The classical SEIR model

SEIR models are continuous-time, mass conservative compartment-based models of infectious diseases [4, 13]. They assume a homogeneously mixing population (or fully connected graphs) and focus on the evolution of mean properties of the closed system. All of these models are classical and widely used tools to investigate the principal mechanisms governing the spread of infections and their dynamics. There is a broad range of such models, from more conceptual to more realistic versions, e.g. SEIR with delay [14], spatial coupling [15, 16], extended compartments [17], or those that consider progression of treatments and age distribution [18].

Main compartments of SEIR models (see Fig 1, framed) are: susceptible S (the pool of individuals socially active and at risk of infection), exposed E (corresponding to latent carriers of the infection), infectious I (individuals having developed the disease and being contagious) and removed R (those that have processed the disease, being either recovered or dead). The model’s default parameters are the average contact rate β, the inverse of mean incubation period α and the inverse of mean contagious period γ. When focusing on infection dynamics rather than patients’ fate, the latter combines recovery and death rate [19]. From these parameters, epidemiologists calculate the “basic reproduction number” R₀ = β/γ [20] at the epidemic beginning. During the epidemic progression, isolation after diagnosis, vaccination campaigns and active mitigation measures are in action. Hence, we speak of “effective reproduction number” [21].

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Fig 1. Scheme of the SPQEIR model.

The basic SEIR model (framed blue blocks) is extended by the red blocks to the SPQEIR model. Parameters that are linked to mitigation strategies are shown in red. Interpretation and values of parameters are given in Table 2.

https://doi.org/10.1371/journal.pone.0252019.g001

Data and analyzed countries

When investigating the ability of our conceptual model to explain mitigation, we compared it with empirical data. To do this, we considered the main non-pharmaceutical interventions applied by several countries, by integrating multi-disciplinary information. In fact, governments worldwide have issued a number of social measures, including those for public health safeguard, economic support, movement restriction and non-pharmaceutical interventions to hamper disease spreading. Scholars from political sciences and sociology have recorded and classified such measures [22, 23]. Among the resources listed on the World Health Organization “Tracking Public Health and Policy Measures” [9], we used information from the ACAPS database [24] that contains a curated categorization of policy measures. ACAPS is an independent, non-profit information provider helping humanitarian actors to respond more effectively to disasters. The ACAPS analysis team has aggregated and classified interventions from different sources (media, governments and international organizations), for all countries and in time. Mitigation measures against the epidemic are classified under “Movement restrictions”, “Lockdown”, “Social Distancing” and “Monitoring and Surveillance”. Our modelling choice is based on these categories, which are reflected by additional compartments to the classical SEIR model (see next section).

Epidemiological data for all selected countries and regions were obtained from the COVID-19 Data Repository by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University [25]. The data are from 22 Jan 2020 to 08 July 2020. Lombardy data were obtained from the Protezione Civile Italiana data repository “Dati COVID-19 Italia” [26], from 22 Feb 2020 to 08 July 2020. We acknowledge that the quality of data of early detection of COVID-19 cases is often associated with limited testing capabilities, which could bias subsequent analysis. However, across the analysed countries, the share of positive tests was similar (see e.g. [27]), and no significant deviation from expected dynamics was observed by studies applying Benford’s law [28, 29], possibly indicating that these data still capture to a reasonable extent the dynamics of the epidemic wave. In addition, the analysed countries were selected based on the fact that they sufficiently met the other model assumptions, e.g. low spatial heterogeneity and large amount of cases to fulfill the mean field assumption.

This study analyses the effect of mitigation measures in flattening the curve. Despite having a precise starting date, such measures take some days to be fully effective. We estimate an average delay using the Google Mobility Reports [27, 30] for the selected countries. Google provides changes in mobility with respect to a monthly baseline, w.r.t. 6 locations: Retail & Recreation, Grocery & Pharmacy, Transit stations, Workplaces, Residential, Parks. We average the decrease in mobility at the first four locations (corresponding to those where social mixing happens more frequently [31]) to get a proxy of the time needed for hard lockdown to be fully effective (cf. Fig 4c).

The extended SPQEIR model to reflect suppression strategies

SEIR models reproduce the typical bell-shaped epidemic curves for the number of infected people. The dynamics of this curve is of high importance for practical policy making. Not only it relates to the main stressors for the health system [17, 18, 31], but it also has an impact on the economic system [32–34], e.g. because it takes some time () to mitigate the curve, until the number of new infections is below an accepted threshold. Commonly, mitigation measures against epidemics aim at flattening the curve of new infections [10]. However, the classical SEIR model is not granular enough to investigate mitigation measures when they need to be considered or should be sequentially reduced if already in place. Therefore, we extend the classical SEIR model as in Fig 1 (red insertions) into the SPQEIR model, to reflect the intervention categories described above. We particularly focus on the control of Susceptible and Exposed people, given by preventive isolation, contact tracing or social distancing measures, but we also include the control of infectious people by isolation. The model can be summarized as follows:

• The classical blocks S, E, I, R are maintained;
• A social distancing parameter ρ is included to tune the contact rate β;
• Two new compartments are introduced where:
  • Protected P includes individuals that are removed from the susceptible pool and are thus protected from the virus. This can happen through full isolation as in China in early 2020 [35] or by different vaccination strategies which reduce the susceptible pool;
  • Quarantined Q describes latent carriers that are identified and quarantined after monitoring and tracing of contacts.

We do not explicitly introduce a second quarantined state for isolation of confirmed cases after the Infectious state [17, 36] but consider this together with the Removed state, by tuning the removal rate with an extra parameter (see [37] and references therein). Quarantining infected symptomatic patients is a necessary first step in every epidemic [38]. An additional link from Q to R, even though realistic, is neglected as both compartments are already outside the “contagion system” and would therefore be redundant from the perspective of evolution of the infection. In general, protected individuals can get back to the pool of susceptible after a while, but here we neglect this transition, to focus on simulating mitigation programs alone at their early stage. Long-term predictions could be modelled even more realistically by considering such link, that would lead to an additional parameter to be estimated and is beyond the scope of the present paper.

The model has in total 7 parameters. Three of them (β, α, γ introduced in Fig 1) are based on the classical SEIR model. The new parameters ρ, μ, χ, η account for alternative mitigation programs to control the infectious curve (see Table 2 for details). Commonly, social distancing is modelled by the parameter ρ. In a closed-system setting where all individuals belong to the susceptible pool, but interact less intensively with each other, ρ tunes the contact rate parameter β, resulting in the effective reproduction number . The parameter μ stably decreases the susceptible population by introducing an active protection rate. This accounts for improvements of public health, e.g. stricter lockdown of communities, or reduction of the pool of susceptible people after reduced commuters’ activity, or vaccination. The parameter χ introduces an active removal rate of latent carriers. Intensive early contact tracing and improved methods to detect asymptomatic latent carriers may enhance the removal of exposed subjects from the infectious network. Following earlier works [39, 40] and adjusting the current parameters, can be then expressed as . Finally, η models the isolation of contagious individuals by handling the removal rate. This would correspond to identifying infectious individuals before they recover or die, and prevent them from infecting other susceptibles. Consequently, for this parameter alone . Parameter values that are not related to mitigation strategies are set from COVID-19 epidemic literature [37, 41], as the main focus of the present model lies on sensitivity analysis of mitigation parameters. Our model can be further extended by time dependent parameters [38]. Default values for mitigation parameters are {ρ, μ, χ, η} = {1,0,0,0}, corresponding to the classical SEIR model.

The dynamics of our SPQEIR model is described by the following system of differential equations: Here, with N = S + E + I + R + P + Q, implying the conservation of the total number of individuals. As value for the qualitative study, we used N = 10,000. For the cross-country assessment, N is adjusted to true population values for each country. Overall, the effective reproductive number becomes (1)

Mitigation measures are initiated several days after the first infection case. Hence, we activate non-default parameter values after a delay τ. For data fitting, we fit and compare τ to the official date when measures are initialized (cf. Table 1). To integrate the model numerically, we use the odeint function from scipy.integrate Python library.

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Table 1. Test countries, with measures implemented.

https://doi.org/10.1371/journal.pone.0252019.t001

Model fitting

To show how our conceptual analysis is able to reproduce and explain empirical data, we fit the model to the official number of currently infected (active) cases of the first epidemic wave (winter-spring 2020), for each considered country. The choice is corroborated by the fact that all considered countries applied rapid, population-wide measures [24]. Model fitting to the infectious curves is performed in two steps, using the parameters known to be active (cf. Table 1). First, we estimate the “model consistent” date of first infection, so that the simulated curve matches the reported data of active infections. This initial step corresponds to setting the time initial conditions of the SEIR model [17]. The fitting is performed with default parameter values, on a subset of data corresponding to the first outbreak, from first case until when measures are implemented (cf. Table 1). We use a grid search method for least squares, sufficient to fit a single parameter: (2) where t₀ is the “model consistent” estimated date of first infection, t_m refers to the date measures are implemented, and x are respectively reported and model-predicted data, and n is the number of points between t and t_m.

The second step estimates a reasonable set of the mitigation parameters that yield the best fitting of the simulated SPQEIR curve on reported data, during the first phase with implemented measures. This period is identified between the starting date t_m (also included in the fitting) and the phase-out date t_p, cf. Table 1. Holding the epidemic parameters to literature values to achieve cross-country comparison on intervention parameters alone, the fitting is performed for a set of mitigation parameters relative to each country, as reported by policy databases (cf. Table 1). The fit is performed with the widely used lmfit Python library. In S1 Text, we discuss such fitted parameters set and alternative ones.

We also perform a comparative quantitative analysis between our extended model and the simplest SEIR that lumps parameters under a single “social distancing” ρ. This allows comparing the estimate reproduction number and shows the similarity or divergence of different control strategies in explaining the data. To assess how well they allow to fit the data, we employ the classical reduced χ² statistics to evaluate the goodness-of-fit for each of the two models, considering the degrees of freedom [43]: (3) where n′ is the number of data points until phase-out, k is the number of parameters in the model, y_j are estimated values (from data) and the expected ones (from model simulations).

Simulations of single suppression measures

Only social distancing.

The parameter ρ captures social distancing effects, taking values in the interval [0, 1], where 0 indicates no contacts among individuals while 1 is equivalent to no action taken. To perform the current simulations, we assume a delay τ in implementing the measures of 10 days. Such value does not modify the qualitative behavior of the epidemic dynamics but influences the quantitative estimations of peak height and mitigation timing. We refer to S1 Text for further discussion. Overall, Fig 2 reports simulation results about the effects of ρ. The curve of infectious is progressively flattened by social distancing (Fig 2a) and its peak mitigated (Fig 2b). However, the time to mitigation gets delayed for decreasing ρ, until a threshold yielding a disease-free equilibrium rapidly (Fig 2c). In this case, the critical value for ρ is 0.4, leading to . Fig 2c reveals that the dependence of on ρ is not monotonous. With the current settings, values of ρ ≤ 0.3 are best effective to minimise the mitigation timing. In general, the optimal ρ value that minimises mitigation timing depends on τ, as discussed in S1 Text. In fact, longer delays in issuing interventions are not only associated with higher peaks in the infection curves, but also in more stringent parameter values that are necessary to obtain minimal . This fact further stresses the importance of prompt interventions to control the quantitative aspects of epidemic mitigation.

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Fig 2. Effect of social distancing.

(a) Effects of social distancing on the epidemic curve. The grey area indicates when measures are not yet in place. (b) The peak is progressively flattened until a mitigation is reached for sufficiently small ρ. For these settings, the critical value for ρ is 0.4 (it pushes below 1). (c) Unless ρ is small enough, stronger measures of this kind might delay the mitigation time of the epidemic.

https://doi.org/10.1371/journal.pone.0252019.g002

Only active protection.

As discussed above and in S1 Text, our simulations take into account 10 days delay from the first infection to the initiation of active protection. Small values can reflect continuous improvement of protection measures (as people learnt better how to deal with the virus) or different vaccination strategies (thus going beyond non-pharmaceutical strategies). Higher values are considered to model certain effects of a step-wise hard lockdown (see following paragraph). The results are reported in Fig 3. We see that small precautions can make an initial difference (Fig 3a and 3b). The time to zero infectious is decreased with higher values of active protection (Fig 3a and 3b). In particular, μ = 0.01d⁻¹ mitigates the epidemics in about 6 months by protecting 70% of the population. Higher values of μ achieve mitigation faster, while protecting almost 100% of the population. It is probably not fully realistic to consider that these protection rates are obtained only by isolation. Instead, they could represent improved hygiene routines or vaccination strategies and are thus worthy to consider.

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Fig 3. Effect of lock-down.

(a) Effects of active protection on the infectious curve. The grey area indicates when measures are not yet in place. μ is expressed in d⁻¹. (b) Dependency of peak height on μ: the peak is rapidly flattened for increasing μ, then it is smoothly reduced for higher parameter values. (c) High μ values are effective in anticipating the mitigation of the epidemic, but require protecting more than 90% of the population.

https://doi.org/10.1371/journal.pone.0252019.g003

In addition to what analysed above, we also consider strategies which isolate many people at once [44]. This corresponds to reducing S to a relatively small fraction rapidly. Since μ is a rate, we mimic what could happen during a step-wise hard lockdown: large values of μ, but whose effect only lasts for a short period of time (Fig 4b). We thus use the notation μ_ld. In the figure, an example shows how to rapidly protect about 68% of the population with a step-wise μ_ld function. In particular, we use an average four-days long step-wise μ_ld function (Fig 4b) to mimic the rapid, but not abrupt, change in mobility observed in many countries by Google Mobility Reports [30] (Fig 4c). The effects of strong, rapid protection are reported in Fig 4a, showing that such strategy is effective in mitigating the epidemic curve and in reducing the time to mitigation.

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Fig 4. Effect of step-wise hard lock-down.

(a) Flattening the infectious curve by hard lockdown. Rapidly isolating a large population fraction is effective in mitigating the epidemic spreading. (b) Modeling hard lockdown: high μ_ld (orange) is active for four days to isolate and protect a large population fraction rapidly (blue). As an example, we show μ_ld = 0.28d⁻¹ if t ∈ [10, 14]. It results in protecting about 68% of the population in two days. Higher values, e.g. μ_ld = 0.65d⁻¹ would protect 93% of the population at once. (c) Google Mobility Report visualization [30] for analysed countries, around the date of measures setting. Each line reports the mean in mobility change across Retail & Recreation, Grocery & Pharmacy, Transit stations, and Workplaces, around the date of implementation of the measures. A minimum of four days (from top to bottom of steep decrease) is required for measures to be fully effective. Abbreviations explanation: AT = Austria, CH = Switzerland, DK = Denmark, IL = Israel, IR = Ireland, LO = Lombardy.

https://doi.org/10.1371/journal.pone.0252019.g004

Only active quarantining.

Controlling latent carriers before symptom onset is an important strategy to limit transmission. We here consider how mitigation is achieved by targeted interventions, e.g. by contact tracing, and we quantify the interplay between precision and delay in tracing, thus expanding [45]. As above, not only we consider the impact on but on the whole infectious curve, its height and its time evolution.

The simulations in this part are based on realistic assumptions: testing a person is effective only after a few days that that person has been exposed (to have a viral charge that is detectable). This induces a maximal quarantining rate θ, which we set θ = 0.33d⁻¹ as testing is often considered effective after about three days from contagion [46]. Therefore, we get the active quarantining rate χ = χ′ ⋅ θ, where χ′ is a tuning parameter associated e.g. to contact tracing. As θ is fixed, we focus our analysis on χ′. As above, we also assume that testing starts after the epidemic is seen in the population, i.e. some infectious are identified with 10 days delay in the activation of measures.

The corresponding results are reported in Fig 5. The curve is progressively flattened by latent carriers quarantining and its peak mitigated, but the time to mitigation gets delayed for increasing χ′. This happens until a threshold value of that pushes below 1. This value holds if we accept a strategy based on testing, with θ = 0.33. If preventive quarantine of suspected cases does not need testing (for instance, when it is achieved by contact tracing apps), the critical χ′ value could be drastically lower. In particular, if θ = 1d⁻¹, i.e. latent carriers are quarantined the day after a contact.

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Fig 5. Effect of latent carriers quarantining.

(a) Effects of active latent carriers quarantining on the epidemic curve. The grey area indicates when measures are not yet in place. (b) The peak is progressively flattened until a disease-free equilibrium is reached for sufficiently large χ′. (c) Unless χ′ is large enough, stronger measures of this kind might delay the mitigation of the epidemic. Note that the critical χ′ can be lowered for higher θ, e.g. if preventive quarantine does not wait for a positive test.

https://doi.org/10.1371/journal.pone.0252019.g005

The parameter χ′ tunes the rate of removing latent carriers. Hence, it combines tracing and testing capacities, i.e. probability of finding latent carriers (P_find) and probability that their tests are positive (P₊). The latter depends on the false negative rate δ₋ as (4)

So, χ′ = P_find ⋅ P₊. Hence, mitigating the peak of infectious requires an adequate balance of accurate tests and good tracing success as reported in Fig 6. Further quantifying the latter would drastically improve our understanding of the current capabilities and of bottlenecks, towards a more comprehensive feasibility analysis.

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Fig 6. Dependence of latent carriers quarantining on control parameters.

Assessing the impact of P_find and P₊ on the peak of infectious separately. This way, we separate the contribution of those factors to look at resources needed from different fields, e.g. network engineering or wet lab biology. Solutions to boost the testing capacity like [47] could impact both terms.

https://doi.org/10.1371/journal.pone.0252019.g006

Only isolation of infectious.

Isolating contagious individuals is a first step to contrast the pandemic, on top of preventive measures. In this section, we consider its effect alone, to be compared with that of other single parameters shown above. As discussed above, we here consider simulations that include a delay of 10 days from the first infection to the initiation of the measures. Quantitative changes associated with different τ are discussed in the S1 Text. The results are reported in Fig 7. Targeting the infectious population means that fewer people can spread the contagion. The curve of infections is progressively flattened, the more rapidly contagious people are identified and isolated, until a threshold value η = 0.51 (for our initial parameters). In turn, the mitigation time gets longer if η is increased, but has not yet crossed the threshold value. These findings point to the importance of complementing the control of contagious individuals with additional preventive measures such as the ones presented above. We acknowledge that these results are valid on average, but that breaking the infectious chain at specific links can have additional benefits in heterogeneous social networks.

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Fig 7. Dependence of infectious isolation on control parameters.

(a) Effects of isolation of contagious individuals on the epidemic curve. The grey area indicates when measures are not yet in place. (b) The peak is progressively flattened until a disease-free equilibrium is reached for sufficiently large η. (c) Unless η is large enough, stronger measures of this kind might delay the mitigation of the epidemic. Note that the critical η can be higher if there is delay in intervening, i.e. if infectious individuals are isolated after several days and can thus spread the infection.

https://doi.org/10.1371/journal.pone.0252019.g007

Synergistic scenarios

Fully enhanced active quarantining and active protection might not be always feasible, e.g. because of limited resources, technological limitations or welfare restrictions. On the other hand, the isolation of a limited portion of contagious individuals could not be sufficient. Therefore a synergistic approach is very attractive as it can flatten the curve with combinations of interventions that target different population groups and require distinct resources. This section shows a number of possible synergies, concentrating as before on abstract scenarios to investigate how combining different mitigation programs impact the control parameter (cf. Eq (1), the infection curves and the mitigation timing.

As case studies, we consider the 6 synergistic scenarios listed below. Parameters are set without being specific to real measures taken: their value is so far conceptual and meaningful when compared across scenarios. Just like above, we consider a 10 days delay from the first infection to issuing measures; as suggested in other studies [48], delaying action could worsen the situation. To differentiate between a rapid isolation and a constant protection, we use μ_ld (associated to “hard lockdown strategies”, see Section “Only active protection”) separated from μ. To get when measures are initiated, we follow Eq 1, considering χ = χ′ ⋅ θ as in Section “Only active quarantining”. Our scenarios are the following:

1. During the first COVID-19 wave, many European countries opted for a lockdown strategy. A quite large fraction of the population was isolated, individuals were recommended to self-quarantine in case of suspected positiveness, social distancing got mandatory but was sometimes not fully followed, masks and sprays were suggested for protection. So, we set an initial “rapid protection” μ_ld = 0.12 to protect around 38% of the population quickly. Then we chose ρ = 0.7, χ′ = 0.12, η = 0.12 and μ = 0. This yields .
2. In case that isolation of contagious individuals fails, an alternative procedure is to rapidly protect only the population fraction at high risk (μ_ld = 0.06, driving 15% of initial S to P). Social distancing and latent carrier quarantine should then be enforced (ρ = 0.65, χ′ = 0.55). This gives .
3. In case both preventive quarantine of latent carriers and isolation of contagious are not greatly effective (χ′ = 0.03, η = 0.07), and in case of low protection rate and scarce isolation (μ = 0, μ_ld = 0.08), we rise social distancing for all individuals doing business as usual (ρ = 0.45). In this case, .
4. If there are no safety devices that provide an adequate protection (μ = 0) and no isolation is foreseen (μ_ld = 0), we set ρ = 0.6, χ′ = 0.2, η = 0.25 to get .
5. This case has higher than the previous ones, namely . The corresponding parameters are μ_ld = 0.1, μ = 0.002, ρ = 0.7, η = 0.1. This shows that even low enforcement of single interventions can achieve , even thought the corresponding mitigation is slower.
6. Finally, we consider “draconian” [49] measures such that only through isolation and massive screening, that targets Exposed and Infectious individuals. So, μ_ld = 0.3, χ′ = 0.1, η = 0.2 while ρ = 1 and μ = 0. This points to the importance of tracing capacities to minimise the total isolation period.

Simulation results in Fig 8 show that different synergies can lead to different timing, even though the peak is contained similarly (Fig 8a). This has an impact on the cumulative number of cases (Fig 8b) that will be reflected on the death toll. This holds even when the values are very close, as in scenarios 1 to 4: even though is the main driver of the epidemic, the contribution of finer-grained parameters is relevant for the fine-tuning of interventions. Focusing on scenarios 2 and 3, we notice that prevention measures and latent quarantine accelerate the mitigation, even when isolating only vulnerable people. This achieves similar effects as strong social distancing. In addition, active protective measures with relatively low values further concur in mitigating the peak. This finding asks for rapid assessment of masks and sanitising routines.

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Fig 8. Synergistic scenarios.

Simulations of the 6 synergistic scenarios. (a) Curves of infectious Individuals, (b) Cumulative cases. The grey area indicates when measures are not yet in place. It is evident that scenarios leading to similar could show different patterns and mitigation timing. (c) Distribution of times to zero infections for different scenarios.

https://doi.org/10.1371/journal.pone.0252019.g008

Overall, the strength of mitigation measures influences how and how fast the epidemic is flattened. μ_ld mostly governs the peak height after measures are implemented, ρ mainly tunes the curve steepness together with μ, while χ shifts the decaying slope up and down. Overall, a suffices to avoid breakdown of the health system, but its effects could be too slow. Decreasing its value with additional synergistic interventions could speed up epidemic mitigation. A careful assessment of measures’ strength is thus recommended for cross-country comparison.

Model fitting and interventions assessment

In this section, we test our results on several datasets, to estimate the likely impact of different strategies and to show which combination could have yield a similar . This way, we show how countries could achieve mitigation through a synergy of control measures with similar impact on the epidemic but different management and possibly socio-economic impact.

Model fitting.

As described in the Methods section, we first estimate the “model consistent” date of first infection t₀, i.e. the temporal initial condition for the SPQEIR model. Comparing this date with the starting date for intervention measures (Table 1) corresponds to estimating τ for each country. We do not claim this to be the true date of first infection in a country; it is the starting date of infections in case of homogeneous transmission, under the assumption of no superspreading events [50], and with the hypothesis of coherent R₀ (cf. Table 2). During the second fitting step, we also estimate the date at which mitigation measures start having effect on the infectious curve, t_m. Comparing t_m with official intervention dates from Table 1, we notice that about 8 days are necessary to register lockdown effects. This is consistent to early findings on lockdown effectiveness [51]. Estimated dates are reported in Table 3.

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Table 2. SPQEIR model parameters.

https://doi.org/10.1371/journal.pone.0252019.t002

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Table 3. COVID-19 significant dates.

https://doi.org/10.1371/journal.pone.0252019.t003

Then, we fit mitigation parameters to data of active cases, from the estimated starting date of control measures t_m to phase-out t_p (cf. Table 1). The active parameters for the fit are reported in the same table. For the protection parameter, we used μ_ld acting on 4 days (as introduced in Fig 4) since it better reflects the rapid protection of certain individuals that happened during the first COVID-19 wave. Since S ∼ N, its quantitative impact is anyway greater on the S compartment. Other compartments are impacted by the remaining parameters. The results of the model fitting are reported in Fig 9. The SPQEIR model, with appropriate parameters for each country (cf. Table 1), is fitted to reported infection curves and, overall, model fitting have good agreement with data. This supports the model structure as very simple yet realistic enough to capture the main dynamical behaviour of the infection curves in multiple countries. In addition, it allows for each country to obtain multiple sets of parameters representing different strategies. We notice that the effect of social distancing (ρ) is predominant as it homogeneously prevents the big pool of Susceptible individuals to stream into the Exposed compartment. However, also tracing and isolation can have a considerable effect in complementing population-wide interventions. The values associated to the fitted parameters correspond to non-negligible numbers of individuals affected by the interventions. Such values are discussed in S1 Text. This is informative about how synergistic approaches can realistically explain the mitigation of the infectious curve, and highlight the potential advantages associated with modifying the combination of strategies in subsequent epidemic waves. Finally, it allows a comparison between different countries through the corresponding best fit parameters. For Ireland, although initial social distancing advises were issued on 13th March (cf. Table 1), fitting the complete curve was only possible when considering the lockdown date (28th March) as the major driver of the mitigation.

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Fig 9. Results of model fitting.

Results of model fitting. Infection curves for the considered countries (dotted) are fitted with the SPQEIR model with appropriate parameters (red curves). We also show a comparison with the fitted curve obtained from the “basic” SEIR model with only social distancing (turquoise curves). Parameter values are reported for each country, as well as the corresponding (for the grey area, following Eq 1) and . The period of measures enforcement, from t_m to t_p, is highlighted by the grey region. Time progresses from the estimated day of first infection t₀ (cf. Table 3). Population fraction refers to country-specific populations (cf. Table 1). After phase-out, we prolong the fitted curve (parameter values unchanged) to compare observed data with what could have been if measures had not been lifted (dashed lines). From the data, we can observe a resurgence of cases that points to possible “second outbreaks” (particularly in Israel).

https://doi.org/10.1371/journal.pone.0252019.g009

Model fitting is slightly hindered by data quality. For instance, Ireland reported intermittent data, while Lombardy is not perfectly represented, probably because of some data reporting issues and larger heterogeneity in its spacial patterns.

Finally, the reduced χ² metric (Eq 3) reports that the complete SPQEIR model and the simple social distancing one attain similar goodness of fit, although values for the SPQEIR are slightly lower in all cases. Country-specific extra parameters (cf. Table 1) are thus useful to fine-tune the reproduction of epidemic curves, as noticed in the conceptual analysis, Sec. “Synergistic scenarios”. This shows that synergistic measures are able to provide a similar mitigation of the curve of infections, and an analogous , as the pure social distancing scenario. In turn, synergistic approaches allow lower social distancing values, possibly having less severe social and psychological impacts on the population. This in turn supports the use of several interventions to control the epidemic curve in an effective and timely manner, while balancing social benefits. In addition, the SPQEIR is confirmed to be informative, on top of being fully interpretable and linked to recognised social policy categories.