SUIHTER: A new mathematical model for COVID-19. Application to the analysis of the second epidemic outbreak in Italy
Nicola Parolini, Luca Dede', Paola F. Antonietti, Giovanni Ardenghi, Andrea Manzoni, Edie Miglio, Andrea Pugliese, Marco Verani, Alfio Quarteroni
SSUIHTER : A new mathematical model for COVID-19. Applicationto the analysis of the second epidemic outbreak in Italy
Nicola Parolini , Luca Dede’ , Paola F. Antonietti , Giovanni Ardenghi , AndreaManzoni , Edie Miglio , Andrea Pugliese , Marco Verani , and Alfio Quarteroni MOX, Department of Mathematics, Politecnico di Milano, Italy Department of Mathematics, University of Trento, Italy Institute of Mathematics, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Switzerland(Professor Emeritus)January 13, 2021
Abstract
The COVID-19 epidemic is the last of a long list of pandemics that have affected humankindin the last century. In this paper, we propose a novel mathematical epidemiological model named
SUIHTER from the names of the seven compartments that it comprises: susceptible uninfectedindividuals (S), undetected (both asymptomatic and symptomatic) infected (U), isolated (I),hospitalized (H), threatened (T), extinct (E), and recovered (R). A suitable parameter calibra-tion that is based on the combined use of least squares method and Markov Chain Monte Carlo(MCMC) method is proposed with the aim of reproducing the past history of the epidemic inItaly, surfaced in late February and still ongoing to date, and of validating
SUIHTER in terms ofits predicting capabilities. A distinctive feature of the new model is that it allows a one-to-onecalibration strategy between the model compartments and the data that are daily made avail-able from the Italian Civil Protection. The new model is then applied to the analysis of theItalian epidemic with emphasis on the second outbreak emerged in Fall 2020. In particular, weshow that the epidemiological model
SUIHTER can be suitably used in a predictive manner toperform scenario analysis at national level.
The Coronavirus pandemic of coronavirus disease 2019 (COVID-19) is a tremendous threat toglobal health. Since the outbreak in early December 2019 in China, more than 1 834 573 globaldeaths have been registered, while the estimated total number of confirmed cases is 84 511 153 up toJanuary 2nd, 2020 [1]. The real number of people infected is unknown, but probably much higher.In this scenario, predicting the trend of the epidemic is of paramount importance to mitigatethe pressure on the health systems and activate control strategies (e.g. quarantines, lock-downs,and suspension of travel) aiming at containing the disease and delaying the spread.As these predictions have vital consequences on the different actions taken from governmentsto limit and control the COVID-19 pandemic, the recent period has seen considerable floweringof epidemiological mathematical models; see, e.g, [6, 13, 19, 20, 26, 32]. However, estimates and1 a r X i v : . [ q - b i o . P E ] J a n cenarios emerging from modeling highly depend on different factors, ranging from epidemiologicalassumptions to, perhaps most importantly, the completeness and quality of the data based onwhich models are calibrated. Since the beginning of the COVID-19 emergency, the quality of dataon infections, deaths, tests, and other factors have been spoiled by under-detection or inconsistentdetection of cases, reporting delays, and poor documentation. This inconvenient has affected, andstill is to date hampering, the intrinsic predictive capability of mathematical models.Despite the lack or incompleteness of the available data, which makes modeling the currentCOVID-19 outbreak challenging, mathematical models are still vital to establish predictions withinreasonable ranges, and can be adapted to incorporate the effects of public health authority in-terventions in order to estimate in advance their effectiveness and their impact on the COVID-19spread. Building upon the celebrated SIR (susceptible (S), infectious (I), and recovered (R)) modelproposed in 1927 by Kermack and McKendrick [22], several generalizations have been formulatedover the years by enriching the number of compartments, e.g. Susceptible – Exposed – Infectious– Recovered (SEIR), Susceptible - Infectious - Susceptible (SIS), Susceptible - Exposed - Infected- Recovered - Deceased (SEIRD), Susceptible – Exposed – Infectious – Asymptomatic – Recovered(SEIAR), Susceptible - Infectious - Susceptible - Recovered (SIRS), Susceptible - Exposed - Infec-tious - Quarantined - Recovered (SEIQR), Maternally - derived immunity - Susceptible – Exposed– Infectious – Recovered (MSEIR), ... ; we refer to, e.g., [7, 21, 28] for an overview. Overall, thesemodels have been abundantly applied to locally analyze COVID-19 outbreak dynamics in variouscountries (see, e.g., [23, 25, 26, 27]).However, the peculiar epidemiological traits of the COVID-19 ask for models better able toaccurately portray the mutable dynamic characteristics of the ongoing epidemic, with particularemphasis on two critical aspects: (i) the crucial rˆole played by the undetected (both asymptomaticand symptomatic) individuals; (ii) the number of individuals that require Intensive Care Unit(ICU) admission . This latter aspect is of paramount importance in designing realistic scenariosthat incorporate the pressure of the epidemic on the national health systems.In this paper we introduce a new mathematical model, named SUIHTER , based on the initialsof the seven compartments that it comprises: susceptible uninfected individuals (S), undetected(both asymptomatic and symptomatic) infected (U), isolated (I), hospitalized (H), threatened (T),extinct (E), recovered (R). It is a system of coupled ordinary differential equations (ODEs) thatare driven by a set of parameters that are indeed piecewise constant time dependent functions.A first set of parameters denote the transmission rates due to contacts between susceptible andundetected, quarantined or hospitalized subjects. A second set of parameters mimics the ratesat which I (isolated) and H (hospitalized) individuals develop clinically relevant or life-threateningsymptoms. A further parameter indicates the probability rate of detection of previously undetectedinfected individuals. Another set of parameters indicates the rate of recovery for the four classesof infected subjects. Finally, a last parameter denotes the mortality rate.This new model has been conceived to face some of the limitations that can be found in exist-ing epidemiological models applied to the COVID-19 pandemic. On the one hand, some studiesadopt simple SIR-like models [23, 26, 27], which have the advantage of having a limited number ofparameters to be calibrated, but pay the price of being unable to track the dynamics of differentcategories of infected individuals. On the other hand, the sophisticated multi-compartmental mod-els (see e.g. [6, 20]) have been proposed to account for the state-of-the-art knowledge of the clinicalcharacterization for different classes of infected individuals according to the actual level of diseaseseverity. However, it is not always possible (and, even when possible, it is not easy) to associate2he multiple infected compartments to the available data. The
SUIHTER model has been designedwith the objective of creating the most compact model able to predict the different categories ofinfectious individuals which are considered relevant by the policy makers. The model adopts atwo-step calibration process based on a preliminary estimation of the model parameters that uses aLeast Squares minimization, followed by a Bayesian calibration performed through a Markov ChainMonte Carlo algorithm. The model has been adopted to simulate the second COVID-19 epidemicoutbreak in Italy arisen in Fall 2020 (and still ongoing). In particular, we have investigated thecapability of the model in forecasting with an adequate advance notice the activation of exponentialgrowth at the beginning of a new outbreak, as well as the occurrence of a peak for the most relevantcompartments. Results of the calibration, simulation by
SUIHTER and predictions for few Italianregions, namely Lombardy, Emilia-Romagna and Lazio, are also reported.The outline of the paper is as follows: in Section 2 we introduce the
SUIHTER mathematicalmodel; Section 3 is devoted to the description of the calibration procedure, Section 4 contains thenumerical results along with their discussion. In Section 5, we draw our conclusions and we discusssome model’s limitations.
The spread of COVID-19 had made it clear that it is of paramount importance to include inepidemiological models a compartment describing the dynamics of infected individuals that arestill undetected. This is, e.g., the case of [20]. However, some compartments presented in [20](undetected asymptomatic infected and undetected symptomatic infected) are virtually impossibleto be validated since these classes of individuals cannot be traced in public databases (cf. [2]). Forthis reason, building upon [20] we propose a new model more suited to taking full advantage ofpublicly available data. In particular, our model is described by the following system of ordinarydifferential equations˙ S ( t ) = − S ( t ) β U U ( t ) + β I I ( t ) + β H H ( t ) N , ˙ U ( t ) = S ( t ) β U U ( t ) + β I I ( t ) + β H H ( t ) N − ( δ + ρ U ) U ( t ) , ˙ I ( t ) = δU ( t ) − ( ρ I + ω I + γ I ) I ( t ) + θ H H ( t ) , ˙ H ( t ) = ω I I ( t ) − ( ρ H + ω H + θ H + γ H ) H ( t )+ θ T T ( t ) , ˙ T ( t ) = ω H H ( t ) − ( ρ T + θ T + γ T ) T ( t ) , ˙ E ( t ) = γ I I ( t )+ γ H H ( t )+ γ T T ( t ) , ˙ R ( t ) = ρ U U ( t ) + ρ I I ( t ) + ρ H H ( t ) + ρ T T ( t ) (1)where the compartments of the model are defined as follows (see Figure 1): • S : number of susceptible (uninfected) individuals; • U : number of undetected (both asymptomatic and symptomatic) infected individuals; • I : number of isolated (quarantined) individuals;3igure 1: Interactions among compartments in SUIHTER model • H : number of hospitalized individuals, respectively; • T : number of threatened (acutely symptomatic infected, detected) individuals; • E : number of extinct individuals; • R : number of recovered individuals,and N = S + U + I + H + T + E + R denotes the total population (assumed constant).The model is characterized by the following 15 parameters, some of which are possibly chosenas time dependent piece-wise polynomial functions: • β U , β I , β H denote the transmission rates due to contacts between a susceptible subject andan undetected infected, a quarantined, or a hospitalized subject, respectively; • ω I denotes the rate at which I -individuals develop clinically relevant symptoms, while ω H denotes the rate at which H -individuals develop life-threatening symptoms; • θ H and θ T denote the rates at which H and T -individuals improve their health conditionsand return to the less critical I and H compartments, respectively; • δ denotes the probability rate of detection, relative to undetected infected individuals; • ρ U , ρ I , ρ H and ρ T denote the rate of recovery for the four classes of infected subjects;4 γ I , γ H and γ T denote the mortality rates for the individuals isolated at home, hospitalizedand hosted in ICUs, respectively.In mathematical epidemiology a fundamental quantity is the basic reproduction number (denotedby R ), which is used to measure the transmission potential of a disease. It represents the averagenumber of secondary infections produced by a typical case of an infection in a population whereeveryone is susceptible (see [7, 28]). For our model, by using a similar argument to the one adoptedin the proof of Proposition 1 in [20], we find R = β U r + δr (cid:18) β I ( r r − θ T ω H ) + β H ω I r r r r − r θ H ω I − r θ T ω H (cid:19) , (2)where r = δ + ρ U , r = ρ I + ω I + γ I , r = ρ H + ω H + θ H + γ H , and r = ρ T + θ T + γ T . For thesake of comparison (cf. Eq. (32) in [20]), we observe that in the present context the characteristicpolynomial q ( s ) of the Jacobian matrix associated to the linearization of (1) around the equilibriumconfiguration ( ¯ S, , , , , ¯ E, ¯ R ) with ¯ S + ¯ E + ¯ R = N is q ( s ) = s p ( s ) with p ( s ) = D ( s ) − ¯ SN ( s )where D ( s ) = ( s + r )( s + r )( s + r )( s + r ) − ( s + r ) θ H ω I − ( s + r )( s + r ) θ T ω H and N ( s ) = ( s + r ) { β U [( s + r )( s + r ) − ω I θ H ] + β I δ ( s + r ) + β H δω I } − β U ω H θ T ( s + r ) − β I δθ T ω H . From the mathematical point of view, the reproduction number R plays the role of a thresholdvalue at the outset of the epidemic. If R >
1, the disease spreads in the population; if R < θ H = θ T = β H = 0.Our SUIHTER model, as other compartmental models, corresponds to a particular case of anintegral model with arbitrary distribution of infectious time, for which R is well-known [14]. Model calibration through data fitting is essential to reproduce the past history of the epidemicand to perform short-term forecasts by inferring the epidemiological characteristics of COVID-19.Here we use reported isolated, hospitalized, threatened and extinct cases data to estimate theparameters of the proposed
SUIHTER model. In particular, we perform the calibration in twosteps. Firstly, we find a set of parameter values using an (ordinary) least squares (LS) estimator.Then, we perform a Bayesian calibration using a Markov Chain Monte Carlo (MCMC) algorithm,starting from a prior distribution of the parameters centered about the LS estimate. Calibrationof epidemiological models has been already performed in a Bayesian framework, following thepioneering paper by O’Neill and Roberts [31], for several infectious diseases [9, 15, 24]. In thecase of COVID-19 epidemic, Bayesian inference has been performed using simpler SIR [33, 37],meta-community SEIR-like [6, 17, 19, 25] and SEIAR [32] models, in this latter case aiming at5stimating nine parameters – including a dynamic, time-dependent contact rate β ( t ) – during thefirst outbreak of the COVID-19 epidemic. In addition to model calibration, our analysis alsoprovides a numerical assessment of the predictive capability of the model, in forecasting with anadequate advance notice both (i) the activation of an exponential growth at the beginning of theoutbreak, and (ii) the occurrence of a peak for the most relevant compartments.System (1) can be recast in the following general form describing a system of ODEs for astate vector Y with n e components (or compartments): find Y ( t ) : [ t I , t F ] → R n e with Y ( t ) =[ Y ( t ) , . . . , Y n e ( t )] T such that Y (cid:48) ( t ) = F ( t, Y ( t ); p ( t )) t ∈ ( t I , t F ] (3) Y ( t I ) = Y . (4)The evolution of the system depends on n par time-dependent parameters, collected into the function p ( t ) : ( t I , t F ] → R n p . The initial conditions Y ∈ R n e are assumed to be known.Let us partition the interval I = [ t I , t F ] into n ph phases, corresponding to different epidemicstages due to, e.g., partial restrictions (such as lock-down measures) or different containment rulesintroduced by the Government or by the local Authorities. Moreover, assume that on each phase,the value of the n par model parameters is constant (but unknown), so that we can introduce thefollowing set of admissible parameters P ad = { p ( t ) : p ( t ) | I k = p k ∈ [ p L,k , p U,k ] , k = 1 , . . . , n ph } (5)where p L,k , p U,k are given constant vectors. For the sake of notation, let us denote by p ∈ R n p the vectors of unknown parameters to be estimated, with n p = n par n ph , and let Y = Y ( t, p )highlight the dependence of the states on the parameters. Consequently, P ad is the n p -dimensionalhypercube delimited by the constraints (5). Additional constraints on the parameters are assumed,by imposing that some of them are constant on all phases.We consider n me measurements of n com = 4 < n e compartments at equispaced times t j = j ∆ t , j = 0 , . . . , n me − I = [ t I , t F ], with t = t I , t n me − = t F ; in total, we have n com × n me = 4 × n me reported data, say ˆ D ( t ) = { ˆ Y I,H,T,E ( t j ) } n me − j =0 ∈ R × n me , that is,ˆ D ( t ) = { ( ˆ I ( t ) , ˆ H ( t ) , ˆ T ( t ) , ˆ E ( t )) T , . . . , ( ˆ I ( t n me − ) , ˆ H ( t n me − )) , ˆ T ( t n me − ) , ˆ E ( t n me − )) T } . The first stage of the calibration process is then performed by seeking a LS estimate of theparameters vector, given by the solution of the following minimization problem,ˆ p = arg min p ∈P ad {J ( p ) } (6)where J ( p ) := n me − (cid:88) j =0 (cid:88) k = { I,H,T,E } α k ( t j ) (cid:107) Y k ( t j , p ) − ˆ Y k ( t j ) (cid:107) (7)being Y ( t j ) the solution of (3)-(4) evaluated at a certain given instant t j , j = 0 , . . . , n me − (cid:107) · (cid:107) the usual Euclidean vector norm. Here, we denote by Y k , k = { I, H, T, E } the com-ponents of the vector Y corresponding to the compartments I, H, T , and E , respectively, and by D ( t, p ) = { Y k ( t, p ) , k = { I, H, T, E }} the model outcome used for its calibration. For a balanced6istribution of the error across the different compartments, whose amplitudes vary along time, thedynamical weight coefficients are defined as α k ( t j ) = 1 / ˆ Y k ( t j ).We considered the official epidemiological data supplied daily by the Italian Civil Protection,hereafter called “raw data” and freely available at https://github.com/pcm-dpc/COVID-19 , [2].The accuracy of these data is highly questioned, in particular concerning the estimate of the totalnumber of infection (strongly dependent on the daily screening effort). The n com = 4 time seriesselected for model calibration (Isolated, Hospitalized, Threatened and Extincts) are those consid-ered more reliable among the data daily supplied by the authorities. One of the key features ofthe proposed SUIHTER model is indeed the one-to-one correspondence of the compartments withthe categories for which reliable data, as the ones provided on a daily-basis by the Italian CivilProtection, are available [2].When n ph phases are considered, equation (6) leads to the optimization of n p = 15 n ph pa-rameters in total. Namely, for each phase of the epidemic, we have the 15 parameters given by[ β U , β I , β H , ω I , ω H , δ, ρ U , ρ I , ρ H , ρ T , θ H , θ T , γ I , γ H , γ T ].Unfortunately, so many parameters make the calibration process problematic. In what follows,we calibrate our model under the following simplifying assumptions: • β I is taken proportional to β U , i.e. β I = αβ U , α ∈ R being an additional constant parameterto be calibrated; • β H , θ H , θ T and γ H are set to zero; • δ , ρ U , ρ I , ρ H , ρ T , γ I ∈ R are constant on [ t I , t F ].With these restrictions, the total number of parameters to be calibrated is reduced to 4 n ph + 7.The first stage of the calibration process has been performed by solving the minimization prob-lem (6) numerically. We have used a parallel version of the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm with box constraints (L-BFGS-B), see [39] for details.The second stage of the calibration process aims at quantifying uncertainties and has beencarried out employing a Bayesian framework, since the latter provides probability densities of theinput parameters that can be propagated through the model.Bayesian inference allows us to construct a probability distribution function (PDF) for theunknown parameters merging prior information and available data, these latter entering in theexpression of the likelihood function. The posterior PDF can then be obtained through the Bayestheorem on conditional probabilities. For the case at hand, we quantify the likelihood of theparameter vector p and model outcome D ( t, p ) in correlation to the reported cases ˆ D ( t ) as π ( ˆ D ( t ) | p ) ∼ N ( D ( t, p ) , σ I )where I ∈ R × is the identity matrix and the (unknown) variance σ is assumed to be constantfor each compartment.Using Bayes’ theorem, we obtain the posterior distribution of the parameters p accounting forthe prior knowledge on the parameters and the reported cases, as π ( p | ˆ D ( t )) = π ( ˆ D ( t ) | p ) π ( p ) π ( ˆ D ( t )) = π ( ˆ D ( t ) | p ) π ( p ) (cid:82) P π ( ˆ D ( t ) | p ) π ( p ) d p , where π ( p ) denotes the prior distribution for the parameters. Here, we assume that the prior PDFfor p is uniform, centered at the LS estimate ˆ p j obtained during the former calibration stage, on a7ange [0 . p j , . p j ]. An alternative, more common and rigorous procedure, would require to specifyinformative priors for the parameters, starting from key epidemiological features, as done, e.g., in[19]. However, given the large numbers of parameters to be estimated – some of which do notfind explicit counterparts in epidemiological literature – we have assumed uniform priors, centeredabout the LS estimates, as a practical shortcut to overcome the difficulty in specifying the priordistribution. In terms of predictive capability of the model, numerical results provided in Section4 allows us to assess the proposed approach.Since we cannot obtain the posterior distribution over the model parameters p analytically,we adopt approximate-inference techniques based on Monte Carlo (MC) methods, which aim atgenerating a sequence of random samples from a Markov chain whose distribution approaches theposterior distribution asymptotically, whence the name of Markov chain Monte Carlo (MCMC)[35]. In particular, we have used the delayed rejection adaptive Metropolis (DRAM) algorithmimplemented in pymcmcstat , see [29] for the details. The first 10 000 samples of the chain serveto tune the sampler and are later discarded (burn-in period). We use the next 90 000 samples toapproximate the posterior distribution for the parameters p .From the generated chains, we draw N MC samples of the parameters p , . . . , p N MC that weuse to perform forward propagation of uncertainty through the model, and to compute predictiveenvelopes of the SUIHTER model compartments (or predictive distributions).We report the MC samples of the trajectories on the time interval ( t I , t for ], including a forecastwindow ( t F , t for ] that extends beyond the time window ( t I , t F ] where data have been reported, toassess the predictive capability of the model. In this section we present three batteries of numerical results assessing the forecasting capabilitiesof the
SUIHTER model. Our analysis focuses on the second wave of the epidemic that started at theend of the Summer of 2020 and, at the time of this writing, is still affecting Italy. In Section 4.1, wepresent the simulation of the second wave obtained with the
SUIHTER model using for its calibrationall the data between August 20th and December 31st. By limiting the time range of the data usedfor the calibration, we also investigate the model capability in forecasting the peaks of the differentcompartments (see Section 4.2) and the exponential outbreak in the early phase of the second wave(see Section 4.3).Our results at the national level for the second outbreak have been obtained by initializingthe
Isolated , Hospitalized , Threatened and
Extinct compartments with the data provided by theDipartimento della Protezione Civile [2] at August 20th. The remaining compartments – i.e.
Sus-ceptible , Undetected and
Recovered for which data are unavailable – have been instead initializedwith the values obtained by running (and calibrating) the
SUIHTER model from February 24th upto August 20th (first outbreak and its tail) with initial values set as S = 60 483 174, U = 500, I = 94, H = 101, T = 96, E = 7 and R = 1 on February 24th. The initial values for the secondoutbreak are therefore S = 57 630 019, U = 9 286, I = 15 063, H = 883, T = 68, E = 35 418 and R = 2 793 236 on August 20th. Note that this would imply that, by the end of the first wave,around 4 .
6% of the Italian population had been infected. A serosurvey organized by ISTAT andISS had estimated that 2 .
5% of the Italian population had been infected [30, 36]; the survey how-ever had a low compliance, so that its results may be biased. A corresponding survey in Spain[34] with a much higher compliance rate estimated seropositivity to 4 .
6% or 5%, depending on the8ethodology used for the seroprevalence analysis. Thus, the value obtained for August 20th looksrather reasonable.
The
SUIHTER model has been used to simulate the second epidemic outbreak, starting from August20th until December 31st, 2020. The different phases in which the parameters can take differentvalues have been identified according to the occurrence of some critical events: • September 24th: all schools at the national level reopened after the summer (and springlockdown) closure (schools calendars vary by grades and by region level in Italy); • October 8th: new rules imposing the mandatory use of masks in all locations (either indooror outdoor) accessible to public; • October 26th: confinement rules including distance learning for most secondary schools, lim-itations on the activity of shops, bars and restaurants, strong limitation of sport and leisureactivities ; • November 6th: stricter confinement rules including distance learning from 9th grade, furtherrestrictions on commercial activities, limitations on the circulation outside the own munici-pality (for some Italian regions, classified as red regions) ; • November 15th: additional confinement rules as more regions turned to red color ; • November 19th: additional confinement rules as more regions turned to red color ; • November 29th: relaxation of confinement rules in some regions turned to orange color ; • December 11th: relaxation of confinement rules in some regions turned to yellow color ; • December 21st: stricter confinement rules are introduced for Christmas holidays .Considering a time lag of 4 days (to account for the incubation period) [18], the correspondingphases on which the model parameters are defined and possibly changing) are: • Phase 1: August 20th - September 28th; • Phase 2: September 29th - October 11th; • Phase 3: October 12th - October 29th; • Phase 4: October 30th - November 9th; DPCM October 24, 2020, DPCM November 4, 2020, DPCM December 21, 2020, Phase 5: November 10th - November 18th; • Phase 6: November 29th - November 23rd; • Phase 7: November 24th - December 3rd; • Phase 8: December 4th - December 15th; • Phase 9: December 16th - December 25th; • Phase 10: December 26th - December 31st.As mentioned in Section 3, the compartments employed for calibration are only those with morereliable data, namely
Isolated (I),
Hospitalized (H),
Threatened (T) and
Extinct (E) individuals.We performed the model calibration by employing the MCMC parameter estimation proceduredescribed in Section 3, over the 10 phases using the data over the full time range between August20th and December 31st. The simulations were run for the subsequent 30 days beyond the dateassociated to the last set of data used for the calibration forecasting the evolution of the epidemicuntil end of January 2021. For the new additional phase the values of the parameters are obtainedby linearly extrapolating the two (constant) values of the corresponding parameter of the last twophases, located at the final day of each phase, namely phases 9 and 10.In Figure 2, we report the expected values for the time evolution of the 7 compartments ofthe
SUIHTER model as well as the time evolution of additional compartment of the
Daily newpositive , which corresponds to δ U ( t ), and the corresponding 90% prediction intervals obtained bypropagating input uncertainties through the model.We can notice that the calibrated compartments ( Isolated , Hospitalized , Threatened and
Ex-tinct ) fit very accurately the data time series. Two additional compartments for which data areavailable but not used for the calibration, namely the
Recovered and the
Daily new positives , areused to assess the accuracy of the model. In particular, since the available data for the
Recovered cases do not include the undetected cases that recovered before being detected, a novel compartment R D ( t ) = (cid:90) tt I ( ρ I I ( τ ) + ρ H H ( τ ) + ρ T T ( τ )) dτ, collecting the individuals recovered after being detected is used for comparison, showing a goodmatch with the data. Moreover, the time history of the Daily new positives is also in reasonableagreement with the data, proving that the model is able to capture the main dynamics of the systemalso for those quantities that are not directly driven by the data calibration. Our calibrationindicates that from the rise of the second outbreak to December 31st, 2020, 4 410 025 ±
113 231individuals have been infected, of which 66 . ± .
7% has been detected. In addition, the casefatality ratio (the ratio between the total number of deceased and diagnosed individuals over theperiod) is 2 . ± . . ± . .
23% in [8] for the firstItalian outbreak. We also observe that our calculated estimates are likely to be underestimated asthe second outbreak is still ongoing at the present time and compartments of isolated and extinctindividuals become populated at different time scales.10igure 2: Expected values (solid lines) and 90% prediction intervals (shaded areas) for the 7compartments of the
SUIHTER model plus the additional
Daily new positives compartment.The mean values and the standard deviations computed by the MCMC calibration are reportedin Table 1 for the parameters that are constant over the simulation and in Table 2 for the param-eters that are free to change in each phase. The former parameters and time dependent functionsrepresent rates that can be used to interpret the dynamics of the second Italian outbreak. Forexample, large values of β U indicate sustained transmission rates at the corresponding phases. Val-ues of healing rates ρ I , ρ H and ρ T are proportional to the probability of healing for individuals inthe compartments I, H and T, but are inversely proportional to the corresponding average time ofhealing; the rate ρ I also incorporates the healing on isolated individuals who are however asymp-tomatic. To better understand the role of the parameters, note that if they were constant, ρ I ρ I + ω I would represent the probability for an isolated individual to recover without being hospitalized, and11 ean Std Dev α δ γ I ρ U ρ I ρ H ρ T Table 1: Mean values and standard deviations of the constant parameters β U ω I ω H γ T R Phase Mean Std Dev Mean Std Dev Mean Std Dev Mean Std Dev Mean Std Dev1 0.26402 0.002019 0.00642 0.000317 0.01517 0.000804 0.07238 0.002805 1.119 0.00522 0.35072 0.003113 0.00843 0.000460 0.02251 0.001167 0.12388 0.004607 1.482 0.01443 0.34635 0.002448 0.00999 0.000402 0.02494 0.001156 0.09046 0.003339 1.460 0.01154 0.27296 0.004197 0.00753 0.000339 0.02983 0.001260 0.15781 0.002524 1.154 0.01965 0.24914 0.003912 0.00540 0.000264 0.02872 0.001226 0.17076 0.004714 1.058 0.01836 0.17528 0.004656 0.00481 0.000266 0.03088 0.001476 0.19419 0.006230 0.743 0.02197 0.21801 0.003305 0.00388 0.000192 0.02985 0.001292 0.19134 0.005199 0.926 0.01568 0.19450 0.003049 0.00370 0.000194 0.02859 0.001210 0.19086 0.004291 0.827 0.01439 0.26871 0.005721 0.00349 0.000183 0.02782 0.001275 0.19319 0.005484 1.143 0.027110 0.28086 0.004685 0.00402 0.000219 0.02889 0.001400 0.19082 0.003435 1.193 0.0222
Table 2: Mean values and standard deviations of the parameters that changes over the phases andthe corresponding R similarly ρ H ρ H + ω H represents the probability for a hospitalized individual to recover without beingtransferred to ICUs. In the same way, γ T γ T + ρ T represents the probability of dying for an individualin ICUs, and δδ + ρ U represents the probability that an infected individual is detected.Finally, Table 2 also reports the value of the basic reproduction number R calculated as inEq.(2) for the SUIHTER model. The calculation uses the model parameters reported in Tables 1and 2 (columns 1 − ± R obtained by the calibration reflects the full reopening ofeducational activities and work restart after holidays, as well as the public health measures andrestrictions later introduced by authorities to contain the second epidemic outbreak. In particular,the rise of R in Phases 2 and 3 follows the full schools reopening and restart of working activitiesfrom mid September, and probably accounts for seasonality effects too. Restrictions on mobility,schools, businesses and partial lock-downs were introduced in late October at regional and nationallevels, as reflected by the decrease of R from Phase 4 to 6, when R became smaller than one.Partial reopening and easing restrictions were gradually introduced in some regions and at thenational level from late November, as the new increment of R from Phase 7 indicates. The results obtained simulating the epidemic at the national scale can indeed hide specific localoutbreaks. The
SUIHTER model can also simulate the evolution of the epidemic for everyone of the12ombardy Emilia-Romagna LazioFigure 3: Expected values (solid lines) and 90% prediction intervals (shaded areas) for the
Isolated , Hospitalized , Threatened and
Extinct compartments in three Italian regions, from left to rightLombardy, Emilia-Romagna and Lazio20 Italian Regions for which the same data time series as those used for the national calibration areavailable. Unfortunately, this is not true for the finer geographical level (the 107 provinces) sinceonly the number of total cases from the beginning of the epidemic is provided.Following the same initialization and calibration strategies adopted for the national level, wehave carried out the simulation of the second epidemic outbreak in three Italian regions, namelyLombardy, Emilia-Romagna and Lazio. In Figure 3, the expected value for the time evolutionof the four compartments used for the calibration and the corresponding 90% prediction intervalsare reported for the former three regions. We can observe, for instance, that: i) the peaks ofthe compartments have been reached earlier in Lombardy; ii) after the peaks have been reached,the decrease of the curves is much slower in Lazio than in the other regions. In all the cases theresults obtained by numerical simulations stand in very good agreement with the real data, withthe only exception of the threatened compartment in Lazio where a slight discrepancy (within 10%in relative terms) is observed. Predictions realized for the former three regions indicate differentepidemic trends at the regional level. 13igure 4: Peak forecast obtained by the
SUIHTER model with different data ranges for the
Isolated , Hospitalized , Threatened and
Extinct compartments
Predicting the peak of an epidemic outbreak is a tremendous challenge for an epidemiologicalmodel. Yet, the predictive capability of epidemiological models is of paramount importance toinform policymakers about the dynamics of the disease and foresee timing and level of peaks ofinfected, hospitalized and ICU treated individuals, as well as the potential effects of policy responses.With the goal of investigating to which extent our
SUIHTER model is able to predict the oc-currence of the epidemic wave peak, we repeated the calibration using the data over limited timeranges.In particular, we have considered three different cases: in
Case 0 we used all the data timehistories available until December 3rd, while in
Cases 1 , and , the data employed for thecalibration were limited to November 23rd, November 19th, and November 9th, respectively. Foreach case, the simulations were run for the subsequent 30 days beyond the date associated to thelast set of data used for the calibration and the linear extrapolation carried out as indicated before.In Figure 4, we report the expected value for the time evolution of the four compartments usedfor the calibration, and the 90% prediction intervals obtained by propagating input uncertaintiesthrough the model. The accuracy of the forecast, as expected, improves as far as a richer set ofdata are employed in the calibration. Our simulations show the occurrence of a peak for each ofthe three compartments, not only for Case 0 in which the time lapse of the data used for thecalibration covers the peaks, but also for
Case 1, 2 , and , when the data time-series employed forthe calibration are still rising. However, we should remark that if the model is calibrated with a14horter time series, namely available data stop more than 30 days before the peak, the occurrenceof the peak cannot be correctly predicted.As already noticed, because of the overall complexity of the problem and the limited dataavailable for its calibration, by no means we intend here to certify in rigorous terms the actualvalues of the future compartments. However, in spite of the widths of the predictive intervals(which depend, at some extent, on the widths of the chosen prior distributions), we nonethelessobserve that the expected values (solid lines in Figure 4) carry meaningful prediction capabilities.To further quantify the prediction accuracy, it is interesting to assess this peak forecasting withrespect to the actual day and value that have been observed for the different compartments at theend of November 2021. Moreover, we propose a comparison with the predictions obtained using thedifferent strategies based on data fitting recently presented in [1]. Namely, for each compartment,we have considered both a simple polynomial (quadratic) extrapolation of their time history (astrategy that is known to be unreliable on large time intervals), as well as a model based on theregistration of the second outbreak with the curve of the corresponding first outbreak occurred inSpring 2020.A comparison between the peak forecast obtained with the SUIHTER model, the quadratic ex-trapolation (based on the last 10 days), and the registration approach is displayed in Figure 5, forthe
Isolated , Hospitalized and
Threatened compartments. The curves show how the prediction interms of day of peak occurrence and peak value changes as far as an increasing number of data areused (the last data day is reported on the horizontal axis).By comparing the peak predictions with the day and value of the actual measured data peak(reported with a dashed line in Figure 5), we should first remark that the
SUIHTER predictionlargely outperforms those obtained with polynomial extrapolation. Moreover, even when comparedwith predictions based on the registration with the first epidemic wave, the
SUIHTER model is moreaccurate for most of the considered quantities. When making this comparison, it is worthy noticingthat, while prediction based on the registration strongly depends on the evolution of the differentcompartment during the first epidemic wave, the predictions based on
SUIHTER do not require anya-priori knowledge of previous epidemic waves.
In this section we discuss the capability of the
SUIHTER model of predicting the occurrence of theexponential outbreak of the COVID-19 epidemic. To this aim, a second set of simulations havebeen performed by focusing on the early stages of the second wave.Similarly to the previous section, we have considered different calibrations based on employingdifferent subsets of the data available until the end of October (when the epidemic reached itsmaximum rate of growth). In particular, we performed three different calibrations by employingdata until October 29th, October 11th and September 30th, respectively.The results displayed in Figure 6, in particular the evolution of the three infected compartment,indicate that the onset of an exponential growth activating the second wave could have beenpredicted using the data available until October 11th. Also in this case, we report the expectedvalue for the time evolution of the 4 compartments used for the calibration, and the 90% predictionintervals obtained by propagating input uncertainties through the model. Although in this casethe model prediction detaches from reported data at later stages – showing its extreme sensitivityto data, typical of any exponential growth – it is remarkable that our calibration procedure wouldhave predicted a dramatic variation of the epidemic trend from September 30th to October 11th.15solated (peak day) Isolated (peak value)Hospitalized (peak day) Hospitalized (peak value)Threatened (peak day) Threatened (peak value)Figure 5: Peak day (left) and peak value (right) vs. last used data by day for the three compartments
Isolated (top),
Hospitalized (middle) and
Threatened (bottom), estimated with data extrapolation,data registration and
SUIHTER model 16igure 6: Outbreak forecast obtained by the
SUIHTER model with different data ranges for the
Isolated , Hospitalized , Threatened and
Extinct compartments
In this paper, we have introduced a new mathematical model, named
SUIHTER , to describe the ongo-ing pandemic of coronavirus disease 2019 (COVID-19). This epidemiological model is constructedon seven compartments – susceptible uninfected individuals (S), undetected (both asymptomaticand symptomatic) infected (U), isolated (I), hospitalized (H), threatened (T), extinct (E) and re-covered (R) – and we exploit it to study and analyse the second Italian outbreak emerged in Fall2020 and still ongoing. In particular, our model is suited for calibration against data made avail-able daily by the Italian Civil Protection [2]. On the basis of these data at the national level,our calibration populates the compartments I, H, T and E, which we purposely use to determinetransmission rates, rates of recovery, infection fatality rates, etc. In particular,
SUIHTER is able ofdetermine the infected, but undetected population, a compartment (U) that is crucial for studyingand understanding the epidemic, especially considering that large shares of infected individualswent uncounted during the first and even the second outbreaks in Italy. Moreover, thanks to ourapproach transmission rates, and thus the basic reproduction number R , can be estimated on adaily basis. Finally, our calibration is made robust by exploiting Bayesian estimation using theMarkov Chain Monte Carlo method.The SUIHTER model calibrated at the Italian national level is validated against data related tothe last part of the second outbreak. Comparisons are made against basic statistical models, namelyquadratic regression and registration of the first epidemic wave. The comparison demonstrates thebetter accuracy of
SUIHTER for predictive purposes. This is made possible by using extrapolatedtransmission rates that are calibrated at earlier times through regression models, a feature that17llows capturing peaks of the second Italian outbreak correctly, and enables using
SUIHTER in apredictive fashion by leveraging data available at the current date. This novel approach attemptsto circumvent a common issue of the use of epidemiological compartmental models for forecasting[32], that is accurately capturing transmission rates. However, as our approach is based on inter-polating values of these transitions rates, the accuracy of their extrapolation and, consequently,their exploitation for prediction within
SUIHTER can only be limited to restricted time windows,especially when government interventions and citizen behaviours are changing. Note that, althoughthe calibration procedure did not make any assumptions about the temporal changes in parameters,the estimates accurately reflect the policy changes: estimates of R decrease as control measuresare tightened and increase when they are relaxed.A further limitation of our approach is that we are currently calibrating the Italian epidemicoutbreaks at the national level, that is as a whole, without summing up the different contributions atthe level of the 20 Italian regions for which data are available [2]; we indeed performed the calibrationonly for few of the Italian regions, namely Lombardy, Emilia-Romagna and Lazio. Populatingcompartments at the national level by summing up results obtained by tailored calibrations of eachItalian region would allow a better capturing of the spatio-temporal heterogeneity of the Italianoutbreaks, which reflects different mobility patterns and density of population. In this respect,several different approaches have been proposed in literature, see, e.g. [11] and the referencestherein, ranging from the use of network based models [5, 12], to systems of ordinary differentialequations on network [3, 4], as well as non-local partial differential equations [38]. Among thecontributions appeared during the COVID-19 pandemic, we also recall the recent papers [6, 19, 25],where a meta-community SEIR-like model has been proposed and employed to reproduce thecontagion in Italy. Still, calibrating our SUIHTER at the regional level, and for all the regions,would require a more sophisticated design due to the intrinsic ill-posedness of the inverse problem,especially when taking mobility patterns into account. Nevertheless, we plan to better addressspatio-temporal heterogeneity of the Italian outbreaks in the future by generalizing our
SUIHTER model to incorporate suitable spatial-multicity mobility terms at the regional level. Even if a morespatially detailed compartment model is surely desirable, to act, for example, at the province level(Italy is comprised of 107 provinces), at the time being no detailed data for its calibration havebeen made available.Albeit the
SUIHTER is namely very sophisticated and involves 15 time-dependent parametersand functions to be determined based on available data, we limited our calibration to a subset ofthe possible control variables, by forcibly setting to zero some parameters that we deemed to beless relevant for the transmission of the epidemic and by assuming as piece-constant over time someother ones. We also neglected incubation tine, and we implicitly assumed that all distributions inthe states are exponential, which is far from correct [16]. Still, we believe that this qualifies as anacceptable compromise among the complexity of the
SUIHTER model and its calibration procedure,the associated computational costs, and the accuracy of the results. Some of the calibrated param-eters assume values that are able to compensate for those parameters prescribed a priori, even iftheir interpretation may not result straightforward in explaining the outbreak. In this respect, weplan to assess the robustness of our approach by allowing the calibration of additional parameters.Further, our multi-compartment
SUIHTER model does not consider stratification of ages groupswithin the compartments. This is namely an important aspect as some compartments like H, Tand E are mostly populated by the elderly, while the transmission mechanisms widely differ byage and context of infection (workplace, school, family, etc.). We also plan to improve
SUIHTER by18onsidering age stratification within its compartments.Finally, in consideration of the ongoing emergency situation amidst the second Italian outbreak,we believe that our
SUIHTER model is well suited to be used in a predictive manner to support andmotivate public health measures. To the best of our knowledge, apart from [10] wherein a SEIRDmodel is used at the regional level,
SUIHTER stands as one of the first models to analyze the secondItalian COVID-19 outbreak and that can readily serve the purpose of predicting the epidemic trend.
Acknowledgements
The authors would like to thank Prof. Luca Formaggia for his insightful suggestions and carefulreading of the manuscript.
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