2.1 Compartmental epidemiological model

In this work, the evolution of COVID-19 is modeled for the entire population of a region, which is assumed to be isolated. The model is an extension of a basic SEIR model [23], applied to a closed population (i.e. no births, deaths, immigration or emigration) divided into n age groups. This model classifies individuals of a population into the following mutually exclusive categories: (susceptible), (exposed but not infectious), (infected), (mild symptoms), (severe symptoms), (critical symptoms), (recovered) and (dead). The index j = 1 , . . . , n denotes the corresponding age group.

The flow between epidemiological categories of the model is shown in Fig 1. Infected individuals in the age group j can interact with susceptible individuals in the age group k with a transmission rate . Susceptible individuals exposed to the disease move to the exposed compartment . Individuals in this compartment do not transmit the virus. After a mean incubation time , exposed individuals move to the infected group . At this stage, individuals can spread the virus to susceptible persons during the period . Subsequently, individuals move to the compartments , or with probabilities , and , respectively. The group comprises individuals with severe cases that require hospitalization and, after a time , recover from the disease and move to the recovered compartment . The compartment (critical) represents the individuals with severe cases that require hospitalizations and, after a time , die and move to the dead compartment . The compartment consists of the individuals who present mild symptoms and require no hospitalization, and after a time , they transit to the recovered compartment. The model does not incorporate transitions between , and , because these compartments do not represent the illness progression but rather indicate the worst state that an individual has reached. If an individual is misclassified in one of these compartments, the data assimilation system will make the required adjustments to correct it, as explained in the next section. After a period , individuals from the recovered compartment become susceptible again, given that SARS-COV-2 immunity diminishes substantially after 5-7 months [24]. The compartments are designed to characterize the dynamics of COVID-19 infection. Individuals are unable to transmit the virus in the initial incubation phase becoming infectious afterward. They are also expected to be isolated once the symptoms are apparent (or tested positive). Therefore, individuals in , , or are expected to be isolated and do not spread the disease, only individuals in the compartment do.

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Fig 1. Diagram of the compartmental model.

An individual moves to the next compartment after a period , which depends on the compartment X.

https://doi.org/10.1371/journal.pone.0318426.g001

The model parameters are the transmission matrix parameters (which is the number of contacts that a person in group j have with persons in group k, in a period of time Δt, multiplied by the probability of that contact resulting in an infection), the average time an individual stays in each of the epidemiological states , , , , and , and the fractions of infections and of moving to and , respectively. The population of each age compartment is constrained by the total population of the age group .

The resulting model equations are

(1)

Table 1 summarizes the variables and the parameters and Table 2 shows the numeric values of all the fixed parameters. In [25], is reported, and in [26] , while and are set both to 15 days following [14]. The fractions of hospitalizations, , in Table 2 were estimated from the early stages of the pandemic using the available data. Eqs (1) are integrated with the Euler method using a time step of 1h.

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Table 1. Model variables and parameters.

https://doi.org/10.1371/journal.pone.0318426.t001

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Table 2. Numeric values of the model parameters. All time scales (τ’s) are expressed in days.

https://doi.org/10.1371/journal.pone.0318426.t002

The most important parameters controlling the spread of a disease in a meta-population model are the elements of the transmission matrix. In a population divided into age-compartments, these elements represent the interaction between the infected and susceptible age groups, hence they are the main driver of the disease evolution. One of the central objectives of this work is to parameterize and estimate, from observed data, the transmission matrix from observations to obtain a better representation of the propagation between age groups. In Eq (1), the elements of the transition matrix are not independent [20]: the total number of contacts that individuals in the group j have with those in group k has to be equal to the total amount of contacts that individuals in group k have with those in group j:

(2)

The most relevant parameter in epidemiological modeling is the basic reproduction number. It represents the mean number of new infected individuals caused by one infected person in a totally susceptible population. The basic reproduction number may be estimated in compartmental models by linearizing the dynamics of the infected differential subsystem, which is the part of the model that governs the production of new infections when all individuals are susceptible (in a SEIR model, for example, this subsystem are the compartments SEI). The resulting Jacobian matrix is known as next generation matrix [27], whose spectral radius corresponds to the basic reproduction number R₀. If the linearization of the infected subsystem is conducted at time t, the spectral radius of the resulting matrix is known as the effective reproduction number . This represents the number of secondary cases that an infected individual produces at time t, assuming the remaining non-infected or recovered population is susceptible. A review of the topic can be found in [28].

2.2 State-parameter estimation with ensemble-based data assimilation

The evolution of epidemiological variables can be modeled as a partially observed time-evolving process, i.e. a hidden Markov model [29]. Within this framework, the evolution of the state of the system can be written as

(3)

where x_k is the state of the system at time k, M ( )  is the dynamical model and η_k is the model error, representing simplifications and numerical approximations in the model. It is assumed to be a realization of (normal distribution of mean 0 and covariance matrix ). In our case, the dynamical model M ( )  is (1). The second equation set forming the hidden Markov model corresponds to the observational map (in other words, the relationship between state variables and the observed quantities that can be measured such as the daily number of new infections). The observations are related to the state x_k by the observation operator which maps the space of state variables to the observational space:

(4)

where ϵ_k is the observation error assumed to be a realization of and Ř is the observation covariance matrix (assumed known). This represents the uncertainty in the observed values such as those produced by limitations in testing strategies, delays in the transmission of reports, etc. The observation covariance matrix is assumed diagonal

(5)

where is the variance of the observed variables. In Section 3, we define . The observation covariance may be estimated with expectation-maximization [30], however here we consider a fixed observation covariance.

If the the hidden Markov model (3) and (4) satisfies the conditions of detailed balance, the inference may be greatly simplified. Here we assume a general framework and use filtering theory for the inference.

In filtering theory, the estimation problem involves obtaining the conditional probability density function (pdf) of x_k knowing the current and past observations , denoted by p ( x_k | Y_k )  (a.k.a. filtering or analysis distribution). We can obtain the prediction pdf by performing a forecast step

(6)

then, using Bayes theorem, the posterior density conditioned on the set of observations is derived,

(7)

Eqs. (6) and (7) can be solved sequentially every time new observations y_k are available, but they have to be integrated over the entire state space, which is usually computationally intractable. However, using a sample-based representation of the distributions, the forecast step can be approximated by a Monte Carlo approach by simply evolving every sample point forward with the model M ( ) . In this work we employ the EnKF, which is a Monte Carlo non-linear extension of the Kalman Filter [31]. The analysis distribution is represented by an ensemble of possible states. The resulting analysis state members are of the form

(8)

where the supra-index i denotes the i-th ensemble member. Each forecast state member is obtained by evolving the model (1) from the previous analysis state: , and then corrected with an innovation term based on the difference between the observation and the forecast state. In Eq (8), the observation vector is perturbed with Gaussian noise: , where . This is required to obtain a sample covariance of the analysis state members with the expected analysis covariance [32]. The matrix K_k, referred to as Kalman gain, gives the weight to the innovation term, and is the rate between the forecast uncertainty and the total uncertainty, namely

(9)

where is the forecast error covariance and Ȟ is the tangent linear observational operator defined by

(10)

In an ensemble Kalman filter, is estimated from the ensemble of forecasted state vectors at time k:

(11)

The analysis mean state, , provides a point estimate of the state of the system.

2.3 Observation operator

Observations are assumed to be cumulative cases () and deaths () both disaggregated by ages. The map from state to observation space is as follows: we assume that the cases are partially documented. This is achieved with a time-dependent parameter in the observational operator, H, which accounts for the sub-detection of cases. In other words, we assume there is a sub-detection bias in some observational variables. This parameter depends on the age group, since the symptoms may increase with age so that the amount of undocumented cases is larger for children. In the age group j, the relation between the cumulative observed cases () and observed deaths () and the state variables at time k is

(12)

where , and

(13)

To avoid index overclutter, we do not include temporal index k in this equation, but note that γ is a time-varying parameter.

For parameter estimation with the ensemble-based data assimilation technique, the model parameters γ to be estimated and the state variables , , , , , , , are concatenated together into an augmented state vector x. Then, the model parameters are estimated in the same way as the state variables, using the EnKF. This parameter estimation methodology is known as augmented state. A review of parameter estimation using various data assimilation methods based on the state augmentation approach can be found in [33]. The fractions of detected cases γ_j are also estimated in this way. The prior density at the initial time is assumed Gaussian for both model variables and parameters. This is coherent with the Gaussian assumption of the Kalman filter. Although the parameters are not part of the model equations, their estimation can be conducted in the same way as for the model variables. They are estimated through their correlations with the observed variables. Therefore, parameter estimation depends crucially on an accurate quantification of the augmented forecast covariance matrix (11).

While chaotic dynamics drive the evolution of state variables leading to an increase in their ensemble spread, persistence is assumed for the time evolution of the parameters. Because of this, an inflation method is required to prevent the parameter ensemble spread from collapsing during the recurrence (e.g. Ruiz et al. 2013) [33].

We conducted preliminary experiments to evaluate the use of multiplicative inflation in the EnKF framework. Despite we use two independent inflation factors, one for the parameters and one for the state variables [34], we were not able to find a suitable set of inflation factors. The attempted combinations resulted in either filter divergence or poor estimation performance. Consequently, we opted for the stochastic approach originally proposed in [35]. The parameter evolution of each ensemble member is modeled as an independent auto-regressive process or correlated random walk with correlation ρ and standard deviation σ, this is

(14)

where ξ is a random vector sampled from a Gaussian distribution with zero mean and identity covariance matrix. The random perturbation is added before the analysis step and is only applied to the parameters ; no inflation is applied to the model state variables.

During the analysis update, the EnKF can result in non-physical values for some model parameters and ensemble members (e.g. negative values for the transmission matrix elements). This is a consequence of the assumption of Gaussian forecast error in the EnKF. To avoid this complication, we force the lower limit of all the estimated parameters to 0 in each ensemble member.

To summarize our estimation method, the EnKF methodology is represented concisely in Algorithm 1.

Algorithm 1: Stochastic ensemble Kalman Filter

3.1 Transmission matrix parameterizations

For an n × n transmission matrix there are independent parameters to be estimated instead of because of the restriction (2). In our case we use three age groups, so the resulting transmission matrix is

(15)

where parameters depend on time.

As shown in the experiments in Section 4, the parameters of (15) are not all identifiable when only information of the accumulated infection cases in each group is available, without details regarding the specific age group responsible for the new exposed individuals.

To overcome this limitation, we propose a parameterization for the transmission matrix with fewer parameters:

(16)

from now on, we call this matrix the parameterized transmission matrix.

This parameterization is a particular case of (15) where the upper diagonal parameters ij are defined as a function of the diagonal elements of the row i and column j: , and the lower diagonal parameters are defined by the constraint (2). The parameter α controls the relative importance of inter-age group and intra-age group infections, with lower values giving more weight to the latter.

3.2 Data

We use three age groups in the range of  [ 0 , 30 ) ,  [ 30 , 65 )  and  [ 65 , − ]  years. This division is motivated because we want to represent age groups with different activities, so that children and young individuals’ activities are mainly school and universities, adults are the working age group and the senior population is assumed to be mainly retired. At the same time, these groups grossly represent different health profiles, with the senior population being the ones that most likely will develop severe symptoms, while the first age group are more likely to have minor symptoms. The total population is assumed to be 44.8 million divided into the three age groups by , , and 4.8 × 10⁶, which represent the approximate number of people within the aforementioned age groups in Argentina (taken from the 2010 population census).

Synthetic observations.

In a data assimilation context, a “twin” experiment is an idealized simulation in which we use a known epidemiological model with a set of “true” parameters to generate synthetic observations by first computing the model state evolution, and then adding random noise in the observation space to simulate the observation error (4). In this way, we generate cumulative cases and deaths for each age group using (1) and add random normal noise to them to generate the synthetic observations. Then, the EnKF is used to reconstruct the evolution of the system and the parameters from the syntethic observations. In this way, we can assess the estimated parameters given by the technique comparing them to the “true” parameters (i.e., the ones used to generate the synthetic observations). The objective of the twin experiments is to evaluate the data assimilation-based parameter estimation in a context in which the true parameters are known and errors in the estimation can be accurately computed. We refer to a “true” state variable or parameter to the state variable or parameter used in the model to generate the synthetic observations.

The model used to simulate the observations uses a “true” transmission matrix which has the form (15).

The “true” parameters are defined as

(17)

The decrease in the transmission matrix parameters at t = 80 d  mimics the effect of a lockdown, and the increase at time 140 d  represents a relaxation to normal conditions but with some sanitary measures (e.g. social distancing, mandatory use of masks in public spaces, etc). These conditions result in a double outbreak situation as observed in Argentina (and several other countries) in the first year of the pandemic.

This time-dependent “true” transmission matrix must be estimated by the assimilation in the twin experiment. Note that the relative changes in the parameters are different for different age groups (i.e. not proportional). We chose on purpose a transmission matrix that cannot be fully represented by the parameterization (16), so that the model used in the estimation is not perfect (some structural uncertainty is introduced in the parameterization process). Another motivation was to represent the resulting different levels of mobility that were found in different age groups.

The true values of the fraction of detected cases , are taken to be 0 . 15, 0 . 2, and 0 . 3 corresponding to the young, adult, and senior age groups. For reference, a well-mixed population detection fraction was estimated in [12] to be 0.16 in the early stage of the pandemic. Intuitively, we expect a higher fraction of symptomatic cases for the elderly age group, as it is the most vulnerable population. The fraction of deaths of each age group is assumed to be 0 . 002, 0 . 05, and 0 . 1 . We used the values showed in [36], weighted by the corresponding population age group in Argentina.

Cumulative infected cases and deaths segmented by age groups are assumed to be daily observed during the time period. For the observational error, we set the observation standard deviation of the accumulated cases to , where indicates the observed cumulative cases for every age group j. We assume that deaths are well documented so the standard deviation of the deaths observational error is . The way we define the observational error means that eventually, all the observations will have the upper limit standard deviation after some time. These observation standard deviations are used to generate the synthetic observations and are also used as the observation variances in the ensemble Kalman filter (9).

4.1 Experiments with synthetic observations

The EnKF estimates all the variables of the system and the parameters of the transmission matrix, which are augmented to the system state vector. The dimension of the state vector is 24 (eight variables in each of the three age groups). From these, 21 variables are independent since there are 3 constraints (last Eq in (1)). The amount of estimated parameters is six in the case of the parameterized transmission matrix: three belonging to the parameterized transmission matrix and three corresponding to the fractions of detected cases, namely the parameter vector is so the augmented state vector dimension is 30. In the case of the full transmission matrix (15), there are nine estimated parameters: six from the matrix and three from the fraction of detected cases, i.e. θ = (λ₁₁, λ₂₂, λ₃₃, λ₁₂, λ₁₃, λ₂₃, γ₁, γ₂, γ₃) so that the augmented state vector dimension is 33.

As mentioned, the EnKF for parameter estimations requires an inflation approach for the parameter spread [33]. The filter exhibits convergence using the correlated random walk, (14), for high values of ρ and σ in the range  [ 0 . 001 , 0 . 2 ] . We performed preliminary experiments to determine optimal values by minimizing the root-mean-square error (RMSE, i.e. ) for the additive inflation parameter values, which we found to be σ = 0 . 05 and ρ = 0 . 999. The same random walk parameter values are used in the real data experiments (Section 4.2).

Fig 2 shows the estimated parameters for the synthetic observation experiment (Section 3.2.1) employing the full transmission matrix (15). There is some delay in the estimated transmission matrix parameters compared to the true parameters at the abrupt changes due to the lockdown measure (both in the beginning and end). Estimated parameters start to adjust to these abrupt changes a few days after the change and they converge to a new value 20-30 days later. The reason for this is that parameters in ensemble-based assimilation systems are estimated through the correlation with observed variables, so these state-parameter correlations take some cycles to adapt to abrupt changes. This behavior can be reduced by tuning up the amount of inflation, at the expense of having an increased spread in the ensemble of estimated parameters and state variables. Overall, the amplitude of the abrupt changes is rather well estimated beyond the mentioned delay.

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Fig 2. Estimated diagonal values (left panels) and off-diagonal parameters (right panels) of the full transmission matrix are shown for the synthetic observation experiments using different initial conditions in the parameters (color lines, IC1, IC2 and IC3).

True parameter values are shown with black lines. Shading around the curves indicates the parameter spread.

https://doi.org/10.1371/journal.pone.0318426.g002

To examine the identifiability and sensitivity to initial conditions of the estimated parameters, three independent experiments with different initial mean parameters (transmission matrix and fraction of detected cases) at t = 0 are shown in Fig 2 denoted as IC1, IC2 and IC3. Some of the time variability of the true parameters is captured. However, the different experiments converge to different estimated parameter values. Different initial conditions of the parameters result in different estimations, and neither of the three experiments is able to estimate precisely the true parameters (Fig 2).

The reason for this lack of identifiability is that an increase in the rate of cases say in the age group 1, can be attributed by the assimilation system to either a change in the parameter or a change in λ₁₁ and . Both scenarios result in the same infection rates so the information provided by the observations is not enough to identify the actual scenario. In the experiment, green curves in Fig 2 (IC2) present a greater underestimation at the end of the lockdown, compared to the orange and blue estimations (IC1 and IC3). This underestimation is balanced with the overestimation of λ₁₂, leading to an evolution of the number of cases consistent with the observations.

Fig 3 shows the estimated daily cases (left panels) and deaths (right panels) of the young (upper panels), adult (middle panels), and senior (lower panels) age groups, using the full transmission matrix (15). Although the assimilated observations are cumulative cases and deaths, we show daily cases, the new cases produced in one day, because differences are more visible. The term “daily cases” was taken from [12]. The three experiments with different initial conditions in the parameters give similar results (curves of the three experiments are indistinguishable in Fig 3). In the three experiments, the EnKF is able to keep track of the observations of cases and deaths in all the age groups, even though the transmission matrix parameters are not identifiable. The ensemble dispersion in the senior age group is relatively larger because the population is almost five times lower than in the other age groups, and all the age groups have the same observation error upper limit so that the relative error of the estimation is higher.

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Fig 3. Estimated daily cases (left panels) and daily deaths (right panels) of the young (upper panels), adult (middle panels), and senior (lower panels) age groups for the full transmission matrix experiment using synthetic observations using different initial conditions in the parameters (color lines, IC1, IC2 and IC3).

The observations are shown with red dots and black lines represent the true variable values (they are almost indistinguishable). Shaded areas around colored curves indicate the corresponding variable spread.

https://doi.org/10.1371/journal.pone.0318426.g003

Given that the transmission matrix parameters are not identifiable using the matrix form (15), we conducted a second set of estimation experiments using the proposed parameterization (16) and the same set of synthetic observations. We took α = 0 . 25 in (15), which represents the degree of intra-group contagious. Fig 4 shows the estimated parameters of the parameterized transmission matrix. The right panels show the values of the diagonal, and the left ones show the values of the upper off-diagonal.

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Fig 4. Estimated diagonal values of the parameterized transmission matrix (left panels) and off-diagonal values (right panels) using synthetic observations for different initial conditions (colored lines, IC1, IC2 and IC3), and the true parameter values (black lines).

Shaded areas around colored curves indicate the parameter ensemble.

https://doi.org/10.1371/journal.pone.0318426.g004

The three experiments converge to the same estimated parameter values, independently of the initial condition. The true values of the parameters cannot be estimated precisely because this parameterization is not able to fit the structure of the prescribed true transmission matrix (17). The parameter estimates in Fig 4 also show a delay in the representation of the sudden parameter changes found at the beginning and at the end of the lockdown period, as found in Fig 2.

Fig 5 shows the effective reproduction number calculated with the next generation matrix for the experiment corresponding to the parameterized transmission matrix (left panel) and to the full transmission matrix (right panel). The true values of can be accurately estimated with both parameterizations (apart from the delay in parameter changes), even when the true transmission matrix is non-reproducible by the parameterized transmission matrix. This result suggests that our parameterized transmission matrix is sufficiently flexible to capture the system’s and its temporal evolution. At the same time, its low enough dimensionality ensures identifiability of its parameters from the available observations.

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Fig 5. Estimated effective reproduction number using the parameterized transmission matrix (left panel) and the full transmission matrix (right panel).

The two panels correspond to different assimilation experiments. Colored curves represent estimations with different initial conditions (IC1, IC2 and IC3), and black curves represent the true parameter values. Shading areas around colored curves indicate the parameter spread.

https://doi.org/10.1371/journal.pone.0318426.g005

Fig 6 shows the fraction of detected cases of each age group (right panels). We expect these parameters to be correlated to the observed accumulated cases and deaths. Therefore, the system should be able to constrain them. The true values of are accurately estimated by the assimilation system, regardless of the initial condition. The spurious peaks estimated in the parameterized transmission matrix at the lockdown transitions are also found in the parameters around time 80 d  and, with much less intensity, at 170 d .

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Fig 6. Estimated fraction of deaths for each age group (left panels) and estimated fraction of detected cases for each age group (right panels).

Left and right panels correspond to different assimilation experiments (either detected fractions or fractions of deaths are estimated). Estimations with different initial conditions are shown with colored curves (IC1, IC2 and IC3), and black curves indicate the true parameter values. Shading around colored curves represent the parameter spread.

https://doi.org/10.1371/journal.pone.0318426.g006

In the previous experiment, we estimated a parameterized transmission matrix and the fraction of detected cases . The cases, deaths, and the parameters can also be estimated alongside the fractions of deaths instead of . To illustrate this, we fix equal to the true values and perform three experiments that estimate the transmission matrix and the fraction of deaths. The parameters are similar to the ones shown in Fig 4. The obtained estimates are shown in the left panels of Fig 6. In all experiments, the estimated parameters converge to the true values, and the sudden change in the estimations is again observed at the times when the true transmission matrix parameters change. The shading in the upper right panel of Fig 6, parameter , is limited to zero because this is the lower bound imposed to parameters in the different ensemble members.

4.2 Experiments with the Argentinian COVID-19 data

An experiment was conducted with the same assimilation system as in the previous section but using the real COVID-19 data from Argentina (Section 3.2.2). In contrast to the twin experiments, the observations in this non-synthetic case may be biased and the observation error covariance is unknown. Indeed, the observed cases are highly noisy. One of the sources of the noise is due to the fact that testing and reports diminish on weekends and increases on Mondays and Tuesdays due to delayed reports, resulting in an spurious weekly cycle in the observations.

We estimate the time-dependent fraction of deaths and the parameterized transmission matrix (16) with α = 0 . 25 in the real observation experiments. The parameter vector in this case is . The estimation must account for a time interval of almost 1.5 years, time in which the SARS-Cov-2 virus underwent multiple mutations, changing the severity of the symptoms, there were advancements in treatments within the healthcare system and it includes the start of the vaccination period. This turns crucial the use of time-dependent parameters.

Fig 7 shows the daily cases (left panels) and daily deaths (right panels) of the young (top panels), adult (middle panels), and senior (bottom panels) age groups. The filter is able to keep track of the observations of each age group. As in the twin experiments, we use three sets of initial conditions, and they yield the same estimation of cases and deaths (in Fig 7 only IC3 estimates are visible). The high-frequency cycle found in the estimations of daily cases and deaths corresponds to the above mentioned spurious weekly cycle. If required, this effect can be mitigated by increasing the observational error of the cases, at the expense of an increase of the uncertainty of the estimations.

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Fig 7. Estimated daily cases (left-side panels) and deaths (right-side panels) using the parameterized transmission matrix in the Argentinian data experiments for the young (upper panels), adult (middle panels) and senior age groups (bottom panels).

Observations are in red dots. Shades around the curves represent the estimated variable uncertainty. The lines of the three estimations, IC1, IC2 and IC3, are indistinguishable.

https://doi.org/10.1371/journal.pone.0318426.g007

The estimated deaths in the young age group have a high dispersion because of the relatively few observed cases compared to the other age groups and their relative errors. Although the estimated accumulated number of deaths is always positive, the daily changes in the number of deaths are sometimes negative for some ensemble members in the young compartment. This non-physical behavior is a consequence of the updates introduced by the observations which may eventually result in the reduction of the estimated number of deaths in order to better fit the observed values.

Fig 8 shows the three independent parameters , i = 1 , 2 , 3 of the parameterized transmission matrix (left panels), and the upper off-diagonal parameters (right panels). All different initial conditions yield the same estimations of the parameters. There is a predominance of the parameters λ₁₁ and given that the majority of the cases occur in the first two age groups. Consequently the interaction parameter between young people and adults is higher compared to and λ₂₃. The filter takes a couple of months to estimate parameters, with a spinup time longer than for the synthetic experiments, so that the estimates during the first two months depend on the chosen initial mean parameters. This parameter behaviour is consistent with the results of Sauer et al [37], where it is shown that an optimal set of parameters of a SEIR model can be estimated using the complete time-series of cumulative cases, but parameter identifiability problems occur when using only cumulative cases in the pre-peak interval. Besides this work compares the use of Markov Chain Monte Carlo and data assimilation methods for inference.

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Fig 8. Estimated diagonal values of the parameterized transmission matrix (left panels) and diagonal values (right panels) for different initial conditions (color lines IC1, IC2, IC3).

Shading around colored lines represent the parameter spread.

https://doi.org/10.1371/journal.pone.0318426.g008

Fig 9 shows the estimated effective reproduction number . The estimated parameter does not depend on the chosen initial conditions. The peak at the start of the pandemic corresponds to the unrestricted spread of the virus until the lockdown starts on March 27, 2020. The filter takes some days to fully estimate the reproduction number up until the lockdown start given the inertia of the system, and after some days, it start to decreace. The estimated periods where corresponds to an increase in the cases up to the peaks. The initial condition IC1 (IC2) with a large (small) initial reproduction number reachs a higher (lower) peak on the lockdown date than the others. The lockdown start date is estimated earlier (later) than observed. The initial condition IC3 estimates correctly the time of the lockdown start. After this spinup time, all estimations converge to the same values.

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Fig 9. Daily cases (upper panel) estimated with different initial conditions and estimated reproduction number (lower panel) using COVID-19 data of Argentina.

Shades around colored curves represent the corresponding variable spread. Vertical grey lines point out periods where , and the vertical red line points out when the lockdown started.

https://doi.org/10.1371/journal.pone.0318426.g009

Fig 10 shows the estimated fractions of deaths of the young (top plot), adult (middle plot) and senior (bottom plot) people age groups. The estimated parameters are slightly higher than the reference values 0 . 002, 0 . 05, and 0 . 1 [36] of the young, adult, and senior age groups. The death fraction of the senior age-group exhibits a large value in the first six months, about 0.175, but then it decreases substantially below 0.1. This may be caused by an improvement in the healthcare system: more hospital beds and artificial ventilators, early detection of low blood oxygen, to name a few.

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Fig 10. Estimated fraction of deaths for young (upper), adults (medium) and senior (lower panel) age groups for different initial conditions (color lines IC1, IC2, IC3), and shades around colored curves represent the corresponding variable spread.

https://doi.org/10.1371/journal.pone.0318426.g010

4.3 Forecasts

To evaluate the potential use of the estimated parameters for decision making, we conducted an evaluation of the performance of the resulting forecasts using the estimated parameterized transmission matrix on the Argentinian COVID-19 data. Once assimilation of the last observations are conducted, individual constant (c), linear (l) and quadratic (q) fits are performed on the last 15 days of the estimated parameterized transmission matrix values to obtain the parameter tendencies. Then, these transmission matrix tendencies are projected 30 days forward, starting from the last value of the analysis state (current day). The extrapolation of the regressed parameters is asssumed to have a standard deviation of , where d is the lead time so that the parameter uncertainty increases over the lead time. Finally, 30-day forecasts are conducted with the free evolution of the model using the projected parameterized transmission matrix and starting from the current (last) analysis state. We compare the forecasts to the analysis daily cases using the entire set of observations over time as the reference. Fig 11 shows some selected forecasts up to 20 day lead time which are started over different dates (every 30 days) of the pandemic and evolved using the linear regression extrapolation of the estimated parameterized transmission matrix. Some forecasts are accurate but some diverge from what actually happened during the tendency changes of the pandemic. The orange shading indicates forecast uncertainty as given by the forecast ensemble members.

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Fig 11. Selected 20-day forecasts (orange curves) conducted at different stages of the pandemic in Argentina using the linear regression.

Initial conditions are separated over 30 days (to avoid overcluttering). Observed cumulative cases are showed as daily new cases as red dots, and analysis of cases in blue using the EnKF with multiplicative inflation. Shades around curves represent ensemble members.

https://doi.org/10.1371/journal.pone.0318426.g011

To assess the accuracy of the different forecasts, a total of 400 forecasts are conducted each started everyday and for a maximum lead time of 30 days. The forecasts cover the time window from June 2020 to the end of August 2021, featuring two peaks of the pandemic so that there is a wide variety of pandemic behaviors. From June 2020, a large amount of cases is detected every day so that we can safely assume a mean-field dynamics in the data assimilation framework. To examine the impact of considering the interactions among different age groups on the forecasts, we repeat the forecast experiments using a well-mixed SEIR model without age-group divisions. This model is obtained setting n = 1 in Eq (1). The initial condition of the well-mixed forecast is the sum along the age groups of the meta-population model.

Fig 12 shows the relative RMSE as a function of the lead time. The behavior of the forecasts is similar along the age groups. There is a clear advantage of the meta-population model forecasts over the well-mixed ones in all age groups. In the meta-population model, the forecast using the extrapolation of the linear regression for the parameters is the most accurate, but in the well-mixed model the constant-fit forecast outperforms the others. In both cases, the quadratic-fit forecast is less accurate.

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Fig 12. Relative average root mean squared error of the quadratic (orange), linear (blue), and constant (red) forecasts for the young (upper panel), adult (middle panel) and senior people (lower panel) using the parameterized transmission matrix and the well-mixed model (labeled ‘wm’).

The different color lines corresponds to the quadratic, linear and constant transmission matrix tendencies (denoted q, l, c in the labels, respectively).

https://doi.org/10.1371/journal.pone.0318426.g012