Fig 1.
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.
Table 1.
Test countries, with measures implemented.
Fig 2.
(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.
Fig 3.
(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.
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 μld (orange) is active for four days to isolate and protect a large population fraction rapidly (blue). As an example, we show μld = 0.28d−1 if t ∈ [10, 14]. It results in protecting about 68% of the population in two days. Higher values, e.g. μld = 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.
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.
Fig 6.
Dependence of latent carriers quarantining on control parameters.
Assessing the impact of Pfind 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.
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.
Fig 8.
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.
Table 2.
SPQEIR model parameters.
Table 3.
COVID-19 significant dates.
Fig 9.
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 tm to tp, 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).