The popIBM utilized here is an individual-based stochastic model designed to align with the demographic distributions of Denmark. This means the individuals in the model are not matched to specific real-world individuals, but rather the model generates a synthetic population with comparable age, vaccination target group, and place of residence to the Danish population.PopulationThe synthetic population was initialized by assigning each individual to an age group and a vaccination target group, followed by distributing them across municipalities and parishes. This assignment was based on data from the Danish central person registry (CPR), which contains information about the birth year and a unique personal identification number that can be matched with other registries. The population was stratified into nine age groups (0-9, 10-19, …, 80+ years). Each individual belongs to one of the 2100+ parishes, which are nested in the 98 municipalities of Denmark.By July 2021, the Danish health authorities had prioritized vaccination for 17 distinct target groups. These vaccination target groups represent the prioritized rollout of vaccines in Denmark. This rollout initially targeted the elderly and healthcare personnel, gradually extending by age to 50+ year-olds, after which a pronged approach was used (see the Supplementary Methods A.1 for more details). By the end of the study period, 26% of the Danish population was fully vaccinated. The model effectively has 18 vaccination target groups, as those aged 0-11 years were not eligible for vaccination during the study period, and therefore not included in the official target groups.Individuals in the simulation belongs to both an age group and a vaccination target group. The age group defines the number of contacts through the contact matrices (see Supplementary Methods A.2 for details), the vaccination target group determines the timing of receiving vaccines (see Supplementary Methods A.1 for details), and the combination of age group and vaccination target group determines the risk of going to hospital (see Supplementary Fig. A2 for details).For each vaccination target group, vaccines were distributed weekly, as reported in10. This was replicated in the model by pre-assigning vaccination dates to all relevant individuals. We assumed that vaccine doses were evenly distributed across the seven days of the week during which they were supplied.In the model each vaccinated individual had a countdown timer set to the date when the vaccination would take effect; this was 21 days after the first dose given according to the vaccination roll out for the Comirnaty and Spikevax, and 28 days for the Vaxzevria. This period was based on the expectation that individuals would receive their second dose in line with the prevailing guidelines. From our data, we found that 98.3% of people who enrolled in the vaccination program during the study period eventually attained “fully vaccinated” status. In effect, we assume that all vaccinated persons in the model follow the vaccination guideline and obtain the effect of the vaccine immediately after the second dose.InitializationSimulation runs began on Monday, January 4, 2021, and were initialized with 4% of the infected individuals harboring the SARS-CoV-2 Alpha strain, reflecting the situation observed in Denmark at that time12,13. We did not initialize the simulation before this date due to the difficulty of accurately determining activity levels over the holiday period. This was followed by a one-week ‘burn-in’ period. The study period under consideration was January 11 to June 30, 2021.Disease spreadIn the model, individuals could occupy one of four disease states, corresponding to the classic SEIR model: susceptible, exposed, infectious, or recovered. Disregarding the effects of local lockdowns, the probability of an individual becoming exposed during a time step was calculated in two phases, in which in- and outgoing transmissions from individuals were separately evaluated:The first step was to compute outgoing transmission, equivalent to the force of infection F, per age group at both the national and municipal levels. Since some infected individuals were in (self-)isolation due to a positive test or local lockdowns, the effective number of infectious individuals was reduced. The number of infectious people capable of transmitting the disease, \({I}_{am}^{{{{\rm{eff}}}}}\), varied across age groups, a, and municipalities, m. The force of infection was determined by multiplying the number of effectively infectious individuals with the time-varying activity matrix, A(t), which outlined activity in Danish society on a national scale (see Supplementary Methods A.2 for details):$${I}_{am}^{{{{\rm{eff}}}}}={\sum}_{i}{{{\rm{I}}}}({{{{\rm{disease}}}}}_{ampi}=={{{\rm{infectious}}}})\min ({\Delta }^{m}({{{{\rm{inc}}}}}_{m}),{\Delta }^{p}({{{{\rm{inc}}}}}_{p})){\xi }_{i}$$
(1)
$${{{{\bf{F}}}}}_{\cdot m}={\beta }_{0}\beta (T){{{\bf{A}}}}(t){{{{\bf{I}}}}}_{\cdot m}^{{{{\rm{eff}}}}}$$
(2)
$${{{{\bf{F}}}}}_{\cdot }^{{{{\rm{DK}}}}}={\sum}_{m}{{{{\bf{F}}}}}_{\cdot m}$$
(3)
Here F⋅m is the per-municipality force of infection vector that includes the nine age groups (implicit a) for the m’th municipality and \({{{{\bf{F}}}}}_{\cdot }^{{{{\rm{DK}}}}}\) is the nation-wide force of infection from each age group. For all parameters, the subscript i refers to the individual. β0 is the overall transmission rate, β(T) is a temperature-dependent scaling of the transmission rate per contact (see Supplementary Methods A.6.2), and \({{{{\bf{I}}}}}_{\cdot m}^{{{{\rm{eff}}}}}\) is the vector of number of effective infectious individuals that includes the nine age groups (implicit a represented by ⋅ ) for the m’th municipality. The functions Δm(incm) and Δp(incp) describe the effects of local lockdown in municipality m and parish p respectively as a function of incidence in the entity (see Supplementary Methods A.6.1 for details), and ξi represents individuals self-isolating at home due to a positive test. When an individual is not in local lockdown or self-isolated due to a positive test the Δx(incx) and ξi parameters take a value of 1. They take a value in the range \(\left[0:1\right[\) if an individual is in an area under local lockdown or if they are in self-isolation due to a positive test. Please note that national lockdown measures are included in the A(t) matrix.When the incidence of the previous week in a given period exceeds a specific threshold the area will enter into a lockdown similar to the strict lockdown of January 2021 (one threshold for municipalities and another, higher threshold for parishes. Lockdowns would be lifted if the area had been consistently under the limit for seven days14. Thresholds were incrementally increased throughout the study period. See Supplementary Methods A.6.1 for details). This results in both in-going and out-going contacts being reduced. These reductions are approximated by modifying the probability of becoming exposed or transmitting the disease by a factor of \(\delta (t)=\sqrt{\lambda (0)/\lambda (t)}\), where λ represent the dominant eigenvalue of the activity matrix, A(t), at times t and t = 0. The latter corresponds to the beginning of the study period under strict lockdown. Note that instead of counting every infectious individual with weight 1, individuals under lockdown will be weighted by Δx(incx). This modification of in-going and out-going contacts has the asymptotic behavior, that if everybody is under local lockdown at any given time, t, the force of infection is approximately the same as at t = 0. It does not match exactly, due to the lockdown at t = 0 having a different impact across age groups than the re-scaled lockdowns at times t. Due to the introduction of soft restrictions when municipalities and parishes were nearing the incidence limits, the Δx(incx) functions would change in a piecewise linear way towards δ(t) (see Supplementary Methods A.6.1 for details).Test-positive individuals isolate themselves which set their outgoing contacts to ξi = 0. In reality, this number is likely higher than zero, as not all individuals were expected to fully comply with guidelines. However, there was generally high compliance in Danish society, and free accommodation was offered by the Danish government for any person not able to isolate at home.Combined, the probability for an individual i of moving from a susceptible to an exposed state at each time step, Pi(S → E), depends on the local, age-group dependent probability Pam(S → E) modified by the vaccination and lockdown status of the individual. Mathematically, this probability becomes:$${P}_{am}(S\to E)=1-\exp \left(-(1-\alpha )\frac{{F}_{a}^{{{{\rm{DK}}}}}}{{N}_{a}^{{{{\rm{DK}}}}}}-\alpha \frac{{F}_{am}}{{N}_{am}}\right)$$
(4)
$${P}_{i}(S\to E)={P}_{am}(S\to E)\min ({\Delta }^{m}({{{{\rm{inc}}}}}_{m}),{\Delta }^{p}({{{{\rm{inc}}}}}_{p})){V}_{ki}{\rho }_{p}$$
(5)
where α is the fraction of transmission going on at the municipality level (throughout this study set to 90%), and Nam is the population of each age group a in each municipality m, with \({N}_{a}^{{{{\rm{DK}}}}}={\sum }_{m}{N}_{am}\). The parameter Vki represents an individual factor describing whether an individual has been fully vaccinated with vaccine type k.Furthermore, each parish, p, has a risk factor, ρp, in the model. This factor is based on the observed cumulative incidence in parishes in Denmark up til the beginning of the study period (see Supplementary Methods A.3.1). This parish-level risk factor has the range of [0.49; 1.57] and is a proxy for many unobserved risk factors within the parish (i.e. population density, socioeconomic status, etc). The parish risk factor was introduced in the model to match the observed spatial heterogeneity in the epidemic. If parish risk factors are not introduced, incidences will be very homogeneous across geographic entities.For the SARS-CoV-2 Alpha variant, the vaccine effectiveness were high15, and thus, in the model, the probability of being infected is reduced by a vaccine-dependent factor Vk. This factor is assumed to be 90% 21 days following the first dose of the Comirnaty and Spikevax vaccines and 60% 28 days following the first dose of Vaxzevria vaccination (see Supplementary Methods A.5 for details). The delay until effect is based on the assumption that individuals receive the second dose according to the vaccination guidelines available at the time.Note that the above equations assume that no one self-isolates due to a false-positive SARS-CoV-2 test (due to high test specificity16).When individuals enter the exposed and infectious stages, the time to move to the next stage (respectively infectious and recovered) is drawn from a gamma distribution with shape parameter k = 2 and a mean time 1/γE = 2.5 days and 1/γI = 5.3 days respectively. Furthermore, 50% of exposed individuals (based on estimates by Johansson et al.17) will draw a time until onset of symptoms also from a k = 2 gamma distribution 1/γsymp = 2 days. Upon symptom onset, individuals will be tested and isolate themselves.In addition, the general population and infected without symptoms are tested randomly following the observed time-varying test incidence stratified by age and vaccination status. The tests were furthermore spatially distributed according to the time-varying incidence in the municipality so that areas with high incidence would be tested more – in accordance with observed behavior (see Supplementary Methods A.3.2 for details).HospitalizationThe risk of being admitted to a hospital follows the observed risk in the period up to the study period stratified by age and vaccination target group and then multiplied with the observed risk factor of 1.42 for the Alpha variant7. This way, the risk is representative of Danish practices and likely also how these groups have contact with the health system and the general health status of these groups. There is not taken into consideration further factors such as specific comorbidities for individuals in these target groups. Vaccination also reduces the risk of going to hospital independently of reducing the risk of getting infected. See Supplementary Table A2 and Supplementary Fig. A2 for details.Societal opennessDuring the study period, Denmark went from a strict national lockdown in January 2021 to a very open society by July 2021. To describe this in simple terms, it was chosen to use the dominant eigenvector, λ(t), of the activity matrix, A(t), as a proxy for openness in the Danish society, as all national level restrictions are incorporated into the activity matrix. Openness is then defined relative to summer 2021, and the lockdown days reported are relative to the openness of society in summer 2021. Lockdown days reported in the results are relative to the societal activity as it was in the summer of 2021. Using this definition, one lockdown day in summer 2021 where no restrictions are in place would reduce the openness of society as much as roughly two lockdown days in the middle of April were some restrictions are still in place.Seasonal effectTo investigate the seasonal effect β(T), a SEIR model was fitted on hospital admission in 18 regions of Sweden in the period of April 2020 to October 2020. Sweden was chosen as case study, since in this period it had minimal changes in mitigation strategy, and because Sweden is assumed to have societal behavior similar to Denmark (see Supplementary Methods A.6.2 for details). Later, a similar model fitted to Denmark directly showed a similar shape and size of effect18 as the model fitted to Sweden.Tests for COVID-19The weekly number of tests in EU member states was downloaded from European Centre for Disease Prevention and Control’s webpage2. The number of tests was aggregated for weeks 1 through 26 of 2021 (until July 4, 2021), at the country level and divided with population size. Austria, Belgium, Hungary, Latvia, and Portugal had incomplete observations and were excluded from the study. The mean and median of the total number of tests divided by population were calculated excluding Denmark. The ratio of the number of tests per inhabitant performed in Denmark compared to the other EU member states was 13 to the median and 8 to the mean. It was therefore decided that the limited test scenarios should reduce the number of tests by a factor of 10 compared to the actual performed number of tests.The cost of testing is estimated by using a price of 16.1 EUR (120 DKK) per PCR test and 20.2 EUR (150 DKK) per antigen test, which is not an official price but estimated by the Danish national broadcaster (DR) on basis of information from various Danish authorities19. Both figures are rounded to the nearest 10 DKK, the DKK/EUR conversion rate used throughout this paper is 7.44, which is the euro-cent rounded average of the rate during 2021. The price is the total price including salaries for public employed workers, and overhead for private companies. In Denmark, almost all PCR tests conducted were done by the public sector, while most antigen tests were performed by the private sector. Prices per inhabitant were then found by multiplying by the number of tests in the scenarios and dividing by the number of inhabitants and days in the study period.Mass testing has the effect in the model that infected individuals may be found prior to being symptomatic with the disease, with a probability related to the test frequency in the group which the individual belong, hereafter they are isolated and cannot transmit the disease. In the ‘Limited test’ scenarios at least 50% of cases are still found due to being symptomatic, but they will typically be found after having transmitted for longer periods. Local lockdowns reduce to varying degree in- and out-going contacts of all individuals belonging to a geographic entity as a function of the proximity of the observed case incidence to the limits. Local lockdowns last at least seven days, but even when lifted high observed case incidences may still reduce activity in the entity (See Supplementary Methods A.6.1 for details on incidence limits).Supplementary Methods contains Supplementary Figs. A1–A10 and Supplementary Tables A1–A4.Reporting summaryFurther information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
Digital twin simulation modelling shows that mass testing and local lockdowns effectively controlled COVID-19 in Denmark
