measures announced at the end of February and the beginning of March, and that they were further reduced in the following weeks as the country was effectively entering full lockdown. As a by-product, these time-varying transmission rates can also at least partially account for seasonal effects on disease transmission. Due to the emerging nature of the pathogen, seasonality has not been given further consideration in this work; however, it may become a key component of future modeling efforts aimed at studying post-pandemic SARS-CoV-2 transmission dynamics3, i.e., if/when the pathogen establishes as endemic. Spatial connectivity too was modified with respect to the baseline scenario to reflect the disruption of mobility patterns induced by the pandemic and the associated containment measures14. Specifically, between-province mobility was progressively reduced as the epidemic unfolded according to estimates obtained through mobility data from mobile applications53,57.Spatially explicit SEPIAR with distributed controlsWe consider a set of n communities connected by human mobility fluxes. In each community, the human population is subdivided according to infection status into the epidemiological compartments of susceptible, exposed (latently infected), post-latent (incubating infectious, also termed pre-symptomatic7), symptomatic infectious, asymptomatic infectious (including paucisymptomatic), and recovered individuals. The present model utilizes previous work aimed to describe the first wave of COVID-19 infections7,14. In particular, it allows us to account for three widely adopted types of containment measures: reduction of local transmission (as a result of the use of personal protections, social distancing, and local mobility restriction), travel restriction, and isolation of infected individuals. To describe the effects of isolation, each infected compartment (exposed, post-latent, symptomatic and asymptomatic) is actually split into two, which allows keeping track of the abundances of infected individuals who are still in the community vs. those who are removed from it (i.e., either in isolation at a hospital, if symptomatic, or quarantined at home, if exposed, post-latent, or asymptomatic). The state variables of the model are summarized in Table 1. Supplementary Figure 1 recapitulates the structure of the model.COVID-19 transmission dynamics are thus described by the following set of ordinary differential equations:$${dot{S}}_{i} =mu ({N}_{i}-{S}_{i})-{lambda }_{i}{S}_{i}\ {dot{E}}_{i} ={lambda }_{i}{S}_{i}-(mu +{delta }^{E}+{chi }_{i}^{E}){E}_{i}\ {dot{P}}_{i} ={delta }^{E}{E}_{i}-(mu +{delta }^{P}+{chi }_{i}^{P}){P}_{i}\ {dot{I}}_{i} =sigma {delta }^{P}{P}_{i}-(mu +alpha +{gamma }^{I}+eta +{chi }_{i}^{I}){I}_{i}\ {dot{A}}_{i} =(1-sigma ){delta }^{P}{P}_{i}-(mu +{gamma }^{A}+{chi }_{i}^{A}){A}_{i}\ {dot{E}}_{i}^{{rm{q}}} ={chi }_{i}^{E}{E}_{i}-(mu +{delta }^{E}){E}_{i}^{{rm{q}}}\ {dot{P}}_{i}^{{rm{q}}} ={chi }_{i}^{P}{P}_{i}+{delta }^{E}{E}_{i}^{{rm{q}}}-(mu +{delta }^{P}){P}_{i}^{{rm{q}}}\ {dot{I}}_{i}^{{rm{h}}} =(eta +{chi }_{i}^{I}){I}_{i}+sigma {delta }^{P}{P}_{i}^{{rm{q}}}-(mu +alpha +{gamma }^{I}){I}_{i}^{{rm{h}}}\ {dot{A}}_{i}^{{rm{q}}} ={chi }_{i}^{A}{A}_{i}+(1-sigma ){delta }^{P}{P}_{i}^{{rm{q}}}-(mu +{gamma }^{A}){A}_{i}^{{rm{q}}}\ {dot{R}}_{i} ={gamma }^{I}({I}_{i}+{I}_{i}^{{rm{h}}})+{gamma }^{A}({A}_{i}+{A}_{i}^{{rm{q}}})-mu {R}_{i}.$$
(3)
Susceptible individuals are recruited into community i (i = 1…n) at a constant rate μNi, with μ and Ni being the average mortality rate of the population and the size of the community in the absence of disease, respectively, and die at rate μ. In this way, the equilibrium size of community i without disease amounts to Ni. Susceptible individuals get exposed to the pathogen at rate λi, corresponding to the force of infection for community i (detailed below), thus becoming latently infected (but not infectious yet). Exposed individuals die at rate μ and transition to the post-latent, infectious stage at rate δE. If containment measures including mass testing and preventive isolation of positive cases are in place, exposed individuals may be removed from the general population and quarantined at rate ({chi }_{i}^{E}). Post-latent individuals die at rate μ, progress to the next infectious classes at rate ηP, developing an infection that can be either symptomatic—with probability σ—or asymptomatic, including the case in which only mild symptoms are present—with probability 1 − σ, and may be tested and quarantined at rate ({chi }_{i}^{P}). Symptomatic infectious individuals die at rate μ + α, with α being an extra-mortality term associated with disease-related complications, recover from infection at rate γI, may spontaneously seek treatment at a hospital at rate η, and may be identified through mass screening and hospitalized at rate ({chi }_{i}^{I}). Asymptomatic individuals die at rate μ, recover at rate γA, and may be quarantined at rate ({chi }_{i}^{A}). Infected individuals who are either hospitalized or quarantined at home are subject to the same epidemiological dynamics as those who are still in the community, but are considered to be effectively removed from it, thus not contributing to disease transmission. Individuals who recover from the infection die at rate μ, and are assumed to have permanent immunity to reinfection. This last assumption is not fundamental, as loss of immunity can be easily included in the model. However, immunity to SARS-CoV-2 reinfection is reported to be relatively long-lasting (a few months at least), hence its loss cannot alter transmission dynamics over epidemic timescales14.The cornerstone of model (Eq. (3)) is the force of infection, λi, which in a spatially explicit setting must account not only for locally acquired infections but also for the role played by human mobility. We assume that, at the spatiotemporal scales of interest for our problem, human mobility mostly depicts daily commuting flows (also coherently with the data available for parameterization; see above) and does not actually entail a permanent relocation of individuals. We thus describe human mobility (and the associated social contacts possibly conducive to disease transmission) by means of instantaneous spatial-mixing matrices ({M}_{c,ij}^{X}) (with X ∈ {S, E, P, I, A, R}), i.e.,$${M}_{c,ij}^{X}=left{begin{array}{ll}{r}^{X}{p}_{i}{q}_{ij}(1-{xi }_{ij})hfill&,{text{if}},i,ne, jhfill\ (1-{p}_{i})+(1-{r}^{X}){p}_{i}+{r}^{X}{p}_{i}{q}_{ij}(1-{xi }_{ij})&,{text{if}},i=j,end{array}right.$$
(4)
where pi (0 ≤ pi ≤ 1 for all i’s) is the fraction of mobile people in community i, qij (0 ≤ qij ≤ 1 for all i’s and j’s) represents the fraction of people moving between i and j (including j = i, (mathop{sum }nolimits_{j = 1}^{n}{q}_{ij}=1) for all i’s), rX (0 ≤ rX ≤ 1 for all X’s) quantifies the fraction of contacts occurring while individuals in epidemiological compartment X are traveling, and ξij (0 ≤ ξij ≤ 1 for all i’s and j’s) represents the effects of travel restrictions that may be imposed between any two communities i and j as a part of the containment response. Therefore, the probability that residents from i have social contacts while being in j (independently of with whom) is assumed to be proportional to the fraction rX of the mobility-related contacts of the individuals in epidemiological compartment X, multiplied by the probability pi that people from i travel (independently of the destination) and the probability qij that the travel occurs between i and j, possibly reduced by a factor 1 − ξij accounting for travel restrictions. All other contacts contribute to mixing within the local community (i in this case). Note also that if ξij = 0 for all i’s and j’s, then ({M}_{c,ij}^{X}) reduces to ({M}_{ij}^{X}), i.e., to the mixing matrix in the absence of disease-containment measures. In this case, (mathop{sum }nolimits_{j = 1}^{n}{M}_{ij}^{X}=1) for all i’s and X’s. It is important to remark, though, that the epidemiologically relevant contacts between the residents of two different communities, say i and j, may not necessarily occur in either i or j; in fact, they could happen anywhere else, say in community k, between residents of i and j simultaneously traveling to k. On this basis, we define the force of infection as$${lambda }_{i}=mathop{sum }limits_{j=1}^{n}{M}_{c,ij}^{S}frac{(1-{epsilon }_{j})left({beta }_{j}^{P}mathop{sum }nolimits_{k = 1}^{n}{M}_{c,kj}^{P}{P}_{k}+{beta }_{j}^{I}mathop{sum }nolimits_{k = 1}^{n}{M}_{c,kj}^{I}{I}_{k}+{beta }_{j}^{A}mathop{sum }nolimits_{k = 1}^{n}{M}_{c,kj}^{A}{A}_{k}right)}{mathop{sum }nolimits_{k = 1}^{n}left({M}_{c,kj}^{S}{S}_{k}+{M}_{c,kj}^{E}{E}_{k}+{M}_{c,kj}^{P}{P}_{k}+{M}_{c,kj}^{I}{I}_{k}+{M}_{c,kj}^{A}{A}_{k}+{M}_{c,kj}^{R}{R}_{k}right)},$$
(5)
where the parameters ({beta }_{j}^{X}) (X ∈ {P, I, A}) are the community-dependent rates of disease transmission from the three infectious classes, ϵj (0 ≤ ϵj ≤ 1 for all j’s) represents the reduction of transmission induced by social distancing, the use of personal protective equipment, and local mobility restrictions if such containment measures are in fact in place, and the terms ({M}_{c,ij}^{X}) (with X ∈ {S, E, P, I, A, R}) describe the epidemiological effects of mobility between i and j in the presence of disease-containment measures. Note that transmission has been assumed to be frequency-dependent.The parameters μ, δX (X ∈ {E, P}), σ, α, η, γX (X ∈ {I, A}), and rX (X ∈ {S, E, P, I, A, R}) are assumed to be community-independent, for they pertain to population demography at the country scale or the clinical course of the disease. By contrast, the transmission rates ({beta }_{i}^{X}) (X ∈ {P, I, A}) and the control parameters, namely the isolation rates ({chi }_{i}^{X}) (X ∈ {E, P, I, A}), the reductions of transmission due to personal protection, social distancing, and local mobility restriction ϵi, and the travel restrictions ξij, are assumed to be possibly community-dependent, thereby reflecting spatial heterogeneities in disease transmission prior to the implementation of containment measures (({beta }_{i}^{X})), testing effort and/or strategy (({chi }_{i}^{X})), local transmission reduction (ϵi), and travel restriction (ξij).Derivation of the basic and control reproduction numbersClose to the DFE, a state in which all individuals are susceptible to the disease (Si = Ni, with Ni being the baseline population size of community i) and all the other epidemiological compartments are empty (({E}_{i}={P}_{i}={I}_{i}={A}_{i}={E}_{i}^{{rm{q}}}={P}_{i}^{{rm{q}}}={I}_{i}^{{rm{h}}}={A}_{i}^{{rm{q}}}={R}_{i}=0) for all i’s), the dynamics of model (Eq. (3)) is described by the linearized system (dot{{bf{x}}}={{bf{J}}}_{{bf{c}}}{bf{x}}), where ({bf{x}}={[{S}_{i},{E}_{i},{P}_{i},{I}_{i},{A}_{i},{E}_{i}^{{rm{q}}},{P}_{i}^{{rm{q}}},{I}_{i}^{{rm{h}}},{A}_{i}^{{rm{q}}},{R}_{i}]}^{T}) (where i = 1…n and the superscript T denotes matrix transposition) and Jc is the spatial Jacobian matrix$${{bf{J}}}_{{bf{c}}}=left[begin{array}{llllllllll}-mu {bf{I}}&{bf{0}}&-{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{P}}}&-{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{I}}}&-{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{A}}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{E}}}&{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{P}}}&{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{I}}}&{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{A}}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{delta }^{E}{bf{I}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{P}}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&sigma {delta }^{P}{bf{I}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{I}}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&(1-sigma ){delta }^{P}{bf{I}}&{bf{0}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{A}}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{{boldsymbol{chi }}}^{{bf{E}}}&{bf{0}}&{bf{0}}&{bf{0}}&-(mu +{delta }^{E}){bf{I}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&{{boldsymbol{chi }}}^{{bf{P}}}&{bf{0}}&{bf{0}}&{delta }^{E}{bf{I}}&-(mu +{delta }^{P}){bf{I}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&{bf{0}}&eta {bf{I}}+{{boldsymbol{chi }}}^{{bf{I}}}&{bf{0}}&{bf{0}}&sigma {delta }^{P}{bf{I}}&-(mu +alpha +{gamma }^{I}){bf{I}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{{boldsymbol{chi }}}^{{bf{A}}}&{bf{0}}&(1-sigma ){delta }^{P}{bf{I}}&{bf{0}}&-(mu +{gamma }^{A}){bf{I}}&{bf{0}}\ {bf{0}}&{bf{0}}&{bf{0}}&{gamma }^{I}{bf{I}}&{gamma }^{A}{bf{I}}&{bf{0}}&{bf{0}}&{gamma }^{I}{bf{I}}&{gamma }^{A}{bf{I}}&-mu {bf{I}}end{array}right],$$
(6)
where I and 0 are the identity and null matrices of size n, respectively, ({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{X}}}) (X ∈ {E, P, I, A}) are diagonal matrices whose non-zero elements are (mu +{delta }^{E}+{chi }_{i}^{E}) (for ({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{E}}})), (mu +{delta }^{P}+{chi }_{i}^{P}) (for ({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{P}}})), (mu +alpha +eta +{gamma }^{I}+{chi }_{i}^{I}) (for ({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{I}}})), and (mu +{gamma }^{A}+{chi }_{i}^{A}) (for ({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{A}}})), and the matrices ({{boldsymbol{theta }}}_{{bf{c}}}^{{bf{X}}}) (X ∈ {P, I, A}) are given by$${{boldsymbol{theta }}}_{{bf{c}}}^{{bf{X}}}={bf{N}}{{bf{M}}}_{{bf{c}}}^{{bf{S}}}({bf{I}}-{boldsymbol{epsilon }}){{boldsymbol{beta }}}^{{bf{X}}}{({{boldsymbol{Delta }}}_{{bf{c}}})}^{-1}{({{bf{M}}}_{{bf{c}}}^{{bf{X}}})}^{T},$$
(7)
where N is a diagonal matrix whose non-zero elements are the population sizes Ni, ({{bf{M}}}_{{bf{c}}}^{{bf{X}}}=[{M}_{c,ij}^{X}]) (X ∈ {S, P, I, A}) are sub-stochastic matrices representing the spatially explicit contact terms in the presence of containment measures, ϵ is a diagonal matrix whose non-zero entries are the transmission reductions ϵi, βX (X ∈ {P, I, A}) are diagonal matrices whose non-zero elements are the contact rates ({beta }_{i}^{X}), and Δc is a diagonal matrix whose non-zero entries are the elements of vector ({bf{u}}{bf{N}}{{bf{M}}}_{{bf{c}}}^{{bf{S}}}), with u being a unitary row vector of size n.Because of its block-triangular structure, it is immediate to see that Jc has 6n strictly negative eigenvalues, namely −μ, with multiplicity 2n, and −(μ + δE),−(μ + δP), −(μ + α + γI), and −(μ + γA), each with multiplicity n. Therefore, the asymptotic stability properties of the DFE of model (Eq. (3)), which determine whether long-term disease circulation in the presence of controls is possible, are linked to the eigenvalues of a reduced-order spatial Jacobian associated with the infection subsystem, i.e., the subset of state variables directly related to disease transmission, in this case {E1, …, En, P1, …, Pn, I1, …, In, A1, …, An}. Note that introducing waning immunity would not change the spectral properties of the Jacobian matrix evaluated at the DFE. The reduced-order Jacobian ({{bf{J}}}_{{bf{c}}}^{* }) thus reads$${{bf{J}}}_{{bf{c}}}^{* }=left[begin{array}{llll}-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{E}}}&{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{P}}}&{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{I}}}&{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{A}}}\ {delta }^{E}{bf{I}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{P}}}&{bf{0}}&{bf{0}}\ {bf{0}}&sigma {delta }^{P}{bf{I}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{I}}}&{bf{0}}\ {bf{0}}&(1-sigma ){delta }^{P}{bf{I}}&{bf{0}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{A}}}end{array}right].$$
(8)
The asymptotic stability properties of the DFE can be assessed through a NGM approach22,37. In fact, the spectral radius of the NGM provides an estimate of the so-called control reproduction number58, ({{mathcal{R}}}_{{rm{c}}}), which can be thought of as the average number of secondary infections produced by one infected individual in a completely susceptible population in the presence of disease-containment measures. Clearly, if ({{mathcal{R}}}_{{rm{c}}}, > , 1) the pathogen can invade the population in the long run, and endemic transmission will eventually be established despite the implementation of disease-containment measures. To evaluate ({{mathcal{R}}}_{{rm{c}}}) for model (Eq. (3)), the Jacobian of the infection subsystem can be decomposed into a spatial transmission matrix$${{bf{T}}}_{{bf{c}}}=left[begin{array}{llll}{bf{0}}&{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{P}}}&{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{I}}}&{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{A}}}\ {bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}end{array}right],$$
(9)
and a transition matrix$${{boldsymbol{Sigma }}}_{{bf{c}}}=left[begin{array}{llll}-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{E}}}&{bf{0}}&{bf{0}}&{bf{0}}\ {delta }^{E}{bf{I}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{P}}}&{bf{0}}&{bf{0}}\ {bf{0}}&sigma {delta }^{P}{bf{I}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{I}}}&{bf{0}}\ {bf{0}}&(1-sigma ){delta }^{P}{bf{I}}&{bf{0}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{A}}}end{array}right],$$
(10)
so that Jc = Tc + Σc. The spatial NGM with large domain ({{bf{K}}}_{{bf{c}}}^{{bf{L}}}), including variables other than the states-at-infection59 (i.e., the exposed individuals Ei) thus reads$${{bf{K}}}_{{bf{c}}}^{{bf{L}}}=-{{bf{T}}}_{{bf{c}}}{({{mathbf{Sigma }}}_{{bf{c}}})}^{-1}=left[begin{array}{llll}{{bf{K}}}_{{bf{c}}}^{{bf{1}}}&{{bf{K}}}_{{bf{c}}}^{{bf{2}}}&{{bf{K}}}_{{bf{c}}}^{{bf{3}}}&{{bf{K}}}_{{bf{c}}}^{{bf{4}}}\ {bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}end{array}right],$$
(11)
with$${{bf{K}}}_{{bf{c}}}^{{bf{1}}} ={delta }^{E}left[{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{P}}}+sigma {delta }^{P}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{I}}}{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{I}}})}^{-1}+(1-sigma ){delta }^{P}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{A}}}{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{A}}})}^{-1}right]{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{E}}})}^{-1}{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{P}}})}^{-1}\ {{bf{K}}}_{{bf{c}}}^{{bf{2}}} =left[{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{P}}}+sigma {delta }^{P}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{I}}}{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{I}}})}^{-1}+(1-sigma ){delta }^{P}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{A}}}{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{A}}})}^{-1}right]{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{P}}})}^{-1}\ {{bf{K}}}_{{bf{c}}}^{{bf{3}}} ={{boldsymbol{theta }}}_{{bf{c}}}^{{bf{I}}}{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{I}}})}^{-1}\ {{bf{K}}}_{{bf{c}}}^{{bf{4}}} ={{boldsymbol{theta }}}_{{bf{c}}}^{{bf{A}}}{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{A}}})}^{-1}.$$
(12)
Because of the peculiar block-triangular structure of ({{bf{K}}}_{{bf{c}}}^{{bf{L}}}), the spatial NGM with small domain (Kc, accounting only for Ei) is simply ({{bf{K}}}_{{bf{c}}}^{{bf{1}}}) (see again Diekmann et al.59). The control reproduction number can thus be found as the spectral radius of the NGM (with either large or small domain), i.e.,$${{mathcal{R}}}_{{rm{c}}}=rho ({{bf{K}}}_{{bf{c}}}^{{bf{L}}})=rho ({{bf{K}}}_{{bf{c}}})=rho ({{bf{G}}}_{{bf{c}}}^{{bf{P}}}+{{bf{G}}}_{{bf{c}}}^{{bf{I}}}+{{bf{G}}}_{{bf{c}}}^{{bf{A}}}),$$
(13)
where$${{bf{G}}}_{{bf{c}}}^{{bf{P}}} ={delta }^{E}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{P}}}{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{E}}}{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{P}}})}^{-1}\ {{bf{G}}}_{{bf{c}}}^{{bf{I}}} =sigma {delta }^{E}{delta }^{P}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{I}}}{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{E}}}{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{P}}}{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{I}}})}^{-1}\ {{bf{G}}}_{{bf{c}}}^{{bf{A}}} =(1-sigma ){delta }^{E}{delta }^{P}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{A}}}{({{boldsymbol{phi }}}_{{bf{c}}}^{{bf{E}}}{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{P}}}{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{A}}})}^{-1}$$
(14)
are three spatially explicit generation matrices describing the contributions of post-latent infectious people, infectious symptomatic people, and asymptomatic/paucisymptomatic infectious people to the next generation of infections in a neighborhood of the DFE in the presence of disease-containment measures.In the absence of controls, i.e., if the isolation rates ({chi }_{i}^{X}) (X ∈ {E, P, I, A}), the transmission reductions ϵi, and the travel restrictions ξij are equal to zero for all i’s and j’s, then the control reproduction number ({{mathcal{R}}}_{{rm{c}}}) reduces to the basic reproduction number ({{mathcal{R}}}_{0}), defined as the average number of secondary infections produced by one infected individual in a population that is completely susceptible to the disease and where no containment measures are in place. ({{mathcal{R}}}_{0}) can be evaluated as the spectral radius of matrix GP + GI + GA, where$${{bf{G}}}^{{bf{P}}} ={delta }^{E}{{boldsymbol{theta }}}^{{bf{P}}}{({{boldsymbol{phi }}}^{{bf{E}}}{{boldsymbol{phi }}}^{{bf{P}}})}^{-1}\ {{bf{G}}}^{{bf{I}}} =sigma {delta }^{E}{delta }^{P}{{boldsymbol{theta }}}^{{bf{I}}}{({{boldsymbol{phi }}}^{{bf{E}}}{{boldsymbol{phi }}}^{{bf{P}}}{{boldsymbol{phi }}}^{{bf{I}}})}^{-1}\ {{bf{G}}}^{{bf{A}}} =(1-sigma ){delta }^{E}{delta }^{P}{{boldsymbol{theta }}}^{{bf{A}}}{({{boldsymbol{phi }}}^{{bf{E}}}{{boldsymbol{phi }}}^{{bf{P}}}{{boldsymbol{phi }}}^{{bf{A}}})}^{-1}.$$
(15)
In the previous set of expressions, ϕX (X ∈ {E, P, I, A}) are diagonal matrices whose non-zero elements are μ + δE (for ϕE), μ + δP (for ϕP), μ + α + η + γI (for ϕI), and μ + γA (for ϕA), while matrices θX (X ∈ {P, I, A}) are given by ({bf{N}}{{bf{M}}}^{{bf{S}}}{{boldsymbol{beta }}}^{{bf{X}}}{({boldsymbol{Delta }})}^{-1}{({{bf{M}}}^{{bf{X}}})}^{T}), with ({{bf{M}}}^{{bf{X}}}=[{M}_{ij}^{X}]) (X ∈ {S, P, I, A}) and ({M}_{ij}^{X}={M}_{c,ij}^{X}) evaluated with ξij = 0 for all i’s and j’s, and Δ is a diagonal matrix whose non-zero entries are the elements of vector uNMS.Derivation of basic and control epidemicity indicesThe concept of epidemicity26 extends previous work24,25 where a reactivity index was defined and applied to study the transient dynamics of ecological systems characterized by steady-state behavior. To explain, in physical terms, the meaning of reactivity and of the Hermitian matrix used to derive it, consider a linear system dx/dt = Ax, where ({bf{x}}={({x}_{1},ldots ,{x}_{n})}^{T}) is the state vector and A is a n × n real state matrix. The system is subject to pulse perturbations x(0) = x0  > 0. Reactivity is defined as the gradient of the Euclidean norm (| | {bf{x}}| | =sqrt{{x}_{1}^{2}+cdots +{x}_{n}^{2}}=sqrt{{{bf{x}}}^{T}{bf{x}}}) of the state vector, evaluated for the fastest-growing initial perturbation, and corresponds to the spectral abscissa ({{{Lambda }}}_{max }^{{rm{Re}}}(cdot )) of the Hermitian part (A + AT)/2 of matrix A24. Following Mari et al.25, an asymptotically stable equilibrium is characterized by positive generalized reactivity if there exist small perturbations that can lead to a transient growth in the Euclidean norm of a suitable system output y = Wx, with matrix W describing a linear transformation of the system state.In epidemiological applications, W should include the variables of the infection subsystem26. Therefore, a suitable output transformation for the problem at hand is$${bf{W}}=left[begin{array}{llllllllll}{bf{0}}&{w}^{E}{bf{I}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&{w}^{P}{bf{I}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&{bf{0}}&{w}^{I}{bf{I}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}\ {bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{w}^{A}{bf{I}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}&{bf{0}}end{array}right],$$
(16)
where wE, wP, wI, wA are the weights assigned to the variables of the infection subsystem in the output ({bf{y}}=[{w}^{E}{E}_{1},ldots ,{w}^{E}{E}_{n},{w}^{P}{P}_{1},ldots ,{w}^{P}{P}_{n},{w}^{I}{I}_{1},ldots ,{w}^{I}{I}_{n},{w}^{A}{A}_{1},ldots ,{w}^{A}{A}_{n}]^{T}). Generalized reactivity for the DFE of system (Eq. (3)) is positive if the spectral abscissa of a suitable Hermitian matrix (either H0 or Hc, depending on whether the spread of disease is uncontrolled or some containment measures are in place) is also positive. In SEPIAR, the expressions of matrices H0 and Hc are far from trivial, as shown below, and the evaluation of spectral abscissae typically requires numerical techniques. Note also that, since recovered individuals are not accounted for in the system output, including waning immunity would not alter the epidemicity properties of the DFE.Let us consider the most general case of disease-containment measures being in place (which includes as a limit case also uncontrolled pathogen spread). If we note that (ker ({bf{W}})=ker ({bf{W}}{{bf{J}}}_{{bf{c}}})), with Jc being the Jacobian of SEPIAR at the DFE in the presence of controls, matrix Hc can be defined25,27 as the Hermitian part of WJc(W)+, i.e.,$${{bf{H}}}_{{bf{c}}}=H({bf{W}}{{bf{J}}}_{{bf{c}}}{({bf{W}})}^{+})=frac{1}{2}left{{bf{W}}{{bf{J}}}_{{bf{c}}}{({bf{W}})}^{+}+{[{({bf{W}})}^{+}]}^{T}{({{bf{J}}}_{{bf{c}}})}^{T}{({bf{W}})}^{T}right},$$
(17)
where (W)+ is the right pseudo-inverse (a generalization of the concept of inverse for non-square matrices) of W, and can be evaluated as$${({bf{W}})}^{+}={({bf{W}})}^{T}{[{bf{W}}{({bf{W}})}^{T}]}^{-1}.$$
(18)
Matrix$${{bf{H}}}_{{bf{c}}}=left[begin{array}{llll}-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{E}}}&frac{{w}^{P}}{2{w}^{E}}{delta }^{E}{bf{I}}+frac{{w}^{E}}{2{w}^{P}}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{P}}}&frac{{w}^{E}}{2{w}^{I}}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{I}}}&frac{{w}^{E}}{2{w}^{A}}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{A}}}\ frac{{w}^{P}}{2{w}^{E}}{delta }^{E}{bf{I}}+frac{{w}^{E}}{2{w}^{P}}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{P}}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{P}}}&frac{{w}^{I}}{2{w}^{P}}sigma {delta }^{P}{bf{I}}&frac{{w}^{A}}{2{w}^{P}}(1-sigma ){delta }^{P}{bf{I}}\ frac{{w}^{E}}{2{w}^{I}}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{I}}}&frac{{w}^{I}}{2{w}^{P}}sigma {delta }^{P}{bf{I}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{I}}}&{bf{0}}\ frac{{w}^{E}}{2{w}^{A}}{{boldsymbol{theta }}}_{{bf{c}}}^{{bf{A}}}&frac{{w}^{A}}{2{w}^{P}}(1-sigma ){delta }^{P}{bf{I}}&{bf{0}}&-{{boldsymbol{phi }}}_{{bf{c}}}^{{bf{A}}}end{array}right]$$
(19)
is Hermitian, hence real and symmetric. Therefore all eigenvalues are real and the spectral abscissa ({e}_{{rm{c}}}={{{Lambda }}}_{max }^{{rm{Re}}}({{bf{H}}}_{{bf{c}}})) coincides with the largest eigenvalue, which corresponds to the fastest-growing perturbation in the system output. Thus, ec can be interpreted as a control epidemicity index: if ec  > 0, there must exist some small perturbations to the DFE that are temporarily amplified in the system output, thus generating a transient, subthreshold epidemic wave.Absent any containment measures, the control epidemicity index, ec, reduces to the basic epidemicity index, ({e}_{0}={{{Lambda }}}_{max }^{{rm{Re}}}({{bf{H}}}_{{bf{0}}})), where$${{bf{H}}}_{{bf{0}}}=H({bf{W}}{{bf{J}}}_{{bf{0}}}{({bf{W}})}^{+})=frac{1}{2}left{{bf{W}}{{bf{J}}}_{{bf{0}}}{({bf{W}})}^{+}+{[{({bf{W}})}^{+}]}^{T}{({{bf{J}}}_{{bf{0}}})}^{T}{({bf{W}})}^{T}right}$$
(20)
and the Jacobian matrix J0 can be obtained from Jc by setting equal to zero the isolation rates ({chi }_{i}^{X}) (X ∈ {E, P, I, A}), the transmission reductions ϵi, and the travel restrictions ξij for all i’s and j’s.The effective reproduction number and the effective epidemicity indexThe reproduction numbers and the epidemicity indices defined above can be rigorously applied only to characterize the spread of disease in a fully naïve population (Si = Ni ∀ i). As soon as the pathogen begins to circulate within the population, the state of the system gradually departs from the DFE. Under these circumstances, it is customary19,21 to define a time-dependent, effective reproduction number, ({mathcal{R}}(t)), to track the number of secondary infections caused by a single infectious individual in a population in which the pool of susceptible individuals is progressively depleted, and control measures are possibly in place58. Similarly, it is possible to define an effective epidemicity index, e(t), to evaluate the likelihood that transient epidemic waves may occur even if ({mathcal{R}}(t), More

The area of study (Fig. 6) was the Greater North Sea ecoregion, which includes the EEZs of six countries (England, Scotland, the Netherlands, Denmark, Norway and Germany). The Kattegat area, the English Channel, and the Belgium EEZ were omitted from the study area. The North Sea Marine Ecosystem is a large semi-closed continental sea situated on the continental shelf of North-western Europe, with a dominant physical division between the comparatively deep northern part (50–200 m, with the Norwegian Trench dropping to 700 m) and the shallower southern part (20–50 m)48. The North Sea is one of the most varied coastal regions in the world, which is characterised by, among others, rocky, fjord and mountainous shores as well as sandy beaches with dunes48. Apart from the marine seabirds feeding primarily in the coastal areas, under 5 km from the coast (e.g., terns, sea-ducks, grebes), the North Sea basin also hosts pelagic birds feeding further offshore, with some also diving for food (guillemot, razorbill, etc.). The North Sea basin is also a major habitat for four marine mammal species, of which the harbour porpoise and harbour seal are the most common. Moreover, fish ecology has been a widely studied topic, especially for commercial species, due to evidence of a decline in the fish stock, such as sprat, whiting, bib, and mackerel. Fish communities, and in particular the small pelagic fish group (such as European sprat, European pilchard), play also a key ecologic role, constituting the main pray for most piscivorous fishes, cetacean and seabirds49, Based on early surveys, the predominant species divided by the three North Sea fish communities are: saithe (43.6% in the shelf edge), haddock (42.4% in the central North Sea, 11.6% in the shelf edge), whiting (21.6% in the eastern North Sea, 13.9% central North Sea), and dab (21.8% in the eastern North Sea)34. More recent assessments of North Sea fish community are emphasizing the clear geographical distinction between the fish species living in the southern part of the North Sea, a shallow area with high primary production and pronounced seasonality, and northern part, a deeper area with lower primary production and lower seasonal variation in temperature and salinity. The southern North Sea fish community is represented by fish species such as lesser weever, while the northern North Sea fish community is represented by species such as saithe, with species like whitting, haddock representative for the North–West subdivision, and the European plaice having the highest abundance in the South–East community50. The future fish stock and spatial distribution is however uncertain due to impacts of climate change related factors (e.g., growing temperatures)49 and overexploitation.Figure 6Offshore wind farm prospects (existing/authorised/planned) in the North Sea basin.Full size imageThe most prominent human activities in the North Sea basin are fishing, coastal construction, maritime transport, oil and gas exploration and production, tourism, military, and OWF construction38. Within this list, the construction of OWFs has seen a rapid increase, aiming to reach a total cumulative installed capacity of 61.8–66.8 GW by 203051. As indicated in Fig. 6, the new designated/search/scoping areas for the location of future OWFs will significantly increase the current space reserved for the offshore production of renewable energy in the North Sea basin.Spatio-temporal database of OWF developments in the North Sea basinFor the input of the geo-spatial layers with the location of OWF areas we compiled a comprehensive spatial data repository in QGIS containing the shapefiles of analysed OWF, from 1999 to 2027 (last year of available official information on OWF development, Appendix D). The analysis was performed for the North Sea geographic area, referred here as the basin scale, taking into account the cumulative pressures from individual OWF projects (project scale). The main data sources for geospatial information for OWF, for the entire North Sea basin, are EMODnet (Human Activities data portal) and OSPAR, which were complemented by data on the country level, where needed; i.e. from Crown Estate Scotland (Energy infrastructure, Legal Agreements), Rijkswaterstaat for the Netherlands. From the available geo-spatial data for OWF, we selected the OWF in our area of study (Fig. 6) with the status of consent-authorised, authorised, pre-construction, under construction, or fully commissioned (operational). Therefore, planned OWF such as Vesterhavet Syd and Vesterhavet Nord, for which the start date of construction is still unknown, were not included in the analysis. Similarly, for the Horns Rev 3 OWF no geo-referenced spatial footprint was available in the open-access data sets, and therefore it was not included in the analysis.The collected OWF geospatial data was aggregated to create a geospatial database, for the studied period of 1999–2050, composed by the following attributes: code name, country, name, production capacity (MW), area (({mathrm{km}}^{2})), number of turbines, start operation (year), installation time, and status in the period 1999–2050 (construction, operation, decommissioning). The created geospatial dataset was additionally cross-checked for integrity with the information provided through the online platform 4coffshore.com.The lack of data regarding the construction time was complemented with the methodology proposed by Lacal-Arántegui et al.36. Based on this research, we calculated the time required for OWF construction phase related activities multiplying 1.06 days by the known production capacity (total MW) for each analysed OWF.The average time of operation is considered to be 20 years, probably profitably extendable to 25 years, as stated in a number of studies on the cycle of offshore wind farms52. For this case study, the operation time considered is 20 years (subject to change). Since there is little experience with the decommissioning of offshore wind farms (only a few OWFs have so far been decommissioned in the UK and Denmark), the decommissioning time is not yet clear. There are a number of parameters that influence the decommissioning time, which are: the number of turbines, the foundation type, the distance to port, etc. It is estimated that the time taken for decommissioning should be around 50–60% less than the installation time37. Our study considers the decommissioning time as 50% of the construction time.Time-aware cumulative effects assessmentIn this study, Tools4MSP53,54, a Python-based Free and Open Source Software (FOSS) for geospatial analysis in support of Maritime Spatial Planning and marine environmental management, was used for the assessment of the impacts of OWFs on the marine ecosystem, in the three development stages. We applied the Tools4MSP CEA module to the OWF of the North Sea basin for the period 1999–2050, taking into account the full life cycle of the OWF development, namely the construction, operation and decommissioning phases. The modified methodology from Menegon et al.31 and subsequent implementation55, proposes to calculate the CEA score for each cell of analysis as follows (Eqs. 1, 2):$$CEA=sum_{k=1}^{n}d({E}_{k}) sum_{j=1}^{m}{s}_{i,j} eff({P}_{j}{E}_{k})$$
(1)
where eff is the effect of pressure P over the environmental component E and is defined as follows:$$eff left({P}_{j}{E}_{k}right)=(sum_{i=1}^{l}{w}_{i,j} i({U}_{i},{M}_{i,j,k})){^{prime}}$$
(2)
whereas,

({U}_{i}) defines the human activity, namely the OWF activity in the study area

({E}_{k}) defines the environmental components of the study area described in the Table 1

({d(E}_{k})) defines intensity or presence/absence of the k-th environmental component

({P}_{j}) defines the pressures exerted by human activities dependent on the three different OWF development phases (Annex B)

({w}_{i,j}) refers to the specific pressure weight according to the OWF phase

({s(P}_{j}, {E}_{k})) is the sensitivity of the k-th environmental component to the j-th pressure

({i({U}_{i, }M({U}_{i, }P}_{j}, {E}_{k}))) is the distance model propagating j-th pressure caused by i-th activity over the k-th environmental component

({M(U}_{i}, {P}_{j})) is the 2D Gaussian kernel function used for convolution, which considers buffer distances at 1 km, 5 km, 10 km, 20 km, and 50 km56.

Table 1 Primary sources for the environmental component data sets.Full size tableIn Eq. (3), the CEA 1999–2050 describes the modelling over the time frame 1999–2050, whereas ({CEA}_{t}) is the cumulative effect of year t within the timeframe 1999–2050:$${CEA}_{1999-2050}= sum_{t=1999}^{2050}{CEA}_{t}$$In this study, each final CEA score was normalised. To normalise the value of each initial CEA score obtained using the Eq. (1), we calculated its percentage of the sum of all CEA scores for all OWFs in the three development phases, period spanning the period 1999–2050 (({CEA}_{1999-2050})).Environmental componentsThe selection of the environmental components (receptors) impacted by the identified pressures is an essential part of the scoping phase for OWF location, as monitoring the status (distribution, abundance) of different identified species represents a relevant indicator for the ecosystem status. For the evaluation of the habitats and species that can be affected by the cumulative ecological effects of OWF, we adapted the methodology of Meissl et al.14. Therefore, we selected the environmental components based first on their: (1) ecological value, supported by legal documents identifying species protected by law or through various national and international agreements (e.g. EU Habitats Directive, Wild Mammals (Protection) Act (UK), see Table 1 in Appendix E), to which we added species with (2) commercial value, but also with a (3) broad geographic-scale habitat occurrence of the species in the studied area, based on previous studies35 and on 35 EIA studies for OWF in the North Sea basin.Among the five fish species selected, sprat and sandeel play key roles in the marine food web (small pelagic fish), as prey source for piscivorous fish, cetacean and birds. The ecological value of sandeel, sprat, whiting and saither is also highlighted through EU or national protection agreements such as Priority Marine Features—PMF or Scottish/UK Biodiversity list (see Appendix E, Table 2). The list is completed by haddock, one of the fish species with commercial importance, highly dominant in the Central North Sea. With regards to the spatial occurrence at the basin level, the fish species selected are representative for both of the two distinct North Sea communities50, the southern part of the North Sea (sprat), and the northern and north-west part (haddock, whiting, saithe).The three selected seabird species are of ecological importance for the marine ecosystem, as indicated through the European, national and international protection agreements, such as the EU Birds Directive Migratory Species or the IUCN Red List (see Appendix E, Table 1). While razorbill and guillemot have similar feeding and flying patterns (low flight, catch pray underwater), there is evidence of different behaviors towards OWFs, with relatively more avoidance from razorbill compared to guillemot. In relation to the spatial distribution of the three selected species, there is a clear distinction between razorbill, highly present in the coastal areas of west North Sea basin, guillemot, with a relatively even distribution across the marine basin, and fulmar, one of the 4 most common seabirds in the studied area, in particular in the central and N–E parts.In the marine mammals category we selected the harbor porpoise, indicated to be one of the most impacted species in this category57, with a high occurrence in the North Sea basin. Its ecological value is emphasized by its presence in European and international lists for habitat protection, such as EU Habitats Directive58, OSPAR List of Threatened and/or Declining Species59, the Agreement on the Conservation of Small Cetaceans in the Baltic and the North Seas (ASCOBANS)60. The harbor porpoise is the protected species in numerous Natura 2000 areas in the North Sea basin, such as the Spatial Area of Conservation Southern North Sea61 (British EEZ) or The Special area of Protection Kleverbank62 (Dutch EEZ).Among the selected fish species, sandeel had the highest occurrence in EIA studies of OWF developments (23 out of 35), while guillemot had the highest occurrence among seabird species (25 out of 35). With an occurrence of 26 out of the 35 analysed EIA document, the harbour porpoise is the most studied mammal in relation to the impact of OWF.As a result, we selected three EUNIS marine seabed habitat types (European Union Nature Information System)58 (Appendix E, Table 2), three seabird species, one mammal species and five fish species (Appendix E, Table 1). The list can be extended; however, for this exercise we considered it sufficient.The data sets used to represent the spatial distribution (presence/absence, intensity) of the environmental components in the studied area were obtained from multiple sources and were used in the Tools4MSP model either directly (EUNIS habitats, marine mammals, seabirds) or further processed using a predictive distribution model (fish species). In the case of EUNIS marine habitats, the data source was the online geo-portal EMODnet, through the Seabed Habitat service (Table 1), which provided GIS polygon layers for each habitat type and was further used to indicate presence/absence of a specific habitat.For the distribution of the selected mammal species, the harbour porpoise, we used the modelling results of Waggit et al.16, translated into maps for the prediction of densities (nr. animals/({mathrm{km}}^{2})). The mapping approach starts with collating data from available surveys, which are further standardised with regards to transect length, number of platform sides, and the effective strip width. Finally, the standardised data sets were used in a binomial and a Poisson model, in association with environmental conditions (Table 1), in order to deliver a homogenous cover of species distribution maps, on 10 km × 10 km spatial resolution grid16.For the distribution of the selected seabird species (razorbill, fulmar, guillemot), we used the results of the SEAPOP program (http://www.seapop.no/en/distribution-status/), through the open-source data portal (https://www2.nina.no/seapop/seapophtml/). The proposed methodology for creating the occurrence density prediction maps, on a 10 × 10 km spatial resolution grid, starts with the modelling of the presence/absence of birds using a binomial distribution and “logit link”. This was followed by the modelling of the number of birds using a Gamma distribution with a “log link” function, which also took into account geographically fixed explanatory variables (geographic position, water depth, and distance to coast).The predictive model for the spatial distribution of fish species biomass (haddock, sandeel, whiting, saithe, sprat) was developed using AI4Blue software, an open-source, python-based library for Artificial Intelligence based geospatial analysis of Blue Growth settings (AI4Blue, 2021)63. The model was based on two types of inputs: (1) the observation data on the presence of species and (2) data on the absence of species (absence data) for the period 2000–2019. Both data types were extracted by the ICES North Sea International Bottom Trawl Survey (NSI-IBTS, extracted survey year 2000–2019 including all available quarters) for commercial fish species, which was accessed on the online ICES-DATRAS database64. Data was extracted using two DATRAS web service Application Programming Interfaces (APIs): (1) the HHData, that returns detailed haul-based meta-data of the survey (e.g. haul position, sampling method etc.) and (2) the CPUEPerLengthPerHaulPerHour for the catch/unit of effort per length of sampled species.The presence data were represented by the catch/unit of effort (CPUE), expressed in kg of biomass of the specified species per one hour of hauling. The biomass was estimated by using the SAMLK (sex-maturity-age-length keys) dataset for ICES standard species. This approach is a viable alternative to presence-only data models, as it tackles the biased outcomes resulting from an non-uniform marine coverage of the data sets (mainly along the shipping routes)65. The absence data were estimated using the methodology presented by Coro et al.65, which detects absence location for the chosen species as the locations in which repeated surveys (with the selected species on the survey’s species target list) report information only on other species.Additionally, the predictive model automatically correlates the presence/absence data with environmental conditions (Appendix E, Table 3) data to more accurately estimate the likelihood of species presence in the North Sea basin. Intersecting a large number of surveys containing observation data on the presence of selected species can return the true absence data locations, which represent a valuable indicator for geographical areas with unsuitable habitat (see methodology by Coro et al.65). Those locations were estimated from abiotic and biotic parameters and differed to the sampling absences which were estimated from surveys without presence data65. The environmental conditions (Appendix E, Table 2) data were accessed through direct queries using the MOTU Client option from the Marine Copernicus database. In order to input the layers to the CEA calculation, the input layer for the biomass was transformed using log[x + 1] to avoid an over-dominance of extreme values and all datasets rescaled from 0 to 1 in order to allow direct comparison on a single, unit-less scale55.The rescaled special distribution of biomass for the selected species are presented in Appendix F (Fig. a–j).OWF pressures and relative weightsA systematic literature review was conducted to reach a first quantification of the OWF pressure weights (({w}_{i,j}),) in the construction, operation, and decommissioning phases (({U}_{i})). The OWF-related pressures specific to each of the phases of the OWF life cycle were based on the comprehensive analysis of all the existing Environmental Impact Assessment (EIA) methodologies used in the North Sea countries14. The review enabled the collection of 18 pressures that were subsequently compared and merged with the pressures established in the Marine Strategy Framework Directive, applied by the EU countries in the assessment of environmental impacts66. Figure 7 illustrates the impact chain linking the three OWF development phases with the exerted 18 pressures and the 12 selected environmental components impacted.Figure 7Impact chain defining OWF phases-pressure-environmental components analysed in the North Sea (the strength of the link between pressures and environmental components is proportional to the sensitivity scores. The order is descending from the pressures with highest impact, as well as from the environmental components most affected).Full size imageSensitivity in this research is defined as the likelihood of change when a pressure is applied to a receptor (environmental component) and is a function of the ability of the receptor to adapt, tolerate or resist change and its ability to recover from the impact67. The criteria for assessing the sensitivities of environmental components is based on MarLIN (Marine Life Information Network) detailed criteria (https://www.marlin.ac.uk/sensitivity/sensitivity_rationale).We validated the weights of pressures (({w}_{i,j}) from 0 to 5) and scores of environmental components sensitivities (({s(P}_{j}, {E}_{k})) from 0 to 5), as well as the distance of pressure propagation (≤1000 m to ≥ 25,000 m), through a series of 4 questionnaires for the marine mammals, seabirds, fish and seabed habitats. The compiled questionnaires were further validated through semi-interviews of 9 experts in the field of marine ecology, spatial planning, environmental impact assessment and offshore wind energy development. The expert-based questionnaires also included a confidence level for the proposed scores, which ranged between 0.2 (very low confidence: based on expert judgement; proxy assessment) and 1 (very high confidence: based on peer reviewed papers, report, assessment on the same receptor). The confidence level was used in determining the final scores for the pressure weights and species sensitivities. The final scores for weights and sensitivity scores were identified either by calculating the mean value (for cases where literature review scores and expert scores differed by  > 2 units) or selecting the higher value—precautionary principle (for cases where scores from different sources differed by  More

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Study areaChina is one of the countries with the poorest per capita water resources in the world while also having the largest water consumption in the world. In 2018, China’s total water consumption was 601.55 billion m3, with 369.31 billion m3 of water used in agriculture, accounting for 61.4% of the total water2. Agriculture is the most important industrial sector in water resource consumption. However, due to regional and climate differences, the distribution of agricultural water resources is uneven, and the shortage of water resources seriously affects agricultural development in water-deficient areas.Figure 1 shows the agricultural water consumption in China by province for 2007 and 2018. The agricultural water consumption includes farmland irrigation water consumption (classified as paddy field, irrigated land, vegetable field, groundwater exploitation), forest, animal husbandry, fishery, and livestock (classified as forest and fruit, grassland, fish pond, animal husbandry, groundwater exploitation), domestic water consumption of rural residents and rural ecological environment water consumption. Previous studies have mainly considered the irrigation water consumption of the planting industry as the research object at the provincial or regional levels (e.g., eastern, central, and western regions). Few were able to consider all 31 provinces in China and have comprehensively assessed water consumption and water use efficiency in the various types of agricultural production3,4,5,6,10,16,17,22,23,24,25,30. In this study, the agricultural water use efficiency and its influencing factors are assessed based on the agricultural water consumption of agriculture, forestry, animal husbandry, and fishery in China.Figure 1Agricultural water consumption in China by province for (a) 2007 and (b) 2018. Note: Map created using ArcGIS [10.2], (http://www.esri.com/software/arcgis).Full size imageResearch methodIn this study, the agricultural water use efficiency (under the common frontier and the group frontier) is calculated using the super-efficiency slacks-based measure (Super-SBM) model. The significant factors affecting water-use efficiency are then analyzed through the threshold regression model.Super-efficiency SBM modelData envelopment analysis (DEA) is an efficiency evaluation method proposed by Charnes31, a famous American operational research scientist. While traditional radial and angular DEA models do not require the specific form of the estimation function, they ignore the relaxation of variables and result in efficiency values in the range of 0 to 1. If there are multiple efficiency value of decision making units(DMUs) with an efficiency value of 1, these values cannot be compared. The efficiency of the super efficiency DEA model can be greater than 1, which means that the efficiency level of all decision-making units can be compared.To avoid the problem of slack variables, Tone (2001) proposed the SBM model, which is a non-radial and non-angular DEA analysis method based on the relaxation variable measure16,17,18,19,20,32. The SBM model of unexpected output solves the slack problem of input and output variables, minimizing deviations in the efficiency measurement. The super-efficiency SBM model combines the super-efficiency DEA model and the SBM model. It is also one of the methods based on data envelopment analysis, which can measure the efficiency of all decision-making units and the slack of input and output variables.Assume n to be the decision-making units, each of which has m inputs, expected output r1, and unexpected output r2. Let X (X ∈ Rm), Yd (Yd ∈ Rs1), and Yu (Yu ∈ Rs2) be matrices, such that (X=[{x}_{1},dots ,{x}_{n}]in {R}^{m*n}) and (Y=[{y}_{1}^{d}, dots ,{ y}_{n}^{d}in {R}^{{r}_{1}*n}). The form of the super-efficiency SBM model is as follows1,17,19,54:$$min=frac{frac{1}{m}sum_{i=1}^{m}(overline{x}/{x}_{ik})}{1/left({r}_{1}+{r}_{2}right)*(sum_{r=1}^{{r}_{1}}overline{{y}^{d}}/{y}_{rk}^{d}+sum_{q=1}^{{r}_{2}}overline{{y}^{u}}/{y}_{qk}^{u}}.$$
(1)
Among them,$$overline{x}ge sum_{j=1ne k}^{n}{x}_{ij}{lambda }_{j}, i=1,dots ,m;$$
(2)
$$overline{{y}^{d}}le sum_{j=1,ne k}^{n}{y}_{rj}^{d}{lambda }_{j}, r=1,dots ,{s}_{1};$$
(3)
$$overline{{y}^{d}}ge sum_{j=1,ne k}^{n}{y}_{qj}^{u}{lambda }_{j}, q=1,dots ,{s}_{2};$$
(4)
$${lambda }_{y}ge 0,j=1,dots ,n;jne 0;$$
(5)
$$overline{x}ge {x}_{k},k=1,dots ,m;$$
(6)
$$overline{{y}^{d}}le {y}_{k}^{d},d=1,dots ,{r}_{1};$$
(7)
$$overline{{y}^{u}}ge {y}_{k}^{u},b=1,dots ,{r}_{2}.$$
(8)
Based on the Super-SBM model (Eq. 1) and its constraint formula, the agricultural water use efficiency for the different provinces was calculated for the period 2007–2018 using Maxdea 8 ultra software.Threshold effectConsidering the differences in economic development and technical levels, the agricultural water use in different regions of China shows characteristics of time-series evolution, spatial heterogeneity, and unbalanced spatial distribution. There is a non-linear relationship between the influencing factors of agricultural water use efficiency, which suggests the existence of certain threshold characteristics33,34. This means that for a particular determinant, agricultural water use efficiency would be affected differently depending on whether the parameter has crossed the threshold. In this study, the threshold panel model proposed by Hansen is used. The threshold value of the threshold variable is taken as the critical point, and the regression equation is divided into different stage intervals to analyze the influence of threshold variables on the explained variables at different stages . Therefore, according to the relationship between agricultural water use efficiency and its influencing factors in different regions, the following single threshold regression model is set:$${Y}_{it}=alpha {X}_{it}+{beta }_{1}{T}_{it}Ileft({T}_{it}le {gamma }_{1}right)+{beta }_{2}{T}_{it}Ileft({T}_{it} >{gamma }_{1}right)+C+{varepsilon }_{it},$$
(9)
such that i is the province; t is the year; Yit and Tit are the explanatory variables and explained variables, respectively; Xit is the control variable that has a significant impact on the explained variables; Tit is threshold variable, which changes with the different explanatory variables; γ is a specific threshold value; α is the corresponding coefficient vector; β1 and β2 represent the influence coefficients of the threshold variable Tit on the explained variable Yit in the case of ({T}_{it}le {gamma }_{1}) and ({T}_{it} >{gamma }_{1}) , respectively; C is a constant; ε is random disturbance term, ({varepsilon }_{it}sim i.i.d.N(0,{sigma }^{2})); and, I (·) is an indicative function. After obtaining the estimated value of each parameter, two tests need to be carried out: (1) establish whether the threshold effect is significant; and (2) determine whether the estimated threshold value is equal to the true value. In addition, the above equation assumes that only one threshold exists. For two or more thresholds, the model would have to be adjusted according to the data.Based on the panel data of 31 provinces in China from 2007 to 201844,45,46, Stata15.0 software was used to perform threshold regression on seven variables: per capita water resources, rural labor force, disposable income, government’s attention, foreign trade dependence, industrial structure, and gross domestic product (GDP). The threshold effect of each factor can be analyzed, and the impact on agricultural water consumption can be assessed using the threshold value.Variable selection and data sourceThe super-efficiency SBM model was used in calculating the agricultural water use efficiency for the 31 provinces in China from 2007 to 2018. The input–output indicators were defined before the calculations, as shown in Extended Data Table 1.The selection of input–output factors to measure the utilization efficiency of agricultural water resources follows the principles of availability and operability. The input variables included: (1) agricultural water consumption, (2) the number of employees in agriculture, forestry, animal husbandry, and fishery, (3) the total power of agricultural machinery, and (4) the expenditure of local finance on agriculture, forestry, and water affairs. In terms of output, the added value in agriculture, forestry, animal husbandry, and fishery (based on 2007) was used as the expected output, while ammonia nitrogen emission, agricultural chemical oxygen demand emission, and agricultural carbon emission comprised the unexpected output.This study considered the scale of carbon emissions released by the agricultural system. According to existing research, agricultural carbon emissions are associated with rural environmental pollution35. The main consequence of agricultural pollutant emissions is soil pollution, which leads to rural groundwater pollution36,37,39,40,41,41. The deterioration of groundwater quality adversely affects the development of the agricultural economy and threatens the safety of the drinking water supply for rural residents.The threshold regression model was used to investigate the convergence of agricultural water use efficiency and observe the changes in agricultural water consumption under different influencing factors. The control variables include the following: water resource endowment, the number of agricultural labor, the income level of rural residents, industrial structure, the degree of government’s attention, the degree of dependence on foreign trade, and the level of economic development, as shown in Extended Data Table 2. For water resource endowment (WR), WR is expressed in per capita water resource (m3 / person). Zhang Lixiao45,46 and previous studies have shown a negative correlation between water resource endowment and water resource utilization. For agricultural labor (ah), the variable is expressed by the number of people engaged in agriculture, forestry, animal husbandry, and fishery (10,000 people). Past studies suggest rural population affects the consumption of agricultural water resources47,50,51,52,53,52. For income levels, rural residents’ income level is indicated by the per capita disposable income of rural households. Wang Xueyuan et al.3 and Han Qing et al.53 argue that the increase in the rural residents’ income would limit agricultural water consumption. For industrial structure (× 2), which is expressed by the proportion of industrial added value in GDP, research has shown water resource efficiency would vary under different industrial structures54,57,56. For the government’s attention degree (GA), the variable is expressed by the proportion of agriculture, water affairs, and forestry spending in the total financial expenditure. The government’s support for comprehensive agricultural development and infrastructure and technology upgrading for agricultural, forestry, and water conservation significantly affects water resource utilization efficiency16,56,59,58. For the degree of dependence on foreign trade (open), the parameter is indicated by the proportion of the total import and export of agricultural and sideline products in the GDP. Changes in import demand can reduce or increase the consumption and pollution of water resources. Likewise, export demand changes, especially in high water-consuming and high polluting products, can significantly improve or degrade water resource efficiency. And for the level of economic development, expressed in terms of GDP, the level of regional economic development plays a positive role in promoting the efficiency of water resource utilization59,62,61. More

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