key: cord-0049406-6cv8n2bf
authors: de Weck, Olivier; Krob, Daniel; Lefei, Li; Lui, Pao Chuen; Rauzy, Antoine; Zhang, Xinguo
title: Handling the COVID‐19 crisis: Toward an agile model‐based systems approach
date: 2020-08-27
journal: nan
DOI: 10.1002/sys.21557
sha: a5260c6020e5e61eaa0b4faf8dbe190464e6be95
doc_id: 49406
cord_uid: 6cv8n2bf

The COVID‐19 pandemic has caught many nations by surprise and has already caused millions of infections and hundreds of thousands of deaths worldwide. It has also exposed a deep crisis in modeling and exposed a lack of systems thinking by focusing mainly on only the short term and thinking of this event as only a health crisis. In this paper, authors from several of the key countries involved in COVID‐19 propose a holistic systems model that views the problem from a perspective of human society including the natural environment, human population, health system, and economic system. We model the crisis theoretically as a feedback control problem with delay, and partial controllability and observability. Using a quantitative model of the human population allows us to test different assumptions such as detection threshold, delay to take action, fraction of the population infected, effectiveness and length of confinement strategies, and impact of earlier lifting of social distancing restrictions. Each conceptual scenario is subject to 1000+ Monte‐Carlo simulations and yields both expected and surprising results. For example, we demonstrate through computational experiments that maintaining strict confinement policies for longer than 60 days may indeed be able to suppress lethality below 1% and yield the best health outcomes, but cause economic damages due to lost work that could turn out to be counterproductive in the long term. We conclude by proposing a hierarchical Computerized, Command, Control, and Communications (C4) information system and enterprise architecture for COVID‐19 with real‐time measurements and control actions taken at each level.

F I G U R E 1 Confirmed deaths per million people as of July 21, 2020 makes it a systemic crisis and not only a pure health crisis. The closest analog we have at a global scale is the H1N1 influenza pandemic of 1917-1919 ("Spanish flu") which killed between 17 and 50 million people worldwide. 6 Thus, handling the current COVID-19 crisis requires a holistic approach taking into consideration an extremely complex system, ie., society as a whole.

Another important aspect of the COVID-19 crisis is that the pandemic propagation has been very fast, thus demanding rapid decisionmaking. Moreover, and we think that this is a structuring feature of this crisis, the incubation time of the disease introduces a delay-that has been estimated as being up to two weeks according to epidemiologists (Refs. 7, 8, or 9 )-between the implementation of countermeasures and the observation of their effects. This is compounded by the fact that a significant fraction of the virus carriers appear to be asymptomatic, causing a large difference between the numbers of actual cases and of known or confirmed cases (see Refs. 1, 3, or 5) . This explains why the problem of monitoring the COVID-19 crisis can be seen as a controltheoretic problem with delay in the feedback loop used to stabilize the situation in addition to the problem of low or only partial observability of the true system states. We shall elaborate further on this point.

From a system-theoretic perspective, the above characteristics raise several difficult problems. The first one, which is rather expected, regards scalability: can our current systems engineering and modeling methods (cf. for instance, Refs. 10-16, or 17) be extended to a system, or more precisely a system-of-systems (cf. Ref. 18 or 19) , as large and as complex as human society as a whole? This question is clearly not easy to solve and appears moreover poorly addressed by the only known models of such scope, ie., the so-called World models, based on generalized Volterra equations, that followed the seminal work of Forrester in the 1970s (see Refs. 20, 21, and 22) .

A second problem is caused by the emergence of local and partial solutions which is significant since the COVID-19 crisis impacts all sectors of society, including the medical, financial, transportation, manufacturing, and overall economic systems. Society therefore needs fast and innovative solutions in order to mitigate as much as possible the consequences of the crisis. Time pressure favors local and partial solutions, but also a strong coordination among actors in order to avoid contradictory strategies. A central question is therefore how to favor the emergence of bottom-up local actions while, at the same time, ensuring top-down monitoring and coordination of such actions, with short feedback loops. This calls for an agile approach (see Refs. 23, 24, or 25) to the global COVID-19 crisis.

Stating the above problems, we made a clear choice in this paper: we do strongly believe in the use of models, and more precisely of systemic models to think through and manage the crisis. Models as we consider them here are, however, not Platonic ideals, but observational models which rely on the observation of the reality of the COVID-19 crisis, including the effects of the decisions made based on them. Such models are intended to capture the systemic nature of the crisis in order to achieve a better understanding of the situation and to allow a better communication among stakeholders. In that respect, models have two main roles: first, the concrete calculation of key performance indicators to support the decision-making process through experiments in silico;

and a second more metaphorical one, to help us think better about the dynamic evolution of the systems at stake. The remainder of this paper is organized as follows. In Section 2, we discuss which systemic models may support better management of the COVID-19 crisis. Then, in Section 3, we advocate for an agile approach for crisis management. Section 4 completes the paper with several recommendations.

The general impression which emerges from the large and rapidly expanding literature dedicated to the COVID-19 pandemic is that this crisis was first and foremost analyzed as primarily a health crisis (cf.

Ref. 26 or 2) . Economic impacts of the crisis were of course quickly understood, but, as far as we could observe, they were rather considered as an inevitable consequence of the health crisis that has to be managed as a second priority. 27 However, the aggressive mitigation measures that were set up in many countries were and are at the same time quite efficient from a health-preservation point of view (see, for instance, Ref. 28 or 29) and highly inefficient from an economical perspective due to their global economic impact on all of society (see Ref. 30 or 31) . In this matter, there is-to the best of our knowledge-no rational discussion in the scientific literature on what could be the best tradeoff for jointly minimizing both the health impact and the economic impact of the COVID-19 crisis. Perhaps the biggest ethical issue around such trade-offs is that it would require placing an explicit economic value on human lives, as discussed for instance in Ref. 32 . This is something that no national or regional government in the world has apparently been willing to do.

Moreover, what shall one do if the health crisis remains endemic in the near future which is one of the possible scenarios (cf. Section 2.2.2)?

As one can see, thinking from a global rather than a purely local perspective can deeply change the way one addresses the crisis and its consequences.

This situation is probably the consequence of the fact that the crisis is mainly observed on daily basis, through for instance the daily COVID-19 reports provided by the World Health Organization, 5 by other institutions, 3 and by each local government, leading to a rather shortterm vision of the crisis. However, changing the time scale of observation gives us immediately a totally different point-of-view on the COVID-19 crisis. If we are, for instance, observing the crisis at the time step of a quarter of a year (three months), it becomes almost instantaneous and can be considered as an event-in the classical meaning of synchronous modeling 33 -without any duration. Thus, the choice of time step and sampling frequency is critical as it is for any control system. This perspective change forces us to think what could be the next state of the system under observation, ie, human society, which may be on its way toward a deep economic crisis, at least in Western countries. Continuing the analysis at the same coarse time scale, a possible catastrophic evolution scenario would be a financial crisis result-F I G U R E 2 A possible catastrophic scenario that could result from the initial COVID-19 health crisis ing with some delay from an economic crisis initiated by the health crisis, thus generating the specter of a deep and prolonged recession, as pointed out as a possibility by some economists (Ref. 27 or 31). Moreover, this situation could then also lead to more "classical" health crises in the future (see Figure 2 ) due to the two-sided coupled interaction between the public health system and the economic system.

In such a catastrophic future scenario, extending the duration of people's confinement in Western countries in order to minimize the short-term health impact during the initial crisis could, for instance, result in deeply debilitating the health of more or less the same population in the mid-to long-term future. Such a possible paradox is typical in optimal control theory where the optimal trajectory of any nonlinear system can never be obtained through local optimizations alone. 34 In order to take into account and to avoid such paradoxical consequences, one must choose a systems approach to analyze the COVID-19 crisis, integrating all existing domains of knowledge into a common understanding of the crisis, in order to obtain a global vision, both in space and time and at different possible observation scales, and thus giving a chance to find the global optimum for human society as a whole.

We can thus see that there is another crisis, hidden within the COVID-19 crisis, which is a crisis of models. The global community is indeed focusing on short-term health-specific models to better master the crisis, but these models are inadequate as soon as one wants to address the crisis from a longer-term society-wide perspective which requires systemic models.

In this matter, let us recall that a model is an abstraction (in the meaning of abstract interpretation theory 35 ) of reality, but not reality itself, as expressed, for instance, by the famous assertion "A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness," popularized by Korzybski 36 or the well-known "All models are wrong, some are useful" by Box. 37 "Models" which are not actually reflecting reality within some error bounds are in fact not models in that observational definition and may even have negative impacts on reality since they will lead to wrong decisions or control actions. These negative impacts of wrong "models" can of course be amplified in the context of a systemic crisis such as COVID-19.

Our point of view is clearly supported by an analysis of the 2020 scientific literature to date. A search of the keyword "COVID-19" on Google Scholar 38 in April 2020, revealed that, at this moment of time, only 10 papers-ie., around 1%-of the first 900 most cited papers on COVID-19 were not discussing primarily health issues (health covering here biology, epidemiology, medicine, and health policy and management), but rather focusing on the societal and economic consequences of the crisis. Moreover, in terms of citations, most of these 10 papers were poorly cited: two were cited around 20 times, three around 10 times, and the remaining ones less than 5 times, while the average number of citations per paper was 15 in our sample. Only very few health-oriented papers, such as Ref. 39 , also discuss mixed strategies involving economic or psychological considerations to fight the coronavirus. It seems therefore that the majority of the scientific effort is focused on the short-term, without taking into account what might be the mid-and long-term societal consequences of the COVID-19 crisis.

One may also notice that there is probably another crisis of medical models that can be observed due to the COVID-19 crisis. This other crisis focuses around the merits of hydroxychloroquine and azithromycin as a possible treatment of COVID-19, as proposed by Raoult and his team. 40 This has since then been shown to be a proposal which was not supported by a rigorous methodological approach according to medical methodologists. 41 However, medical statistical methodology (see Ref. 42 for an introduction to this domain) appears also to be questionable from a modeling perspective: the frequency-based models used in methodological medicine usually cannot have probabilistic interpretations due to a lack of large series of experiments required to apply the law of large numbers; 43 hence such frequency-based models can only find correlations between proposed medications and observe effects on structurally limited series due to the high costs of clinical studies. 44 But since correlation is not causation, it is just not possible, without any understanding of the underlying biological mechanisms, to scientifically deduce anything from such studies, as long as we agree on the fact that science deals with causal explanations, which does however not prevent using correlation-based results from a practical perspective as soon as they are established in a sound way. 72 In this analysis, the debate around the rigor of the pragmatic and agile approach followed by Raoult may just be a new Popperian debate 45 opposing different medical methods for addressing an infectious health crisis, similar to the debates that existed in physics around Aristotelian theory in the 16th century 46 or aether theory in the 19th century.

To conclude this initial discussion on the crisis of models, we point out that if the scenario that we highlighted in Figure 2 comes true, we may also eventually be forced to deal with another crisis of models, namely, the crisis of mathematical models used in finance. These other "models" are not necessarily models in the observational sense that we are using in this paper since they suffer from many well-known issues such as reflexivity, 47 which refers to the fact that mathematical financial models are essentially observing other mathematical financial models, or more deeply the lack of evidence for the market equilibrium hypothesis, 48 which is at the heart of the probabilistic framework used in mathematical finance, but which is in fact rarely observed in practice (see, for instance, Ref. 48 or 49), especially in a financial crisis situation where the market is of course highly unbalanced and volatile and therefore out of equilibrium, as pointed out by several researchers.

The COVID-19 crisis is thus forcing us to open our eyes and to look for the "right" models to use for effectively managing human society.

One should use models that are effectively capturing the reality as it is and not as we would like it to be, if we want to make nondominated decisions in the face of a crisis of such magnitude and have a chance to tackle it successfully.

As stated above, there is a crucial need for constructing a realistic observational system model of the COVID-19 crisis. We shall now present the main ingredients of such a systemic model.

Taking a systems approach leads us naturally to construct first a systemic framework for modeling the COVID-19 crisis. The first step toward that objective is to understand what are the main systems 10 involved in or impacted by the crisis. In that respect, the following ones are quite obvious:

• the natural environment from which the coronavirus which initiated the crisis is coming, • the social system, which contains the population that is or can be infected by the coronavirus, • the health system which attempts to cure the people infected by the coronavirus, • the governance system which has to choose the optimal health policy to face the pandemic, • the economic system which may be indirectly impacted by the COVID-19 crisis.

Note that the impact of the COVID-19 crisis on the economic system depends of course on the health policy choosen by the governance (political) system. If a health policy recommends or forces-as often done 5 -a large fraction of its population to stay home, it causes a double shock, 31 first on the supply side since economic actors which are lacking a work force must reduce their production and secondly on the demand side since people who are not working anymore are usually paid less or not at all and thus are also consuming less.

We can now sketch the first item of our generic COVID-19 systemic framework which is the high-level environment 10 that we modeled in Figure 3 . This first system view exposes the exchanges of matter, people, information, and money-plus coronavirus here-that exist between the main systems involved in the COVID-19 crisis. Note that the overall system taken into account here, ie., human society as a whole, including its natural environment, is a closed system on our home planet Earth. As a consequence, the only levers to solve the crisis are internal to this global system. Using that technique, the point is thus to be able to construct realistic domain-specific lifecycle scenarios for each system involved in the COVID-19 environment. We first focus only on the social and economic systems, since we are considering here the situation that occurs after the end of the COVID-19 health crisis (see Figure 2 ). We can then see that:

• The lifecycle of the social system can be analyzed to first order in terms of wealth and health, where these features can be, respectively, In a systems approach, we will thus have to construct the different possible global lifecycle scenarios that can be achieved in this way (see Figure 4 for an illustration of this classical process), to evaluate their probabilities and to define means to mitigate the worst consequences.

To obtain more detailed models, we shall moreover refine them in terms of space, to capture the geographic dimension of human society, and time, and to make optimal trade-off decisions between the shortand long-term impact of the COVID-19 crisis. Note also that these lifecycle scenarios are of course highly country-dependent due to the central role of the governance system in the resolution of the COVID-19 crisis, as well as the susceptibility of the population which is an initial condition.

The last element of our COVID-19 systemic framework is finally a mission statement, 10 ie, the core high level requirement regarding human society which expresses the objective that the governance system wants to fulfill. One can indeed understand that the behavior of our system of interest-human society-will be different depending on whether one wants to minimize the impact of the COVID-19 crisis on the social, health, or economic system or to find the best balance between the impacts on these three systems. This is a multiobjective optimization problem for which we provide a sample result below, and that we intend to explore more in details in a forthcoming paper. It is therefore of high importance-as system theory tells us (see Refs. 10, 14, or 16)-to be able to clearly define the mission to achieve.

Taking a systems approach to the COVID-19 crisis requires instantiating our systemic framework per country. Each country has its own specificities, associated with its own history and culture, that one must consider in any systems approach: for instance, Chinese traditional medicine and rigorous group behaviors are specific to China, while a centralized governance system and poorly followed health rules are specific to France, while a heterogeneous health system that favors more affluent consumers and differentiated laws and policies by state are specific to the United States of America. These types of compartmental models have significant limitations since they only consider the human population in a macroscopic way, reacting globally in a uniform manner to an epidemic, which is not the case in reality. Furthermore, in a classic SIRD model, eventually 100% of the population is infected, which is never observed in practice. In the COVID-19 pandemic, one can also observe clusters where the epidemic seems to recursively focus, 5 which rather suggests a fractal epidemic propagation, as also mentioned in an older paper by Jansse et al in 1999 55 which did not seem to have been further explored by the epidemiology community. Such fractal behavior is however not at all captured by the classical SIRD-like compartmental models. Note also that, quite surprisingly, we did not find significant scientific papers studying the geometric multiscale structure of the geography of the COVID-19 pandemic, which also suggests that this dimension has not yet been analyzed in depth.

In order to better integrate geography, which is one of the most important features of the human population system, we choose a as, for instance, in Ref. 56 . In such an approach, the human population is modeled as a network, that is to say a nondirected graph, 57 where each node of the network represents an individual or a group of people, eg., a family, and each edge represents a connection between people. For the purpose of our study, we used networks randomly generated according to the Barabási-Albert model, 58 which is believed to capture the most important features of real social networks. We shall recall that the Barabási-Albert model generates networks by introducing nodes one by one (after an initial step). A degree d is chosen for each new node, which is then connected to d other nodes chosen at random from the nodes already in the network. To simulate a social network, the average value of the degree d is usually chosen between 2 and 3.

The Barabási-Albert model produces randomized scale-free networks in which most of the nodes have a low degree (below 10), but some may have a very high degree. In order to understand how an epidemic propagates in a population modeled in this way, we used networks with 100 000 nodes and an average degree d for new nodes of 2.1. With these features, the degree of nodes in a social network is typically distributed as shown in Table 1 . Potential "superspreaders" are individuals with large degree >20.

To model the propagation of an epidemic in this network, we discretized a classical SIRD-like model (see Refs. 52 and 54 and Figure 5 ) which leads us to represent the evolution of the state of each F I G U R E 6 Stochastic state automaton modeling the possible evolution of a node in the social network node of the social network that models the human population by a stochastic finite automaton whose possible transitions are described in typically Gaussian-for incubation and sickness times. We, however, think that our experiments can give us a better qualitative understanding of epidemic propagation since we believe that this social-network approach better captures the fundamentals of the social system, compared to the simpler compartment-type models. It may thus be helpful for constructing more realistic epidemic propagation models, even if it would require a very significant amount of data collection and fine tuning. The use of contact tracers in health systems is, for instance, a direct, but laborious, way to reconstruct such social networks to quickly identify infected people and to isolate them before they infect others. 60 Our first experiment consisted of simulating increasingly virulent epidemics by assuming increasing values of the probability ρ of infecting somebody (1000 trials were done per value of ρ). Our results are described in Table 2 . They show a remarkably interesting phenomenon:

for all values of the probability ρ, only a tiny fraction π of the population is eventually infected in most of the number ν of simulations (less than 1 of 1000 persons in more than 90% of the cases), or when a significant proportion (greater than 1%) is infected, the fraction of infected people π depends on ρ. In simpler terms, this can be stated as follows:

there are a lot of viruses circulating in the population, but only a few of them give rise to epidemic outbreaks. The reasons for which a virus gives rise to an epidemic outbreak are intrinsic to the virus itself, but TA B L E 2 Proportion π of the population that is infected, for different values of the propagation probability ρ. also dependent on external factors such as who is infected first, eg, a person with few contacts and low nodal degree or a superspreader with high nodal degree as shown in Table 1 , and also the behavior of the population which impacts ρ. This may explain, at least to some extent, why some countries or regions are more stricken than others, which suggests again a fractal interpretation of the geographical scope of an epidemic, as already mentioned above.

The second experiment that we shall report on in this section aimed at studying the effects of the deconfinement of a confined population that has been ordered to shelter-in-place. We studied here different proportions τ of the population that becomes sick before the epidemic becomes observable (ie., roughly between day 10 and 20 in Figure 5) and different values of the duration γ in terms of days of confinement.

We considered that there was a delay δ of 20 days before confinement was put in place and took ρ = 0.015. We also simulated the efficiency of the confinement by reducing the capacity of edges in the social network to propagate the disease by a factor 1 − ε with 0 ≤ ε ≤ 1. This factor represents the degree of adherence of the population to sanitary guidelines for social distancing, wearing face masks, and so forth.

At each step of the simulation, representing one day, an infected node has thus probability ρ × (1 − ε) to infect an adjacent healthy node. In our experiment, we took ε = 0.66 (= 2/3). We then reported the computed values of the resulting lethality in Table 3 . We assumed that γ is as large as necessary, which is clearly not realistic since confinement cannot be maintained too long for both economic and psychological reasons, but the results give the underlying trend. Each measure reported was obtained by means of a Monte-Carlo simulation of 2000 trials.

As expected, the longer the confinement, the fewer deaths. Note however that, to be fully efficient, the confinement must be rather long, several months (>90 days) in our virtual experiment. The most interesting part of this experiment comes however from the observation of the total duration of the epidemic outbreak. Table 4 shows these durations for the same values of τ and γ as in Table 3 . If the confinement is sufficiently long, the lethality drops significantly (in some cases below 1%), but also the total duration of the epidemic outbreak is shortened. If the confinement is not maintained sufficiently long, it is still partially effective, in that it reduces the lethality, but it has a quite paradoxical consequence: the epidemic outbreak lasts longer than if no countermeasures were taken at all. A short confinement does not prevent the disease from significantly propagating: it just slows down the propagation and avoids the sharp peak of infected shown in Figure 5 around day 40, which seems to be its main motivation in order not to overwhelm the capacity of the health care system. For this reason, when the population is deconfined too early, the disease is still present and remains endemic.

The above experiments do not pretend to fully represent reality, but are just to motivate the use of social-network models for epidemic modeling. As pointed out by Stattner and Vidot, 56 "network models turn out to be a more realistic approach than simple models like compartment or metapopulation models, since they are more suited to the complexity of real relationships." One of the limitations of existing network models is, however, that they do not distinguish between recurring social links with family members and coworkers and casual links based on one-time encounters such as in public transportation or at large events. They should therefore be further refined and integrated into a model-based agile approach for crisis management, while taking into account their limitations.

In this section, we model the potential impact of the epidemic as a function of different actions of the governance system on the economic system (see Figure 3 ). In order to do so we must expand the prior analysis by not only considering lethality in terms of deaths (see Table 3 ), but also the value of lost economic activity during confinement. Reverting back to the simplified SIRD model in Figure 5 , but now accounting for the fraction of population ε actually adhering to confinement during a lockdown of duration γ, which is ordered with some delay δ after a critical cumulative threshold τ of the population has become infected, we run a set of simulations. The baseline run of the model shown in Figure 5 is considered as scenario 0 with no countermeasures and it is gradually modified using the one-factor-at-a-time (OFAT) technique to test a number of actions by the governance system, including reducing the delay δ to order a lockdown, increasing the level of rigor ε of the confinement, as well as its duration γ . Table 5 shows the results of a number of numerical experiments to probe these trade-off in terms of the value of human lives lost, versus productive work lost in the economic system. In order to estimate the economic impact of the epidemic a number of assumptions were made: In the baseline scenario 0, we do not take any countermeasures and the bulk of the $4.38B total loss is due to the deaths of 4% of the population. The $312M in lost work are due to the inability of the infected and sick population to perform work during their illness, which is assumed to last for 14 days. This is the kind of situation we would expect to see in a country with a government that is either unable or unwilling to intervene in the crisis. Scenarios 1-3 institute a partial lockdown (ε = 66%) after either 10 or 20 days delay after recognizing the onset of the epidemic and the confinement lasts either 30 or 60 days. The results are not satisfactory, since the total damages exceed the baseline case where no action is taken. This outcome is due to the fact that one-third of the population does not adhere to the confinement and continues to be infected, making the disease endemic. A prolonged partial lockdown for 60 days with only 66% effectiveness as shown in scenario 3 is the worst case and leads to both a high number of deaths (about 4000) as well as high economic damages totaling $6.1B due to the prolonged shutdown, which ultimately is ineffective. This scenario is representative of the overall situation in the United States in mid2020. In scenarios 4-7, we shorten the reaction time to trigger the confinement after only five days (quick government action) and we gradually increase the rigor of the confinement to 90% (strong government enforcement). It turns out that these actions are highly effective, yielding a best-case scenario 7 with only 66 deaths, a short epidemic duration of 61 days and only $740M in damages, mainly due to the strict but short 30-day confinement in which 90% of the population participates. This essentially prevents the epidemic from blossoming and quickly snuffs out the disease. The TA B L E 5 Scenario analysis with SIRD model for assessing total human and economic damages: n, number of daily contacts, ρ, probability of infection, τ, fraction of population infected to trigger confinement, ε, fraction of population adhering to confinement, δ, delay to confinement start, γ, confinement duration, t, duration of epidemic, total number of deaths, lost work in millions $M, and total damages including lost human lives and lost work in billions $B, N = 100,000 population size Figure 7 . This may reflect the situation in countries that The economic analysis shows that the initial conditions, speed of response, and rigor of response by the governance system are crucial in determining the outcome. Figures 8 and 9 , respectively, show the sharp contrast between the ratio of human loss (deaths) and economic work loss for scenarios 0 (do nothing) and scenario 5 (rapid and strong government response).

There is indeed a trade-off between deaths and lost work, as in sce- 

In the previous section, we identified a deep crisis of models that has been exposed by the COVID-19 pandemic and proposed to mitigate this issue by constructing a systemic model of the crisis. In this section, we shall deal with some possible solutions to master the crisis using a systems approach.

As is well known in any scientific discipline, the solution of a problem highly depends on the clarity and rigor of the way the problem is framed. We will therefore dedicate this short section to the statement of the problem that we need to solve in the context of the COVID-19 crisis.

A first characteristic of the COVID-19 crisis is its global impact on human society. This crisis can thus be considered as a common cause failure-in the meaning of system safety theory 61 -for all main systems forming human society. If we are taking a safety approach, the first problem to solve is thus to mitigate the impacts of the crisis on the vulnerable systems forming human society, that is to say the social, health, and economic systems, as results from the system analysis of Section 2.2.1.

A second characteristic of the COVID-19 crisis comes from the need to take into account strong feedback delays. In this matter, a first type of delay comes from the fact that it is most of the time too late for deploying mitigation actions to limit the epidemic propagation when significant numbers of infections are observed somewhere, since the effects of these actions will only be observable two weeks later. This was clearly shown in Table 5 in scenarios 8-10. Moreover, a secondtotally different type of delay comes from the fact that focusing on short-term health impacts of the crisis may lead to long-term issues of an economic nature, which forces to arbitrate between short-and longterm consequences of a given action.

Finally, a last characteristic of the COVID-19 crisis is uncertainty.

Due to the global nature of the crisis and the rather short period of time on which it is concentrated, uncertainty is everywhere. Clinical data about the infection are permanently partial, so difficult to interpret. Understanding of the real social system network structure is never easy to capture. The exact nature and size of the impact on the economic system are difficult to evaluate. Precise data on the capabilities on which to rely may be tricky to obtain. Last, but not least, the crisis also results in a massive, heterogeneous and often contradictory amount of data in which the really interesting signals may be either weak or hidden.

Synthesizing these three features of the crisis, the problem to solve in our context can now be clearly stated: how to optimally mitigate the short-and long-term consequences of the COVID-19 pandemic on human society, taking into account delays and uncertainties that are specific to this crisis?

One can notice that this statement is a typical control problemin the sense of control theory 62 -integrating here delay and uncertainty, which can be addressed by many existing techniques (see Refs. 63 and 64) . Consequently, the objective should be to design a new system that can support this controllability objective. Based on the closed-loop control principle, which is the only one that allows to achieve a given target behavior along the time axis, 62 such a F I G U R E 1 0 High-level COVID-19 environment integrating a specific decision-aid system that has yet to be designed COVID-19 decision-aid system (shown as the gray box at the top of Figure 10 ) will have to measure the current state of the main systems forming human society in order to provide effective feedback actions on the social system through the governance system, the only legitimate one to make decisions and take control actions. Figure 10 depicts how such a decision-aid system could be integrated into the high-level COVID-19 environment.

There is at least one domain where making decisions under structural uncertainties on an underlying geographic scope is quite well known since a long time in human history, which is the military domain. Architecting a COVID-19 decision-aid system using the typical architectural pattern of a Computerized, Command, Control, and Communications (C4) system (see Ref. 65 or 66) , used in the defense area, seems thus quite a natural idea, as it is also quite often used in a system-of-systems engineering context (see Ref. 18 or 19) . This leads us to propose an organization for a COVID-19 decision-aid system based on the following three hierarchical layers, that correspond to three natural levels of abstraction associated with a given geographic scope (that may be either the international, country, and local levels or country, region, and city levels in practice), exactly like C4 systems are organized:

1. The strategic layer is the place where global situational awareness is required to master the crisis on a given large-scale geographic scope: its mission is to monitor at a high level the crisis and to elaborate strategic decisions based on an overall vision, fed by tactical information;

2. The operation layer is intended to master the crisis on a given medium-scale geographic scope: it is thus a distributed system which has to capture and synthesize tactical information and make operational decisions on their basis in accordance with the upper strategic decisions;

3. The tactical layer is intended to master the crisis on a local geographic scope: it is thus again a distributed system which has to capture and synthetize field information and make tactical decisions on their basis in accordance with the upper operational decisions.

Note that this architecture shown in Figure 11 shall be understood as a hierarchical enterprise architecture, which defines how an organizational system, supported by suitable information systems and systemic models as discussed previously, shall be organized and behave.

The main idea underpinning it is the principle of subsidiarity: decisions should be taken as close as possible to the level that is the most appropriate for their resolution. This principle means in particular that an upper level shall avoid to make decisions that are too intrusive at a lower level in order to let each local level take always the more appropriate actions depending on the real local conditions that it can observe, while following at the same time global orientations when locally relevant. This is crucial in the military sphere, but even more so in the context of the COVID-19 crisis where speed of decision making is fundamental due to the latency of the epidemic propagation as seen in Section 2.

Note that one shall also capture weak signals of systemic importance at each level of the proposed architecture: to illustrate that point, the fact that a police officer is infected in a certain area is, for instance, a typical weak signal since we may infer from it that there is a certain probability that the whole police force in the concerned area is or will be infected, at least in the near future (since the number of daily contacts or nodal degree of police officers may exceed n > 10, see Tables 1   and 5 ).

Proposing the previous hierarchical architectural pattern is, however, of course not enough to specify how a COVID-19 decision-aid system shall work. In this matter, the first point is to organize the systemic model that we sketched out in Section 2.2 according to the hierarchy that we just presented and which is used to organize the proposed decision-aid system. Hence, such a model shall not be mono- Last, but not least, the COVID-19 decision-aid system that we sketched here shall behave in an agile way, in the meaning of agility in software or industrial development (see Refs. 68-70, 71 or 24) . A pending problem is to have a plan, do, check, and act process that can quickly adapt to a quite fast-changing reality. Agility allows to solve that issue by structuring in a very rigorous way the analysis, decision, and action processes, while providing a lot of flexibility to all involved actors, which are two mandatory features for addressing a complex crisis like COVID-19. In practice, an agile COVID-19 decision-aid process has typically to be organized around regular agile rituals-managed 

In this paper, we draw attention to the core importance of having realistic system models to manage and to mitigate a systemic crisis of the order of magnitude such as the COVID-19 crisis. We also sketched out what could be an agile approach to use in this kind of crisis. Our purpose was of course not to propose some definitive solution which is probably impossible. We do, however, think that the ideas contained in this paper are valuable contributions that may be of interest in the context of the COVID-19 crisis, especially due to the fact the underlying health crisis will probably be endemic for a certain period of time (at least for 200-300 days according to most of our simulation runs) and be coupled with future short-and mid-term economic outcomes. While there are economic impacts due to strong mitigation actions such as mandated confinements (causing lost economic activity), the value loss due to human deaths at an estimated lethality rate of 4% would far exceed the economic losses. We have shown that this depends strongly on the average valuation of a human life, which is in itself a highly controversial issue.

There are of course many detailed aspects of the proposed COVID-19 decision support system that require further detail and elaboration. We focused on the issue of delay and rigor of action in the overall epidemic control system in this paper. However, as we discover more about the particular nature of this particular coronavirus, the issue of observability of human society (testing) may for instance be an even larger one. 

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The authors would like to thank the three anonymous referees for their valuable comments which contributed to a substantial improvement of the paper. 

We would finally like to stress the fact that systems engineering has an important role to play in the COVID-19 context since it can enable the necessary collaboration of the various disciplines-such as biology, economics, engineering, epidemiology, finance, geography, health policy management, immunology, logistics, manufacturing, medicine, safety, sociology, urban systems, and so forth-that are all providing a piece of the complex puzzle posed by the global COVID-19 crisis.