key: cord-0719405-d4bdpsmy
authors: Cai, Yinan; Golay, Michael W.
title: A dynamic Bayesian network‐based emergency decision‐making framework highlighting emergency propagations: Illustrated using the Fukushima nuclear accidents and the Covid‐19 pandemic
date: 2022-04-26
journal: Risk Anal
DOI: 10.1111/risa.13928
sha: 7978ce97a9006ecc0aa717555106fb1e923f15d6
doc_id: 719405
cord_uid: d4bdpsmy

When facing public emergencies, human societies need to make decisions rapidly in order to mitigate the problems. However, this process can be difficult due to complexity of the emergency scenarios and lack of systematic methods for analyzing them. In the work reported here, we develop a framework based upon dynamic Bayesian networks in order to simulate emergency scenarios and support corresponding decisions. In this framework, we highlight the importance of emergency propagation, which is a critical factor often ignored by decisionmakers. We illustrate that failure of considering emergency propagation can lead to suboptimal mitigation strategies. By incorporating this critical factor, our framework enables decisionmakers to identify optimal response strategies minimizing emergency impacts. Scenarios developed from two public emergencies: the 2011 Fukushima nuclear power plant accidents and the Covid‐19 pandemic, are utilized to illustrate the framework in this paper. Capabilities of the framework in supporting decision making in both events illustrate its generality and adaptability when dealing with complex real‐world situations. Our analysis results reveal many similarities between these two seemingly distinct events. This indicates that seemingly unrelated emergencies can share many common features beyond their idiosyncratic characteristics. Valuable mitigation insights can be obtained by analyzing a broad range of past emergencies systematically.

When a public emergency occurs, human societies need to determine mitigation strategies in a timely manner in order to control the overall situation. However, this can be difficult due to the complexity of the problems and interdependency of multiple factors. Such problems occurred during the 2011 Fukushima nuclear power plant accidents 1 (Fukushima accidents) as well as the 2020 Covid-19 pandemic. During the challenging due to psychological and technical reasons. In these processes, decisionmakers need to collect and process a large amount of highly uncertain information under high pressure and harsh environmental conditions (Aven, 2016) . Moreover, the dynamic nature of emergencies imposes additional difficulties in anticipating potential future progresses of emergency scenarios (Sanderson et al., 2020) . Given these challenges, incorporating all relevant information into response decisions is important yet difficult for decisionmakers.

The importance of incorporating as many relevant factors as possible has been widely recognized. Zio (Zio, 2018) emphasizes the importance of managing risks in a systematic way considering all potential progresses of emergencies. Zio (Zio, 2016) elaborates the value of considering all interdependencies of systems when formulating emergency risk scenarios. Son et al. (Son et al., 2020) highlights the importance of developing integrated decision support systems incorporating all relevant information. In light of these recognitions, a variety of methods, including case-based reasoning (Feng & Xiang-Yang, 2018; Shimin et al., 2007; D. Wang et al., 2020) , network methods (Buzna et al., 2006; Helbing et al., 2006; Li & Chen, 2014; Ouyang et al., 2008) , and the fuzzy decision methods (Camastra et al., 2015; Hao et al., 2018; Peng et al., 2019) have been utilized in analyzing and mitigating emergency risks. Even though they all made valuable contributions to this problem, an important factor, emergency propagation between multiple entities and its effect upon mitigation strategies, has been neglected by previous researchers.

Traditional risk management mostly focus upon a single entity. For nuclear accident, studies focus upon a single nuclear reactor (Fleming, 2005; Modarres et al., 2017; United States Nuclear Regulatory Commission, 2003; ) . For Covid, studies focus upon a single, predefined region (Block et al., 2020; R. Li et al., 2020; Zhang et al., 2019) . However, emergency propagation between different nuclear reactors or different regions, which act as a critical risk factor, has not been researched. Past research upon general emergency propagation mostly focuses upon propagation mechanisms (Chen et al., 2019; Deng et al., 2018; N. Feng et al., 2014; Garvey et al., 2015; P. Wang et al., 2021) . Few studies have revealed the effects of emergency propagation upon response decisions. In our work, we focus upon this point and reveal that failure of incorporating emergency propagation leads to suboptimal mitigation strategies and unnecessary losses. In this work, a decision-making framework based upon dynamic Bayesian networks (DBNs) is proposed to support emergency decisions. Event propagation between different entities is highlighted in our framework to provide more accurate analysis of potential emergency scenarios. DBN is selected in our framework due to its capabilities of representing complex and dynamic interactions between different factors in emergencies (Khakzad et al., 2011; Mkrtchyan et al., 2015; Weber et al., 2012) . This enables us to dynamically model emergency propagation between multiple entities and anticipate their effects upon future progresses of emergency scenarios. Moreover, the capabilities of DBN in han-dling uncertainties (Iqbal et al., 2015; Langseth & Portinale, 2007; Lee et al., 2008) make it suitable to deal with emergency mitigation problems, which often occur under highly uncertain environments (Aven, 2016) .

Scenarios developed from two real-world emergencies: the 2011 Fukushima Nuclear Accidents and the 2020 Covid-19 pandemic are discussed in this work to illustrate the utilization of the framework. These two seemingly unrelated events are selected to illustrate the generality and adaptability of the proposed framework. Moreover, we found that these two events face common challenges due to the nature of public emergencies beyond their idiosyncratic characteristics. The common challenges analyzed in this work are listed in Table 1 below. By developing parallel analysis of the events and comparing corresponding insights, we can understand some fundamental aspects of public emergencies and provide guidance applicable to future events in general. Importance of learning from past experiences has been widely recognized by the emergency management community (Hoffmann & Muttarak, 2017; Salama et al., 2004; Tsui, 2013) . Our work illustrates that in addition to learning from past emergencies in a similar field, one can learn valuable lessons from seemingly unrelated emergencies. This enables the research community and decisionmakers to obtain insights that cannot be revealed by solely focusing upon emergencies in a single field.

Interactions between various factors of a public emergency can be complex. During the Fukushima accidents, failure of a single reactor unit propagated to neighboring units according to interviews with engineers involved in its mitigation (Cai & Golay, 2020a) . A similar situation occurs in the Covid-19 pandemic since the virus can spread between different regions. Moreover, interactions between emergency status and human activities can be complex as well. During the Fukushima accidents, nuclear system status could be changed by implementing restoration activities. Site personnel could also adjust their activities based upon latest perceived nuclear system status. In the Covid-19 case, the severity of the pandemic could be controlled by implementing behavior restrictions. Humans may also change their behaviors when they believe that the pandemic is less severe.

When a public emergency occurs, it can be difficult to respond to it in a timely fashion, especially when it imposes novel challenges beyond the experiences of human societies. Such delays can degrade the effects of implemented response strategies. During the Fukushima accidents, functional restoration delays imposed great challenge to mitigation activities (Cai & Golay, 2020a) . For the Covid-19 pandemic, delays of contact tracing, diagnosis, and laboratory results can decrease the effectiveness of corresponding activities (Kretzschmar et al., 2020; Mögling et al., 2020; Rong et al., 2020) . Moreover, only focusing upon decreasing response delay at a single nuclear reactor unit or a single region is not sufficient due to potential emergency propagation. Therefore, response delay is an important factor needing to be considered systematically when dealing with public emergencies. Public emergencies often create spikes in demand for resources, which can cause resource shortage problems. Human resources and medical resources were and are limited during the Fukushima accidents and Covid-19 pandemic, respectively (Cai & Golay, 2020a; Emanuel et al., 2020) . When resources are insufficient to support all response activities, how they should be allocated to provide greatest benefits to emergency mitigation is an important question. During the Fukushima accidents, multiple nuclear reactors at a site competed for limited personnel. In the Covid-19 pandemic, various regions in a country may compete for limited medical resources. Such competition leads to conflicting interests needing to be balanced carefully.

When dealing with a public emergency, multiple response tasks may need to be accomplished. Identifying the bottleneck problem can be challenging without systematically considering their joint effects. During the Fukushima accidents, false diagnosis of system status and lack of staff were two problems challenging the mitigation process (Cai & Golay, 2020a) . For the Covid-19 pandemic, errors of clinical tests and lack of effective contact tracing methods have imposed challenges in controlling the pandemic (Arevalo-Rodriguez et al., 2020; Kretzschmar et al., 2020) . These tasks may not be equally important in order to control the emergency situation. Identifying and focusing upon the bottleneck problem can mitigate the emergency most effectively.

In the following parts of this work, analyses of the Fukushima accidents and Covid-19 pandemic are developed to illustrate the proposed framework. The framework represents an analytical workflow to anticipate progresses of emergency events and identify optimal response strategies. Decisionmakers first identify a set of interactions believed to be the most relevant to emergency scenarios. Then, a DBN can be constructed to simulate emergency progresses under certain response activities incorporating all these interactions. Finally, by varying response activities and analyzing their effects upon risk metrics, decisionmakers can determine optimal response activities minimizing corresponding losses and formulate mitigation strategies accordingly. In this work, analytical results are discussed emphasizing their implications upon general public emergency responses. As shown in this work, conducting full-scale analysis incorporating emergency propagation can provide valuable guidance for decision making in public emergencies.

In this section, Fukushima accidents scenarios are analyzed using the proposed framework. The models used are plausible ones based upon interviews with Tokyo Electric Power Company (TEPCO) staff who worked on stabilizing the Fukushima nuclear power plants after the Tohoku earthquake and tsunami. Figure 1 illustrates the structure of the DBN utilized for analysis of the Fukushima accidents. In this work, a nuclear power plant having two reactor units is selected as the analysis object. The figure only illustrates the modeling structure for Unit 1 (Unit 2 follows the same structure). In this paper, we focus upon explaining high-level modeling assumptions that are applicable to public emergencies in general. Technical details related to the Fukushima accidents and nuclear power plants are available in Golay (2020b, 2021) . Baseline assumptions for modeling parameters employed in this work are presented in Appendix 1 Table A1 . All scenarios presented in this Section utilize the parameters in the table unless otherwise specified.

We choose to expand the graph in time to decouple various feedback effects. In Figure 1 , green nodes represent system status at the current time, t, while blue nodes represent system status at a previous time, t − dt. Feedback effects between human restoration activities and system status are implicitly encoded by the dependency of a system's status upon its status at a previous time step. The feedback effects between system diagnosis outcomes, as represented by gray nodes, and its status are decoupled in time as well. As shown in the graph, system status at time, t − dt, affects corresponding diagnosis outcomes at the same time, which in turn affects the system's status at the end of the next time step, t. Detailed explanation of the nodes and their dependencies are given in Appendix 2.

In addition to dependencies within a single nuclear reactor unit described in Figure 1 , interactions between multiple nuclear reactor units at the same site is another problem worth emphasizing. According to interviews with TEPCO engineers involved in Fukushima accidents mitigation, such interactions imposed great challenges to response decisions F I G U R E 1 Time dependent modeling graph for interactions between reactor systems and human activities F I G U R E 2 Interactions among two nuclear reactor units at the same site (Cai & Golay, 2020a) . Figure 2 illustrates the interactions between the two nuclear reactor units considered in this section. Such interactions are represented by edges between systems and structures of different reactor units. In this work, in addition to intra-unit interdependencies, DC power and portable reactor cooling restorations are also assumed to be vulnerable to reactor building failure at the neighboring units as occurred during the Fukushima accidents (Cai & Golay, 2020a) . This dependency is expanded in time to decouple potential feedback effects between the two units. When solving the Bayesian network, both intra-unit dependencies encoded in Figure 1 , and inter-unit dependencies encoded in Figure 2 would be considered in our model.

Integrating the dependencies described in Figures 1 and 2 , we obtain a complete DBN model that can be utilized to simulate the status of all systems at all time steps given a specific set of site conditions as input. Figure 3 below represents the output of the DBN defining the status of reactor cores, internal energy, cooling status, and reactor buildings for one specific scenario. In this work, we focus upon maximizing long-term success probabilities of nuclear reactor cores. Thus, we extract and summarize the status of nuclear reactor cores when time goes to infinity. In the example shown in Figure 3 , Unit 1 succeeds while Unit 2 fails ultimately. Running multiple iterations and summarizing reactor core status in all these simulations using the Monte Carlo method, we F I G U R E 3 An example scenario of the dynamic Bayesian network (DBN) output for a single iteration can obtain the success probability of nuclear reactor cores under a specific set of site conditions. Varying input site conditions and analyzing resulted reactor core success probabilities, we can determine the conditions maximizing success probabilities and corresponding mitigation strategies. In this way, decisionmakers can identify optimal response strategies maximizing overall benefits.

Analysis of the Fukushima accidents response strategies

Fukushima accidents restoration delay problem When a nuclear reactor loses normal cooling capabilities, the reactor core will heat up (due to fission product decay) and fail if cooling capabilities are not restored in time. Therefore, it is important to implement restoration activities in a timely fashion. This problem is more complex when multiple reactor units need to be protected. When it is difficult to proceed with restoration tasks at all reactor units in parallel, decisionmakers tend to devote most of their efforts to the reactor unit in the most urgent condition, as during the Fukushima accidents (Cai & Golay, 2020a) . However, this is not always the best strategy due to the interdependencies between different reactor units, as illustrated in the analysis shown in this section.

In the scenarios analyzed in this section, restoration delay of nuclear reactor Unit 1 is kept at zero. The restoration delay for nuclear reactor Unit 2 is varied in order to analyze its effects upon both units. Figure 4 (A) and (B) displays the effects of Unit 2 restoration delay upon success probabilities of two units, with and without incorporating inter-unit failure propagation, respectively. As shown in Figure 4 (A), where inter-unit failure propagation is correctly considered, the increase of Unit 2 restoration delay not only decreases its own success probability, but also decreases the other unit's (Unit 1′s) success probability. In this case, even though Unit 1 has no restoration delay, failure of Unit 2 can propagate to Unit 1 and degrade its safety. Therefore, it is not sufficient to focus upon a single reactor unit in order to protect its integrity. However, if inter-unit accident propagation is not correctly considered, as shown in Figure 4 (B), decisionmakers may mistakenly believe that Unit 1′s success probabilities are not affected by restoration delays at other units. As a result, decisionmakers may fail to properly coordinate restoration tasks at multiple nuclear reactors. This can lead to unnecessary losses in real-world emergencies. During the Fukushima accidents, mitigation focus shifted from one reactor unit to another consecutively due to lack of systematic coordination between them. Long restoration delays at reactor units not being prioritized negatively affected safety of all nuclear reactor units at the site (Cai & Golay, 2020a) . In addition to qualitatively revealing this effect, our framework also quantifies the effects of Unit 2 response delays F I G U R E 4 Effects of Unit 2 (U2) restoration delay on site success probabilities. Delay for Unit 1 (U1) is assumed to be equal to zero F I G U R E 5 Effects of personnel resource distribution on site success probabilities. Delay and false diagnosis probability are assumed to be equal to zero for both units upon Unit 1 success probabilities. Such results can provide more accurate guidance for emergency responses. Similar results are obtained for the Covid-19 pandemic as shown in Section 3.2.1.

Fukushima accidents resource distribution problem When resources are insufficient to support restoration work at all reactors units in parallel, the most tempting decision would be that of allocating all of them to the unit in the most urgent condition, as decisionmakers did during the Fukushima accidents (Cai & Golay, 2020a) . However, as illustrated in this section, this may not be the best choice given the interdependency between multiple reactor units. Figure 5 (A) and (B) displays the effects of resource distribution on site success probabilities, with and without incorporating inter-unit accident propagation, respectively. In these scenarios, the operator population is very few relative to the needs as displayed in Table A1 . Therefore, two reactor units need to compete for the limited human resource. The x-axis represents the resource fraction allocated to Unit 1, increasing its resource fraction decreases resources allocated to Unit 2 since resource fractions of both units sum to unity. An optimal resource distribution exists that maximizes the success probabilities of any single unit as well as of both units. Figure 5 (A), where inter-unit accident propagation is properly considered, even if one wants to maximize the success probability of a single reactor unit, allocating all resources to it is not optimal. Devoting all resources to a single reactor unit would leave the other one unattended and having a high failure probability. The failure could propagate back to the unit getting all resources, and thus decrease its success probability. Such effects could remain unnoticed if decisionmakers were to employ localized views and only focus upon a single reactor unit, as illustrated in Figure 5 (B). If decisionmakers fail to consider inter-unit propagation, they may wrongly determine that allocating all resources to a single nuclear reactor would maximize its success probability. As a result, they may fail to systematically coordinate restoration activities at multiple reactors and formulate suboptimal mitigation strategies. In addition to qualitatively revealing the importance of properly allocating resources among multiple reactors, our framework quantitatively identifies the optimal resource distribution for each reactor unit. As a result, corresponding response decisions can be made more accurately. Analyses of the Covid-19 pandemic reveal similar results as discussed in Section 3.2.2.

During the Fukushima accidents, multiple challenging problems, including resource shortages and false system status diagnosis, needed to be solved to mitigate the situation (Cai & Golay, 2020a) . However, solving each of the problems can require large efforts. Therefore, decisionmakers need to identify and prioritize the bottleneck restoration task to mitigate the accidents most effectively. To achieve this, candidate restoration tasks need to be compared based upon their effects in improving site success probabilities. This could be difficult without a systematic method to analyze the whole situation. In this section, we illustrate the capability of our framework in supporting such decision-making problems.

To quantitatively compare the effects of increasing human resource and decreasing false diagnosis rate, we conduct sensitivity analysis of the two factors jointly. Figure 6 presents site success probabilities under various combinations of the two factors. Even though success probabilities increase with decreased false diagnosis rates under all operator populations, low diagnosis accuracy is not always the bottleneck problem most seriously challenging the site safety. For example, when the human resource level is low (5 operators for each task), site success probability is below 0.1 even if no false diagnosis exists. In such scenarios, increasing human resource level is the bottleneck problem needing to be prioritized. If site decisionmakers focus upon nonbottleneck problems, it would be very difficult to substantially improve the site safety. Therefore, identifying the bottleneck problem in emergency decision making is very important because it determines the highlevel direction of all mitigation tasks. Our framework is capable of providing decision-making guidance for such problems F I G U R E 6 Joint effects of system false diagnosis rate and operator population upon success probability of both cores given its capability of quantitatively analyzing joint effects of multiple restoration tasks. Similar results are also revealed in the Covid-19 pandemic as discussed in Section 3.2.3.

In this section, the proposed framework is illustrated using scenarios adapted from the Covid-19 pandemic. In order to illustrate the interactions between neighboring regions during the Covid-19 pandemic, New York (NY) and New Jersey (NJ) states in the United States are selected as illustrative examples due to their vicinity and high transportation volume between them 2 . Figure 7 presents the DBN structure for NY, the graph for NJ follows a similar structure. Similar to the Fukushima case, the modeling graph is expanded in time in order to decouple potential feedback effects. In this section, response strategies are compared based upon their effects on cumulative fatality number when the pandemic ends. Detailed modeling assumptions are available in Appendix 3.

Two categories of strategies have been implemented in order to control the Covid-19 pandemic: preventive strategies (Güner et al., 2020; Joshi & Mehendale, 2021; Molnar et al., 2021; Sharma et al., 2021) curbing the spread of the virus and medical strategies (Adhikari et al., 2021; Janiaud et al., 2021; Mohamed et al., 2021; Putman et al., 2021) treating infected individuals. In this work, we mainly focus upon the effects of preventive strategies upon severity of the pandemic. Joshi and Mehendale (2021) provide a reasonably complete list of potential preventive strategies. These strategies fall into the 2 The two states are only selected as illustrative examples. Similar analysis can be conducted on any vicinity regions.

Illustrative modeling graph for Covid-19 response in a single state in the United States, selecting New York as an example following categories: mask wearing, social distance, vaccination, and environmental disinfection. In this work, all strategies except for environmental disinfection are incorporated in our models. This treatment is sufficient for our purpose since environment disinfection can be incorporated into the model by adding a multiplier representing its preventive effects, similar to the way that mask wearing and social distance have been modeled.

Detailed explanations of the nodes and their interactions in Figure 7 are available in Appendix 4. In Figure 7 , the feedback effects between human activities and pandemic status are encoded by the dependency of mask wearing and social distance upon the number of perceived Covid-19 cases. Detailed assumptions for this dependency are available in Appendix A3.2. Presumably, more restrictions are implemented when the pandemic is serious. Restrictions can be relaxed when the pandemic is in a stable state. Moreover, the perceived case numbers can deviate from the actual situation due to test errors. In addition to mask wearing and social distancing, case tracking, and vaccine availability can also limit virus spread. Case tracking levels depend upon the number of tests, because only confirmed cases can potentially be tracked. In additional to activities inside the state, virus spread in NY can also be affected by the pandemic status in the neighboring state: NJ. Figure 8 illustrates the virus spread mechanism considered between them.

The major virus spreading mechanism considered in this work between NY and NJ states is by populations commuting between the two states. According to United States Census Bureau (n.d.-a) around half a million people commute daily between them. A severe pandemic in any single state could propagate to the other due to their close connections. This is consistent with the actual situation during the first few months of the pandemic when both states suffered. In this work, we assume that 1000 and 500 people are infected by the virus at NY and NJ, respectively. The virus could spread either way when in the later stage of simulated scenarios.

Utilizing the graph structure represented in Figure 7 and the modeling assumptions described in Appendix 3, Covid cases and fatality numbers at both states for all time steps can be obtained by solving the DBN. In this work, we focus upon the cumulative Covid fatality number of the two states when the pandemic ends. Extracting this information from DBN simulation results and varying the input hyper parameters corresponding to response strategies, decision makers can understand the value of alternative response strategies and identify the ones minimizing the losses caused by the pandemic.

Covid-19 pandemic response delay problem When responding to the Covid-19 pandemic, decisionmakers at each region tend to focus upon decreasing their own response delays. However, delays in the neighboring region can also challenge its safety due to the existence of interregional virus spread. Such effects would remain unnoticed without systematic analysis of the pandemic status involving multiple regions. In this section, we illustrate this phenomenon using delays of testing and behavior restrictions at NY and NJ as examples. Figure 9 shows the effects of the NJ response delay on the total fatality number 3 of the two states, with and without considering interstate virus transmission. In both the scenarios, the NY response delay is set to zero. As shown in the Figure 9 (A), when inter-state virus transmission is incorporated into the model, even though there is no delay at NY, its fatality number increases with increased delays at the F I G U R E 8 Virus spread mechanism between New York and New Jersey F I G U R E 9 Effects of New Jersey response delay on fatality number of two states All populations are assumed to have access to masks in order to eliminate the effects of resource shortage neighboring state. This indicates that it is not sufficient to protect a single state by only decreasing its own response delay. On the contrary, if one fails to consider interstate virus transmission, as shown in Figure 9 (B), NY decisionmakers can falsely believe that coordination between neighboring regions is unnecessary since the NY fatality number is not affected by NJ response delays. This illustrates the importance of considering emergency propagation in real-world scenarios. Failure of doing so can lead to overly optimistic estimation of emergency scenario results and suboptimal mitigation decisions. In addition to qualitatively revealing such vulnerabilities, our framework can quantify such effects. This can better support response decisions given the nonlinear relationship between fatality numbers and response delays.

Covid-19 pandemic resource distribution problem This section explores resource distribution problem for the Covid-19 pandemic using mask resources as an example. Other resources like medical resources and human resources can be modeled following the same logic. In this work, we assume that wearing masks could decrease virus spread by as much as 70%. However, that effect depends upon the availability of masks. In the scenario discussed in this section, we assume that 20% of people in each state have access to their own masks originally. We focus upon the problem of distributing additional mask supplies when they become available. A similar situation occurred when the hospital ship Comfort arrived at the New York region in order to help treat n.d.) . The ship could have chosen to dock in NY or in NJ to provide additional medical resources to the corresponding state. Two states compete for the limited resources since docking the ship at one state would decrease the amounts of benefits provided to the other state. Figure 10 presents the effects of mask resource distribution upon fatality numbers, 4, 5 with and without considering interstate virus transmission. The x-axis corresponds to F I G U R E 1 0 Effects of resource distribution on fatality numbers the fraction of resources allocated to NY. Resource fractions for the two states are assumed to sum to unity. As shown in Figure 10 (A), where interstate virus transmission is considered, as occurred in reality, single state fatality numbers are not minimized when all resources are allocated to it. Similar to the Fukushima accidents implications discussed in Section 2.2.2, allocating all resources to any single state would leave the other one unattended and worsen its pandemic situation. As a result, the safety of the state getting all resources can be threatened due to interstate virus spread. Such effects could often be neglected by decisionmakers. However, if decisionmakers fail to consider inter-state virus transmission, as shown in Figure 10 (B), the erroneous conclusion that allocating all resources to a single state would minimize its fatality number would be drawn. This occurred in reality as the hospital ship Comfort mentioned previously mainly served the NY state instead of serving the two states simultaneously, which corresponds to the situation of allocating all resources to a single state (LaGrone, n.d.) . Therefore, it is important to conduct systematic analysis of the whole situation incorporating emergency propagation in order to reach better response strategies for similar problems. In addition to qualitatively revealing this problem, our framework also provides optimal resource allocation for each scenario quantitatively. This can better guide decision making for such problems during emergencies.

Covid-19 pandemic bottleneck task identification problem During the Covid-19 pandemic, multiple tasks need to be accomplished. How to prioritize between them is an important problem determining the effectiveness of response strategies. Focusing upon other problems instead of the bottleneck ones could lead to waste of resources since they are not the key factors limiting the severity of the pandemic. This section F I G U R E 1 1 Joint effects of test error and case track fraction on fatality numbers explores this problem using increasing clinical test accuracy and improving contact tracing rate as illustrative examples. False negative clinical tests where infectious individuals are missed could impose great challenges upon pandemic mitigation (Arevalo-Rodriguez et al., 2020) , because it is unlikely to track or impose any behavior restrictions upon unidentified individuals. Even if infectious individuals are identified, their contact rate could still be high if no behavior restriction is imposed upon them. In this work, we assume that an infectious individual will not infect others only if he/she is both detected and tracked. Any behavior restriction, including self-quarantine, that can limit the contact rate of infectious individuals is considered a form of tracking. Figure 11 presents the joint effects of the two factors upon fatality numbers. 6 As shown in the figure, even though decreasing test error can generally decrease fatality numbers, the benefits provided by it depends upon case tracking fraction. When no case is tracked, improving test accuracy cannot decrease fatality numbers since no behavior restriction is implemented upon the detected individuals. In such cases, increasing case track fraction is the bottleneck problem needing to be prioritized. On the contrary, if test error is high (0.9), fatality numbers are high even if most of the cases are tracked. Thus, more efforts should be devoted to decreasing false test rate. Identifying and focusing upon the bottleneck problem can mitigate the pandemic most effectively. The capability of our framework in quantifying such effects can be practically useful as well. According to Arevalo-Rodriguez et al. (2020) , most studies report false negative rates ranging between 0.018 and 0.33, which is not particularly high if a sufficient number of individuals are tracked based upon the results in Figure 11 . Therefore, in the regions where few cases are tracked, tracking more cases can contribute greatly to controlling the pandemic.

In this work, we propose a DBN-based emergency decisionmaking framework incorporating event propagation between multiple entities. The proposed framework is illustrated using the Fukushima accidents and the Covid-19 pandemic scenarios. By comparing scenarios with and without incorporating emergency propagation, we illustrate that failure of incorporating this factor can lead to erroneous mitigation insights and suboptimal mitigation strategies. As discussed in Section 1, this is a point that has not been focused upon by previous researchers. Based upon our analysis results and reflection of mitigation strategies implemented in practice, we propose high-level mitigation suggestions that can be utilized for all emergencies facing challenges similar to those discussed in this work. By conducting parallel analysis of the Fukushima accidents and the Covid-19 pandemic, we illustrate that seemingly distinct emergency events can face common challenges and share common features despite their idiosyncratic characteristics. This expands the potential emergency candidates that decisionmakers can learn from. Emergencies from a specific field can be scarce, learning from a broad range of seemingly unrelated events can provide valuable insights for decision makers. Utilization of our framework in the two relatively distinct events illustrates the generality and adaptability of the framework. Applying the proposed framework to future emergencies can support high-quality response decisions when human societies face significant challenges unexpectedly.

The Fukushima related research project is funded by Tokyo Electric Power Company.

Cai https://orcid.org/0000-0001-9870-5989 Total site operators for each task 10

Resource distribution among two units 50% 50%

Description of Bayesian network nodes for Fukushima nuclear accident analysis (Figure 1) TA B L E A 2 Explanation of Bayesian network nodes for Figure 1 Node category Node name Node description Relationship with parent nodes

Site conditions (white node)

Unit 1 Resource, delay, and system false diagnosis rates This node describes site conditions that will affect restoration tasks of all systems.

Site conditions are input as hyperparameters in the model to analyze their effects upon accident scenarios.

System status at the end of a previous time tdt. Utilized as the initial status for the current time t (blue nodes)

Status of Containment at a previous time tdt Initial status of a system at the next time step is the output of the system status of the DBN from the previous time step. For example, suppose that status of DC system at the end of time step t = 1 is success. Then, at the beginning of time step t = 2, its initial status would be success. All other systems follow the same logic. 

In this work, a modified version of susceptible-infected-recovered (SIR) (Harko et al., 2014) model is utilized to describe the epidemiology features of the Covid-19. The population are divided into three compartments: susceptive individuals (S), infectious (I) individuals, and removed (R) individuals. The modified model for a single state is presented in Equation A1 , while corresponding variables and their definitions are presented in Table A3 . All variables listed in the 

As shown in Equation A1, susceptive individuals can either be infected or immunized. Mask wearing and social distance can change the actual infection rate. Infectious individuals commuting from neighboring state can also infect susceptive individuals. We assume that only infectious individuals not being tracked can infect other people. Infectious individuals, no matter whether being tracked or not, can be removed if they are recovered or dead.

Behavior restrictions including mask wearing and social distance can be implemented or relaxed based upon the pandemic severity. In this work, we assume that these methods are implemented for 90 days initially. Later, they are adjusted based upon the number of confirmed Covid-19 cases. Figure A1 illustrates the model utilized in this work to describe the implementation and relaxation of behavior restrictions. When perceived Covid-19 cases in NY consecutively increase for Δt impose days, the behavior restrictions are implemented for Δt restrict days in order to control the pandemic. 7 After that, the behavior restrictions are lifted. This assumption is consistent with the actual case. New York city started Phase 1 reopen in early June 2020 (Cuomo, n.d.-a) , which was around 90 days since a disaster emergency was declared in early March 2020 (Cuomo, n.d.-b) . Later in July 2020, New York Governor Andrew Cuomo indicated that the reopening could rollback to an earlier phase if the Covid-19 f t f t depends upon three parameters:

• fraction of infectious cases being tested, 0.2 in this work (Matrajt & Leung, 2020 × death fraction situation were to get worse (Borter, n.d.) . The dependency of behavior restrictions upon the pandemic status is captured by our model.

8 -9 n

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Bayesian belief networks for human reliability analysis: A review of applications and gaps

Advances in multi-unit nuclear power plant probabilistic risk assessment

Delayed laboratory response to COVID-19 caused by molecular diagnostic contamination

Computational drug discovery and repurposing for the treatment of COVID-19: A systematic review

Communitylevel prevention of childhood maltreatment: Next steps in a world with COVID-19

Emergency response to disaster-struck scale-free network with redundant systems

Generalized orthopair fuzzy weighted distance-based approximation (WDBA) algorithm in emergency decision-making

Antirheumatic disease therapies for the treatment of COVID-19: A systematic review and meta-analysis

Effect of delay in diagnosis on transmission of COVID-19

Lessons learned from complex emergencies over past decade

Individual dynamic risk analysis (iDRA): A systematic review and network model development

COVID-19: A review on the novel coronavirus disease evolution, transmission, detection, control and prevention

Research on case-based reasoning combined with rule-based reasoning for emergency

Investigating resilience in emergency management: An integrative review of literature

Initial response to complex emergencies and natural disasters

2011-2015 5-Year ACS commuting flows

Emergency decision-making model of environmental emergencies based on case-based reasoning method

The analysis of urban flood risk propagation based on the modified susceptible infected recovered model

Overview on Bayesian networks applications for dependability, risk analysis and maintenance areas

Numerical study of the stratification erosion benchmark for NPPs containment using CFD code GASFLOW-MPI

Challenges in the vulnerability and risk analysis of critical infrastructures

The future of risk assessment

A dynamic Bayesian network-based emergency decision-making framework highlighting emergency propagations: Illustrated using the Fukushima nuclear accidents and the Covid-19 pandemic

Description of Bayesian network nodes for Covid-19 analysis (Figure 7) TA B L E A 5 Explanation of Bayesian network nodes for Figure 7 Node category Node name Node description Relationship with parent nodes