Estimates of the proportion of SARS-CoV-2 infected individuals in Sweden
EESTIMATES OF THE PROPORTION OF SARS-COV-2INFECTED INDIVIDUALS IN SWEDEN
HENRIK HULT AND MARTINA FAVERO
Abstract.
In this paper a Bayesian SEIR model is studied to estimate theproportion of the population infected with SARS-CoV-2, the virus responsi-ble for COVID-19. To capture heterogeneity in the population and the effectof interventions to reduce the rate of epidemic spread, the model uses a time-varying contact rate, whose logarithm has a Gaussian process prior. A Poissonpoint process is used to model the occurrence of deaths due to COVID-19 andthe model is calibrated using data of daily death counts in combination witha snapshot of the the proportion of individuals with an active infection, per-formed in Stockholm in late March. The methodology is applied to regions inSweden. The results show that the estimated proportion of the population whohas been infected is around 13 .
5% in Stockholm, by 2020-05-15, and ranges be-tween 2 . − .
6% in the other investigated regions. In Stockholm where thepeak of daily death counts is likely behind us, parameter uncertainty does notheavily influence the expected daily number of deaths, nor the expected cumu-lative number of deaths. It does, however, impact the estimated cumulativenumber of infected individuals. In the other regions, where random samplingof the number of active infections is not available, parameter sharing is usedto improve estimates, but the parameter uncertainty remains substantial. Introduction
To understand the spread of the novel coronavirus, SARS-CoV-2, at an aggregatelevel it is possible to model the dynamic evolution of the epidemic using standardepidemic models. Such models include the (stochastic) Reed-Frost model and moregeneral Markov chain models, or the corresponding (deterministic) law of largenumbers limits such as the general epidemic model, see [3]. There is an extensiveliterature on extensions of the standard epidemic models incorporating various de-grees of heterogeneity in the population, e.g. age groups, demographic information,spatial dependence, etc. These additional characteristics make the models morerealistic. For instance, it is possible to evaluate the effect of various interventionstrategies. More complex models also involve additional parameters that need tobe estimated, contributing to a higher degree of parameter uncertainty.A problem when calibrating, even the standard epidemic models, to COVID-19data is that there are few reliable sources on the number of infected individuals.Publicly available sources provide data on the number of positive tests, the numberof hospitalizations, the number of ICU admission and the number of deaths dueto COVID-19. In some cases, small random samples of an active infection may beavailable. For example, the Swedish Folkh¨alsomyndigheten performed such a test
Department of Mathematics, KTH Royal Institute of Technology
E-mail addresses : [email protected]; [email protected] . Date : May 21, 2020. a r X i v : . [ q - b i o . P E ] M a y ESTIMATES OF COVID-19 INFECTED INDIVIDUALS in Stockholm with about 700 subjects in early April 2020. Moreover, there is stillno consensus in the literature on the value of important parameters such as thebasic reproduction number R and the infection fatality rate.A useful approach to incorporate the parameter uncertainty in the models is toconsider a Bayesian framework. In the Bayesian approach parameter uncertaintyis quantified by prior distributions over the unknown parameters. The impact ofobserved data, in the form of a likelihood, yields, via Bayes’ theorem, the posteriordistribution, which quantifies the effects of parameter uncertainty. The posteriorcan be used to construct estimates on the number of infected individuals, predictionson the future occurrence of infections and deaths, as well as uncertainties in suchestimates.In this paper an SEIR epidemic model with time-varying contact rate will beused to model the evolution of the number of susceptible (S), exposed (E), infected(I), and recovered (R) individuals. A time varying contact rate is used to captureheterogeneity in the population, which causes the rate of the spread of the epidemicto vary as the virus spreads through the population. Moreover, the time varyingcontact rate allows modeling the effect of interventions aimed at reducing the rateof epidemic spread. A Poisson point process is introduced to model the occurrenceand time of deaths. Random samples of tests for active infections are treated asbinomial trials where the success probability is the proportion of the population inthe infectious state.The methods are illustrated on regional data of daily COVID-19 deaths in Swe-den. It is demonstrated that, by combining the information in the observed numberof deaths and random samples of active infections, fairly precise estimates on thenumber of infected individuals can be given. By assuming that some parametersare identical in several regions, estimates for regions outside Stockholm can also beprovided, albeit with greater uncertainty.Our approach is inspired by [4] where the authors considers a Bayesian approachto model an influenza outbreak. The main extensions include the introductionof the Poisson point process to model the occurrence of deaths, the addition ofrandom sampling to test for infection, and an extension to multiple regions. Toevaluate the posterior distribution we employ Markov chain Monte Carlo (MCMC)sampling. Samples from the posterior are obtained using the Hamiltonian MonteCarlo algorithm, NUTS, by [9], implemented in the software Stan, which is an opensource software for MCMC. 2. Background
To model the spread of the epidemic we consider the deterministic SEIR model[1, 5], which is a simple deterministic model describing the evolution of the numberof susceptible, exposed, infected, and recovered individuals in a large homogeneouspopulation with N individuals. The epidemic is modeled by { ( S t , E t , I t ) , t ≥ } ,where S t , E t and I t represent the number of susceptible, exposed and infected in-dividuals at time t , respectively. The total number of recovered and deceasedindividuals at time t ≥ N − S t − E t − I t . The epidemic starts STIMATES OF COVID-19 INFECTED INDIVIDUALS 3 from a state S , E , I with S + E + I = N , and proceeds by updating, S t +1 = S t − β S t I t N ,E t +1 = E t + β S t I t N − νE t ,I t +1 = I t + νE t − γI t . The parameters are the contact rate β >
0, the rate ν of transition from theexposed to the infected state and the recovery rate γ >
0. Note that I t representsthe number of individuals with an active infection at time t , whereas N − S t is thecumulative number of individuals who have been exposed, and possibly infected,recovered or deceased, up until time t .In the context of the COVID-19 epidemic the contact rate cannot be assumed tobe constant, primarily due to interventions implemented in the early stage of theepidemic. Moreover, as the SEIR model describes the evolution at an aggregatelevel, a time varying contact rate may be used to capture inhomogeneities in thepopulation. If, for example, the epidemic is initiated in a rural area the contactrate may be rather low, but as the epidemic reaches major cities the contact ratewill be higher. The resulting SEIR model with time varying contact rate is givenby S t +1 = S t − β t S t I t N ,E t +1 = E t + β t S t I t N − νE t ,I t +1 = I t + νE t − γI t . , (1)Clearly, one needs to put some restriction on the amount of variation of the contactrate. In this paper a Gaussian process prior will be used on the log contact rate,which restricts the amount of variation in time, but is sufficiently flexible to capturethe reduction in contact rate after the interventions.When observations on the number of infected and recovered individuals are avail-able, the model (1) can be fitted to these observations. In the context of COVID-19,observations on the number of infected and recovered individuals are unavailable.There are many symptomatic individuals who are not tested and potentially alarge pool of asymptomatic individuals. In this paper we will rely on the numberof registered deaths due to COVID-19 to calibrate the model. In addition we willincorporate the test results from a random sample that provides a snapshot on thenumber of individuals with an active infection.3. A Poisson point process for the occurrence of deaths
To model the occurrence of deaths due to COVID-19 we consider the followingPoisson point process representation. We refer to [10] for details on Poisson pointprocesses. Let f denote the infection fatality rate, that is, the probability thatan infected individual eventually dies from the infection. Consider the number ofindividuals that enters the infected state on day t , that is, νE t . Each such infectedindividual has probability f to eventually die from the infection. Conditional ondeath due to the infection, the time from infection until death is assumed inde-pendent of everything else and follows a probability distribution with probabilitymass function p s D . Each individual that dies may be represented as a point ( t, τ )in E := { ( t, τ ) ∈ N : τ ≥ t } , where t denotes the time of entry to the infected ESTIMATES OF COVID-19 INFECTED INDIVIDUALS state and τ the time of death of the individual. The number of deaths at time τ can then be computed by counting the number of points on the line ∪ τt =0 ( t, τ ).The number of deaths, and the corresponding time of infection and time of deathis conveniently modelled by a Poisson point process on E . Let ξ be a Poisson pointprocess on E with intensity ν ( t, τ ) = f νE t p s D ( τ − t ) . (2)We may interpret a point at ( t, τ ) of the Poisson point process as the time ofinfection, t , and the time of death, τ , of an individual who dies from the infection.The number of deaths D τ that occurs at time τ is then given by summing up allthe points of the point process on the row corresponding to τ , ξ ( ∪ τt =0 ( t, τ )). Sincethe rows are disjoint this implies that D , D , . . . are independent with each D τ having a Poisson distribution with parameter λ τ = ν ( ∪ τt =0 ( t, τ )) = f ν τ (cid:88) t =0 E t p s D ( τ − t ) . Throughout this paper p s D is the probability mass function of a negative bino-mial distribution with mean s D . More precisely, a parametrization of the negativebinomial distribution with parameters r, s D will be used, where p s D ( n ) = (cid:18) s D r + s D (cid:19) n (cid:18) rr + s D (cid:19) r (cid:18) n + r − r − (cid:19) . The value r = 3 will be used throughout as this fits well with the distribution ofobserved duration from symptoms to death in the study by [15].4. Prior distributions and derivation of the likelihood
In this section we provide the assumptions on the prior distributions and derivethe expression of the likelihood of the model. Note that λ τ is a function of allthe parameters of the model, θ = ( { β t } , ν, γ, s , f, s D ). The parameters, theirinterpretation and prior distribution are summarized in Table 1. Actually, sincethe contact rate is positive, a Gaussian process (GP) prior will be used for thenatural logarithm of the contact rate, denoted log-GP in the sequel. The Gaussianprocess has a constant mean µ and a squared-exponential covariance kernel k withparameters α, ρ, δ such thatCov(log β t , log β t ) = k ( t − t ) = α exp (cid:18) − ρ | t − t | (cid:19) + δ I { t = t } . To compute the likelihood the observed number of daily deaths, d , d , . . . , d T , Parameter Explanation Prior distribution Hyper-parameters { β t } contact rate log-GP( µ, k ) µ, α, ρ, δν rate from exposed to infected Gamma( a ν , b ν ) a ν , b ν γ recovery rate Gamma( a γ , b γ ) a γ , b γ s initial susceptible fraction Beta( a s , b s ) a s , b s f infection fatality rate Beta( a f , b f ) a f , b f s D expected duration Gamma( a s D , b s D ) a s D , b s D Table 1.
Specification of the parameters and prior distributions.
STIMATES OF COVID-19 INFECTED INDIVIDUALS 5 will be used, in combination with a random sample of n tests for active infection,performed at a time t , when such test result is available. The number of individuals Z with positive test result has a Bin( n , I t /N ) distribution. The full likelihood isgiven by: p D,Z | Θ ( d , . . . , d T , z | θ ) = T (cid:89) τ =1 Prob( D τ = d τ ) × Prob( Z = z )= T (cid:89) τ =1 λ d τ τ d τ ! e − λ τ × (cid:18) n z (cid:19) (cid:18) I t N (cid:19) z (cid:18) − I t N (cid:19) n − z . (3)The joint prior is the product of the marginal priors and leads, by Bayes’s theorem,to the posterior, p Θ | D,Z ( θ | d , . . . , d T , z ) ∝ p D,Z | Θ ( d , . . . , d T , z | θ ) p Θ ( θ ) . The expected number of daily deaths λ τ , the cumulative number of deaths andthe cumulative number of infected individuals N − S t are all functions of θ and theirdistribution can therefore be inferred from the posterior p Θ | D,Z . By sampling from p Θ | D and iterating the dynamics (1) estimates of these quantities may be obtainedalong with the effects of parameter uncertainty. Moreover, predictions on the futuredevelopment of the above mentioned quantities can be obtained by extrapolatingthe contact rate into the future.As the posterior distribution is unavailable in explicit form it is necessary toemploy Monte Carlo methods. In the next section Markov chain Monte Carlomethods are briefly described to sample from the posterior.4.1. Multiple regions.
The SEIR model (1) and the derivation of the likelihood(3) considers a single region. In the context of multiple regions it may be reasonableto assume that some parameters are identical. For example, when consideringmultiple regions of Sweden below it will be assumed that the rate, ν , from exposedto infected, the recovery rate, γ , the infection fatality rate, f , and the duration, s D are identical in all regions. It is tempting to include interaction terms between theregions as infected individuals from one region may travel to another region andcause new infections. In this paper, it will be assumed that each region has its owntime varying contact rate that incorporates fluctuations in new infections due toimport cases from other regions.The likelihood from multiple regions is simply the product of the marginal like-lihood for the individual regions and the prior is the product of the marginal priorsfor each parameter. Thus, for two regions the prior will be the product of twoGaussian process priors for the respective log contact rates for the two regions andthe product of the marginal priors for the remaining parameters.5. Markov chain Monte Carlo
Markov chain Monte Carlo (MCMC) methods in Bayesian analysis aims at sam-pling from the posterior distribution. This is non-trivial because the marginaldistribution of the data, which acts as normalizing constant of the posterior is prac-tically impossible to compute. In MCMC algorithms the posterior is represented asa target distribution. The algorithms rely on the construction of a Markov chainwhose invariant distribution is the target distribution. Standard MCMC meth-ods are based on acceptance-rejection steps, where random proposals are accepted
ESTIMATES OF COVID-19 INFECTED INDIVIDUALS or rejected with a probability that does not require knowledge of the normalizingconstant, e.g., Metropolis-Hasting and Gibbs sampling [13, 8, 6]. When the tar-get distribution is complex and multi-modal, standard methods may lead to poormixing of the Markov chain and slow convergence to the target distribution.To overcome slow mixing of the Markov chain gradient-based sampling can beapplied, which adapt the proposal distribution based on gradients of the target,see e.g. [2]. In this paper we will employ a Hamiltonian Monte Carlo sampler, theNo-U-turn sampler (NUTS) by [9] in combination with automatic differentiation tonumerically approximate the gradients [7], which is implemented in the open sourcesoftware Stan.6.
Estimates and predictions for regions of Sweden
In this section the estimates of the number of infected individuals and predictionson the evolution of the number of deaths and number of infected individuals areprovided for ten regions of Sweden. The epidemic is considered to start on 2020-03-01 and interventions in Sweden began on 2020-03-16. The joint prior distribution isthe product of the marginal priors, and the hyper-parameters are specified in Table2. The choices of hyper-parameter values are made in line with existing literatureon the COVID-19 epidemic. As a general principle we have used informative priorson the parameters ν, γ, and s D , whereas the priors on the time-varying contactrate { β t } and the fatality rate f are uninformative. Folkh¨alsomyndigheten reportsthat the incubation period is usually around 5 days, which corresponds to 1 /ν ≈ /γ ≈
14. The overallinfection fatality rate f is estimated to be in the range 0 . − . , { β t } log-GP( µ, k ) µ = 0 , α = 1 , .
13 [0 . , . ρ = 15 , δ = 10 − ν Gamma( a ν , b ν ) a ν = 100 , b ν = 500 0 . . , . γ Gamma( a γ , b γ ) a γ = 100 , b γ = 1400 0 .
071 [0 . , . s Beta( a s , b s ) a s = 100 , b s = 0 . .
99 [0 . , . f Beta( a f , b f ) a f = 1 , b f = 1 0 . . , . s D Gamma( a s D , b s D ) a s D = 900 , b s D = 50 18.0 [16 . , . Table 2.
Prior distributions, hyper-parameters and prior credi-bility intervals.6.1.
Region Stockholm.
The Region Stockholm has N = 2 .
34 million inhabi-tants. The daily death counts in Region Stockholm from 2020-03-01 – 2020-05-15are obtained from the webpage: https://c19.se/ . On 2020-04-09 the Swedishauthority, Folkh¨alsomyndigheten, published the results of a random sample of 707
STIMATES OF COVID-19 INFECTED INDIVIDUALS 7 individuals performed between 2020-03-27 and 2020-04-03 . It showed that 18 in-dividuals carried the SARS-CoV-2 virus. These results are included in the analysisas a binomial sample of size n = 707 and success probability I t /N where the testdate, t , is assumed to be 2020-03-30.A summary of the marginal posterior distributions is provided in Table 3. Theposterior distribution of the time varying contact rate is illustrated in Figure 1.Note that although there is great uncertainty about the initial contact rate, themodel clearly picks up the reduction in contact rate after the interventions began on2020-03-16. The contact rate is gradually reduced around the time of interventionand then remains at a low level. This slow reduction of the contact is, however, notdue to stiffness of the Gaussian process kernel. We have experimented with a sharpbreak-point in the contact rate at the time of intervention, but it did not providemore accurate results. On the contrary, the data suggests that the reduction ofthe contact rate is slow. The contact rate is estimated until 2020-05-01. After thisdate the posterior is unreliable. This is because many of the deaths of individualswho are infected after 2020-05-01 have not yet been observed. For this reason, thecontact rate is only estimated until 2020-05-01.To perform estimates and predictions on the future number of daily and cumula-tive infections and deaths, the contact rate has been extrapolated from its value on2020-05-01. The posterior distribution suggests that the contact rate is constant, ata low rate, since roughly 2020-04-07, which motivates extrapolation into the future,assuming that the interventions remains at the present level. Figure 1.
Estimated contact rate for Region Stockholm until2020-05-01 based on data from 2020-03-01 – 2020-05-15. The graphshows the posterior median and point-wise 95% credibility interval.After 2020-05-01 the contact rate is extrapolated, by assuming itwill remain constant. ESTIMATES OF COVID-19 INFECTED INDIVIDUALS
Figure 2 (top left) shows the observed daily number of deaths (black dots) alongwith the posterior median (dark red) and 95% credibility interval (red) for the ex-pected number of daily deaths. Figure 2 (top right) shows the observed cumulativenumber of deaths (black dots) along with the posterior median (dark red) and 95%credibility interval (red) for the expected cumulative number of deaths.We observe that the parameter uncertainty does not substantially impact theexpected number of daily deaths and the peak of the daily number of deaths appearsto have occurred by mid April. Similarly, the expected cumulative number of deathsin Stockholm is likely to terminate slightly above 2000.We emphasize that this is the expected number of deaths , λ τ . Since we areconsidering a Poisson distribution for the number of daily deaths an approximate95%-prediction interval would be λ τ ± √ λ τ , where λ τ is the Poisson parameter onday τ .Note from the observed number of daily deaths that the empirical distributionof daily deaths appear to be overdispersed, the variance is substantially largerthan the mean. This is likely due to reporting of the data. The data presentedat https://c19.se/ does not correct the reporting of death dates in hindsight.A comparison at the national level with data provided by Folkh¨alsomyndighetenshows that the official records of the daily number of deaths for Sweden does notappear to be overdispersed. Nevertheless, even after smoothing the data from https://c19.se/ by a moving average over a few days, the results of the simula-tions remain essentially the same.Figure 2 (bottom left/right) shows the posterior median (dark red) and 95%credibility interval (red) for the daily/cumulative number of infected individuals.Although the parameter uncertainty has significant impact on the cumulative num-ber of infected individuals, some conclusions are still possible. As of mid May, thecumulative number of infected individuals has almost reached its terminal valueand the spread of the epidemic has slowed down significantly. The estimated cu-mulative number of infected individuals is 13 .
5% of the population in Stockholm.The estimated number of infected individuals by 2020-04-11 is 10 . , which indicated that 10% of the population in Stockholm had developedanti-bodies against the SARS-CoV-2 virus by the first weeks of April.We emphasize that the estimate of the cumulative number of infected individualsin Stockholm relies heavily on the inclusion of results from the random samplingperformed by Folkh¨alsomyndigheten in late March, early April. Without this cru-cial piece of information similar models to the one analyzed here may provide asignificantly higher estimate on the cumulative number of infected. STIMATES OF COVID-19 INFECTED INDIVIDUALS 9
Figure 2.
Data from Region Stockholm until 2020-05-16.
Topleft:
Observed daily number of deaths (black dots), the posteriormedian (dark red) and 95% credibility interval for the expecteddaily number of deaths.
Top right:
Observed cumulative numberof deaths (black dots), the posterior median (dark red) and 95%credibility interval for the expected cumulative number of deaths.
Bottom left:
The posterior median (dark red) and 95% credibilityinterval for the daily number of infected individuals.
Bottom right:
The posterior median (dark red) and 95% credibility interval forthe cumulative number of infected individuals.Parameter Post. mean Post. 95%-C.I. Prior 95%-C.I. ν .
21 [0 . , .
25] [0 . , . γ .
08 [0 . , .
10] [0 . , . s . . , .
0] [0 . , . f .
007 [0 . , . . , . s D . . , .
9] [16 . , . Table 3.
Marginal posterior median and credibility intervals forRegion Stockholm.6.2.
Summary of the results for ten regions of Sweden.
In this section es-timates of the cumulative number of infected individuals are provided for the fol-lowing regions of Sweden:(1) Stockholm (population: 2 . · )(2) V¨astra G¨otaland (population: 1 . · )(3) ¨Osterg¨otland (population: 1 . · )(4) ¨Orebro (population: 3 . · )(5) Sk˚ane (population: 1 . · )(6) J¨onk¨oping (population: 3 . · ) (7) S¨ormland (population: 2 . · )(8) V¨astmanland (population: 2 . · )(9) Uppsala (population: 3 . · )(10) Dalarna (population: 2 . · )The daily death counts for the regions of Sweden until 2020-05-15 are obtainedfrom the webpage: https://c19.se/ . There is no random testing providing in-formation on the proportion of infected individuals outside Region Stockholm. Toestimate the contact rate and the cumulative number of infected individuals in re-gions outside Stockholm, we have implemented the multi-region model pairwise,with two regions in each MCMC simulation, where one region is Stockholm andthe other region is from the list above. It is assumed that the parameters ν, γ, f, and s D are identical in both regions, but the time varying contact rate and theinitial proportion of susceptible individuals are different between the regions. Theposterior of the contact rates for the different regions are provided in Figure 3and the corresponding estimates and predictions for the daily number of deaths,the cumulative number of deaths, the daily number of infected individuals and thecumulative number of infected individuals are provided in Figures 4 - 12. Theparameter uncertainty is generally high and in several regions the number of newinfections and daily deaths may still increase. Estimates on the proportion of in-fected individuals on 2020-05-15 in the different regions are provided in Table 4along with 95% credibility intervals. Overall the proportions are low, far from herdimmunity. Region Prop. infected 95%-C.I.Stockholm 13 .
5% [8 . , . .
3% [3 . , . .
3% [4 . , . .
9% [5 . , . .
5% [1 . , . .
8% [5 . , . .
6% [9 . , . .
8% [6 . , . .
4% [5 . , . .
4% [6 . , . Table 4.
Estimated proportion of the population who have hada SARS-CoV-2 infection by May 15, for ten regions of Sweden.
References [1] R.M. Anderson and R.M. May.
Infectious Diseases of Humans: Dynamics and Control .Oxford University Press, 1992.[2] M. Betancourt, S. Byrne, S. Livingstone, and M. Girolami. The geometric foundations ofHamiltonian Monte Carlo.
Bernoulli
23 (4A), 2257–2298, 2017.[3] T. Britton. Basic estimation-prediction techniques for COVID-19, and a prediction for Stock-holm.
MedRxiv https://doi.org/10.1101/2020.04.15.20066050[4] A. Chatzilena, E. van Leeuwen, O. Ratmann, M. Baguelin, and N. Demiris. Contemporarystatistical inference for infectious disease models using Stan.
Epidemics , 29, 100367, 2019.[5] O. Diekmann, J.A.P. Heesterbeek, and T. Britton.
Mathematical tools for understandinginfectious disease dynamics.
Princeton University Press, 2013.
STIMATES OF COVID-19 INFECTED INDIVIDUALS 11 [6] S. Geman, and D. Geman, D. Stochastic relaxation, Gibbs distributions, and the Bayesianrestoration of images.
IEEE Trans. Pattern Anal. Mach. Intell. (6), 721–741, 1984.[7] A. Griewank, and A. Walther, A.
Evaluating Derivatives: Principles and Techniques ofAlgorithmic Differentiation , SIAM, 2008.[8] W.K. Hastings. Monte Carlo sampling methods using Markov Chains and their applications,
Biometrika , 57(1), 97–109,1970.[9] M.D. Hoffmann and A. Gelman. The No-U-turn sampler: adaptively setting path lengths inHamiltonian Monte Carlo.
J. Mach. Learn. Res.
15 (1), 1593–1623, 2014.[10] O. Kallenberg.
Random Measures, Theory and Applications . Springer, 2017.[11] Q. Li et al. Early transmission dynamics in Wuhan, China, of novel Coronavirus
InfectedPneumonia. N. Engl. J. Med.
Science , 10.1126/science.abb3221[13] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller. Equation of statecalculations by fast computing machines.
J. Chem. Phys.
21 (6), 1087–1092, 1953.[14] R. Verity et al. Estimates of the severity of coronavirus disease 2019: a model-based analysis
The Lancet Infections Diseases , March 30, 2020,[15] J.T. Wu et al. Estimating clinical severity of COVID-19 from the transmission dynamics inWuhan, China.
Nature Medicine
26, 506–510, 2020.
Figure 3.
Estimated contact rate for regions of Sweden until2020-05-01 based on data from 2020-03-01 – 2020-05-15. The graphshows the posterior median and point-wise 95% credibility inter-val. After 2020-05-01 the contact rate is extrapolated, by assumingit will remain constant. From right to left, first row: Stockholm,V¨astra G¨otaland, second row: ¨Osterg¨otland, ¨Orebro, third row:Sk˚ane, J¨onk¨oping, fourth row S¨ormland, V¨astmanland, fifth row:Uppsala, Dalarna.
STIMATES OF COVID-19 INFECTED INDIVIDUALS 13
Figure 4.
Data from Region V¨astra G¨otaland until 2020-05-15.
Top left:
Observed daily number of deaths (black dots), the poste-rior median (dark red) and 95% credibility interval for the expecteddaily number of deaths.
Top right:
Observed cumulative numberof deaths (black dots), the posterior median (dark red) and 95%credibility interval for the expected cumulative number of deaths.
Bottom left:
The posterior median (dark red) and 95% credibilityinterval for the daily number of infected individuals.
Bottom right:
The posterior median (dark red) and 95% credibility interval forthe cumulative number of infected individuals.
Figure 5.
Data from Region ¨Osterg¨otland until 2020-05-15.
Topleft:
Observed daily number of deaths (black dots), the posteriormedian (dark red) and 95% credibility interval for the expecteddaily number of deaths.
Top right:
Observed cumulative numberof deaths (black dots), the posterior median (dark red) and 95%credibility interval for the expected cumulative number of deaths.
Bottom left:
The posterior median (dark red) and 95% credibilityinterval for the daily number of infected individuals.
Bottom right:
The posterior median (dark red) and 95% credibility interval forthe cumulative number of infected individuals.
STIMATES OF COVID-19 INFECTED INDIVIDUALS 15
Figure 6.
Data from Region ¨Orebro until 2020-05-15.
Top left:
Observed daily number of deaths (black dots), the posterior median(dark red) and 95% credibility interval for the expected daily num-ber of deaths.
Top right:
Observed cumulative number of deaths(black dots), the posterior median (dark red) and 95% credibilityinterval for the expected cumulative number of deaths.
Bottomleft:
The posterior median (dark red) and 95% credibility intervalfor the daily number of infected individuals.
Bottom right:
Theposterior median (dark red) and 95% credibility interval for thecumulative number of infected individuals.
Figure 7.
Data from Region Sk˚ane until 2020-05-15.
Top left:
Observed daily number of deaths (black dots), the posterior median(dark red) and 95% credibility interval for the expected daily num-ber of deaths.
Top right:
Observed cumulative number of deaths(black dots), the posterior median (dark red) and 95% credibilityinterval for the expected cumulative number of deaths.
Bottomleft:
The posterior median (dark red) and 95% credibility intervalfor the daily number of infected individuals.
Bottom right:
Theposterior median (dark red) and 95% credibility interval for thecumulative number of infected individuals.
STIMATES OF COVID-19 INFECTED INDIVIDUALS 17
Figure 8.
Data from Region J¨onk¨oping until 2020-05-15.
Topleft:
Observed daily number of deaths (black dots), the posteriormedian (dark red) and 95% credibility interval for the expecteddaily number of deaths.
Top right:
Observed cumulative numberof deaths (black dots), the posterior median (dark red) and 95%credibility interval for the expected cumulative number of deaths.
Bottom left:
The posterior median (dark red) and 95% credibilityinterval for the daily number of infected individuals.
Bottom right:
The posterior median (dark red) and 95% credibility interval forthe cumulative number of infected individuals.
Figure 9.
Data from Region S¨ormland until 2020-05-15.
Topleft:
Observed daily number of deaths (black dots), the posteriormedian (dark red) and 95% credibility interval for the expecteddaily number of deaths.
Top right:
Observed cumulative numberof deaths (black dots), the posterior median (dark red) and 95%credibility interval for the expected cumulative number of deaths.
Bottom left:
The posterior median (dark red) and 95% credibilityinterval for the daily number of infected individuals.
Bottom right:
The posterior median (dark red) and 95% credibility interval forthe cumulative number of infected individuals.
STIMATES OF COVID-19 INFECTED INDIVIDUALS 19
Figure 10.
Data from Region V¨astmanland until 2020-05-15.
Topleft:
Observed daily number of deaths (black dots), the posteriormedian (dark red) and 95% credibility interval for the expecteddaily number of deaths.
Top right:
Observed cumulative numberof deaths (black dots), the posterior median (dark red) and 95%credibility interval for the expected cumulative number of deaths.
Bottom left:
The posterior median (dark red) and 95% credibilityinterval for the daily number of infected individuals.
Bottom right:
The posterior median (dark red) and 95% credibility interval forthe cumulative number of infected individuals.
Figure 11.
Data from Region Uppsala until 2020-05-15.
Top left:
Observed daily number of deaths (black dots), the posterior median(dark red) and 95% credibility interval for the expected daily num-ber of deaths.
Top right:
Observed cumulative number of deaths(black dots), the posterior median (dark red) and 95% credibilityinterval for the expected cumulative number of deaths.
Bottomleft:
The posterior median (dark red) and 95% credibility intervalfor the daily number of infected individuals.
Bottom right:
Theposterior median (dark red) and 95% credibility interval for thecumulative number of infected individuals.
STIMATES OF COVID-19 INFECTED INDIVIDUALS 21
Figure 12.
Data from Region Dalarna until 2020-05-15.
Top left:
Observed daily number of deaths (black dots), the posterior median(dark red) and 95% credibility interval for the expected daily num-ber of deaths.
Top right:
Observed cumulative number of deaths(black dots), the posterior median (dark red) and 95% credibilityinterval for the expected cumulative number of deaths.
Bottomleft:
The posterior median (dark red) and 95% credibility intervalfor the daily number of infected individuals.