Modelling the second wave of COVID-19 infections in France and Italy via a Stochastic SEIR model
AAIP/123-QED
Modelling the second wave of COVID-19 infections in France and Italy via aStochastic SEIR model
Davide Faranda
1, 2, 3, a) and Tommaso Alberti Laboratoire des Sciences du Climat et de l’Environnement,CEA Saclay l’Orme des Merisiers, UMR 8212 CEA-CNRS-UVSQ,Université Paris-Saclay & IPSL, 91191, Gif-sur-Yvette, France London Mathematical Laboratory, 8 Margravine Gardens, London, W6 8RH,UK LMD/IPSL, Ecole Normale Superieure, PSL research University, 75005, Paris,France INAF - Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere 100,00133 Roma, Italy (Dated: 11 June 2020)
COVID-19 has forced quarantine measures in several countries across the world. Thesemeasures have proven to be effective in significantly reducing the prevalence of the virus.To date, no effective treatment or vaccine is available. In the effort of preserving bothpublic health as well as the economical and social textures, France and Italy governmentshave partially released lockdown measures. Here we extrapolate the long-term behavior ofthe epidemics in both countries using a Susceptible-Exposed-Infected-Recovered (SEIR)model where parameters are stochastically perturbed to handle the uncertainty in the esti-mates of COVID-19 prevalence. Our results suggest that uncertainties in both parametersand initial conditions rapidly propagate in the model and can result in different outcomesof the epidemics leading or not to a second wave of infections. Using actual knowledge,asymptotic estimates of COVID-19 prevalence can fluctuate of order of ten millions unitsin both countries. a) Correspondence to [email protected] a r X i v : . [ q - b i o . P E ] J un . LEAD PARAGRAPHCOVID-19 pandemic poses serious threats to public health as well as economic and so-cial stability of many countries. A real time extrapolation of the evolution of COVID-19epidemics is challenging both for the nonlinearities undermining the dynamics and the ig-norance of the initial conditions, i.e., the number of actual infected individuals. Here wefocus on France and Italy, which have partially released initial lockdown measures. Thegoal is to explore sensitivity of COVID-19 epidemic evolution to the release of lockdownmeasures using dynamical (Susceptible-Exposed-Infected-Recovered) stochastic models. Weshow that the large uncertainties arising from both poor data quality and inadequate estima-tions of model parameters (incubation, infection and recovery rates) propagate to long termextrapolations of infections counts. Nonetheless, distinct scenarios can be clearly identified,showing either a second wave or a quasi-linear increase of total infections.II. INTRODUCTION SARS-CoV-2 is a zoonotic virus of the coronavirus family emerged in Wuhan (China) at theend of 2019 and rapidly propagated across the world until it has been declared a pandemic bythe World Health Organization on March 11, 2020 . SARS-CoV-2 virus provokes an infectiousdisease known as COVID-19 that has an incredibly large spectrum of symptoms or none dependingon the age, health status and the immune defenses of each individuals . SARS-CoV-2 causespotentially life-threatening form of pneumonia and/or cardiac injuries in a non-negligible patientsfraction .To date, no treatment of vaccine is available for COVID-19 . Efforts to contain the virus andto not overwhelm intensive care facilities are based on quarantine measures which have provenvery effective in several countries . Despite this, lockdown measures entail enormous econom-ical, social and psychological costs. Recent estimates of the International Monetary Fund recentlyannounced a global recession that will drag global GDP lower by 3% in 2020, although contin-uously developing and changing as well as significantly depending country-by-country . Morethan 20 million people have lost their job in United States and a large percentage of Italianshave developed psychological disturbances such as insomnia or anxiety due to the strict lockdownmeasures . Those measures have been taken on the basis of epidemics models, which are fitted2n the available data . In Italy, initial lockdown measures started on February 23rd for 11 mu-nicipalities in both Lombardia and Veneto which were identified as the two main Italian clusters.After the initial spread of the epidemics into different regions all Italian territory was placed into aquarantine on March 9th, with total lockdown measures including all commercial activities (apartsupermarkets and pharmacies), non-essential businesses and industries, and severe restrictions totransports and movements of people at regional, national, and extra-national levels . People wereasked to stay at home or near for sporting activities and dog hygiene (within 200 m from home),to reduce as much as possible their movements (only for food shopping and care reasons), andsmart-working was especially encouraged in both public and private administrations and compa-nies. At the early stages of epidemics intensive cares were almost saturated with a peak of 4000people on April 3rd and a peak of hospitalisations of 30000 on April 4th, significantly reducingafter these dates, reaching 1500 and 17000, respectively, at the beginning of phase 2 on May 4th,and 750 and 1000 on May 18th when lockdown measures on commercial activities were relaxed.These numbers, continuously declining during the next days and weeks, confirmed the benefit oflockdown measures .Alarmed by the exponential growth of new infections and the saturation of the intensive care beds,also France introduced strict lockdown measures on March 17th . The French government re-stricted travels to food shopping, care and work when teleworking was not possible, outings nearhome for individual sporting activity and/or dog hygiene, and it imposed the closure of the Schen-gen area borders as well as the postponement of the second round of municipal elections. Thenumber of patients in intensive care, like the number of hospitalisations overall peaked in earlyApril and then started to decline, showing the benefits of lockdown measures. On Monday, May11th, France began a gradual easing of COVID-19 lockdown measures . Trips of up to 100 kilo-metres from home are allowed without justification, as will gatherings of up to 10 people. Longertrips will still be allowed only for work or for compelling family reasons, as justified by a signedform. Guiding the government’s plans for easing the lockdown is the division of the country intotwo zones, green and red, based on health indicators. Paris region (Ile de France), with about 12milions inhabitants is flagged, to date, as an orange zone.In both countries, the release of lockdown measures has been authorised by authorities afterconsulting scientific committees which were monitoring the behavior of the curve of infectionsusing COVID-19 data. Those data are provided daily, following a request of the WHO. To date,the WHO guidelines require countries to report, at each day t , the total number of infected patients3 ( t ) as well as the number of deaths D ( t ) . Large uncertainties have been documented in the countof I ( t ) . Whereas in the early stage of the epidemic several countries tested asymptomatic indi-viduals to track back the infection chain, recent policies to estimate I ( t ) have changed. Most ofthe western countries have previously tested only patients displaying severe SARS-CoV-2 symp-toms . In an effort of tracking all the chain of infections, Italy and France are now testing allindividuals displaying COVID-19 symptoms and those who had strict contacts with infected indi-viduals. The importance of tracking asymptomatic patients has been proven in a recent study .The authors have estimated that an enormous part of total infections were undocumented (80% to90%) and that those undetected infections were the source for 79% of documented cases in China.Tracking strategies have proven effective in supporting actions to reduce the rate of new infections,without the need of lockdown measures, as in South Korea .The goal of this paper is to explore possible future epidemics scenarios of the long term behav-ior of the COVID-19 epidemic but taking into account the role of uncertainties in both the pa-rameters value and the infection counts to investigate different outcomes of the epidemics leadingor not to a second wave of infections. To this purpose we use a stochastic Susceptible-Exposed-Infected-Recovered (SEIR) model which consist in a set of ordinary differential equations wherecontrol parameters are time-dependent modelled via a stochastic process. This allows to mimic thedependence on control parameters on some additional/external factors as super-spreaders and theenforcing/relaxing of confinement measures . As for the classical SEIR models the populationis divided into four compartmental groups, i.e., Susceptible, Exposed, Infected, and Recoveredindividuals. The stochastic SEIR model shows that long-term extrapolation is sensitive to both theinitial conditions and the value of control parameters , with asymptotic estimates fluctuating onthe order of ten millions units in both countries, leading or not a second wave of infections. Thissensitivity arising from both poor data quality and inadequate estimations of model parametershas been also recently investigated by means of a statistical model based on a generalized logisticdistribution . The paper is organised as follows: in Section III we discuss the various sourcesof data for COVID-19 and their shortcomings, and then we discuss in detail the SEIR model andits statistical modelling. In Section IV we discuss the results focusing on the statistical sensitivityof the modelling, and apply it to data from France and Italy. We finish, in Section V, with someremarks and point out some limitiations of our study.4 II. DATA AND MODELLINGA. Data
This paper relies on data stored into the Visual Dashboard repository of the Johns Hopkins Uni-versity Center for Systems Science and Engineering (JHU CSSE) supported by ESRI Living AtlasTeam and the Johns Hopkins University Applied Physics Lab (JHU APL). Data can be freelyaccessed and downloaded at https://systems.jhu.edu/research/public-health/ncov/ ,and refers to the confirmed cases by means of a laboratory test . Nevertheless there are someinconsistencies between countries due to different protocols in testing patients (suspected symp-toms, tracing-back procedures, wide range tests) , as well as, to local management of healthinfrastructures and institutions. As an example due to the regional-level system of Italian health-care data are collected at a regional level and then reported to the National level via the ProtezioneCivile transferring them to WHO. These processes could be affected by some inconsistencies anddelays , especially during the most critical phase of the epidemic diffusion that could introduceerrors and biases into the daily data. These incongruities mostly affected the period between Febru-ary 23rd and March 10th, particularly regarding the counts of deaths due to a protocol change fromthe Italian Ministry of Health . A similar situation occurs in France where the initial testing strat-egy was based only on detecting those individuals experiencing severe COVID19 symptoms .In the post lockdown phase, France has extended its testing capacity to asymptomatic individualswho have been in contact with infected patients . B. A Stochastic epidemiological Susceptible-Exposed-Infected-Recovered model
One of the most used epidemiological models is the so-called Susceptible-Exposed-Infected-Recovered (SEIR) model belonging to the class of compartmental models . It assumes that thetotal population N can be divided into four classes of individuals that are susceptible S , exposed E , infected I , and recovered or dead R (assumed to be not susceptible to reinfection). The modelis based on the following assumptions:1. the total population does not vary in time, e.g., dN / dt = dS / dt + dE / dt + dI / dt + dR / dt = , ∀ t ≥ S → I → R ;5. exposed individuals E encountered an infected person but are not themselves infectious;4. recovered or died individuals R are forever immune.Thus, the model reads as dSdt = − λ S ( t ) I ( t ) , (1) dEdt = λ S ( t ) I ( t ) − α E ( t ) , (2) dIdt = α E ( t ) − γ I ( t ) , (3) dRdt = γ I ( t ) , (4)where γ > λ = λ / S ( ) > S ( ) , and α is the inverse of the incubation period. Its discreteversion can be simply obtained via an Euler Scheme as S ( t + ) = S ( t ) − λ S ( t ) I ( t ) , (5) E ( t + ) = ( − α ) E ( t ) + λ S ( t ) I ( t ) , (6) I ( t + ) = ( − γ ) I ( t ) + α E ( t ) , (7) R ( t + ) = R ( t ) + γ I ( t ) . (8)in which we fixed dt = γ and λ the model also allows to derived the so-called R parameter, e.g., R = λ / γ , representing theaverage reproduction number of the virus. It is related to the number of cases that can potentially(on average) caused from an infected individual during its infectious period ( τ in f = γ − ). Earlyestimates in Wuhan on January 2020 reported R = . . . which lead to γ = .
37 fixing λ (cid:39) and a 95% confidence level range for the incubation period between 2 and 11 days .However, the R parameter as well as models parameters λ , γ , and α can vary in time duringthe epidemics due to different factors as the possible presence of the so-called super-spreaders ,intrinsic changes of the SARS-CoV-2 features, lockdown measures, asymptomatic individuals whoare not tracked out, counting procedures and protocols, and so on .To deal with uncertainties in long-term extrapolations and with the time-dependency of controlparameters a stochastic approach could provide new insights in modeling epidemics , espe-cially when epidemics show a wide range of spatial and temporal variability . However,6nstead of investigating how to get a realistic behavior by stochastically perturbing control pa-rameters, here we investigate how uncertainties into the final counts C ( t ) are controlled by modelparameters . Thus, we use a stochastic version of the SEIR model in which the set of controlparameters κ ∈ { α , λ , γ } is modelled via a stochastic process κ ( t ) = | κ + σ · ξ ( t ) | , (9)with being σ the intensity of the perturbation, ξ ( t ) a random variable from a collection ofnormally distributed N ( , ) values at each time t , and since κ ( t ) ≥ than a continuous one . IV. RESULTSA. Model validation: first wave
We begin this section by validating the SEIR stochastic model on the first wave of infections.We have therefore to chose the initial conditions, and then introduce the lockdown measures in theparameters. a. France
In France, the first documented case of COVID-19 infections goes back to December 27th, 2019.Doctors at a hospital in the northern suburbs of Paris retested samples from patients between De-cember 2nd, 2019, and January 16th, 2020. Of the 14 patient samples retested, one sample, froma 42-year-old man came back positive . As initial condition for the SEIR model, we therefore set I ( t = ) = t = R = . . . whichlead to γ = .
37 fixing λ (cid:39)
1. Strict lockdown measures are introduced at t =
80 (i.e., March17th, 2020). First wave modelling results are shown in Figure 1. Figure 1a) shows the modelledvalue of R . During confinement, we let λ fluctuates by 20% of its value around 1/4. We basethis new infection rate on the mobility data for France during confinement, which have shown adrop by ∼
75% according to the INSERM report . The resulting confinement R = . . . ,a range of values compatible with that published by the Pasteur Institute . The cumulative num-ber of infections is shown in Figure 1b) and shows, on average, that 8 millions people have been7nfected by SARS-CoV-2 in France. The uncertainty range is extremely large, according to theerror propagation given by the stochastic fluctuations of the parameters (see for explanations). Itextends from few hundred thousands individuals up to 18 millions. The average is however closeto the value proposed by the authors in , who estimate a prevalence of ∼
6% of COVID-19 inthe French population. Another realistic feature of the model is the presence of an asymmetricbehavior of the right tail of daily infections distributions (Figure 1c) that has also been observedin real COVID-19 published data . b. Italy For Italy, the first suspect COVID-19 case goes back to December 22nd, 2019, a 41-year-oldwoman who could only be tested positive for SARS-CoV-2 antibodies in April 2020 . As initialcondition we therefore set I ( t = ) = t = R = . . . leading to γ = .
37 if fixing λ (cid:39)
1. A first semi-lockdown was setin Italy on March 9th, 2020 ( t =
78) and enforced on March 22nd, 2020 ( t = R on the mobility data for Italy whichshow for the first part of the confinement a reduction of about 50 % and a similar reduction toFrance (75%) for the strict lockdown phase. Figure 2 shows the results for the first wave byletting λ = . ± ∆ λ , where ∆ λ represents a 20% fluctuations around the mean value, and byfixing an initial condition on susceptible individuals S ( ) = . · corresponding to the estimateof the Italian population. A clear difference emerges with respect to the case of France in thebehavior of R which shows an intermediate reduction near t =
80, corresponding to March 11th,2020, to R = . . . before reaching the final value of R = . . . . This sort of "step" into the R time behavior corresponds to the time interval between semi- and full-lockdown measures,whose efficiency significantly increases after March 24th, 2020, also corresponding to the peakvalue of infections. This is confirmed by looking at daily infections distributions (Figure 2c) thatshows a peak value near March 24th, 2020, also observed in real COVID-19 data . Finally, thecumulative number of infections (Figure 2b) shows that, on average, almost 10 millions peoplehave been infected by SARS-CoV-2 in Italy, ranging between few hundred thousands up to 25millions due to the the error propagation by the stochastic fluctuations of model parameters (see for explanations). Nevertheless the wide range of uncertainty the average value is close to thevalue estimated from a team of experts of the Imperial College London according to which the9.6% of Italian population has been infected, with a 95% confidence level ranging between 3.2%and 26% . These estimates correspond to cumulative infections of ∼ ∼ ∼
16 millions, well in agreement with our model and other statistical estimates . B. Future epidemics scenarios
After lockdown measures are released, for both countries, we model three different scenarios: afirst one where all restrictions are lifted (back to normality), a second one where strict measures aretaken (semi-lockdown) and a third one where the population remains mostly confined (lockdown). a. France
Results for France are shown in Figure 3. Lockdown is released at t = R (cid:39)
1, that can be achievedby imposing strict distancing measures, contact tracking as well as reduction in mobility. It resultsin a linear modest increase of the total number of infections that does not produce a proper wave ofinfections. As in the first wave modelling, large uncertainties are also present in future scenariosalthough the three distinct behaviors clearly appear. b. Italy
Figure 4 shows the results for modeling future epidemic scenarios for Italy. The first relaxationof lockdown measures started at t = t = t = R re-approachingthe initial value ( R = . ) . The semi-lockdown (green) scenario produces a second wave mostlysimilar, in terms of intensity, as the first wave, but occurring at t = R (cid:39)
1, resulting from strict distancingmeasures and reduced mobility, and does not produce a proper wave of infections. However, allscenarios are clearly characterized by a wide range of uncertainties, although producing three welldistinct behaviors in both cumulative and daily infections.
C. Phase Diagrams
In the previous section we have seen that increasing R above 1 can or not produce a secondwave of infections and introduce also a time delay in the appearance of a second wave of infec-tions. We now analyse this effect in a complete phase diagram fashion. Figures 5-6 show thephase diagrams for France and for Italy, respectively. The diagrams are built in terms of ensembleaverages of number of infections per day I ( t ) versus the average value of R after the confine-ment (panels a), and the errors (represented as standard deviation of the average I ( t ) over the 30realisations) are shown in panels b. First we note that despite some small differences in the de-lay of the COVID-19 second wave of infections peak, the diagrams are very similar. In order toavoid a second wave, R could fluctuate on values even slightly larger than one. Furthermore, for1 . < R <
2, the second wave is delayed in Autumn or Winter 2020/2021 months. The uncer-tainty follows the same behavior as the average and it peaks when the number of daily infectionsis maximum. This means that the ability to control the outcome of the epidemics is highly reducedif R is too high. V. DISCUSSION
France and Italy have faced a long phase of lockdown with severe restrictions in mobility andsocial contacts. They have managed to reduce the number of daily COVID-19 infections drasti-cally and released almost simultaneously lockdown measures. This paper addresses the possiblefuture scenarios of COVID-19 infections in those countries by using one of the simplest possiblemodel capable to reproduce the first wave of infections and to take into account uncertainties,namely a stochastic SEIR model with fluctuating parameters.We have first verified that the model is capable to reproduce the behavior of the first wave of10nfections and provide an estimate of COVID-19 prevalence that is coherent with clinical testsand other studies. The introduction of stochasticity accounts for the large uncertainties in boththe initial conditions as well as the fluctuations in the basic reproduction number R originatingfrom changes in virus characteristics, mobility or misapplication in confinement measures. 30realisations of the model have been produced and they show very different COVID-19 prevalenceafter the first wave. The range goes from thousands of infected to tens of millions of infections inboth countries. Average values are compatible with those found in other studies .Then, we have modelled future epidemics scenarios by choosing specific fluctuating behaviorsfor R and performing again, 30 realisations of the stochastic SEIR model. Despite the very largeuncertainties, distinct scenarios clearly appear from the noise. In particular, they suggest that asecond wave can be avoided even with R values slightly larger than one. This means that actualdistancing measures which include the use of surgical masks, the reduction in mobility and theactive contact tracking can be effective in avoiding a second peak of infections without the needof imposing further strict lockdown measures. The analysis of phase diagrams show that there isa sharp transition between observing or not a second wave of infections when the value of R isclose to 1.5. Moreover, the models show that the higher R , the lower the ability to control thenumber of infections in the epidemics.This model has also evident deficiencies in representing the COVID-19 infections. First of all,the choice of the initial conditions is conditioned by our ignorance on the diffusion of the virus inFrance and Italy in December 2019. Furthermore, we are unable to verify on an extensive datasetthe outcome of the first wave: on one side antibodies blood tests have still a lower reliability and on the other they have not been applied on an extensive number of individuals to get reliableestimates. On top of the data-driven limitations, we have those introduced by the use of compart-ment models, as there are geographic, social and age differences in the spread of the COVID-19disease in both countries . Furthermore, we also assume that fluctuations on the parameters ofthe SEIR model are Gaussian, although there are good reasons to think that they should be heavytailed distributions . We would like to remark however that, to overcome these limitations,one would need to fit more complex models and introduce additional parameters which can, at thepresent stage, barely inferred by the data.Our choice to stick the stochastic SEIR model is indeed driven by its simplicity and the possi-11ility of modeling realistic the uncertainties with the stochastic fluctuations instead of adding newparameters whose inference may affect the results. This study can be applied to other countries,and this is why we publish along the code of our analysis alongside with the paper. To date, North-ern Europe, UK, US and other American countries are still facing the first wave of infections, sothat future scenarios cannot be devised with the same clarity as those outlined in this study forFrance and Italy. VI. ACKNOWLEDGMENTS
DF acknowledges All the London Mathematical Laboratory fellows, B Dubrulle, F Pons, NBartolo, F Daviaud, P Yiou, M Kagayema, S Fromang and G Ramstein for useful discussions.
VII. DATA AVAILABILITY
The data that support the findings of this study are openly available in https://systems.jhu.edu/research/public-health/ncov/ , maintained by Johns Hopkins University Centerfor Systems Science.
VIII. APPENDIX A: NUMERICAL CODE % This appendix contains the MATLAB code used to perform% the analysis contained in the paper via a stochasitc% SEIR model%% PARAMETER DEFINITIONS%tmax: number of day of integrationstmax=500;%nrel: number of realisations of the modelnrel=30;%tconf: lockdown daytconf=50%tconf2: lockdown releasetconf2=100 % LOOP ON DIFFERENT VALUES OF LAMBDA, INFECTION RATEfor la=1:50lambdaconf=0.25;lambdares=la.*0.02;%% LOOP ON REALIZATIONSfor rel=1:nrelS=zeros(1,tmax);E=zeros(1,tmax);I=zeros(1,tmax);R=zeros(1,tmax);C=zeros(1,tmax);lambda=zeros(1,tmax);%S Susceptible individuals (France population)S(1)=67000000;%I Infected individualsI(1)=585;% RecoveredR(1)=0;% Inital timeT(1)=0;% Cumulative infectionsC(1)=0;% alpha is the inverse of the incubation period (1/t_incubation)alpha0=0.27;% R0 is equal to 2.68R0=2.68;% gamma is the inverse of the mean infectious periodgamma0=lambda0./R0;% uncertainty in gamma and lambdacoeff_gamma=0.5; oeff_lambda=0.005;%% LOOP ON TIME, INTEGRATION OF SEIR MODELSfor t=1:1:tmax%gamma=1/Tr where Tr is the recovery time (2 weeks)%Stochastic gammagamma=gamma0+gamma0./5*randn;%Change lambda for confinementif t==tconflambda0=lambdaconf;endif t==tconf2lambda0=lambdares;end%Stochastic lambdalambda(t+1)=(lambda0+lambda0./5*randn)./S(1);%Stochastic alphaalpha=alpha0+alpha0./5*randn;%Computation of R0R0(t+1)=lambda(t+1)./gamma0;%Iteration of the modelT(t+1)=t;S(t+1)=S(t)-(lambda(t+1)*S(t)*I(t));E(t+1)=E(t)+(lambda(t+1)*S(t)*I(t))-alpha*E(t);I(t+1)=I(t) +alpha*E(t) -gamma*I(t);R(t+1)=R(t)+(gamma*I(t));%cumulative infectedC(t+1)=gamma0.*sum(I);%Variables for different realisationsIrel(rel,t+1)=I(t+1);lambdarel(rel,t+1)=lambda(t+1);end nd%% AVERAGING OVER DIFFERENT REALIZATIONSlambdamoy(la,:)=mean(lambdarel,1);Imoy(la,:)=mean(Irel,1);Istd(la,:)=std(Irel,1);lambdavec(la)=lambdares;R0moy(la,:)=lambdamoy(la,:)./gamma0.*S(1);end REFERENCES E. R. Gaunt, A. Hardie, E. C. Claas, P. Simmonds, and K. E. Templeton, “Epidemiology andclinical presentations of the four human coronaviruses 229e, hku1, nl63, and oc43 detectedover 3 years using a novel multiplex real-time pcr method,” Journal of clinical microbiology ,2940–2947 (2010). J. Wu, W. Cai, D. Watkins, and J. Glanz, “How the virus got out,” The New York Times (2020). W. H. Organization et al. , “Coronavirus disease 2019 (covid-19): situation report, 51,” (2020). C. COVID and R. Team, “Severe outcomes among patients with coronavirus disease 2019(covid-19)—united states, february 12–march 16, 2020,” MMWR Morb Mortal Wkly Rep ,343–346 (2020). Y.-Y. Zheng, Y.-T. Ma, J.-Y. Zhang, and X. Xie, “Covid-19 and the cardiovascular system,”Nature Reviews Cardiology , 259–260 (2020). C. Huang, Y. Wang, X. Li, L. Ren, J. Zhao, Y. Hu, L. Zhang, G. Fan, J. Xu, X. Gu, et al. ,“Clinical features of patients infected with 2019 novel coronavirus in wuhan, china,” The Lancet , 497–506 (2020). M. Cascella, M. Rajnik, A. Cuomo, S. C. Dulebohn, and R. Di Napoli, “Features, evaluationand treatment coronavirus (covid-19),” in
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IG. 1. Susceptible-Exposed-Infected-Recovered (SEIR) model of COVID-19 for France (Eqs 5-8) with λ = ./ S ( ) , α = . γ = .
37. Initial conditions are set to I ( ) = S ( ) = . · , E ( ) = R ( ) = t = t =
78 (Mar 17, 2020). a) Time evolutionfor the basic reproduction number R , b) Time evolution for the cumulative number of infections C ( t ) , c)Time evolution for the daily infected individuals I ( t ) . Solid line shows the average for 30 realisation of theSEIR stochatic models, shading extends to 3 standard deviations of the mean. IG. 2. Susceptible-Exposed-Infected-Recovered (SEIR) model of COVID-19 for Italy (Eqs 5-8) with λ = ./ S ( ) , α = . γ = .
37. Initial conditions are set to I ( ) = S ( ) = . · , E ( ) = R ( ) = t = t =
78 (Mar 9, 2020)and enforced at t =
89 (Mar 22, 2020). a) Time evolution for the basic reproduction number R , b) Timeevolution for the cumulative number of infections C ( t ) , c) Time evolution for the daily infected individuals I ( t ) . Solid line shows the average for 30 realisation of the SEIR stochatic models, shading extends to 3standard deviations of the mean. IG. 3. Susceptible-Exposed-Infected-Recovered (SEIR) model of COVID-19 for the second wave inFrance. Initial conditions are set as in Figure 1. After the confinement is released ( t = R , b) Time evolution for the cumulative number of infections C ( t ) , c) Time evolution for the daily infected individuals I ( t ) . Solid line shows the average for 30 realisationof the SEIR stochatic models, shading extends to 3 standard deviations of the mean. IG. 4. Susceptible-Exposed-Infected-Recovered (SEIR) model of COVID-19 for the second wave in Italy.Initial conditions are set as in Figure 2. After the confinement is released ( t = t = R , b) Time evolution for the cumulativenumber of infections C ( t ) , c) Time evolution for the daily infected individuals I ( t ) . Solid line shows theaverage for 30 realisations of the SEIR stochatic models, shading extends to 3 standard deviations of themean. IG. 5. Phase diagram for the Susceptible-Exposed-Infected-Recovered (SEIR) model of COVID-19 for thesecond wave in France. Initial conditions are set as in Figure 1. After the confinement is released ( t = R modelled. a) Average of daily infected individuals I ( t ) . b) Standard deviationof daily infected individuals. Diagrams are obtained using 30 realisations of the SEIR models. IG. 6. Phase diagram for the Susceptible-Exposed-Infected-Recovered (SEIR) model of COVID-19 for thesecond wave in Italy. Initial conditions are set as in Figure 2. After the confinement is released ( t = t =
146 May 18, 2020) all possible R modelled. a) Average of daily infected individuals I ( t ) . b) Standard deviation of daily infected individuals. Diagrams are obtained using 30 realisations of theSEIR models.. b) Standard deviation of daily infected individuals. Diagrams are obtained using 30 realisations of theSEIR models.