key: cord-339820-x8r27w14 authors: Guan, L.; Prieur, C.; Zhang, L.; Georges, D.; Bellemain, P. title: Transport effect of COVID-19 pandemic in France date: 2020-07-29 journal: nan DOI: 10.1101/2020.07.27.20161430 sha: doc_id: 339820 cord_uid: x8r27w14 An extension of the classical pandemic SIRD model is considered for the regional spread of COVID-19 in France under lockdown strategies. This compartment model divides the infected and the recovered individuals into undetected and detected compartments respectively. By fitting the extended model to the real detected data during the lockdown, an optimization algorithm is used to derive the optimal parameters, the initial condition and the epidemics start date of regions in France. Considering all the age classes together, a network model of the pandemic transport between regions in France is presented on the basis of the regional extended model and is simulated to reveal the transport effect of COVID-19 pandemic after lockdown. Using the the measured values of displacement of people mobilizing between each city, the pandemic network of all cities in France is simulated by using the same model and method as the pandemic network of regions. Finally, a discussion on an integro-differential equation is given and a new model for the network pandemic model of each age class is provided. Up to now, COVID-19 has widely spread over the world and is much more contagious than expected. The outbreak of COVID-19 has resulted in a huge pressure of hospital capacity and a massive death of population in the world. Quarantine and lockdown measures have been taken in many countries to con- 5 trol the spread of the infection, and has proved the amazingly effectiveness of these measures for the outbreak of COVID-19, in particular in China (see [1] ). Quarantine is a rather old technique to prevent the spread of diseases. It is used at the individual level to constrain the movement of all the population and encourage them stay at home. Lockdown measures reduce the pandemic trans-10 mission by increasing social distance and limiting the contacts and mobility of people, e.g. with cancellation of public gatherings, the closure of public transportation, the closure of borders. COVID-19 may yield a very large number of asymptomatic infected individuals, as mentioned in [2] and [3] . Therefore, most countries have implemented indiscriminate lockdown. But the long time 15 of duration of lockdown can cause inestimable financial costs, many job losses, and particularly psychological panic of people and social instability of some countries. As declared by some governments (see [4] ), testing is crucial to exit lockdown, mitigate the health harm and decrease the economic expensation. In this paper, 20 we consider two classes of active detection. The first one is the short range test: molecular or Polymerase Chain Reaction (PCR) test, that is used to detect whether one person has been infected in the past. The second test is the long range test: serology or immunity test, that allows to determine whether one person is immune to COVID-19 now. This test is used to identify the individuals 25 that cannot be infected again. For our research on COVID-19, we aim to evaluate the effect of lockdown 2 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint within a given geographical scale in France, such as the largest cities, or urban agglomerations, or French departments, or one of the 13 Metropolitan Regions (to go from the finest geographical scale to the largest one). The estimations 30 of effect are also considered on different age-classes, such as early childhood, scholar childhood, working class groups, or the elderly. Besides, we propose to understand the effect of partial lockdown or other confinement strategies depending on some geographical perimeters or some age groups (as the one that Lyon experienced very recently, see [5] ) 35 In the context of COVID-19, there have been many papers that focus on estimating the effect of lockdown strategies on the spread of the pandemic (e.g. [6] and [7] ). In [8] , the lockdown effect is estimated using stochastic approximation, expectation maximization and an estimation of basic reproductive numbers. In this work, we aim at evaluating the dynamics of the pandemic after the lockdown 40 by looking on the transport effect. In this paper, one contribution is that an extension of the typical SIRD pandemic model is presented for characterizing the regional spread of COVID-19 in France before and after the lockdown strategies. Taking into account the detection ratios of infected and immune persons, this extended compartment 45 model integrates all the related features of the transmission of COVID-19 in the regional level. In order to estimate the effect of lockdown strategies and understand the evolution of the undetected compartments for each region in France, an optimization algorithm is used to derive the optimal parameters for regions by fitting the extended model to real reported data during the lockdown. 50 Based on regional model analysis before and after the lockdown, we present a network model to characterize the pandemic transmission between regions in France after lockdown and evaluate the transport effect of COVID-19 pandemic, when considering all age classes together. The most interesting point is the chosen exponential transmission rate function β(t), in order to incorpo-55 rate the complex effect of lockdown and unlockdown strategies and the delay of incubation. This paper is organized as follows. In Section 2, the extended model is de-pandemic network of all cities in France. In the 'Discussion' section, considering the age classification, an integro-differential model is presented for the pandemic network transmission, at any geographical scale, and for any set of age classes. In this paper, the scenario we consider is a large safe population into which 70 a low level of infectious agent is introduced and a closed population with neither birth, nor natural death, nor migration. There is one basic model of modelling pandemic transmission which is well known as susceptible-infected-recovereddead (SIRD) model in [9] . This mathematical compartmental model is described where S(t) is the number of susceptible people at time t, I(t) is the number of infected people at time t, R(t) is the number of recovered people at time t, D(t) is the number of deaths due to pandemic until time t, with constant parameters: β is transmission rate per infected, δ is the removal or recovery rate, α is the disease mortality rate. The compartment variables S(t), I(t), R(t), D(t) satisfy 80 at any time instant t, here N is the total number of population of the considered area. From the differential equations (1)-(4), it is obvious that at any time instant t, the total rate βI(t) of transmission from entire susceptible compartment to infected compartment is proportional to the infected I; the infected individuals 85 recover at a constant rate δ; the infected go to death compartment at a constant rate α. In fact, with the exception of the detected well-known data, there are some undetected data that cannot be measured but are significantly important for the analysis of the evolution of COVID-19 in France under lockdown policy. Moreover they are useful to provide efficient social policies, such as optimal management of limited healthcare resources, the ideal decision of the duration and level of lockdown or re-lockdown, and so on. Inspired by [10] , the basic SIRD model is extended to a more sophisticated has recovered from the pandemic and is immune. The flow diagram of this model is sketched out in Figure 1 . 5 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint The evolution of each compartments is modelled by the following equations, dI with 6 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint The other parameters in equation (6)-(13) are defined as follows: • γ IR is the daily individual transition rate from I to R, and γ IR = (1 − All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint • γ IH is the daily individual transition rate from I to H, and γ IH = (1 − • γ IU is the daily individual transition rate from I to U , and γ IU = (1 − • γ HR is the daily individual transition rate from H to R, and γ HR = (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. The infection transmission rate β(t) is the rate of the pandemic transmission from an undetected infected person to susceptible individuals at time instant t. As in [12] , in order to combine the complex effects of lockdown strategy, a time-dependent exponentially decreasing function can be used to model the 170 transmission rate β(t), with constant parameters β 0 , µ and κ. Note that β(t) is constant during the initial stage of implementing effective lockdown strategies such as social distance, quarantine, healthcare, and mask worn. The transmission rate exponentially decreases at rate µ after these lockdown strategies take effect. The transmission 175 rate β(t) can be illustrated in Figure 2 . As one of the most critical epidemiological parameters, the basic reproductive ratio R 0 defines the average number of secondary cases an average primary case produces in a totally susceptible population (see [13] ). As for the model in [10] , for the considered model in this paper, only the I − individuals transmit 180 9 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint the disease to the susceptible individuals during the early phase of outbreak. , the initial number of susceptible individuals exceeds the critical threshold to allow the pandemic to spread. Thus the initial basic reproductive rate is When the transmission rate β(t) and S(t) evolve as time goes by, one dynamic reproductive rate that depends on time is introduced and known as effective reproduction number R(t) in [14] . In this model, it is defined as, for Similarly, when R(t) < 1, the number of secondary cases infected by a 190 primary undetected infected case on day t, dies out over time, leading to a delay in the number of infected individuals. But when R(t) > 1, the number of undetected infected individuals grows over time. Therefore, by the control of the transmission rate β(t) that can constrain R(t) to be less than 1, the number of infected individuals grows slowly to ease the pressure on medical resources. When S(t) is bellow a threshold, the epidemic goes to extinction (see e.g., [15] ). The required level of vaccination to eradicate the infection is also attained from the effective reproduction number. The compartmental model introduced in Figure 1 exhibits a large number of unknown parameters (20 if we consider λ 2 = 0). The uncertainty on these 200 parameters can not be neglected. As an example, let us propagate uncertainty at the scale of the region Auvergne-Rhône-Alpes. The vector of unknown parameters is: We take into account the uncertainties on these parameters by considering that each parameter is uniformly distributed with bounds consistent with typical 205 10 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . reported values (see, e.g., [10] and references therein). Lower and upper bounds for each parameter are reported in Table 1 hereafter. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint Figure 3 shows that the prior uncertainty is pretty high, since for example the difference between the 75 % and the 25 % quantiles for the number of people in hospital is more than 50000 at the end of the lockdown period. On Figure 4 we propagate the parameter uncertainty on the maximum number of people in intensive care units, on the date at which this maximum value is attained and on the total number of reported cases. Note that the total number of reported cases is obtained from the daily number of reported cases, DR, which is driven by the following equation: The We see fpr example on these boxplots that the median for the maximum number 225 of people in intensive care is more than 8000 with the IQR greater than 20000. In view of the importance of uncertainties propagated from the model parameters to the quantities of interest (e.g., number of infected people at hospitals), it appears necessary to calibrate the model. Our calibration procedure is described in the next section. In this section, regional scales of France are considered and all age classes are summed to calibrate the parameters of the pandemic model (6)-(13) during 12 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint The following weighted least square cost function is minimized for parameters optimization: where p is a vector which consists of calibrated parameters; Z meas (t i ) is the measured values of the corresponding observed state vector Z(p, t i ) at time t i , i = 1, . . . , n, with n the number of days considered for calibration. This optimization problem is solved using Levenberg-Marquardt algorithm (see [17] ). Since it is a local algorithm, we adopt, as in [11, Chapter 6] , a multi-start ap- (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. is calibrated on daily data for H, U , D and R + on the lockdown period 2020-250 03-18 to 2020-05-11 from two data sources: the first one is a public and governmental data source [19] and the second one is a dedicated national platform with a privileged access [20] . The time step is chosen as ten percentage of one day for the numerical discretization. A general solver for ordinary differential equations is used to In order to characterize the dynamics of the pandemic transmission processes 260 during the confinement, the epidemiological model (6)-(13) was described in the 14 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint 15 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint 16 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. previous section. We now consider the government action of unlockdown after confinement, there is a pandemic transmission effect between each region in France. Considering N a age groups, the following pandemic network model of 17 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. 18 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. where transmission rate β ijk (t) depends only on (i, j), and is piecewise continuous depending on the scenario: lockdown or no-lockdown, for all t; for age group j, L kij (t) is the proportion of individuals moving from region k to region i in the age class j; the other parameters depend on the location, and also on the age group j; σ(j, k, t) is periodic (space dependent period T j,k ), satisfies 270 T j,k 0 σ(j, k, t)dt = 0, and takes value in the interval [−1, 1]; C i is the set of all regions that have pandemic transmission with region i. As the fast periodic switching policy in [21] , we consider the inverse of the (same) exponential function of infection transmission rate β(t) in (18) to denote β ijk (t). Even though the end of confinement, the social strategies still go on, so a 275 continuous function β(t) is used for the whole transmission process of COVID-19 from the start date of infection, with the end time of lockdown t end . The transmission rate β(t) for the whole transmission process is illustrated in (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. Table 8 : Third part of components L ki in the mobility matrix L. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint is 11th of May in France. In this section, we use the parameter identification method developed in where L ki (t) is the proportion of individuals moving from city k to city i, and is derived from the real data of INSEE, and C i is the set of all cities that have 22 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint 23 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint 24 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint pandemic transmission with city i. All the other parameters are chosen as the 305 ones of the region to which each city belongs. To simulate this system of 8 * 36.000 differential equations, we now specify initial conditions. To simplify, the epidemic start date of each city is taken as the same as the epidemic start date of the region to which it belongs, and the initial condition for the undetected infected individuals I − 0 for the capital of (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint Figure 11 : The maps of the transport effect between cities in France (undetected infected plus detected infected from 0% (blue) to 2% (magenta) of the population for each commune): the date for the map on the left is 2020-05-01 and the one for the map on the right is 2020-06-01. Figure 12 : The maps of the transport effect between cities in France(undetected infected plus detected infected from 0% (blue) to 2% (magenta) of the population for each commune): the date for the map on the left is 2020-07-01 and the one for the map on the right is 2020-08-01. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint which could be included in the modelling of the transmission rate β(t). 6. Discussion and a new integro-differential model In this section, the general form of an integro-differential model capable of integrating different age classes and areas is introduced to discuss the transport effect of COVID-19 in France after lockdown. By "areas" we mean a given 325 geographical scale as the set of 13 Metropolitan regions (as considered in Section 4), or the set or all 101 French departments, or all cities (as considered in Section 5), or other geographical areas. For each age class a ∈ ages in area x ∈ areas, we consider the following integro-differential equations, for any time t ≥ 0 after confinement, ∂ t X(a, x, t) = f a (X(., x, t)) + areas σ(a, x, y, t)(Λ in (a, x, y, t) − Λ out (a, x, y, t))X(a, y, t)dy • areas, the set of different areas of population under study, depending on the considered geographical scale. As an example, considering all 27 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. • X(a, x, t) ∈ R 8 is the 8-vector consisting of compartments of the age class a, in the area x, at time t; • For all age class a, f a (X(., x, t)) is the pandemic transmission dynamics for age class a from all other age classes in the area x at time t. Without considering the age effect, it is given by the right-hand side of systems (6)- (13) . Inspired by the contact matrix approach developed in e.g. [23, Chapter 3, Page 76], by considering multiple age classes, the transmission term is the following integral where β a,b,x (t) is the contact function between age classes a and b, in the 28 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint area x, and at time t. Therefore the function f a is given by where all parameters depend on the age class a and the area x; • Λ in (a, x, y, t) ∈ R is the density of people coming (in) area x from area y ∈ areas at time t, for age class a; • Λ out (a, x, y, t) ∈ R is the density of people going to (out) area y ∈ areas from area x at time t, for age class a; • σ(a, x, y, t) is the lockdown function for the age class a, between the areas x and y at time t. As an example, before the 11th of May, it was forbidden to travel for more than 100km in France. Such a policy could depend on the age classes and on the areas, e.g., to control so called "clusters" of COVID-19; • areas σ(a, x, y, t)Λ in (a, x, y, t)X(a, y, t)dy provides the total number of people coming into area x from all the other areas. 29 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint Equation (40) describes the network dynamics of COVID-19 pandemic after lockdown and the transport effect on different age class on the basis of the regional pandemic transmission dynamics during lockdown. The proposed 360 structure makes it easier to understand different forms of the kernel. The interest of this model is that it could be adapated to any geographical scales, and to all age classes. For a control point of view, the most important term is σ(a, x, y, t) which defines the lockdown policy that defines the mobility between areas x and y at time t for the age class a. Many control problems could be 365 studied for this model, as optimal control to reduced the pandemic effect, or to minimize the mortality in particular. It is of great importance for the mobility dynamics of the pandemic. Beyond that, inspired by advection-diffusion modelling of population dynamics (as considered in [24]), it is natural to model the displacement inside a 370 given area by a diffusion term (see [25] ). The corresponding model is formulated as follows: ∂ t X(a, x, t) = f a (X(., x, t)) + d(a, x, t)∂ xx X(a, x, t) + areas (Λ in (a, x, y, t) − Λ out (a, x, y, t))X(b, y, t)dy +F ext (a, x, t), where the diffusion coefficient d(a, x, t) is a function that depends on age class a, areas x and time t. This 2-order partial differential equation predicts that for age class a in the 375 area x, how diffusion causes the number of individuals in the different compartments, especially undetected infectives and deaths, to change with respect to time t after lockdown. As long as one susceptible person is infected after directly or indirectly contacting disease carriers in the area x, diffusion takes place. When the number of infectious individuals in a local area is low compared 380 to the surrounding areas, the pandemic will diffuse in from the surroundings, so the number of infectives in this area will increase. Conversely, the pandemic will diffuse out and the number of infectives will increase in the surrounding areas. The process of diffusion is influenced by distance, nearby individuals or 30 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint areas have higher probability of contact than remote individuals or areas. 385 Finally, gender differentiation or other properties may be taken into account to characterize types of populations and to study the optimal lockdown control of pandemic dynamics based on our previous work. It is worth stressing that, in the long run, optimal lockdown strategies should consider the balance between the lower number of deaths and minimum healthcare and social costs. In this paper, we investigated an extended model of the classical SIRD pandemic model to characterize the regional transmission of COVID-19 after lockdown in France. Incorporating the time delays arising from incubation, testing and the complex effects of government measures, an exponential function of 395 the transmission rate β(t) was presented for the regional model. By fitting the regional model to the real data, the optimal parameters of this regional model for each region in France were derived. Based on the previous results of the extended model, we introduced and simulated a network model of pandemic transmission between regions after confinement in France while considering age 400 classes. Regarding the transmission rate β(t) for the network model, we selected the inverse function of the previous β(t) to contribute to the transport effect after lockdown. By using the same model and method, we simulated the pandemic network for all cities in Franc to visualize the transport effects of the pandemic between cities. Considering age classes, we discussed an integro-differential 405 equation for modelling the network of infectious diseases in the discussion part. Because of the large volumes of data and complicated calculations needed for parameters calibration and simulation when considering many geographical areas and many age classes, the requirements in terms of computer hardware and software are rather high. In order to achieve accurate results, appropriate and 410 efficient data processing methods will be applied. Moreover appropriate dedicated theoretical work is needed to study the integro-differential model derived in Section 6. 31 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint In future works, we will formulate and study optimal control problems in order to balance the induced sanitary and economic costs. The lockdown strate-415 gies implemented in France should be evaluated and compared to the proposed optimal strategies. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 29, 2020. . https://doi.org/10.1101/2020.07.27.20161430 doi: medRxiv preprint Early dynamics of transmission and control of COVID-19: a mathematical modelling study Asymptomatic coronavirus infection: MERS-CoV and SARS-CoV-2 (COVID-19) Covid-19: identifying and isolating asymptomatic people helped 430 eliminate virus in Italian village Estimated effectiveness of symptom and risk screening to prevent the spread of COVID-19 Coronavirus : 450 personnes en quatorzaine après des cas de Covid-19 déclarés dans uneécole de Lyon An SEIR infectious disease model with testing and conditional quarantine Effect of a 445 one-month lockdown on the epidemic dynamics of COVID-19 in France Population modeling of early COVID-19 epidemic dynamics in French regions and estimation of the lockdown impact on infection rate COVID-19 pandemic 455 control: balancing detection policy and lockdown intervention under ICU sustainability Basics and trends in sensitivity analysis Predicting the number of reported and unreported 460 cases for the COVID-19 epidemic in South Korea Modeling infectious diseases in humans and animals Infectious diseases of humans: Dynamics and control Stochastic Epidemic Models with Inference On the distribution of points in a cube and the approximate evaluation of integrals The Levenberg-Marquardt algorithm: implementation and theory A comparison of three methods for selecting values of input variables in the analysis of output from a computer code Gouvernement français On fast multi-shot COVID-19 interventions for post lock-down mitigation Modeling infectious diseases in humans and animals Modeling fish population movements: from an individual-based representation to an advection-diffusion equation The Mathematics of diffusion The authors are very greatfull to Sébastien Da Veiga, Senior Expert in Statistics and Optimization at SafranTech (France) for the R codes used for 420 calibration and uncertainty calibration.