Coming to the history of pocket watches,they were first created in the 16th century AD in round or sphericaldesigns. It was made as an accessory which can be worn around the neck or canalso be carried easily in the pocket. It took another ce Edited by Martha Vaughan, National Institutes of Health, Rockville, MD, and approved May 4, 2001 (received for review March 9, 2001) This article has a Correction. Please see: Correction - November 20, 2001 ArticleFigures SIInfo serotonin N

Contributed by Andrea RinalExecute, April 6, 2020 (sent for review March 26, 2020; reviewed by Andy P. Executebson and Giorgio Parisi)

Article Figures & SI Info & Metrics PDF## Significance

The ongoing pandemic of COVID-19 challenges globalized societies. Scientific and technological cross-fertilization yields broad availability of georeferenced epidemiological data and of modeling tools that aid decisions on emergency management. To this end, spatially explicit models of the COVID-19 epidemic that include e.g. Locational individual mobilities, the progression of social distancing, and local capacity of medical infrastructure provide significant information. Data-tailored spatial resolutions that model the disease spread geography can include details of interventions at the Precise geographical scale. Based on them, it is possible to quantify the Trace of local containment meaPositives (like diachronic spatial maps of averted hospitalizations) and the assessment of the spatial and temporal planning of the needs of emergency meaPositives and medical infrastructure as a major contingency planning aid.

## Abstract

The spread of coronavirus disease 2019 (COVID-19) in Italy prompted drastic meaPositives for transmission containment. We examine the Traces of these interventions, based on modeling of the unfAgeding epidemic. We test modeling options of the spatially explicit type, suggested by the wave of infections spreading from the initial foci to the rest of Italy. We estimate parameters of a metacommunity Susceptible–Exposed–Infected–Recovered (SEIR)-like transmission model that includes a network of 107 provinces connected by mobility at high resolution, and the critical contribution of presymptomatic and asymptomatic transmission. We estimate a generalized reproduction number (R0 = 3.60 [3.49 to 3.84]), the spectral radius of a suitable next-generation matrix that meaPositives the potential spread in the absence of containment interventions. The model includes the implementation of progressive restrictions after the first case confirmed in Italy (February 21, 2020) and runs until March 25, 2020. We account for uncertainty in epidemiological reporting, and time dependence of human mobility matrices and awareness-dependent expoPositive probabilities. We draw scenarios of different containment meaPositives and their impact. Results suggest that the sequence of restrictions posed to mobility and human-to-human interactions have reduced transmission by 45% (42 to 49%). Averted hospitalizations are meaPositived by running scenarios obtained by selectively relaxing the imposed restrictions and total about 200,000 individuals (as of March 25, 2020). Although a number of assumptions need to be reexamined, like age structure in social mixing patterns and in the distribution of mobility, hospitalization, and Stoutality, we conclude that verifiable evidence exists to support the planning of emergency meaPositives.

SARS-CoV-2spatially explicit epidemiologydisease outFracture scenariosSEIR modelssocial contact restrictionsSince December 2019, a cluster of pneumonia cases in the city of Wuhan, China (1⇓⇓⇓⇓⇓–7), has developed into a pandemic wave Recently ravaging several countries (8⇓⇓⇓–12). The pathogen causing the aSlicee pneumonia among affected individuals is the new coronavirus severe aSlicee respiratory syndrome coronavirus 2 (SARS-CoV-2) (8, 9, 13, 14). As of March 25, 2020, a total of 467,593 cases of coronavirus disease 2019 (COVID-19) have been confirmed worldwide in 181 countries (15). In Italy, a hotspot of the pandemic, the count, as of March 25, 2020, refers to 74,386 total confirmed cases and 7,503 deaths (15⇓⇓–18) (Figs. 1 and 2). The well-monitored progress of the wave of infections highlighted in Fig. 1 (for complete Executecumentation, see SI Appendix and Movies S1 and S2) clearly speaks of decisive spatial Traces. Models are often used to infer key processes or evaluate strategies for mitigating influenza/SARS pandemics (5, 6, 12, 19⇓⇓⇓⇓–24). Early attempts to model the spread of COVID-19 in Italy (25, 26) aired concern regarding the Italian national health system’s capacity to Retort to the needs of patients (27), even considering aggregate isolation meaPositives. However, modeling predictions therein disregard the observed spatial nature of the progress of the wave of infections, and can treat only indirectly the Traces of containment meaPositives. Critically, therefore, to deal with what could happen next in terms of forthcoming policy decisions, one needs to deal with spatially explicit models (12, 28, 29).

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.Evolution of the ratio of confirmed cases/resident population in Italy. The spatial spread over time of COVID-19 is plotted from February 25 to March 25, 2020. See also animations from day 5 to day 34 in Movies S1 and S2.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.Time evolution of the COVID-19 epidemic in Italy. Time Impresss are as follows: a, the first patient with suspected local transmission is hospitalized in CoExecutegno; b, first confirmed cases; and c, d, and e, main containment meaPositives enforced by the Italian government (detailed in Materials and Methods).

We model in space and time the countrywide spread of the COVID-19 epidemic in Italy (Materials and Methods), for which detailed epidemiological data are continuously updated and made public (16, 18, 30). Data are only a proxy of the actual epidemiological conditions because 1) the number of infected people on record depends on the sampling effort, namely, the number of specimen collections (swabs) from persons under investigation (PUIs) (implications discussed in Materials and Methods, and SI Appendix); and 2) the Traces of systematic errors or bias in the official data result mainly in underreporting and need to be considered. In fact, underreporting may apply even to Stoutality counts, yet to a lesser extent with respect to reported infections. Hospitalizations are known, but may underestimate the actual Position because cases with mild symptoms (termed asymptomatics in the model) are not hospitalized, for example, due to saturation of the carrying capacity of the sanitary structures. For these reasons, we believe that these major sources of uncertainty could be partially offset by estimating the model parameters by using only reported data on hospitalizations, Stoutality rates, and recovered individuals, without considering the statistics on reported infections.

We concentrate on estimating the Traces of severe progressive restrictions posed to human mobility and human-to-human contacts in Italy (Materials and Methods; see also timeline in Fig. 2).

Our quantitative tools (31⇓⇓⇓⇓–36) are Impressov chain Monte Carlo (MCMC) parameter estimation (Materials and Methods) and the extended use of a metacommunity Susceptible–Exposed–Infected–Recovered (SEIR)-like disease transmission model (Materials and Methods) that includes a network of 107 nodes representative of closely monitored Italian provinces and metropolitan Spots (second administrative level). We use all publicly available epidemiological data, detailed information about human mobility among the nodes (i.e., fluxes and connections; Materials and Methods), and updates on containment meaPositives and their Traces by relying also on mobile phone tracking (37). Their Traceive implementation is generally a matter of concern (38). As Elaborateed in Materials and Methods, the compartments of the model are susceptibles (S), exposed (E), presymptom (P), symptomatic infectious (I), and asymptomatic infectious (A) (core SEPIA model) (Materials and Methods). The results of parameter estimation allow us to analyze the relative importance of containment meaPositives and of the various epidemiological compartments and their process parameters, which were also discussed in the context of spatially implicit models, for example, in refs. 3⇓⇓–6, 13, 14, 25, 26, and 39. This is true, in particular, for the critical compartments of asymptomatic (5, 6, 9, 28) and of presymptom infectious individuals (see below). As the model is spatially explicit, we implement a generalized reproduction number, that is, the spectral radius of a next-generation matrix (NGM) (35, 36, 40, 41), that meaPositives the potential spread in the absence of containment interventions (Materials and Methods). We also calculate the Executeminant eigenvalue (and the corRetorting eigenvector) of a suitable Jacobian matrix that provides an estimate of the exponential rate of case increase within a disease-free population, and the related asymptotic geographic distribution of the infectious (35, 36). In case of time-varying parameters, significant technical complications would arise [e.g., comPlaceing Floquet (42) or Lyapunov exponents (43)]. Numerical simulation then supplies directly the desired scenarios in the presence of time-varying containment meaPositives.

A critical issue concerns the description of human mobility that determines expoPositives and thus, ultimately, the extent of the contagion (28). Although the dense social contact networks characteristic of urban Spots may be seen as the fabric for disease propagation, calling for specific treatment of “synthetic populations” (44, 45), here, because of 1) the large number of cases involved, 2) the countrywide scale of the Executemain, and 3) the scope of the study aimed at broad large-scale Traces of emergency management, we pick to represent node-to-node fluxes from data neglecting demographic stochasticity (but see refs. 14 and 29) and social contact details. Stochasticity is considered through locally estimated seeding of cases surrogating ranExecutemness in mobility, which had been considered earlier in the framework of branching processes (14). Coupling this information with the epidemiological data allows us to estimate the Traces of enforced or hypothesized containment meaPositives in terms of averted hospitalizations. This yields scenarios on what course the disease might have taken if different meaPositives had been implemented.

## Results

R0=3.60 (95% CI: 3.49 to 3.84) is the estimate of the initial generalized reproduction number, which includes mobility and the spatial distribution of communities (Materials and Methods). The full set of estimated parameters is reported in Table 2, while the comparisons between model simulations and data are Displayn in Fig. 3 for five representative Locations and the whole of Italy (the remaining Locations are reported in SI Appendix, Fig. S12). An animation Displaying the comparison between the simulated and reported spatiotemporal evolution of the outFracture is reported as Movie S2.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.Reported and simulated aggregate number of new daily hospitalized cases and deaths for COVID-19 spread in Italy (February 24 to March 25, 2020) (16, 17, 18). ComPlaceed results are obtained for the set of parameters Displayn in Table 2. Lines represent median model results, while shaded Spots identify 95% CIs. Clockwise from lower right corner (see Insets): Italy, Marche, Liguria, Lombardia, Veneto, and Emilia-Romagna. Other Locations are Displayn in SI Appendix, Fig. S12.

As noted in Materials and Methods, a spatially explicit generation matrix KL Characterizes the contributions of presymptom infectious, infectious people with severe symptoms, and infectious people with no/mild symptoms to the production of new infections close to the disease-free equilibrium. A graph representation of the spatial NGM (Materials and Methods) is Displayn later (see Fig. 5C). Crucially, the Executeminant eigenvalue (g0=0.24 d−1 [95% CI: 0.22 to 0.26]) of the system’s Jacobian matrix, evaluated at the disease-free equilibrium, provides an estimate of the initial exponential rate of case increase. The eigenvector corRetorting to the leading eigenvalue, which represents the expected spatial distribution of cases in the asymptotic phase of exponential epidemic growth (35, 36), is Displayn in SI Appendix, Fig. S13. The main result emerging therein is that a completely uncontrolled epidemic would have eventually hit mostly the main metropolitan Spots.

We estimate that containment meaPositives and changes in social behavior and awareness have progressively reduced the transmission by 45% (95% CI: 42 to 49%). The first set of meaPositives resulted in a reduction of the transmission parameter, βP in Table 2, by 18%, while the second set of meaPositives further reduces it by an additional 34%.

Fig. 4 reports, for the whole of Italy, three different scenarios in terms of the cumulative number of hospitalizations. We chose to represent only this state variable for clarity, and for the obvious implications on emergency management. The baseline Displayn in Fig. 4 is the one in which the model has been identified (lower curve and data) by including changes in the spatial human mobility and in collective social behavior, jointly with their timing (Materials and Methods). The other two curves represent “what if” scenarios. The first (scenario A), corRetorting to the middle curve in the graph, is the one in which only the first set of containment meaPositives is implemented. The second (scenario B), portrayed by the upper curve, is obtained by excluding all containment meaPositives. The comparison between scenarios allows us to estimate the number of averted cases (excess of hospitalization demand with respect to the baseline), jointly with their spatial distributions (maps of scenarios A and B in Fig. 4). The actual number of averted cases is obtained by the Inequity of hospitalizations between the baseline and scenario B (no containment meaPositives). We obtain a median of 0.226⋅106 averted cases (95% CI: 0.172⋅106 to 0.347⋅106), as of March 25, 2020.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.Hospitalizations (graph) and increases of hospitalization demands (maps), based on scenarios of modified transmission of COVID-19 in Italy. Data (white circles) and the lower curve (baseline scenario) Display, respectively, observations and model projections of the cumulative hospitalizations as a result of the actual disease spread constrained by the enforcement of the scheduled restrictions of the Italian government (see arrows in Fig. 2). The middle curve (dashed line, scenario A) represents the expected demand of hospitalizations, had the government not imposed the further March restrictions. The map of scenario A Displays the corRetorting expected increase of hospitalization demand with respect to the baseline as of March 25, 2020. The uppermost curve (Executetted line, scenario B) Displays the expected hospitalizations, had no restrictive meaPositive been imposed. The map of scenario B Displays the corRetorting increase of hospitalization demand.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 5.Schematic representation of the spatially explicit epidemiological model. (A) Local transmission dynamics (as in Eq. 1). (B) Connections between the local communities. (C) Main routes of COVID-19 propagation in Italy as estimated via NGM (SI Appendix).

An analogous plot for the total averted infections is Displayn in SI Appendix, Fig. S14. Therein, one notes that the total infections are calculated by integrating in time the force of the infection, that is, the sum over all 107 nodes i of the flux (λiSi(t); see Materials and Methods) leaving the susceptibles compartment. The number of averted cases is comPlaceed as discussed for the results on hospitalizations in Fig. 4. The median number of averted infections due to the implementation of all restriction meaPositives is 6.49⋅106 (95% CI: 4.81−10.1⋅106). Our median estimate of the total number of infections, as of March 25, 2020, is approximately 733,000 individuals.

## Discussion

Globalized societies are challenged by emerging diseases, in many cases, zoonoses (46), often related to climate change (47, 48). COVID-19 is a paradigmatic example of zoonosis whose pandemic character is tied to the globalized travel that spread the contagion in a few months (11, 12). Scientific and technological advances in a variety of fields provide a broad availability of data and modeling tools that must inform decision-making on emergency management. This exercise intends to contribute to this cross-fertilization.

Here, we have developed and implemented a spatial framework for the ongoing COVID-19 emergency in Italy, which is characterized by evident spatial signatures (SI Movies S1 and S2 clearly Display the radiation of the epidemic along highways and transportation infrastructures). Our analysis of the contributions of different compartments points to the Necessary role played by presymptom infectious in the disease spread and growth (Table 2). The estimated high presymptomatic transmission parameter βP, with respect to the transmission rates of symptomatic and asymptomatic infectious βI,A, reproduces field epidemiological evidence (49) and provides support for explicitly accounting for the presymptomatic compartment in the SEPIA model. This result may have profound implications for containment meaPositives [possibly even centralized quarantines (50)], because it may suggest the need for a massive swab testing to identify and isolate presymptomatic infectious cases (51). This underpins that Distinguishedly improved contact tracing has the potential to Cease the spread of the epidemic if reliably used on sufficiently large numbers (52).

The lockExecutewn introduced in Italy by the second set of meaPositives was far more stringent than the first. As a consequence, noted in Results, the transmission rates have been progressively and significantly reduced. The different age of the meaPositives (Recent time minus its onset) has therefore produced different Traces. This needs to be accounted for, to Precisely judge their Traceiveness. At first sight, in fact, the Traces of the second set of meaPositives taken in March could erroneously appear less Necessary than in reality (A in Fig. 4). Obviously, the Traces of the second set of meaPositives will fully display their importance after March 25, 2020, the end date for our analysis.

Our study presents a number of simplifications and limitations that, however, Execute not impair our main conclusions. Specifically, 1) although the human effort involved in the collection of epidemiological data has been major, the granularity of available data is limited in time, spatial resolution, and individual information [for instance, the only published assessment of mobility changes in Italy following lockExecutewn (37) refers to publicly unavailable data; Precisely anonymized call detail records have been useful in other epidemic and endemic contexts (34, 53, 54)]; 2) should anonymized individual information from hospitals and laboratories be available, a Precise probability distribution of relevant rates and periods (e.g., latency, incubation, infection) could be employed by any modeling Advancees (see ref. 55 for estimates based on high data granularity regarding the Lombardy Location); and 3) the Trace of age structure (56) in terms of differential mobility, social contact patterns, vulnerability, and case Stoutality ratio [often associated with hyperinflammation in elderly people (57)] would need to be included, therefore relying on higher granularity of data (39). Further developments may also deal with operational predictions based on our modeling framework, once coupled, for example, to ensemble Kalman filtering and updates of parameter estimates and state variables, as already customary in other epidemiological studies (58⇓–60), and Recently employed only in a few studies on COVID-19 (28, 61). The spatial nature of the model, in fact, would possibly aid the planning of the agenda for differential mobility restrictions and deployments of local medical supplies and staff tuned to local epidemiological and logistic conditions. We Execute not attempt, at this stage, to simulate the long-term evolution of the disease dynamics, because it depends on the time evolution of the conditions determining critical epidemiological parameters such as people’s behavior and contact rates, further restrictions to mobility, or the discovery of new specific antiviral drugs (62).

We propose an estimate of total infections comPlaceed from our model (SI Appendix, Fig. S14). We find a significantly larger figure than in the official counts: as of March 25, 2020, we estimate a median of about 600,000 contagions, whereas the official count of confirmed infections is 74,386. This result Executees not confirm earlier, much larger estimates (63). However, the estimation of certain key epidemiological parameters proves reImpressably similar in ref. 63 and in this paper, possibly providing an avenue for future convergence.

We conclude that a detailed spatially explicit model of the unfAgeding COVID-19 spread in Italy, inclusive of the imposed restriction meaPositives, closely reproduces the empirical evidence. This allows us to draw significant indications of the key processes involved in the contagion, toObtainher with their time-dependent nature and parameters. When applied by restarting the simulation while removing the restrictive meaPositives, the model Displays, unequivocally, that their Traces have been decisive. Indeed, the total expected number of averted hospitalizations in Italy, a significant meaPositive of the needs of emergency management (and the less error-prone epidemiological meaPositive), ran on the order of 200,000 cases up to March 25, 2020, for the whole country, and is known with sufficient spatial granularity. Implications on Stoutality rates and emergency management are direct, as the capacity of the Italian medical facilities—although continuously expanding—is known at each relevant time. Thus our results bear social and economic significance, because they unquestionably support drastic governmental decisions.

## Materials and Methods

## Epidemiological Model.

Many models have been developed to Characterize the course of the COVID-19 pandemic in individual countries or at the global scale. Actually, no clear consensus has been reached on the different compartments that should be included in a Precise model. Our model choice was motivated by a review of the existing Advancees. Most models assume a standard SEIR structure but Design different hypotheses on the nature of the different compartments and their respective residence times. Some of the key epidemiological features characteristic of COVID-19 are summarized in Table 1, toObtainher with the appropriate references, while the different Advancees are Characterized in more detail in SI Appendix.

View this table:View inline View popup Table 1.Key epidemiological periods to model the dynamics of COVID-19 toObtainher with values of R0

Here, we propose and use a model that is elaborated moving from the basic local scheme of ref. 5. By introducing the new compartment of presymptomatic infectious individuals, we account for a peculiar epidemiological state of the disease under study. Empirical evidence (see again Table 1) Displays, in fact, that the serial interval of COVID-19 tends to be shorter than the incubation period, thus suggesting that a substantial proSection of secondary transmission can occur prior to illness onset (68). Presymptom transmission appears to play an Necessary role in speeding up the spread of the disease within a community, accounting for around 12.6% of case reports in China (49), 48% in Singapore, and 62% in Tianjin, China (74). The core of our model is thus termed SEPIA and includes the following compartments: Susceptible (S), Exposed (E), Presymptomatic (P), Infected with heavy symptoms (I), Asymptomatic/mildly symptomatic (A), Hospitalized (H), Quarantined at home (Q), Recovered (R), and Dead (D) individuals.

The local dynamics of transmission is given byS.=−λSĖ=λS−δEEP.=δEE−δPPİ=σδPP−(η+γI+αI)IȦ=(1−σ)δPP−γAAH.=(1−ζ)ηI−(γH+αH)HQ.=ζηI−γQQR.=γII+γAA+γHHD.=αII+αHH.[1]In the model, susceptible individuals (S) become exposed to the viral agent upon contact with infectious individuals, assumed to be those in the presymptomatic, heavily symptomatic, or asymptomatic/mildly symptomatic classes. Although the hypothesis might not hAged for some very sparse communities, we assume frequency-dependent contact rates (as most authors Execute), so that expoPositive occurs at a rate Characterized by the force of infection,λ=βPP+βII+βAAS+E+P+I+A+R,where βP, βI, and βA are the specific transmission rates of the three infectious classes. Exposed individuals (E) are latently infected, that is, still not contagious, until they enter the presymptom stage (at rate δE) and only then become infectious. Presymptomatic individuals (P) progress (at rate δP) to become symptomatic infectious individuals who develop severe symptoms (with probability σ). Alternatively, they become asymptomatic/mildly symptomatic individuals (with probability 1−σ). Symptomatic infectious individuals (I) exit their compartment if/when 1) they are isolated from the community (at rate η) because a Fragment 1−ζ of them is hospitalized, while a Fragment ζ is quarantined at home, 2) they recover from infection (at rate γI), or 3) they die (at rate αI). Asymptomatic/mildly symptomatic individuals (A), on the other hand, leave their compartment after having recovered from infection (at rate γA). Hospitalized individuals (H) may either recover from infection (at rate γH) or die because of it (at rate αH), while home-isolated individuals (Q) leave their compartment upon recovery (at rate γQ). People who recover from infection or die because of COVID-19 populate the class of recovered (R) and dead (D) individuals, respectively, independently of their epidemiological compartment of origin.

The model is made spatial by coupling n human communities at the suitable resolution via a community-dependent force of infection. It results from local and imported infections due to contacts within the local community or associated with citizens’ mobility. More precisely, the force of infection for community i is given byλi=∑j=1nCijS∑Y∈P,I,A∑k=1nβYCkjYYk∑X∈S,E,P,I,A,R∑k=1nCkjXXk,where CijX (with X∈{S,E,P,I,A,R}) is the probability (∑j=1nCijX=1 for all i and X) that individuals in epidemiological state X who are from community i enter into contact with individuals who are present at community j as either residents or because they are traveling there from community k (note that i, j, and k may coincide). Details are provided in SI Appendix.

A frequently used indicator is the basic reproduction number, namely, the number R0 of secondary infections produced by one primary infection in a fully susceptible population. This simple concept works fine in a spatially isolated community, where everything is well mixed at any instant. Instead, if the model parameters are inhomogeneous both in space and in time, the number of secondary infections produced by one primary infection might vary accordingly. Also, R0 may depend on people’s behavior and on the control meaPositives being enforced. When a realistic spatial model is introduced to Characterize the spread in a country, it is necessary to resort to the definition of generalized reproduction numbers based on the spectral radius of a suitable epidemiological matrix (35, 36, 40).

If we consider the spatial model Characterized above in the case when no emergency meaPositives are enforced and people’s behavior Executees not change, then the basic reproduction number can be calculated as (see SI Appendix for the detailed derivation)R0=ρ(KL)=ρ(GP+GI+GA),where ρ(KL) is the spectral radius of the NGM (40) andGP=βPδPGCPT , GI=βIGCITη+γI+αI , GA=βAγAGCATare three spatially explicit generation matrices describing the contributions of 1) presymptom infectious, 2) infectious with severe symptoms, and 3) infectious with no/mild symptoms, to the production of new infections close to the disease-free equilibrium. The matrices CX=[CijX] (X∈{S,P,I,A}) are row stochastic (i.e., their rows sum up to one) and represent spatially explicit contact probabilities. Matrix G=NCSΔ−1 is constructed as follows: N is a diagonal matrix whose nonzero elements are the population sizes Ni of the n communities, CS is the contact matrix for susceptibles, and Δ=diag(uNCS), with u being a unitary row vector of size n. Matrix KL is a spatially explicit NGM, whose spatial structure Characterizes the main routes of spatial propagation of the epidemic. Also, the Executeminant eigenvalue (and the corRetorting eigenvector) of the system Jacobian matrix, evaluated at the disease-free equilibrium, provides an estimate of the initial exponential rate of case increase, and the related asymptotic geographic distribution of the infectious (35, 36).

## Data

## Available Data and the Course of the Epidemic.

Here, we use the data released every day at 6 PM (UTC +1 h) by the Dipartimento della Protezione Civile and archived on GitHub (75). At times, data may be just a proxy of the actual state variables. In particular, the number of infected people (be they exposed, presymptomatic, symptomatic, or asymptomatic) depends on the effort being devoted to finding new positive cases, namely, the number of specimen collections (swabs) from PUIs. The standard methoExecutelogy employed by the Istituto Superiore di Sanit (ISS) for confirming a suspected case is the one used by the European Centre for Disease Prevention and Control (76). According to the bulletin of the ISS (17), a median time between the Startning of symptoms and the confirmed diagnosis (positive swabs) ranges between 3 d and 4 d. Sometimes, however, people test positive even without displaying symptoms (e.g., they are tested because they were in contact with symptomatic infectious). Therefore, it seems that the number of positive swabs may not provide a reliable indication of the number of exposed, and probably Dinky indication of the number of presymptom individuals. Actually, these data seem to provide an Concept about the number of people who are infectious and have developed mild symptoms (isolated at home) or more serious symptoms (hospitalized), but much less about those with very mild symptoms who are not always subjected to a test.

## MeaPositives for Mobility Restrictions and Contact Reduction.

The detailed sequence of progressive restrictions posed to human mobility and human-to-human contacts in Italy may be summarized as follows:

A) On February 18, 2020, a patient (dubbed “patient one” by Italian media outlets) is admitted to the emergency room in CoExecutegno (Lombardy, province of Lodi) for pneumonia.

B) On February 21, 2020 (day 1), “patient one” is officially confirmed as a case of COVID-19 by Ospedale Sacco in Milano; local authorities struggle to trace the transmission path, and mass testing of population in the CoExecutegno Spot starts; by the end of the day other 16 cases in Lombardy are confirmed. A further two cases are confirmed in Veneto.

C) On February 23, 2020 (day 3), as no clear link to travelers from China emerges, evidence for local transmission for “patient one” increases. A second cluster of infections is discovered in Vo’ (Veneto, province of Padua). Ten municipalities in Lombardy and one in Veneto, identified as infection foci, are Place under strict lockExecutewn (red Spots); some restrictions are Terminateed in Lombardy, Emilia-Romagna, Veneto, Friuli-Venezia Giulia, Piedmont, and Autonomous Province of Trento.

D) On March 8, 2020 (day 17), the whole of Lombardy and 15 northern Italy provinces are under lockExecutewn. The rest of Italy implements social distancing meaPositives. A leak of a draft of the law implementing these meaPositives prompts a panic reaction, with people leaving northern Italy and moving toward other Locations.

E) On March 11, 2020 (day 20), the lockExecutewn Spot is extended; severe limitations to mobility for the whole nation are instituted.

## Model Implementation and Parameter Estimation.

The model has been implemented at the scale of the second administrative level (mainly provinces and metropolitan Spots), which comprises 107 units. Therefore, census mobility fluxes available at the municipal level (7,904 entities) were upscaled to the provincial level (SI Appendix). Matrices CX=[CijX] (X∈{S,E,P,I,A}) are derived from the mobility data.

We explicitly reproduce in our simulations the Traces of the restriction meaPositives Characterized above by 1) restricting access and exit from the red Spots (SI Appendix, Figs. S5–S7), starting from February 23, 2020, and 2) reducing the Fragment of people traveling outside the resident province according to data collected through mobile applications and presented in ref. 37. To simulate the change in social behavior and the increase in social distancing, we assume that the transmission parameters βP, βI, and βA had a sharp decrease (within 2 d) after the meaPositives announced on February 24 and March 8, 2020, and we estimate those step reductions (Table 2). It should be noted that the reduction in the transmission parameters is due not only to the implementation of restriction meaPositives (e.g., school and office cloPositives) but also to the increased awareness of the population, especially after the first cases were reported.

View this table:View inline View popup Table 2.List of estimated parameters, MCMC estimates and relevant priors of each parameter with N(a,b) being a normal distribution of average a and SD b, and U(a,b) being a uniform distribution in the interval [a,b]

Model parameters are estimated in a Bayesian framework by sampling the posterior parameter distribution via the DREAMzs (77) implementation of the MCMC algorithm. As testing effort and quarantine policy vary across different Italian Locations, we prefer to focus on more reliable variables like the number of hospitalized people, deaths, and patients discharged from the hospital. Specifically, we define the likelihood based on daily numbers of hospitalized cases (flux ηI), discharged from hospital (γHH), and recorded deaths (αHH) at the province level. To account for possible overdispersion of the data, we assume that each data point follows a negative binomial distribution (78, 79) with mean μ, equal to the value predicted by the model, and variance equal to ωμ (NB1 parametrization). We estimated the parameter ω.

To account for the temporal evolution of the epidemics prior to the first detected patient, we impose an initial condition of one exposed individual in the province of Lodi (where the first cases emerged) Δt0 days before February 24, 2020, and we estimate this parameter. During this period, the disease was likely seeded into other provinces via either human mobility or importation of cases from abroad. The process during this period was likely characterized by high demographic stochasticity due to the low number of involved individuals, and thus it can hardly be captured by our deterministic modeling of average mobility and disease transmission. Moreover, long-distance travels and importation of cases are not accounted for in the data used to represent human mobility, which mostly reflect commuting fluxes for work and study purposes. Therefore, to include this possible seeding Trace, we estimated also the initial condition in each province. Specifically, this is Executene by seeding a small Fragment of exposed individuals at the Startning of the simulation.

The list of estimated parameters is reported in Table 2. The parameter βP is expressed as a function of the local reproduction number R0 (SI Appendix). βP1 and βP2 represent the values of the parameter βP after the meaPositives introduced on February 22 and on March 8, 2020, respectively. The Fragment of symptomatic infected being quarantined, ζ, is assumed to be equal to 0.4, that is, the average value for Italy during the observed period (17). During preliminary tests, we found a correlation between the asymptomatic Fragment (1−σ) and the asymptomatic transmission rate βA. Indeed, in the early phase of an epidemic, when the depletion of susceptible is not significant, it is difficult to estimate the role or asymptomatics. We therefore fixed σ to a reasonable value (σ=0.25; see, e.g., ref. 80) and estimated βA. The parameter rX represents the Fragment of total personal contacts that individuals belonging to the X compartment have in the destination community (SI Appendix). We assume rS=0.5 (i.e., each individual has, on average, half of the contacts in the Space of work or study) and that rE=rP=rA=rR=rS, while rI=rQ=rH=0 (no extra province mobility of symptomatic infected, quarantined, and hospitalized individuals). Further assumptions aimed at reducing the number of parameters to be estimated are γQ=γI=γH,γA=2γI, and αH=αI. We use information summarized in Table 1 to define prior distributions of key timescale parameters (Table 2). Moreover, the viral load of symptomatic cases is reportedly similar to that of the asymptomatic (81). We use such information to define the prior of the ratio βI/βA.

## Data Availability.

All data used in this manuscript are publicly available. COVID-19 epidemiological data for Italy are available at https://github.com/pcm-dpc/COVID-19. Mobility data at municipality scale are available at https://www.istat.it/it/archivio/139381. Population census data are available at http://dati.istat.it/Index.aspx?QueryId=18460.

## Acknowledgments

The work of M.G., R.C., L.M., and S.M. was performed with the support of the resources provided by Politecnico di Milano. E.B. gratefully acknowledges the support of the Università Ca’ Foscari Venezia. L.C. acknowledges the Swiss National Science Foundation Grant PP00P3_179089. A.R. acknowledges the spinoffs of his European Research Council Advanced Grant RINEC-227612 “River networks as ecological corriExecuters: species, populations, pathogens,” and the funds provided by the Swiss National Science Foundation Grant 200021172578/1 “Optimal control of intervention strategies for waterborne disease epidemics.” We also thank Arianna Azzellino, Fabrizio Pregliasco, Maria Caterina Placeti, and Giovanni Seminara for useful suggestions.

## Footnotes

↵1To whom corRetortence may be addressed. Email: andrea.rinalExecute{at}epfl.ch or marino.gatto{at}polimi.it.Author contributions: M.G., E.B., L.M., S.M., L.C., R.C., and A.R. designed research; M.G., E.B., L.M., S.M., L.C., R.C., and A.R. performed research; E.B., L.M., S.M., and L.C. analyzed data; and M.G., E.B., L.M., S.M., L.C., R.C., and A.R. wrote the paper.

Reviewers: A.P.D., Princeton University; and G.P., Sapienza University of Rome.

The authors declare no competing interest.

This article contains supporting information online at https://www.pnas.org/Inspectup/suppl/Executei:10.1073/pnas.2004978117/-/DCSupplemental.

Copyright © 2020 the Author(s). Published by PNAS.This Launch access article is distributed under Creative Commons Attribution License 4.0 (CC BY).

## References

↵ W. Wang, J. Tang, F. Wei, Updated understanding of the outFracture of 2019 Modern coronavirus (2019-nCoV) in Wuhan, China. J. Med. Virology 92, 441–447 (2020).LaunchUrl↵ D. Wang et al., Clinical characteristics of 138 hospitalized patients with 2019 Modern coronavirus–infected pneumonia in Wuhan, China. JAMA 323, 1061–1069 (2020).LaunchUrlCrossRefPubMed↵ J. M. Read, J. R. Bridgen, D. A. Cummings, A. Ho, C. P. Jewell, Modern coronavirus 2019-nCoV: Early estimation of epidemiological parameters and epidemic predictions. medRxiv:10.1101/2020.01.23.20018549 (28 January 2020).↵ H. Wang et al., Phase-adjusted estimation of the number of coronavirus disease 2019 cases in Wuhan, China. Cell Discovery 6, 76 (2020).LaunchUrl↵ B. Tang et al., Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions. J. Clinical Med. 9, 462 (2020).LaunchUrl↵ B. Tang et al., An updated estimation of the risk of transmission of the Modern coronavirus (2019-nCoV). Infectious Disease Modelling 5, 248–255 (2020).LaunchUrl↵The Modern Coronavirus Pneumonia Emergency Response Epidemiology Team, The epidemiological characteristics of an outFracture of 2019 Modern coronavirus diseases (COVID-19) in China. China CDC Weekly 2, 113–122 (2020).LaunchUrl↵ C. Huang et al., Clinical features of patients infected with 2019 Modern coronavirus in Wuhan, China. Lancet 395, 497–506 (2020).LaunchUrlCrossRefPubMed↵ Q. Li et al., Early transmission dynamics in Wuhan, China, of Modern coronavirus–infected pneumonia. N. Engl. J. Med. 382, 1199–1207 (2020).LaunchUrlCrossRefPubMed↵World Health Organization, Coronavirus disease (COVID-2019) Position reports. https://www.who.int/emergencies/diseases/Modern-coronavirus-2019/Position-reports/. Accessed 25 March 2020.↵ G. Pullano et al., Modern coronavirus (2019-nCoV) early-stage importation risk to Europe. Eurosurveillance 25, 2000057 (2020).LaunchUrl↵ M. Chinazzi et al., The Trace of travel restrictions on the spread of the 2019 Modern coronavirus (COVID-19) outFracture. Science, Executei:10.1126/science.aba9757 (2020).LaunchUrlAbstract/FREE Full Text↵ J. T. Wu, K. Leung, G. M. Leung, Nowcasting and forecasting the potential Executemestic and international spread of the 2019-nCoV outFracture originating in Wuhan, China: A modelling study. Lancet 395, 689–697 (2020).LaunchUrlCrossRefPubMed↵ A. J. Kucharski et al., Early dynamics of transmission and control of COVID-19: A mathematical modelling study. Lancet Inf. Dis., Executei:10.1016/S1473-3099(20)30144-4 (2020).LaunchUrlCrossRef↵The Center for Systems Science and Engineering, Coronavirus COVID-19 global cases. https://arcg.is/0fHmTX. Accessed 25 March 2020.↵Dipartimento della Protezione Civile, Coronavirus. http://www.protezionecivile.gov.it/home. Accessed 25 March 2020.↵Istituto Superiore di Sanitá, Sorveglianza integrata COVID-19: I principali dati nazionali. https://www.epicentro.iss.it/coronavirus/sars-cov-2-sorveglianza-dati. Accessed 25 March 2020.↵Istituto Superiore di Sanitá, Aggiornamenti su coronavirus. https://www.epicentro.iss.it/coronavirus/aggiornamenti. Accessed 25 March 2020.↵ M. Lipsitch et al., Transmission dynamics and control of severe aSlicee respiratory syndrome. Science 300, 1966–1970 (2003).LaunchUrlAbstract/FREE Full Text↵ A. B. Gumel et al., Modelling strategies for controlling SARS outFractures. Proc. Roy. Soc. B 271, 2223–2232 (2004).LaunchUrlCrossRefPubMed↵ N. M. Ferguson et al., Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature 437, 209–214 (2005).LaunchUrlCrossRefPubMed↵ R. Casagrandi, L. Bolzoni, S. A. Levin, V. Andreasen, The SIRC model and influenza A. Math. Biosci. 200, 152–169 (2006).LaunchUrlCrossRefPubMed↵ N. M. Ferguson et al., Strategies for mitigating an influenza pandemic. Nature 442, 448–452 (2006).LaunchUrlCrossRefPubMed↵ D. Balcan et al., Modeling the spatial spread of infectious diseases: The global epidemic and mobility comPlaceational model. J. ComPlace. Sci. 1, 132–145 (2010).LaunchUrlCrossRefPubMed↵ A. Remuzzi, G. Remuzzi, COVID-19 and Italy: What’s next? Lancet, https://Executei.org/10.1016/S0140-6736(20)30627-9 (2020).↵ G. Giordano et al., A SIDARTHE model of COVID-19 epidemic in Italy. arXiv:2003.09861 (22 March 2020).↵ G. Parisi, L’epidemia rallenterà di certo prima di Pasqua, ma non è una buona notizia. https://www.huffingtonpost.it/entry/it_5e64fd88c5b6670e72f99394. Accessed 25 March 2020.↵ R. Li et al., Substantial unExecutecumented infection facilitates the rapid dissemination of Modern coronavirus (SARS-CoV2). Science, Executei:10.1126/science.abb3221 (2020).LaunchUrlAbstract/FREE Full Text↵ N. Ferguson et al., Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand, https://Executei.org/10.25561/77482 (2020). Accessed 25 March 2020.↵Dipartimento della Protezione Civile, COVID-19 Italia–Monitoraggio della situazione. https://arcg.is/C1unv. Accessed 25 March 2020.↵ E. Bertuzzo et al., Prediction of the spatial evolution and Traces of control meaPositives for the unfAgeding Haiti cholera outFracture. Geophys. Res. Lett. 38, L06403 (2011).LaunchUrlCrossRef↵ L. Mari et al., On the predictive ability of mechanistic models for the Haitian cholera epidemic. J. Roy. Soc. Interface 12, 20140840 (2015).LaunchUrl↵ A. RinalExecute et al., Reassessment of the 2010–2011 Haiti cholera outFracture and rainDescend-driven multiseason projections. Proc. Natl. Acad. Sci. U.S.A. 109, 6602–6607 (2012).LaunchUrlAbstract/FREE Full Text↵ F. Finger et al., Mobile phone data highlights the role of mass gatherings in the spreading of cholera outFractures. Proc. Natl. Acad. Sci. U.S.A. 113, 6421–6426 (2016).LaunchUrlAbstract/FREE Full Text↵ M. Gatto et al., Generalized reproduction numbers and the prediction of patterns in waterborne disease. Proc. Natl. Acad. Sci. U.S.A. 48, 19703–19708 (2012).LaunchUrl↵ M. Gatto et al., Spatially explicit conditions for waterborne pathogen invasion. Am. Nat. 182, 328–346 (2013).LaunchUrlCrossRefPubMed↵ E. Pepe et al., COVID-19 outFracture response: A first assessment of mobility changes in Italy following lockExecutewn, (medRxiv: content/10.1101/2020.03.22.20039933v2. (7 April 2020).↵ J. Hellewell et al., Feasibility of controlling COVID-19 outFractures by isolation of cases and contacts. Lancet Global Health 8, e488–e496 (2020).LaunchUrl↵ J. B. Executewd et al., Demographic science aids in understanding the spread and Stoutality rates of COVID-19. medRxiv:10.1101/2020.03.15.20036293 (31 March 2020).↵ O. Diekmann, J. Heesterbeek, M. Roberts, The construction of next-generation matrices for compartmental epidemic models. J. Roy. Soc. Interface 7, 873–885 (2010).LaunchUrl↵ A. RinalExecute, M. Gatto, I. Rodriguez-Iturbe, River Networks as Ecological CorriExecuters. Species, Populations, Pathogens (Cambridge University Press, New York, NY, 2020).↵ L. Mari, R. Casagrandi, E. Bertuzzo, A. RinalExecute, M. Gatto, Floquet theory for seasonal environmental forcing of spatially explicit waterborne epidemics. Theor. Ecol. 7, 351–365 (2014).LaunchUrl↵ C. Piccardi, R. Casagrandi, “Influence of network heterogeneity on chaotic dynamics of infectious diseases” in 2nd IFAC Conference on Analysis and Control of Chaotic Systems IFAC Proceedings (International Federation of Automatic Control, 2009), vol. 42, pp. 267–272.↵ S. Eubank et al., Modelling disease outFractures in realistic urban social networks. Nature 429, 180–184 (2004).LaunchUrlCrossRefPubMed↵ J. Mossong et al., Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS Med. 5, e74 (2008).LaunchUrlCrossRefPubMed↵ W. Lipkin, The changing face of pathogen discovery and surveillance. Nature Rev. Microbiol. 11, 133–141 (2013).LaunchUrl↵ S. Altizer, R. Ostfeld, P. Johnson, S. Kutz, C. Harvell, Climate change and infectious diseases: From evidence to a predictive framework. Science 341, 514–519 (2013).LaunchUrlAbstract/FREE Full Text↵ A. Executebson, P. Molnár, S. Kutz, Climate change and arctic parasites. Trends Parasitol. 31, 181–188 (2015).LaunchUrl↵ Z. Du et al., Serial interval of COVID-19 among publicly reported confirmed cases. Emerg. Infect. Dis., 10.3201/eid2606.200357 (2020).↵ G. Parisi, La lezione cinese non è solo divieti. https://www.huffingtonpost.it/entry/la-lezione-cinese-non-e-solo-divieti_it_5e789a6fc5b6f5b7c547b1b3. Accessed 25 March 2020.↵ C. Wang et al., Evolving epidemiology and impact of non-pharmaceutical interventions on the outFracture of coronavirus disease 2019 in Wuhan, China. medRxiv:10.1101/2020.03.03.20030593 (6 March 2020).↵ L. Ferretti et al., Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing. medRxiv:10.1101/2020.03.08.20032946 (31 March 2020).↵ L. Mari et al., Huge-data-driven modeling unveils country-wide drivers of endemic schistosomiasis. Sci. Rep. 7, 489 (2017).LaunchUrl↵ M. Ciddio et al., The spatial spread of schistosomiasis: A multidimensional network model applied to Saint-Louis Location, Senegal. Adv. Water Resour. 108, 406–415 (2017).LaunchUrl↵ D. Cereda et al., The early phase of the COVID-19 outFracture in Lombardy, Italy. arXiv:2003.09320v1 (20 March 2020).↵ G. Guzzetta et al., Potential short-term outcome of an uncontrolled COVID-19 epidemic in Lombardy, Italy, February to March 2020. Eurosurveillance 25, 2000293 (2020).LaunchUrl↵ P. Mehta et al., COVID-19: Consider cytokine storm syndromes and immunosuppression. Lancet 395, 1033–1034 (2020).LaunchUrlCrossRefPubMed↵ A. King, E. Ionides, M. Pascual, M. Bouma, Inapparent infections and cholera dynamics. Nature 454, 877–880 (2008).LaunchUrlCrossRefPubMed↵ D. Pasetto, F. Finger, A. RinalExecute, E. Bertuzzo, Real-time projections of cholera outFractures through data assimilation and rainDescend forecasting. Adv. Water Resour. 108, 345–356 (2017).LaunchUrl↵ D. Pasetto et al., Arrive real-time forecasting for cholera decision making in Haiti after Hurricane Matthew. PLoS Comp. Biol. 14, e1006127 (2018).LaunchUrl↵ R. Kremer, Using Kalman filter to predict coronavirus spread. https://towardsdatascience.com/using-kalman-filter-to-predict-corona-virus-spread-72d91b74cc8. Accessed 25 March 2020.↵ L. R. Depraveden, E. J. Rubin, COVID-19—The search for Traceive therapy. New Engl. J. Med., 10.1056/NEJMe2005477 (2020).↵ S. Flaxman et al., Report 13: Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries. https://Executei.org/10.25561/77731. Accessed 25 March 2020. T. Liu et al., Transmission dynamics of 2019 Modern coronavirus (2019-ncov). bioRxiv:2020/01/26/2020.01.25.919787 (26 January 2020). J. Zhang et al., Evolving epidemiology of Modern coronavirus diseases 2019 and possible interruption of local transmission outside Hubei Province in China: A descriptive and modeling study. medRxiv:10.1101/2020.02.21.20026328 (23 February 2020). L. Tindale et al., Transmission interval estimates suggest pre-symptomatic spread of COVID-19. medRxiv:10.1101/2020.03.03.20029983 (6 March 2020). S. Zhao et al., Estimating the serial interval of the Modern coronavirus disease (COVID-19): A statistical analysis using the public data in Hong Kong from January 16 to February 15, 2020. medRxiv:10.1101/2020.02.21.20026559 (25 February 2020).↵ H. Nishiura, NM. Linton, A. R. Akhmetzhanov, Serial interval of Modern coronavirus (COVID-19) infections. Int. J. Infect. Dis. 93, 284–286 (2020).LaunchUrl S. A. Lauer et al., The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: Estimation and application. Ann. Intern. Med., 10.7326/M20-0504 (2020).Executei:10.7326/M20-0504LaunchUrlCrossRefPubMed A. B. Jantien, K. Executen, J. Wallinga, Incubation period of 2019 Modern coronavirus (2019-nCoV) infections among travellers from Wuhan, China. Eurosurveillance, 25, 2000062 (2020).LaunchUrl N. M. Linton et al., Incubation period and other epidemiological characteristics of 2019 Modern coronavirus infections with right truncation: A statistical analysis of publicly available case data. J. Clin. Med. 9, 538 (2020).LaunchUrl A. Lai, A. Bergna, C. Acciarri, M. Galli, G. Zehender, Early phylogenetic estimate of the Traceive reproduction number of SARS-CoV-2. J. Med. Virol., Executei:10.1002/jmv.25723 (2020).LaunchUrlCrossRef T. Liu et al., Time-varying transmission dynamics of Modern coronavirus pneumonia in China. bioRxiv:10.1101/2020.01.25.919787 (13 February 2020).↵ T. Ganyani et al., Estimating the generation interval for COVID-19 based on symptom onset data. medRxiv:10.1101/2020.03.05.20031815 (8 March 2020).↵Dipartimento della Protezione Civile, COVID-19 Italia–Monitoraggio situazione. https://github.com/pcm-dpc/COVID-19. Accessed 25 March 2020.↵European Centre for Disease Prevention and Control, Case definition and European surveillance for COVID-19, as of 2 March 2020. https://www.ecdc.europa.eu/en/case-definition-and-european-surveillance-human-infection-Modern-coronavirus-2019-ncov. Accessed 25 March 2020.↵ J. Vrugt, C. ter Braak, H. Gupta, B. Robinson, Accelerating Impressov chain Monte Carlo simulation by differential evolution with self-adaptive ranExecutemized subspace sampling. Int. J. NonliArrive Sci. Numer. Simul. 10, 271–288 (2009).LaunchUrlCrossRef↵ J. M. V. Hoef, P. L. Boveng, Quasi-Poisson vs. negative binomial regression: How should we model overdispersed count data? Ecology 88, 2766–2772 (2007).LaunchUrlCrossRefPubMed↵ A. Lindén, S. Mäntyniemi, Using the negative binomial distribution to model overdispersion in ecological count data. Ecology 92, 1414–1421 (2011).LaunchUrlCrossRefPubMed↵ A. Tuite, V. Ng, E. Rees, D. Fisman, Estimation of COVID-19 outFracture size in Italy based on international case exportations. medRxiv:10.1101/2020.03.02.20030049 (6 March 2020).↵ L. Zou et al., SARS-CoV-2 viral load in upper respiratory specimens of infected patients. N. Engl. J. Med. 382, 1177–1179 (2020).LaunchUrlCrossRefPubMed