Covid-19 Belgium: Extended SEIR-QD model with nursing homes and long-term scenarios-based forecasts
CCovid-19 Belgium: Extended SEIR-QD model with nursing homes andlong-term scenarios-based forecasts
Nicolas Franco
Namur Institute for Complex Systems (naXys) and Department of Mathematics, University of Namur, Namur, BelgiumInteruniversity Institute of Biostatistics and statistical Bioinformatics (I-BioStat) and Data Science Institute, University ofHasselt, Hasselt, Belgium
November 4, 2020
Abstract
We model the evolution of the covid-19 epidemic in Belgium with an age-structured extended SEIR-QDepidemic model with separated consideration for nursing homes. All parameters of the model are estimatedusing a MCMC method, except integrated data on social contacts. The model is calibrated on hospitals’data, number of deaths, nursing homes’ tests and serological tests. We present the current situation inNovember 2020 as well as long-term scenarios-based forecasts concerning the second wave and subsequentlifting of measures.
Keywords:
SARS-CoV-2, age-structured compartmental SEIR model, hospitalisation and mortality data,social contact patterns, Markov Chain Monte Carlo (MCMC)
1. Introduction
While there are many circulating different models concerning the covid-19 pandemic, it is important tohave dedicated models to the specific situation of each country since the evolution of the situation as wellas chosen political measures are different. SEIR-type epidemic models [1] are the most suitable for longterm forecasting and especially SEIR-QD variants concerning the covid-19 pandemic [2, 3]. We present oneof the very few existing extended SEIR-QD model adapted and calibrated on Belgium situation and data.Two similar approaches have been developed by the SIMID COVID-19 team (UHasselt-UAntwerp) [4] andthe BIOMATH team (UGent) [5]. All of those models have their own characteristics and are complemen-tary since it is difficult at this time to exactly know how to model the covid-19 in the best way. Anotheralternative approach has also been developed at the VUB [6] as well as a meta-population model from theSIMID COVID-19 team [7].Here we summarise the main characteristics of our model. The Belgian population is divided into 8compartments in order to take account of the different possible stages of the disease as well as the separa-tion between asymptomatic and symptomatic people with a different infectiousness. Each compartment isdivided into 5 age classes with different characteristics concerning the behaviour and evolution of the disease.The transmission of the coronavirus between all classes is computed using social contact data at differentplaces (home, work, school, leisure) [8]. Except social contact data, all of the 65 parameters of the model areestimated using a Monte Carlo method, hence there is no assumption coming from studies in other countries.Nursing homes are modelled as isolated entities in order to take account of the different spread timing ofthe coronavirus compared to the general population. Specific parameters for the situation in nursing homes
Email address: [email protected] (Nicolas Franco) a r X i v : . [ q - b i o . P E ] N ov ake account of a variable hospitalisation policy based on hospitals load as well as a probability that deathscoming directly from nursing homes are related to the covid-19. There is a specific estimation of potentialreimportations coming from travellers during the holiday period. The model is mainly calibrated usinghospitalisations and deaths using both incidence and prevalence data (depending on which one is the moreappropriate for the considered data) coming from Sciensano’s public raw data [9]. Additional constraintson seroprevalence are coming from Sciensano’s serological studies on blood donors as reported in Sciensanoepidemiological reports [9]. The only positive PCR tests which are taken into consideration are those comingfrom nursing homes from an overall test campaign in April-May.In Section 2, we present a description of the model. Technical details are presented in Subsection 2.2for the equations, Subsection 2.3 for the data and Subsection 2.4 for explanation of the calibration method.Precise details on the timeline used and the full set of estimated parameters of the model are given inAppendix A. The results and discussion Section 3 starts with a presentation of the current estimation fromthe model in Subsection 3.1. Then we present a test on the validity of the model in Subsection 3.2, a middle-term forecast based on estimation of new policy mesures in Subsection 3.3 and finally some scenarios-basedlong-term forecasts in Subsection 3.4.
2. Materials and Methods
We consider the following age classes among the population: 0-24, 25-44, 45-64, 65-74 and 75+. Thoseclasses correspond to public available data, excepted that 75-84 and 85+ classes are merged since it is diffi-cult to know which parts of them are present in nursing homes or homes for the elderly. Hence we assumethat the classes 0-24, 25-44, 45-64, 65-74 are only present among the general population, while the remainingis divided between a general 75+ and a specific class of nursing homes residents.The compartmental model is divided into the following compartments: Susceptible S (people who havenever been infected and are a priori susceptible to be infected), Exposed E (people who have just beeninfected but are without any symptom and still not infectious, hence in latent period), Asymptomatic In-fectious AI (this is the part of the exposed people who fall into a continuously asymptomatic disease, whichare infectious but with a reduced infectious probability due to their asymptomatic status and directly fallinto the recovered status after that period), Presymptomatic Infectious P I (this is the other part of theexposed people who fall into a symptomatic disease, but symptoms do not appear directly, hence there isan intermediate stage where people become infectious but still without any symptom and with an infectiousprobability still reduced), Symptomatic Infectious SI (real disease period where the infectious probability ishigher – people in this compartment will eventually fall either in a recovered status or will be hospitalised,and concerning nursing home, a significant part of them will directly die), Hospitalised Q (hospitalised peo-ple are considered as in quarantine for the model, since their contacts are almost inexistant), Deceased D (deaths from the general population are assumed only coming from hospitalised people – there is a small of exceptions which is not taken into consideration here – however, deaths from nursing homes are takeninto consideration and separated from deaths coming from hospitals), Recovered R (people who recoveredfrom the disease, from asymptomatic ones, symptomatic ones or from the hospital, and are assumed hereimmune for the future). All those compartments exist for every age class. We do not consider in this modelany subdivision inside the hospital compartment.In addition, 2000 isolated nursing homes of similar average size are considered with all those compart-ments. The transmission of infection from the general population to those nursing homes is modelled by arandom infection probability inside each nursing home, which is proportional to the number of infectiouspeople and assumed less important since the lockdown period.A schematic view of the compartments with their relations is presented in Figure 1.2eneral population (age classes i = S i Susceptible E i Exposed AI i Asymptomatic Infectious
P I i PresymptomaticInfectious SI i SymptomaticInfectious Q i Quarantined=Hospitalised D i Deceased R i Recovered (cid:80) j M ij ( λ a ( AI j + P I j ) + λ s SI j ) σ.p ai σ. (1 − p ai ) τ δ i γ ai γ si γ qi ( t ) r i ( t ) nursing homes (2000 separated copies): S h Susceptible E h Exposed AI h Asymptomatic Infectious
P I h PresymptomaticInfectious SI h SymptomaticInfectious Q h Quarantined=Hospitalised D Deceasedfrom hospitals D h Deceasedfrom homes R h Recovered m h ( λ a ( AI h + P I h ) + λ s SI h ) + Random transmissions from visits σ.p ah σ. (1 − p ah ) τ δ h ( t ) γ ah γ sh γ qh ( t ) r h ( t ) P cor ˜ r h ( t )(1 − P cor )˜ r h ( t ) (non covid-19 deaths) (new entrances from S ) Figure 1: Schematic view of the compartmental model duringthe period up to March 14, 2020. Then reduced percentages are estimated by the model for the differentperiods of lockdown and phases of lift of measures. These reduced percentages are the effect at the sametime of mobility restrictions, social distancing, prevention mesures, testing and contact tracing, while it ismathematically impossible to determine the exact part of those effects. Hence new parameters for some orall social contact types are estimated each time there is an important policy change. The timeline and theconsidered coefficients are described in Appendix A with the full list of estimated parameters. Long-termscenarios-based forecasts are constructed assuming a constant policy and compliance to measures during thefuture with different realistic possibilities of percentage of social contacts for still unknown policy effects.This model takes into consideration potential reimportations of covid-19 from abroad during the hol-idays period. No reimportation is assumed in June since borders where barely opened. Reimportationsare estimated during the period July to September using the following method: According to 2019 traveltrends [10], we consider a proportionality of travellers of 36% in July, 26% in August and 21% in September.There is no data available concerning the inhomogeneous repartition inside each month, but we assume ahomogeneous one for July and August while a 2 to 1 ratio between the first half of September and the secondhalf. Only the top five countries of destination are considered with the following proportion: France 23%,Spain 11%, Italy 9% and The Netherlands 7% (Belgium is discarded). Then for each of those countries weconsider the daily ECDC statistics on cumulative numbers for 14 days of covid-19 cases per 100000 [11]. Thereimportation are added using an estimated global coefficient and injected proportionally in the exposedcompartiment of the classes 0-24, 25-44, 45-64 and 65-74 and removed from the corresponding susceptiblecompartments. The estimated reimportations per day are presented in Appendix A.This model does not take into consideration not officially observed effects like seasonality or cross-immunity. The population is only age-structured and not spatially structured. A spatial refinement of sucha model would be really important, but currently the public data officially provided are not of sufficientdetail in order to correctly fit a complex spatial-structured model.4 .2. Equations of the model
Equations of the model for the general population are the following ones, with i = d S i d t = − S i (cid:88) j M ij λ a ( AI j + P I j ) + λ s SI j N j d E i d t = S i (cid:88) j M ij λ a ( AI j + P I j ) + λ s SI j N j − σE i d AI i d t = σp ai E i − γ ai AI i d P I i d t = σ (1 − p ai ) E i − τ P I i d SI i d t = τ P I i − δ i SI i − γ si SI i d Q i d t = δ i SI i − r i ( t ) Q i − γ qi ( t ) Q i d D i d t = r i ( t ) Q i d R i d t = γ ai AI i + γ si SI i + γ qi ( t ) Q i Parameters are explained in Table 1. Parameters without age class index i are assumed similar for allclasses while those with age class index i are class-dependent. Some specific parameters are time-dependent. Parameter class-dep. time-dep. description λ a no no transmission rate from asymptomatic infectious persons λ s no no transmission rate from symptomatic infectious persons σ no no rate at which an exposed person becomes contagious τ no no rate at which an presymptomatic person becomes symptomatic M ij yes no social contact matrices (WAIFW) p ai yes no probability of a completely asymptomatic disease δ i yes no rate at which a symptomatic person develops heavy symptoms and is hospitalised γ ai yes no rate at which a person recovers from asymptomatic disease γ si yes no rate at which a person recovers from light symptomatic disease γ qi ( t ) yes yes rate at which a person recovers from hospital r i ( t ) yes yes rate at which a person dies from hospitalTable 1: Main parameters of the model The transmission is governed by the so-called social contact hypothesis [12]. Social contact matrices M ij (WAIFW) are collected using the SOCRATES online tool [8] and the 2010 dataset [13]. Work and transportcategories are merged as well as leisure and otherplace. 4 different parameters which are adapted dependingon lockdown/policy mesures are used as coefficients, hence the complete contact matrices are: M ij = C home M ij home + C work M ij work + C school M ij school + C leisure M ij leisure During the holiday period (July-September), several elements are removed each day from the S i classesand added to the E i class according to estimated travellers and estimated average infection in the visitedcountries with a global coefficient C reimp (as explained in Section 2.1).The time-dependence of some parameters is computed using a logistic function with estimated parameters P recovery , µ recovery and s recovery : γ qi ( t ) = γ qi (cid:32) P recovery e − t − µ recovery s recovery (cid:33) r i ( t ) = r i (cid:32) − P recovery e − t − µ recovery s recovery (cid:33) N − = 3250000 , N − = 3000000 , N − = 3080000 , N − = 1150000 and N = 870000 outside nursing homes (with an additional N h = 150000 inside nursing homes) for atotal population of N = 11500000 . Those numbers are round numbers coming from the structure of theBelgian population as provided by the Belgian Federal Government on April 2020 [14]. An initial condition p is proportionally distributed between the E i on day 1 among the general population (corresponding toMarch 1 reported situation = February 29 real situation). Nursing homes are assumed not initially infected.Equations of the model for the specific population in nursing homes follow a variation: d S h d t = − S h m h λ a ( AI h + P I h ) + λ s SI h − ˜ r h ( t )(1 − P cor ) I h − Random transmissions from visits + New entrances d E h d t = − d S h d t − σE h d AI h d t = σp ah E h − γ ah AI h d P I h d t = σ (1 − p ah ) E h − τ P I h d SI h d t = τ P I h − δ h SI h − γ sh SI h − ˜ r h ( t ) P cor SI h d Q h d t = δ h SI h − r h ( t ) Q h − γ qh ( t ) Q h d D d t + = r h ( t ) Q h d D h d t = ˜ r h ( t ) Q h d R h d t = γ ah AI h + γ sh SI h + γ qh ( t ) Q h Most of the parameters are similar to the general population (but assumed with different values) withsome additional considerations. There are 2000 nursing homes considered as separated entities, with a con-stant population of 75 inside each one, for a total of N h = 150000 residents. New entrances are consideredin order to fit the empty places up to 75 residents per nursing home and are removed from the 75+ suscep-tible class. Transmissions inside a specific nursing home follow usual SEIR-type transmission with a specificcoefficient m h . Transmissions from the general population is computed in a particular way using a dailyprobability of infection, i.e. each day one additional (integer) infected resident is added with probability P th ¯ S h (cid:80) j λ a ( AI j + P I j )+ λ s SI j N , where the coefficient is distinguished between the initial phase P th and lock-down and subsequent phases P (cid:48) th . ¯ S h means the number of susceptibles within the specific nursing home.Starting from lockdown, transmissions are only considered from the 25-65 population (i.e. with j = 25 − and − ) since transmissions are mainly from nursing homes’ workers. Potential reverse transmissionsare however not monitored here. Deaths from care centres through hospitalisations are counted within the75+ class. Additional deaths from care homes are monitored using a death rate ˜ r h with a coefficient P cor which captures the probability that the death is covid-19 related. Remaining non-covid-19 related deathsare assumed occurring in the susceptible class (or in the recovered class if the first one is empty). For thefirst wave only (March 1 to June 30) a variable hospitalisation policy is computed in order to correspond tothe reality and using variable parameters of constant sum δ h ( t ) + P cor ˜ r h ( t ) = δ h , the proportion being mon-itored over time by a logistic function depending on hospitals load with an additional delay, with estimatedparameters P delay , µ hosp and s hosp : δ h ( t ) = δ h − ˜ r h P cor e − Q ( t − delay ) − µ hosp s hosp ˜ r h ( t ) = ˜ r h e − Q ( t − delay ) − µ hosp s hosp This variable hospitalisation policy is nonexistent for the second wave since most of nursing home residentsare hospitalised during this period. Hence from July 1, those parameters are considered with the value Q = 0 .6he basic reproduction number is estimated by the leading eigenvalue of the next-generation matrix[15, 16] (the eigenvalue is real since this matrix is positive definite): R = maxeigenv (cid:20) λ a (cid:18) p aj γ aj + 1 − p aj τ (cid:19) M ij + λ s (cid:18) − p aj γ sj + δ j (cid:19) M ij (cid:21) ij The effective reproduction number R t = R e is estimated by R t = R (cid:80) i S i ( t ) N − (cid:80) i D i ( t ) . We consider the following data for the calibration of the model coming from Sciensano’s public raw data(October 31, 2020 release):• New hospitalisations (incidence) with an additional corrective estimated parameter SUPP hosp whichestimates the percentage of missing covid-19 patients at the time of admission (hence catching sup-plementary patients not initially hospitalised for covid-19 and transferred from other units)• Hospital load (prevalence)• Released from hospital (cumulative)• Total deaths from hospital (cumulative)• Deaths (incidence) from age classes 45-64, 65-74, 75+ (incidence data are more suitable for thoseclasses since there are a percentage of deaths for which the age class is unknown and prevalence datacould contain an accumulation of errors)• Deaths (cumulative) from all age classes (cumulative data are more suitable for the 0-24, 25-44 classessince incidence data are almost zero)• Deaths from nursing homes (incidence and cumulative)• Total deaths (incidence and cumulative)Deaths reported with a specific date are considered on that specific date while situations reported byhospitals are considered to occur up to 24h before the hospital report hence 2 days before the official datacommunication. Note that graphics are plotted using the dates of Sciensano’s communications (1 day delay).Additional constraints are considered coming from Sciensano’s epidemiological reports [9] (those con-straints determine the set of admissibles parameters). Serological studies on blood donors are considered inthe following way: the ratio between immune people (for the classes 25-44, 45-64 and 65-74) coming directlyfrom the asymptomatic compartment ( (cid:80) i AI → R ) and the total asymptomatic population who has notdeveloped a symptomatic covid-19 disease ( (cid:80) i S + E + AI + [ IA → R ] + P I ) should be respectively between . and . . and . p a − > p a − > . . . in order to reproduce the more sever-ity of the covid-19 on older persons as well as trivial constraints to avoid negative or out-of-bound parameters.Additional constraints are imposed on nursing homes coming from the result of massive PCR test onApril-May: the average percentage of infected people should be ± during the period April 15-30and less than ± during the period May 15-31. Those percentages are estimated from Sciensano’sepidemiological reports using a calculated incidence between each week. Additionally, the average percentageof asymptomatic residents (including presymptomatic ones) among infected should be ± .7 .4. Calibration method All parameters are estimated using a MCMC Metropolis–Hastings algorithm [17]. Two different modesare used for the calibration and the statistical analysis:• Best-fit burning mode: an optimised First-choice hill climbing algorithm using weighted least-squaresperformed on one parameter at a time (i.e. one neighbour = variation of one parameter), with downhillmoves allowed up to 0.2% in order to avoid local optima and with a quick best fit search performedon accepted neighbours• MCMC mode: a Metropolis–Hastings algorithm performed on all parameters (i.e. one neighbour =variation of all parameters) where the likelihood fonction is constructed using the property that theempirical variance from n data follows up to a coefficient a χ distribution (we use nS σ ∼ χ ( n ) ∼ = N ( n, n ) with σ estimated from the burning period)Weights for least-squares are defined in the following way: For each set of data (and each age class), theweight is chosen such that a best fit search considering only those data (hence a best fit discarding all otherdata) gives similar empirical variance. Hence each sort of data gives a similar contribution to the likelihood.For data with both incidence and prevalence numbers considered, the contribution of incidence numbersis favoured at ∼ . . . ).The program is written is C language. The full ODEs are solved by numerical integration using theGNU gsl odeiv2 librairy and a Runge-Kutta-Fehlberg45 integrator. The computation is performed on theHPC cluster Hercules2 .The method of approaching global minima is particular due to the presence of a very high number ofestimated parameters, a large number of numerical integrations to be performed (due to the presence of2000 separated nursing homes) and the fact that the algorithm should not take several days to complete inorder to quickly produce previsions of the epidemic. Hence the method is separated into different steps. Ina first time, a set of best-fit search is performed from a very large distribution (100 times usual standarddeviation) using 5000000 iterations with a 10 times average step and a special trick to increase the rapidityof the algorithm: instead of 2000 different nursing homes, only 100 nursing homes are considered with eachtime 20 copies of each. This approximation is suitable as long as the algorithm is still far from the best-fit. In a second time, the best-fit search is pursued for 20000 iterations using the complete 2000 differentnursing homes in order to affine parameters. A set of at least 250 different priors is collected from thebest obtained results. In a third time, the MCMC algorithm is performed from the 250 priors, with 50000iterations retaining every 5000 iterations. This produces a set of 2500 samples coming from potentiallydifferent local minima zones which avoids a too high autocorrelation of the results. Since the first step isvery time-consuming, further runs of the model are performed without a complete recalibration, by reusingthe previous 2500 samples as a set from which 250 random priors are taken and running a 20000 burningbest-fit period before the MCMC step. This is a kind of data assimilation process. Recalibration shouldhowever be performed each time there is a major change in the model or in the policy. "Plateforme Technologique de calcul Intensif" (PTCI) located at the University of Namur, Belgium. . Results and Discussion We present in this Section the result of the calibration of the model as on November 1, 2020, withconsidered data up to October 31, 2020. Results are presented in the figures with medians, and percentiles, hence with a confidence interval. Figure 2: General view on prevalence data and estimations
In Figure 2, we have a general representation of the evolution of the epidemic in Belgium with hospital-isations, people discharged from hospitals and deaths coming from hospitals and from nursing homes, all inprevalence or cumulative numbers. The interest in modelling the epidemic within nursing homes separatelyfrom the general population can clearly be seen on this figure. Indeed, the form of the death curve fornursing homes is really different from the ones for the general population since the epidemic started a fewtime later in nursing homes but took a bigger proportion. Note that the very small percentage of deathsoccurring at another places is not taken into consideration in this model.The model calibration fits the real prevalence data with a very good exactitude (excluding of course datanoises) despite that fact that the calibration is mostly done on incidence data. The comparison between themodel and some incidence data are presented in Figure 3 for the general incidence data in hospitalisationsand deaths and in Figure 4 for incidence data in deaths with age class repartition. The consideration of acontinuous improvement of care was needed in order to correctly fit the death curves among the differentperiods. However, the figure 2 presents a slightly larger increase than expected for the deaths within theclass 75+. This could be the result of a small decrease in the quality of care during the second wave due tothe huge load of the hospitals (but still better than during the first wave).9 igure 3: Incidence in new hospitalisations (with underreporting correction) and deathsFigure 4: Incidence deaths within each age class
We must remark that in order to obtain a correct fit we needed to introduce specific elements withinthe model. Incidence data on new hospitalisations are underroported. Indeed, somme patients are initiallyadmitted for another reason than covid-19 and only transferred in a covid-19 section afterwards. This resultsin more people going out from hospitals than officially entering. The model estimates those supplementarypatients at . [ . ; . ]. Real data in Figure 3 are plotted with the estimated correction.10eaths coming directly from nursing homes are not all due to covid-19 since many PCR tests are lacking.The model estimates that only . [ . ; . ] of those deaths are really due to covid-19. The ratiobetween deaths coming directly from nursing homes and deceased patients in hospitals coming initially fromnursing homes seems to be not constant, and it was necessary to introduce a variable hospitalisation pol-icy. The best answer found was to monitor hospitalisations from nursing homes through a logistic functiondepending on general hospital load but with a specific delay. Hence, when hospital load starts to becometoo important, less people from nursing homes are hospitalised and the reverse effect occurs when hospitalload gets lower, but each time with a delay estimated at 10.5 days [9.0 ; 12.4]. However, this variable hos-pitalisation policy turned out to be unsuitable for the second wave period since there a is better attentionto residents in nursing homes.Initially the model overestimated the number of deaths from the end of the first wave. It was not possibleto calibrate constant death rates throughout all phases of the epidemic. This is the consequence of botha care improvement in hospitals and a lower aggressiveness of the virus. Hence death and recovered rateswithin each age class are also monitored by a logistic function depending on time. The current improvement(in comparison to the very beginning of the epidemic) is estimated as . [ . ; . ], hence . of patients which should have died in March are now recovering from hospitals. We must remark that it isimpossible to know which part is due to care improvement and which part is due to lower aggressiveness ofthe virus and that the death rate seems to restart becoming a bit higher in October.The basic reproduction number R , representing the average number of cases directly generated by oneinfectious case in a population which is assumed totally susceptible, is estimated in average for each period(we consider this number dependent on lockdown measures) and computed as the leading eigenvalue of thenext-generation matrix (cf. Section 2.2 for details). The effective reproduction number R t (or R e ) representsthe average number of cases directly generated by one infectious case taking account of the already immunepopulation, hence varying over a period. The estimations are presented in Table 2. R R t (at the end of the period)Pre-lockdown: March 1 → March 13 4.13 [3.89 ; 4.39] 4.08 [3.85 ; 4.34]School and leisure closed: March 14 → March 18 2.24 [2.13 ; 2.35] 2.17 [2.07 ; 2.27]Full lockdown: March 19 → May 3 0.65 [0.61 ; 0.72] 0.61 [0.57 ; 0.66]Phase 1-2: May 4 → June 7 0.79 [0.75 ; 0.83] 0.73 [0.70 ; 0.76]Phase 3: June 8 → June 30 0.99 [0.91 ; 1.07] 0.91 [0.84 ; 0.99]Phase 4: July 1 → June 28 1.40 [1.29 ; 1.53] 1.29 [1.19 ; 1.40]Phase 4bis: July 29 → August 31 0.75 [0.63 ; 0.88] 0.69 [0.58 ; 0.81]Second wave: September 1 → October 31 1.73 [1.62 ; 1.85] 1.32 [1.25 ; 1.40]Nursing home (average): March 1 → October 31 0.73 [0.06 ; 1.67] 0.41 [0.04 ; 0.99]Table 2: Estimations of R and R t values The reproduction number of the pre-lockdown period is a bit overestimated. This is probably due tothe fact that the model does not take explicitly account of infections coming from abroad travellers at thisparticular time and this results in an estimated R slightly above 4. For the period phase 1A-1B-2, sincethere were policy changes almost every weeks, we only provide here the estimated R at the end of thisperiod. The second wave R does not take account of the new measure applied in October 19 whose effetsshould only be visible on November. We can see that the estimated R inside nursing homes is very low.This seems logical since nursing homes with only one infected resident do not necessary end with a majorityof infections. This implies however that the important spreading of the covid-19 inside some nursing homesis due in a great part to several importations, probably due to staff members. However, we must remarkthat the uncertainty concerning this reproduction number is very huge, probably the result of disparitiesamong nursing homes. 11he infection fatality rate (IFR) can be estimated using the total set of recovered people according tothe model (hence including untested and asymptomatic people). Due to the consideration of variable deathrates, the IFR is different from the early period of the epidemic and the last months. Estimations arepresented in Table 3. The mean and last period are limited to September since October data need someconsolidation regarding the number of deaths. General IFR March-April period July-September periodOverall population 0.96% [0.84% ; 1.09%] 1.08% [0.93% ; 1.24%] 0.29% [0.28% ; 0.31%]0-24 0.00% [0.00% ; 0.00%] 0.00% [0.00% ; 0.00%] 0.00% [0.00% ; 0.00%]25-44 0.01% [0.01% ; 0.01%] 0.02% [0.01% ; 0.02%] 0.01% [0.00% ; 0.01%]45-64 0.19% [0.17% ; 0.21%] 0.21% [0.19% ; 0.23%] 0.08% [0.07% ; 0.09%]65-74 1.72% [1.54% ; 1.95%] 1.85% [1.67% ; 2.09%] 0.86% [0.75% ; 0.99%]75+ (nurs. homes included) 7.84% [6.85% ; 9.06%] 9.25% [8.01% ; 10.74%] 1.89% [1.69% ; 2.17%]Table 3: Infection fatality rate estimations
Table 4 presents some estimations concerning some characteristics of the disease coming from the model.The set of asymtomatic people probably includes mild symptomatic people. The model cannot really detectthe exact time when symptoms appear, hence the end of the incubation period merely corresponds to theestimated time when the infectiousness becomes more important. The total disease duration for symptomaticpeoples is only for those who are not hospitalised. The hospitalisation duration is the average until dischargedor deceased (no distinction is provided) at the beginning of the epidemic, hence before care improvement.The duration for asymptomatic nursing homes’ residents cannot really be estimated by the model. Indeed,once a single nursing home (or an isolated part of it) is completely infected, asymptomatic infected residentscan remain forever ill without any new possible contamination, hence there is no boundary on such durationcoming from the available data, so this excessive duration must be considered as an outlier.
One way to test the robustness of a model is to confront previous predictions with current reality. Thismodel can provide forecasts in two different ways. When new policy interventions are expected or a specificbehaviour change is planned due to the calendar, it is possible to extrapolate the future transmission of thecovid-19 (monitored here by the number of contacts) using relative percentage of transmission in compara-ison to the pre-lockdown phase. This percentage can only be a vague estimate of what could be the realtransmission and it is sometimes suitable to look at several different scenarios. On the other hand, when nonew policy intervention is expected for a certain time, it is possible to have a more precise forecast basedon current estimated contacts (what we call the current behaviour), forecast which is only valid up to thenext policy intervention. 12e present two previous forecasts from the model. The first one, presented in Figure 5a is a 2 1/2months old forecast based on the specific scenario that the transmission at school from September 1 wouldbe at a level of 75% due to sanitary measures like masks wearing. The second one presented in Figure 5bis a 1 month old forecast based on the current behaviour and the estimation from the model of the numberof contacts, which was estimated at that time to be 69.7% [44.2% ; 88.6%]. Those previous confrontationhighlight the fact that the uncertainty must be taken into consideration for any forecast. (a) Forecast from August 17 (b) Forecast from September 28
Figure 5: Previous forecast from August 17 based on the scenario of a 75% transmission at school from September 1 and fromSeptember 28 based on the continuation of current behaviour. The strong line represents the median, continuous lines representdeciles (10% percentiles) while dashed lines represent ventiles (5% percentiles).
Every forecast is hypothetical. New measures that have not been tested cannot really be estimated onthe level of their impact and it is impossible to predict evolution in compliance to them from the populationas well as future policy changes. This is why any realistic forecast must rely on the assumption of a perfectcontinuity of measures and compliance for elements which are a priori not suspected to change soon and ondifferent hypothetical scenarios for untested modifications of measures.In response to the large second wave in Belgium, authorities have decided to enforce new measures onOctober 19 as closing bars and restaurants, reducing the allowed social contact (known as bubble) to oneperson, promoting teleworking and establishing a curfew during the night. On November 2, a soft lockdownis put into place, with closure of non-essential shops, teleworking mandatory, leisure mostly reduced andsocial contacts even more reduced. Schools are closed during 2 weeks and then reopen with a 5/6 attendance(except for universities).While it is impossible to know with precision the impact from those measures, we can estimate thatthe effect from the soft lockdown could be comparable to the effect of the first lockdown, since the smallremaining liberties could be balanced by generalised sanitary measures like mask wearing. The effects fromOctober 19 measures is more incertain, but should be situated between September behaviour and lockdownbehaviour. Hence the most realistic middle-term scenario is to consider a half reduction on contacts (work,leisure and family) from September situation in comparaison to the first lockdown on October 19, with afull reduction applied from November 2 until the December 13 planned deadline. Schools are considered at0% transmission from November 2 to November 15 and at 5/6 thereafter. Every contacts are assumed tobe restored at September level after December 13 (except for usual school closures).13n Figure 6a, we present the estimated effect on hospital load from those measures. We must note that,according to those measures and to the model, the theoretical maximum capacity of 10000 hospital beds inBelgium should be almost reached but not exceeded, at least in an average national level. Figure 6b presentsthe expected mortality in case of the new measures scenario. We must remark that this expected mortalityrelies on a constant quality of care that may not be maintained. (a) Hospital load for two scenarios (with ventiles) (b) Deaths forecast for new measures scenario (with ventiles)(c) Estimated prevalence (confidence interval 90%) (d) Potential seroprevalence (confidence interval 90%)
Figure 6: Middle-term scenario with potential effects from new measures applied on October 19 and November 2. The firstfigure presents a comparison of the hospital load with or without the effects from the new measures. The others figures presentthe previsions on mortality, prevalence and seroprevalence.
From the model, we can also extrapolate the evolution of the virus through the whole population overtime. In Figure 6c, we present the estimated percentage of infected people over time for each age class.We can clearly see the effect of mid-March lockdown measures on children and working people. The effectof lockdown measures on older people (especially 75+) is less important since the curve is broken in a lesseffective manner. Concerning the second wave, we can see that the virus is really present among the veryyoung population due to two complete months of school opening. This prevalence is completely shut downby the two weeks closure and should be brought at a lower level than other age classes.14n Figure 6d, we present the estimated percentage of recovered people, hence the estimated percentageof immunity acquired within each age class if we make the assumption that a constant immunity is grantedto recovered people. Such a constant immunity is not guaranteed for the moment, but recent studies showthat antibodies should be present after several months for a large majority of the population [18]. Theseroprevalence is calibrated using blood donors tests results (around 1.3% on March 30 and 4.7% on April14) [9]. Since those tests where only performed on an (almost) asymptomatic population which have notdeveloped covid-19 symptoms from the past 4 weeks, the model extrapolates immunity coming also fromthe symptomatic population and from nursing homes. Note that we allow a 7 days delay in our model afterrecovering to be sure of the detectability of the anticorps. Table 5 presents the detail of some seroprevalenceestimation. global immunity among asymptomatic inside nursing homesMarch 30 2.68% [2.26% ; 3.01%] 2.39% [2.00% ; 2.66%] 1.49% [1.22% ; 1.93%]April 14 5.45% [4.68% ; 6.18%] 4.58% [3.91% ; 4.93%] 9.51% [8.29% ; 11.21%]October 31 18.49% [15.42% ; 21.16%] 10.37% [9.09% ; 10.90%] 32.94% [32.44% ; 34.17%]January 1 28.76% [22.92% ; 32.89%] 18.95 % [16.91% ; 20.15%] 56.17% [54.41% ; 56.99%]Table 5: Seroprevalence estimations
The model allows to construct long-term scenarios which are very suitable to study the potential impactfrom a specific measure. The possibilities are numerous but we present in this section a simple study of thepotential impact of an increase in contacts at a specific place (school, family, work and leisure). The increaseis perform from January 4, 2021 up to June 30, 2021, when the risk of an emerging third wave is present.We work here with the assumption that there is no modification on the set of susceptible people exceptfrom natural infection, hence with the assumption that a lasting immunity is granted to recovered people.This hypothesis could be modified negatively in the future if the probability of a reinfection is important orpositively is the immunity is artificially increased by the arrival of a vaccine.The baseline scenario is the restart of all activities on January 4 with similar transmissions/contacts asin September. Those estimated contacts percentage are 91.6% [39.5 %; 99.3%] for school contacts, 51.5%[46.8 %; 54.6%] for family contacts, 8.9% [5.8 %; 13.6%] for work contacts and 30.7% [18.1 %; 56.7%] forleisure contacts. We remind here that those percentages do not correspond to the exact number of con-tact as determined by the attendance, but to the reduced transmission in comparison to the pre-lockdownperiod as the result of decrease of contacts but also of sanitary measures. This could explain while thetransmission is estimated at a very low level at work since sanitary measures and social distancing aremore respected than during leisures or among family. The high transmission percentage at school does notnecessarily mean that schools are the engine of the virus transmission since most of the student are asymp-tomatic with a reduced infectiousness, and the uncertainty is still very important concerning this percentage.The baseline scenario is presented in Figure 7a together with the potential impact of a full transmissionat school, hence a transmission without any sanitary measure as well as without any quarantine imposed bythe testing and tracing process. We can see that the baseline scenario itself provides a non-zero probabilityof a third wave but still particularly low. The full contacts at school scenario increases a bit this probabilityto a reasonable extent.Increases in family contacts, work contacts and leisure contacts are presented in Figures 7b, 7c and7d with each time a hypothetical increase of or . Those increase must be understood as a non-proportional increase (e.g. a work increase of corresponds to .
9% + 10% = 18 . ). We can clearly seethat an increase in leisure contact has the most important effect on the evolution of the epidemic and couldlead to a potentially problematic third wave. Full transmission scenarios for family, work or leisure cannotbe taken as realistic since they would provide a complete explosion in the absence of vaccine.15 a) Increase in school contacts (b) Increase in family contacts(c) Increase in work contacts (d) Increase in leisure contacts Figure 7: Long-term scenarios with potential isolated effect from an increase in contacts at a specific place (with ventiles).
We have presented an age-structured SEIR-QD type model with a large number of ameliorations asa specific consideration for nursing homes, variable parameters and reimportation from travellers. Thoseameliorations were important in order to catch the specificity of the epidemic in Belgium.The model allows to have a good study of the current behaviour of the epidemic, with an estimationof hidden elements like the real prevalence of the virus and the potential evolution of the immunity. Moreimportant, the model allows to construct scenarios-based forecasts in order to estimate the potential impactfrom new policy measures and can explicitly serve, in complement of others models, to policy makers.However, the model suffers from several limitations which would be important to try to solve in order tobetter catch the evolution of the epidemic. In particular, we can say that the lack of spatial considerationis only a huge approximation of the reality, even if the Belgian country is small and very connected. Thecompartmental distinction is limited to asymptomatic and symptomatic while there are several variationsof the severity and hospitals are considered as a unique homogeneous element. Furthermore, the lack of16efinement inside age classes is a brake on the study of interesting scenarios, as e.g. studying the separatedimpact from transmission at primary school, secondary school or university. We must remark however thatsuch a distinction is impossible without sufficiently refined data, and those are not publicly released inBelgium, which is very problematic for quality scientific research.
Acknowledgement
The author wants to acknowledge the different members of the Walloon consortium on mathematicalmodel of the covid-19 epidemic for the numerous discussions, especially Sebastien Clesse, Annick Sartenaer,Alexandre Mauroy, Timoteo Carletti as well as Germain van Bever for statistical discussions. The authorwants also to acknowledge the members of the Flemish consortium for the very useful exchanges, models’comparisons and helps on improvement, especially the members of the SIMID-COVID-19 team (UHasselt-UAntwerp) and the BIOMATH team (UGent).This work was supported by the Namur Institute for Complex Systems (naXys) and the Departmentof Mathematics of the University of Namur, Belgium. Computational resources have been provided by theConsortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifiquede Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region.
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Appendix A. Timeline and estimated parameters
In this Appendix, we provide some technical details concerning the calibration of the model. Table A.6shows the estimated number of reimportations of covid-19 per day during the holidays period. The completeBelgian policy timeline and corresponding social contact matrices coefficients are presented in Table A.8.The complete list of estimated parameters from the calibration on October 31, 2020 data is given in TableA.8.
Date people infected Date people infected Date people infected07/01/20 40.9 [22.3 ; 56.9] 08/01/20 83.7 [45.7 ; 116.4] 09/01/20 180.4 [98.4 ; 250.8]07/02/20 41.6 [22.7 ; 57.9] 08/02/20 84.6 [46.1 ; 117.6] 09/02/20 185.5 [101.2 ; 257.9]07/03/20 42.2 [23.0 ; 58.7] 08/03/20 91.1 [49.7 ; 126.6] 09/03/20 191.9 [104.7 ; 266.8]07/04/20 41.1 [22.4 ; 57.2] 08/04/20 101.1 [55.2 ; 140.6] 09/04/20 196.8 [107.3 ; 273.6]07/05/20 38.3 [20.9 ; 53.3] 08/05/20 105.6 [57.6 ; 146.9] 09/05/20 202.8 [110.6 ; 282.0]07/06/20 39.5 [21.5 ; 54.9] 08/06/20 110.2 [60.1 ; 153.3] 09/06/20 209.3 [114.2 ; 291.0]07/07/20 42.7 [23.3 ; 59.3] 08/07/20 116.1 [63.3 ; 161.5] 09/07/20 218.5 [119.2 ; 303.8]07/08/20 42.6 [23.2 ; 59.2] 08/08/20 119.9 [65.4 ; 166.7] 09/08/20 223.1 [121.7 ; 310.3]07/09/20 44.0 [24.0 ; 61.2] 08/09/20 120.9 [65.9 ; 168.1] 09/09/20 229.4 [125.1 ; 318.9]07/10/20 46.6 [25.4 ; 64.7] 08/10/20 125.5 [68.4 ; 174.5] 09/10/20 234.9 [128.1 ; 326.6]07/11/20 42.3 [23.1 ; 58.9] 08/11/20 134.4 [73.3 ; 186.9] 09/11/20 241.9 [132.0 ; 336.4]07/12/20 41.8 [22.8 ; 58.1] 08/12/20 139.3 [76.0 ; 193.8] 09/12/20 245.5 [133.9 ; 341.4]07/13/20 45.8 [25.0 ; 63.7] 08/13/20 150.4 [82.0 ; 209.1] 09/13/20 252.7 [137.9 ; 351.4]07/14/20 47.6 [26.0 ; 66.2] 08/14/20 157.7 [86.0 ; 219.3] 09/14/20 258.9 [141.2 ; 360.0]07/15/20 46.9 [25.6 ; 65.2] 08/15/20 162.0 [88.4 ; 225.3] 09/15/20 264.8 [144.4 ; 368.2]07/16/20 49.0 [26.7 ; 68.1] 08/16/20 170.3 [92.9 ; 236.8] 09/16/20 135.9 [74.1 ; 189.0]07/17/20 50.8 [27.7 ; 70.6] 08/17/20 189.6 [103.4 ; 263.6] 09/17/20 139.0 [75.8 ; 193.3]07/18/20 51.7 [28.2 ; 71.8] 08/18/20 182.6 [99.6 ; 253.9] 09/18/20 143.7 [78.4 ; 199.8]07/19/20 51.9 [28.3 ; 72.2] 08/19/20 191.0 [104.2 ; 265.6] 09/19/20 147.1 [80.3 ; 204.6]07/20/20 59.3 [32.3 ; 82.4] 08/20/20 200.8 [109.5 ; 279.2] 09/20/20 151.0 [82.4 ; 209.9]07/21/20 64.1 [35.0 ; 89.2] 08/21/20 213.7 [116.5 ; 297.1] 09/21/20 156.4 [85.3 ; 217.5]07/22/20 67.0 [36.5 ; 93.1] 08/22/20 219.2 [119.6 ; 304.8] 09/22/20 158.3 [86.3 ; 220.1]07/23/20 72.9 [39.8 ; 101.4] 08/23/20 228.0 [124.4 ; 317.1] 09/23/20 162.3 [88.5 ; 225.6]07/24/20 77.8 [42.4 ; 108.1] 08/24/20 255.8 [139.5 ; 355.7] 09/24/20 165.9 [90.5 ; 230.6]07/25/20 79.8 [43.5 ; 110.9] 08/25/20 255.3 [139.3 ; 355.0] 09/25/20 170.9 [93.2 ; 237.7]07/26/20 80.1 [43.7 ; 111.3] 08/26/20 265.4 [144.8 ; 369.0] 09/26/20 175.9 [96.0 ; 244.6]07/27/20 89.6 [48.9 ; 124.6] 08/27/20 275.7 [150.4 ; 383.4] 09/27/20 179.4 [97.9 ; 249.4]07/28/20 95.4 [52.1 ; 132.7] 08/28/20 290.7 [158.5 ; 404.1] 09/28/20 185.0 [100.9 ; 257.2]07/29/20 100.9 [55.0 ; 140.3] 08/29/20 301.3 [164.4 ; 418.9] 09/29/20 185.2 [101.0 ; 257.5]07/30/20 106.0 [57.8 ; 147.4] 08/30/20 306.5 [167.2 ; 426.1] 09/30/20 186.5 [101.7 ; 259.3]07/31/20 113.3 [61.8 ; 157.5] 08/31/20 323.9 [176.7 ; 450.3]
Table A.6: Estimation of reimportation per day of covid-19 during the holidays period i m e li n e Su mm a r y H o m e W o r k ( + tr a n s p o rt) S c h oo l L e i s u r e ( + o t h e r s ) P r e - l o c k d o w n : M a r c h → M a r c h e v e r y t h i n g i s o p e n H a l f - l o c k d o w n : M a r c h → M a r c h s c h oo l s a nd a lll e i s u r e s c l o s e d C l e i s u r e l o c k F u lll o c k d o w n : M a r c h → M a y t e l e w o r k i n g + tr a v e l r e s tr i c t i o n s C h o m e l o c k C w o r k l o c k C l e i s u r e l o c k P h a s e A : M a y → M a y s h o p s p a rt i a ll y r e o p e n + f e w c o n t a c t s a ll o w e d C h o m e l o c k + C h o m e un l o c k C w o r k l o c k + C w o r k un l o c k C l e i s u r e l o c k P h a s e B : M a y → M a y ll s h o p s a nd c o m p ag n i e s r e o p e n ( e x ce p t b a r s / r e s t a u r a n t s ) C h o m e l o c k + C h o m e un l o c k C w o r k un l o c k C l e i s u r e l o c k P h a s e : M a y → M a y p r og r e ss i v e p a rt i a l o p e n i n go f s c h oo l s C h o m e l o c k + C h o m e un l o c k C w o r k un l o c k . C s c h oo l un l o c k C l e i s u r e l o c k P h a s e : M a y → J un e p r og r e ss i v e p a rt i a l o p e n i n go f s c h oo l s C h o m e l o c k + C h o m e un l o c k C w o r k un l o c k . C s c h oo l un l o c k C l e i s u r e l o c k P h a s e : J un e → J un e p r og r e ss i v e p a rt i a l o p e n i n go f s c h oo l s C h o m e l o c k + C h o m e un l o c k C w o r k un l o c k . C s c h oo l un l o c k C l e i s u r e l o c k P h a s e : J un e → J un e S c h oo l s p a rt i a ll y o p e n e d + l e i s u r e s + b a r s / r e s t a u r a n t s C h o m e un l o c k C w o r k un l o c k C s c h oo l un l o c k C l e i s u r e j un e P h a s e : J u l y → J un e C u l t u r a l e v e n t s + S o c i a l c o n t a c t s ( p e r s o n s bubb l e ) C h o m e un l o c k − C h o m e l o c k C w o r k un l o c k C l e i s u r e j u l y P h a s e b i s : J u l y → A u g u s t S o c i a l c o n t a c t s r e s tr i c t e d ( p e r s o n s bubb l e ) C h o m e un l o c k C w o r k un l o c k C l e i s u r e a u g S e p t e m b e r → S e p t e m b e r bubb l e o p t i o n a l C h o m e un l o c k − C h o m e l o c k C w o r k un l o c k C s c h oo l s e p t C l e i s u r e s e p t S e p t e m b e r → O c t o b e r bubb l e m a nd a t o r y C h o m e un l o c k C w o r k un l o c k C s c h oo l s e p t C l e i s u r e s e p t T a b l e A . : B e l g i a np o li c y t i m e li n e a nd c o rr e s p o nd i n g s o c i a l c o n t a c t m a t r i ce s c o e ffi c i e n t s arameter Short description Prior (SD) Step (SD) Mean Median 90% confidence interval p initial value . ± × − × − λa transmission (asympt) . ± × − × − λs transmission (sympt) . ± × − × − σ latent period − . ± × − × − τ presympt period − . ± × − × − pa (0 − proba asympt . ± × − × − pa (25 − proba asympt . ± × − × − pa (45 − proba asympt . ± × − × − pa (65 − proba asympt . ± × − × − pa (75+) proba asympt . ± × − × − pah proba asympt . ± × − × − δ (0 − hospitalisation rate . ± × − × − δ (25 − hospitalisation rate . ± × − × − δ (45 − hospitalisation rate . ± × − × − δ (65 − hospitalisation rate . ± × − × − δ (75+) hospitalisation rate . ± × − × − δh hospitalisation rate . ± × − × − γa (0 − recover rate (asympt) . ± × − × − γa (25 − recover rate (asympt) . ± × − × − γa (45 − recover rate (asympt) . ± × − × − γa (65 − recover rate (asympt) . ± × − × − γa (75+) recover rate (asympt) . ± × − × − γah recover rate (asympt) . ± × − × − γs (0 − recover rate (sympt) . ± × − × − γs (25 − recover rate (sympt) . ± × − × − γs (45 − recover rate (sympt) . ± × − × − γs (65 − recover rate (sympt) . ± × − × − γs (75+) recover rate (sympt) . ± × − × − γsh recover rate (sympt) . ± × − × − γq (0 − recover rate (hosp) . ± × − × − γq (25 − recover rate (hosp) . ± × − × − γq (45 − recover rate (hosp) . ± × − × − γq (65 − recover rate (hosp) . ± × − × − γq (75+) recover rate (hosp) . ± × − × − γqh recover rate (hosp) . ± × − × − r (0 − death rate (hosp) . ± × − × − r (25 − death rate (hosp) . ± × − × − r (45 − death rate (hosp) . ± × − × − r (65 − death rate (hosp) . ± × − × − r (75+) death rate (hosp) . ± × − × − rh death rate (hosp) . ± × − × − ˜ rh death rate (homes) . ± × − × − P recovery care improvement . ± × − × − µ recovery care improvement ± × − s recovery care improvement ± × − hosp supplementary entries . ± × − × − µ hosp variable hosp. policy ± ×
102 5 s hosp variable hosp. policy ± ×
102 2 P delay variable hosp. policy ± ×