Enrico, Bibbona

Gianluca, Mastrantonio

Martina, Amongero

English

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

29 November 2022POLITECNICO DI TORINORepository ISTITUZIONALEAnalysing the Covid-19 pandemic in Italy with the SIPRO model / Amongero, Martina; Bibbona, Enrico; Mastrantonio,Gianluca. - 1:(2021), pp. 1568-1573. ((Intervento presentato al convegno SIS 2021.OriginalAnalysing the Covid-19 pandemic in Italy withthe SIPRO modelPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2917494 since: 2021-08-07T21:18:08ZPEARSONAnalysing the Covid-19 pandemic in Italy withthe SIPRO modelAnalizzare la pandemia di Covid-19 in Italia con ilmodello SIPROMartina Amongero, Enrico Bibbona, Gianluca MastrantonioAbstract We propose an epidemic compartmental model that extends the classicalSIR, in order to allow for an estimate of the unobserved infected people that have notbeen tested. The model is then fitted to the epidemic curves of the 20 Italian regionsusing Bayesian methods for mixed-effects models. Despite the interpretability ofthe model, we still face identifiability issues. We explain how we can alleviate them.Abstract Abstract in Italian Proponiamo un modello epidemico compartimentaleche estende il modello SIR classico, allo scopo di permettere la stima del numero(latente) di persone infette che non sono state testate. Il modello è poi stimato sullabase delle curve epidemiche delle 20 regioni italiane, usando un metodo Bayesianoper modelli ad effetti misti. Nonostante l’interpretabilità del modello, si incontranocomunque alcuni problemi di identificabilità. Li illustriamo e indichiamo come pos-sono essere mitigati.Key words: SIR, Covid-19, MCMC, Compartmental models, Epidemic curves,mixed models1 IntroductionIn Italy, as all over the world, the COVID19-pandemic has had a terrible impacton people’s lives, and has caused nearly 100 000 deaths by the end of February2021. The entire scientific community is at work, and statisticians are contributingin several ways. One direction is that of formulating mathematical models that canhelp the surveillance of the epidemics and can be used to evaluate the effectivenessof public policies.Martina Amongero, Enrico Bibbona, Gianluca MastrantonioDISMA - Politecnico di Torinoe-mail: martina.amongero@polito.it, enrico.bibbona@polito.it, gianluca.mastrantonio@polito.it115682 Martina Amongero, Enrico Bibbona, Gianluca MastrantonioTwo main approaches have been proposed: mechanistic models and phenomeno-logical ones. Mechanistic models try to interpret the data by the underlying basicmechanisms. Notable work in this direction are [4, 10, 7, 3]. On the other hand,phenomenological models focus on making the best forecast, no matter if the datagenerating mechanism is interpretable or realistic anyhow. Examples are [2, 5, 1].This work is about a deterministic compartmental model (SIPRO) that falls in thefirst category. The main difficulty in this kind of model is that, the more it is detailedand realistic, the more difficult it becomes to fit it to the available data on epidemiccurves. Indeed, most of the information is unobserved, and hardly reconstructablefrom the available data.The goal of this paper is to propose an extension of the SIR model that includesa compartment of infected people that is not observed directly. When and if thesepeople are tested they enter the positive compartment and, since quarantined, theystop spreading the disease. The motivation is that, as it was shown in [8], the asymp-tomatic part of the infected population gives a significant contribution to the pan-demic’s spread and it is often unreported. This model is applied at the level of the 20Italian regions, with some parameter that is region-specific and some other parame-ters that are common to all regions. Some of the regional parameters are moreoverassumed to be drawn from a common population (random effects).Despite the model is much simpler than other proposals, the inference problem isstill very challenging since the reconstruction of the latent components suffers frompoor identifiability. We illustrate how this can be alleviated by adopting suitableexpedients. For our analysis, we use the Italian public data collected by the Italian”Protezione Civile”, which can be found at the link [6].2 The SIPRO model (ρ(t),µ,α,ν)The population is divided into five compartments: Susceptibles (S), Infected (I),who are infectious, Positives (P) who have been tested and quarantined (no longersource of contagion), Recovered (R), formerly infected that recovered or died with-out having been tested, and, finally, Out (O), the recovered or dead people whotested positive. The state of the system is identified by the proportion of the pop-ulation in each compartment, a vector (i, p,r,o). The proportion of susceptibles is1− i− p− r− o. The evolution of the state variables is described by the functions(I(t), P(t), R(t), O(t)) that solves the following system of ordinary differentialequations (ODE)⎧⎪⎪⎪⎨⎪⎪⎪⎩İ(t) = β (t)I(t)[1− I(t)−P(t)−R(t)−O(t)]−µI(t)−αI(t)Ṗ(t) = αI(t)−νP(t)Ṙ(t) = µI(t)Ȯ(t) = νP(t).(1)1569Analysing the Covid-19 pandemic in Italy with the SIPRO model 3Fig. 1 SIPRO compartmental modelYYYPPYYYPYYνWe can further define S(t) = 1−P(t)−R(t)−O(t) as the proportion of suscepti-bles. A schematic representation is provided in Figure 1. The rate β (t) of infectionis allowed to vary over time to model the impact of social distancing measures. Thevalue 1/(µ +α) represents the mean time an individual spends in compartment I.The quantities 1/µ and 1/α are respectively the time between infection and re-covery and between infection and positivity. Furthermore 1/ν gives the mean timebetween positivity and recovery. Finally ρ(t) = β (t)/(µ+α) gives the reproductionnumber at time t, and we parameterize the model by (ρ(t),µ,α,ν).2.1 The SIPRO model combined with a mixed approachWe build a mixed-model to describe the evolution of the Pandemic in the 20 ItalianWregions. e assume that within each region, the spread of the pandemic follows theSIPRO model (1), with regional reproduction numbers ρi(t), testing policy αi andrecovery rates from positivity i. The recovery rate µ of the unobserved (asymp-tomatic) infections is common to all regions.Let {( 2,i( j), 4,i( j))} j=1,..,Ti=1,...,20be the daily proportions of Positives and Out indi-Wviduals derived from the Protezione Civile database [6]. e assume they are noisySIPRobservations from the regional O variables i( j) and Oi( j), while the othervariables of the model are latent. Since these variables represent proportions, a rea-Wsonable noise model is given by the Dirichlet distribution. e, therefore, assume(2,i( j), 4,i( j), 0,i( j))∼ Dirichlet(γ i( j), γ Oi( j), γ (1− i( j)−Oi( j)))where for all i = 1, ...,20 and j = 1, ...,T , 0,i( j) = 1− 2,i( j)− 4,i( j). The pa-rameter γ Wregulates the noise amplitude (in all regions). e estimate directly theregional reproduction number ρi(t) from which the infection rate is computedas βi(t) = ρi(t) · (µ + αi). For simplicity we assume that the function ρi(t) isi i li b t id f i d t l d { } ithττp ece-w se near e ween a gr o equ space empora no es k k=1,··· ,M wτ1 = 0, · · · , M = T , whose temporal location is specified together with the priors in15704 Martina Amongero, Enrico Bibbona, Gianluca Mastrantoniothe next section. Therefore it is sufficient to estimate the values ρik = ρi(τk) at thenodes. The logarithms of the regional SIPRO parameters ρik and νi are assumed tobe drawn from a Gaussian population (random effects),log(νi)i.i.d.∼ N (log(ν),ω2ν ), log(ρi(k))i.i.d.∼ N (log(ρk),ω2)since they are both affected by national and regional public policies, while the αiare considered as fixed regional effects. The parameter µ should not be affected byany regulation, and it is therefore kept common to all regions. The initial values(I(0), P(0), R(0), O(0)) of the SIPRO equations (1) need also to be estimated.3 Tuning parameters and resultsWe focus on the data from all Italian regions from the 24th of February 2020 tothe 30th of June 2020, corresponding to the so-called first wave. The parametersare estimated using a Bayesian algorithm, but since the model is poorly identifiable,any attempt to estimate the parameter solely based on uninformative priors fails. Alot of information is hidden in the unobserved components, and we need to guidethe algorithm by the use of prior information. The parameters that are linked totransition between two unobserved compartments are the most difficult to estimate,namely ρi(t), αi, and µ . We emphasize that the value of ρi(t) at the first and lastnode is particularly hard since only half of the information on it is available. Thiswill be reflected in the amplitude of the credible intervals. Finally, the inference ismore difficult for regions that had few cases.To obtain sensible values it is necessary to• Use informative prior on αi. In particular we impose 1αi ∈ [0.001,30].• Fix different values of µ ∈ { 15k s.t. k = 1, ...,9} and then choose the best valuesusing the DIC criterion.• Use a quite informative prior on the initial state of the system.The prior on 1/αi is set by constructing an auxiliary variable εi with a Gaussianprior distributionlog(εi)i.i.d.∼ N (−2,0.72),and then by remapping it to the interval [0.001,30] (in days) by the transformation1αi= 0.001 ·1(−∞,0.001)(1εi)+1εi1[0.001,30](1εi)+30 ·1(30,+∞)(1εi)The prior for the initial conditions is specified by the vector d = 100 ·(0.96,0.01,0.01,0.01,0.01),setting for all i(Si(0), Ii(0),Pi(0),Ri(0),Oi(0))i.i.d.∼ D(d)The other priors are1571#! "#$!γAnalysing the Covid-19 pandemic in Italy with the SIPRO model 5log(ν)∼ N (0,100) and log(ω)∼ N (0,100)log(ρk)∼ N (0,100), ∀k ∈ K and ων ∼ U [0,100]log( )∼N (0,100)Table 1 VSIPRO model: Global Parameters Estimated aluesParameter Mean 95% CI Parameter Mean 95%ρ(t1) 2.60 [2.12, 3.17] ρ(t6) 0.59 [0.13, 1.10]ρ(t2) 1.41 [1.15, 1.71] ρ(t7) 0.21 [0.02, 0.82]ρ(t3) 0.87 [0.70, 1.05] 1/ν 31.17 [27.06, 35.84]ρ(t4) 0.33 [0.23, 0.45] 1/µ 10 -ρ(t5) 0.06 [0.02, 0.20] logγ 10.04 [10.00, 10.08]Table 1 reports the estimates of the global parameters with 95% credible inter-difvals. As an example, we present the estimates for two ferent regions: Lombardy,the Italian region that counted more cases, and Abruzzo. Figure 2 graphically rep-resents the results obtained. The functions ρi(t) efclearly reflects the fect of social"#!%"#!&'(#!α"#!)Fig. 2 P(t), O(t) and ρ(t) mean curves, with 95% CI, for Lombarepresent data. The black dots in the last panes represents the gridinterpolation. They are set every three weeks on Mondaysdistancing measures. As expected, the function decreasedestimates of 1αi for Lombardy and Abruzzo, are1L∈ [20.5527 67 and 1 ∈ [7 25 30] with posterior mean 16 19rdy and Abruzzo. Red linesof time nodes of the linearafter the lockdown. The,30] with posterior meanα. A . , . .15726 Martina Amongero, Enrico Bibbona, Gianluca Mastrantonio4 ConclusionWe analyzed and modeled the data collected from February to June (the so-calledfirst phase of the pandemic) using a simple but realistic compartmental model. Wehighlighted the difficulties in the inference and proposed some solutions. Finally,we presented our estimates that are in agreement with what has been reported inother works. Future work will be needed to compare our model and our results tothe ones obtained using other approaches, and to apply the SIPRO-mixed-model tothe data collected in September-January, the so-called second phase, to evaluate theeffectiveness of the new containment measures that are activated on a regional basis.Acknowledgements The authors thankfully acknowledge funding from the Italian Ministeryof Education, University and Research, MIUR, grant Dipartimento di Eccellenza 2018–2022(E11G18000350001) that also provided computational resources, together with HPC@POLITO(http://www.hpc.polito.it).References1. Agosti, A. and Giudici, P. A Poisson Autoregressive Model to Understand COVID-19 Con-tagion Dynamics. Risks, 8 (3), 20202. Farcomeni, A., Maruotti, A., Divino, F., Jona-Lasinio, G., Lovison, G., An ensemble approachto short-term forecast of COVID-19 intensive care occupancy in Italian regions. BiometricalJournal, 20203. Gatto, M., Bertuzzo, E., Mari, L., Miccoli, S., Carraro, L., Casagrandi, R., and Rinaldo, A.(2020). Spread and dynamics of the covid-19 epidemic in italy: Effects of emergency con-tainment measures. Proceedings of the National Academy of Sciences 117, 10484–10491.4. Giordano, G., Blanchini, F., Bruno, R., Colaneri, P., Di Filippo, A., Di Matteo, A., andColaneri, M. (2020). Modelling the covid-19 epidemic and implementation of population-wide interventions in Italy. Nature Medicine pages 1–6.5. Griette, Q. and Demongeot, J. and Magal, P. A robust phenomenological approach to inves-tigate COVID-19 data for France. medRxiv, https://doi.org/10.1101/2021.02.10.212515006. https://github.com/pcm-dpc/COVID-19.7. Jia, W., Han, K., Song, Y., Cao, W., Wang, S., Yang, S., Wang, J., Kou, F., Tai, P., Li, J., et al.(2020). Extended SIR prediction of the epidemics trend of COVID-19 in italy and comparedwith Hunan, China. Frontiers in Medicine, 78. Lavezzo, E., Franchin, E., Ciavarella, C., Cuomo-Dannenburg, G., Barzon, L., Del Vecchio,C., Rossi, L., Manganelli, R., Loregian, A., Navarin, N., et al. (2020). Suppression of aSARS-Cov-2 outbreak in the Italian municipality of Vo’. Nature 584, 425–429 .9. Prague, M., Wittkop, L., Clairon, Q., Dutartre, D., Thiébaut, R., and Hejblum, B. P. (2020).Population modeling of early COVID-19 epidemic dynamics in French regions and estima-tion of the lockdown impact on infection rate. medRxiv .10. Russo, L., Anastassopoulou, C., Tsakris, A., Bifulco, G., Campana, E., Toraldo, G., and Si-ettos, C. (2020). Tracing day-zero and forecasting the fade out of the COVID-19 outbreak inLombardy, Italy: a compartmental modelling and numerical optimization approach. Plos one,2020.1573

Analysing the Covid-19 pandemic in Italy with
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