Ecology

Women in northern Mumbai, India, have planted mangrove saplings to protect the area against rising sea levels.Credit: Mahendra Parikh/Hindustan Times via Getty

Projects that manage, protect and restore ecosystems are widely viewed as win–win strategies for addressing two of this century’s biggest global challenges: climate change and biodiversity loss. Yet the potential contribution of such nature-based solutions to mitigating climate change remains controversial.Decision-makers urgently need to know: what role do nature-based solutions have in the race to net-zero emissions and stop further global temperature increases?Analyses of nature-based solutions often focus on how much carbon they can remove from the atmosphere. Here, we provide a new perspective by modelling how these solutions will affect global temperatures — a crucial metric as humanity attempts to limit global warming.Our analysis shows that nature-based solutions can have a powerful role in reducing temperatures in the long term. Land-use changes will continue to act long past the point at which net-zero emissions are achieved and global temperatures peak (known as peak warming), and will have an important role in planetary cooling in the second half of this century. Before then, nature-based solutions can provide real but limited mitigation benefits. Crucially, the more ambitious the climate target, the shorter the time frame for such solutions to have an effect on peak warming.In other words, nature-based solutions must be designed for longevity. This means paying closer attention to their long-term carbon-sink potential, as well as their impacts on biodiversity, equity and sustainable development goals. It also means continuing to limit global warming through other methods, from decarbonization to geological storage of carbon dioxide.Our model reinforces the conclusion that an ambitious scaling-up of nature-based solutions needs to be implemented fast and thoughtfully — and not at the expense of other measures.Win–winsThe world is currently likely to hit 3 °C of warming above pre-industrial levels by 2100 (although recent policy announcements from the United States and China could reduce this). The 2015 Paris climate agreement aims to limit the global temperature rise this century to well below 2 °C, and, ideally, to 1.5 °C. There is no date for either goal, beyond the “end of this century”. The metric that matters most is the peak temperature, with more-aggressive efforts required to stay below 1.5 °C of warming than for the 2 °C target.
Emissions: world has four times the work or one-third of the time
It is impossible to achieve the needed reduction in peak warming solely through cuts to greenhouse gases, because emissions from certain sectors, such as agriculture and some heavy industry, cannot be driven to zero any time soon. For this reason, we also need to remove greenhouse gases from the atmosphere on an unprecedented scale1.There are various options for doing this. For example, when biomass vegetation is burnt for energy, the emitted CO2 can be retained and stored underground. This process, known as bioenergy with carbon capture and storage (BECCS), requires vast areas of land — compromising food security and biodiversity — as well as time to develop on a large scale. Other options involve industrial machines that capture CO2 from the air; these are currently nascent, expensive technologies.A subset of nature-based solutions can be used specifically to limit warming. These ‘natural climate solutions’ aim to reduce atmospheric greenhouse-gas concentrations in three ways. One is to avoid emissions by protecting ecosystems and thus reducing carbon release; this includes efforts to limit deforestation. Another is to restore ecosystems, such as wetlands, so that they sequester carbon. The third is to improve land management — for timber, crops and grazing — to reduce emissions of carbon, methane and nitrous oxide, as well as to sequester carbon (see ‘Three steps to natural cooling’).

Source: S. Jenkins et al. Geophys. Res. Lett. 45, 2795–2804 (2018).

Decades of work provide strong evidence that nature-based solutions can deliver many local ecological and socio-economic benefits2. Restoring a forest next to a stream, for example, might reduce flooding, improve carbon storage and support fisheries. Growing recognition of such benefits means that interest in nature-based solutions is soaring: they can help people adapt to climate change, achieve sustainable development goals, protect biodiversity and mitigate climate change3.Quantifying nature’s roleThere is still debate around how much nature-based solutions can contribute to achieving net-zero targets by mid-century. This is because results have been estimated across a range of objectives, time frames and model assumptions4,5 (see Supplementary information; SI). Some researchers say that tree restoration is the most effective climate-change solution we have available6 (this in itself has been robustly contested); others argue that nature-based solutions won’t be nearly as fast or as effective as is often stated7.Part of the reason for the impasse is this: many well-known papers discuss the annual carbon uptake possibilities of nature-based solutions; they do not discuss their cooling impact year on year. Because the Paris agreement is framed in terms of temperature, we argue that this gap is critical: researchers need to know how nature-based solutions will affect global temperature.To model this, we consider an ambitious but realistic scenario — an update to previous estimates by one of our co-authors (B.W.G)4,8,9. This scenario considers only those projects for nature-based solutions that are constrained by many factors: they are cost-effective (costing less than US$100 per tonne of CO2 equivalent); ensure adequate global production of food and wood-based products; and involve sufficient biodiversity conservation. They also respect land tenure rights and don’t change the amount of sunlight reflected from Earth, or albedo (see SI). In our scenario, nature-based solutions that avoid emissions ramp up quickly — by 2025 — and absorb carbon while avoiding emissions at a rate of 10 gigatonnes of CO2 per year (Gt CO2 yr−1). This rises to 20 Gt CO2 yr−1 in the most ambitious scenario (peak warming of 1.5 °C by 2055), in which we assume a higher price of carbon. The 10-Gt value is cost-contained. But we also account for 30 years of higher-priced nature-based solutions in the 1.5 °C scenario (up to $200 per tonne of CO2 equivalent; see SI). For comparison, 10 Gt CO2 yr−1 is more than the emissions from the entire global transportation sector.

Instituto Terra, an initiative in Aimorés, Brazil, is restoring a devastated ecosystem.Credit: Christian Ender/Getty

Achieving 10 Gt CO2 yr−1 of mitigation in this way would involve stopping the destruction of ecosystems worldwide (including 270 million hectares of deforestation); restoring 678 million hectares of ecosystems (more than twice the size of India); and improving the management of around 2.5 billion hectares of land by mid-century4. This is ambitious, but it is important to note that the bulk of land required (85%) comes from improving management of existing lands for agriculture, grazing and production forest without displacing yields of food, wood-based products or fuel (see ‘Three steps to natural cooling’).These estimates come with caveats (see SI). The role of nature-based solutions could be larger if one considers, for example, their impacts on other greenhouse gases besides CO2. This could represent an additional amount of roughly 1–3 Gt CO2 equivalent yr−1 of climate mitigation. Alternatively, the contribution of such solutions might be smaller in the long term, if the carbon drawdown from land-based interventions decreased over time. This could happen if these natural sinks became saturated or were affected by climate impacts such as forest fires. These caveats are not included in our estimates.We then modelled how this level of nature-based solutions would affect global temperature up to 2100 (see ‘The long game’ and SI). We looked at illustrative pathways from the Intergovernmental Panel on Climate Change, in which peak warming is constrained to 1.5 °C or 2 °C, and ran these scenarios with the added contribution of nature-based solutions as described. These pathways include BECCS, but no nature-based solutions beyond some avoided deforestation.Taking the temperatureOur analysis shows that implementing this level of nature-based solutions could reduce the peak warming by an additional 0.1 °C under a scenario consistent with a 1.5 °C rise by 2055; 0.3 °C under a scenario consistent with a 2 °C rise by 2085; and 0.3 °C under a 3 °C-by-2100 scenario (see ‘The long game’).

Adapted from Fig. SPM.1 of Ref. 1

The most significant contribution nature-based solutions can make to mitigating the peak temperature is in the 2 °C scenario. In a more ambitious 1.5 °C scenario, there isn’t enough time for nature-based solutions to have as great an impact on peak warming. In the 3 °C scenario, several issues constrain the impact of nature-based solutions, including the limited ability of ecosystems to absorb carbon in a warmer world.Overall, the mitigation potential of nature-based solutions remains small compared to what can be achieved by decarbonizing the economy. Yet, assuming that decarbonization takes place, nature-based solutions can still suppress a chunk of the warming (see SI).Crucially, nature-based solutions cool the planet long after the peak temperature is reached. In the 1.5 °C scenario, they take a total of 0.4 °C off warming by 2100 — four times their suppression to the 2055 peak temperature (see SI, Table S2).
Restoring natural forests is the best way to remove atmospheric carbon
Achieving these significant long-term benefits requires several things. Nature-based solutions of good quality must be scaled up rapidly — and not at the expense of other robust strategies. Long-term geological storage of CO2, for example, will need to be ramped up significantly in the next decade as technologies mature and prices fall. The long-term benefits of nature-based solutions also depend on warming being held in check. The increased frequency and intensity of impacts such as wildfires can undermine ecosystems and their capacity to store carbon or provide other benefits to society.Ecosystems that are protected and carefully managed — such as intact peatlands and old-growth tropical rainforests — are very likely to continue to store carbon for thousands of years. These are also more resilient to climate extremes and pathogens.The right metricsRestoration of forest cover is widely considered the most viable near-term opportunity for carbon removal. Unfortunately, some of this enthusiasm has been used to promote plantation forestry — growing trees of a limited variety of ages and species (for example, in monoculture plantations) does not have the same carbon benefits as maintaining an intact forest ecosystem10.One serious problem is that some nature-based solutions, as currently implemented, can have unintended and unwanted consequences. For example, an area of 34,007 hectares of intact forest ecosystem in Cambodia became a logging concession, with much of it replaced with an acacia monoculture. This was the first large-scale reforestation project to be funded in Cambodia in the context of climate-change mitigation. The project resulted in unethical ecological devastation, affecting 1,900 families in the area11.Similarly, Chilean government subsidies for new plantations of pine and eucalyptus have resulted in plantations expanding by 1.3 million hectares since 1986, with an associated sequestration of about 5.6 million tonnes of carbon. However, regulations stating that expansion cannot happen at the expense of native biodiverse forests were not enforced, resulting in large-scale reductions in native forest cover. Clearing of the original forest has resulted in a net decrease of approximately 0.05 million tonnes of stored carbon since 198612.These examples show how a singular focus on rapid carbon sequestration as the metric of success for land-based climate mitigation can result in perverse outcomes. Activities should be evaluated and monitored with the right metrics, to account for the multitude of benefits they provide in the long term.
Adopt a carbon tax to protect tropical forests
To ensure long-term resilience, projects involving nature-based solutions should adhere to four high-level principles (see nbsguidelines.info). First, nature-based solutions are not an alternative to decarbonization; second, they need to involve a wide range of ecosystems; third, they should be designed in partnership with local communities while respecting Indigenous and other rights; and, finally, they must support biodiversity, from the level of the gene to the ecosystem. In addition, the Oxford principles13 for high-quality offsets call for safe and durable CO2 removal and storage for every tonne of CO2 emitted. Metrics of success should include those for carbon dynamics, biodiversity across multiple trophic levels, and socio-economic factors such as women’s empowerment and youth employment.There are many examples of good-practice projects (see also case studies by the University of Oxford’s Nature-based Solutions Initiative, where N.S. and C.A.J.G. work). For example, mangrove forests in eastern India that have been protected from deforestation since 1985 have been shown to protect coastal regions from the negative impacts of cyclones much better than artificial defences do, while also soaking up carbon14. In Sierra Leone’s tropical rainforest, cocoa agroforestry — where cocoa is planted with trees for shade, alongside pineapples, chillies and maize (corn) as an additional source of food and income — has been shown to produce cocoa sustainably while diminishing forest clearance. One agroforestry project in the Gola Rainforest National Park, initiated 30 years ago, has increased biodiversity and the profitability of crops while saving an estimated 500,000 tonnes of carbon each year through sequestration and avoiding deforestation.Invest wiselyThis much is clear: we urgently need to increase investment in high-quality nature-based solutions. They currently receive a small proportion of existing climate-mitigation financing4,15, which does not reflect their potential.Carbon markets are increasingly relied on to finance nature-based solutions. But carbon offsets on the voluntary market are of variable quality. It can be unclear whether projects really represent a carbon sink, whether they are permanent or if they safeguard social and ecological factors. Offsets that adhere to standards can allow organizations to deliver lower-cost and hence larger climate-mitigation outcomes through nature-based solutions; however, budgets to emit fossil fuels should be ratcheted down rapidly to avoid delaying decarbonization from continued greenhouse-gas emissions.
Account for depreciation of natural capital
Nature-based solutions need both public and private finance; in particular, governments need to reward ecosystem stewardship while taxing polluters and ramping up regulation to ensure that companies meet strict social and environmental safeguards.The United Nations Framework Convention on Climate Change (UNFCCC) needs to provide clear guidelines on national-level accounting for nature-based solutions. This will guide the targets set in the Paris agreement’s Nationally Determined Contributions, and the monitoring, reporting and verification methodologies required to comply with these targets.The next UNFCCC meeting, COP26, is due to be held in Glasgow, UK, this November and provides an opportunity for national reporting systems to tighten national carbon accounting related to nature-based solutions. This would ensure that such solutions make a real, long-term contribution to carbon mitigation and could set metrics to ensure high biodiversity levels and maximize human well-being. One pressing issue for COP26 is Article 6 of the Paris agreement, which established a “mechanism to contribute to the mitigation of greenhouse gas emissions and support sustainable development”. A tightly regulated compliance market defined in Article 6 will provide the grounding for a tightly regulated voluntary offsetting market.COP26 also presents the chance to harmonize the goals of the UNFCCC and those of the Convention on Biological Diversity. For example, nature-based solutions projects are likely to be required to adhere to the principle of free prior informed consent of local people: local communities need to be involved at all stages of project planning and management. Similarly, nature-based solutions should be required to protect and enhance biodiversity. This work can build on existing social and biodiversity standards3.Our economy must be decarbonized at unprecedented rates to achieve net-zero targets by mid-century. Carbon must also be removed from the atmosphere to counter emissions that are hard to eliminate, using nature-based solutions and other means. To transform social and economic systems to deliver resilience in the face of ongoing climate impacts, the world must invest now in nature-based solutions that are ecologically sound, socially equitable and designed to pay dividends over a century or more. Properly managed, these could benefit many generations to come. More

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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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