Flood Evacuation During Pandemic: A multi-objective Framework to Handle Compound Hazard

The evacuation of the population from flood-affected regions is a non-structural measure to mitigate flood hazards. Shelters used for this purpose usually accommodate a large number of flood evacuees for a temporary period. Floods during pandemic result in a compound hazard. Evacuations under such situations are difficult to plan as social distancing is nearly impossible in the highly crowded shelters. This results in a multi-objective problem with conflicting objectives of maximizing the number of evacuees from flood-prone regions and minimizing the number of infections at the end of the shelter's stay. To the best of our knowledge, such a problem is yet to be explored in literature. Here we develop a simulation-optimization framework, where multiple objectives are handled with a max-min approach. The simulation model consists of an extended Susceptible Exposed Infectious Recovered Susceptible (SEIRS) model.We apply the proposed model to the flood-prone Jagatsinghpur district in the state of Odisha, India. We find that the proposed approach can provide an estimate of people required to be evacuated from individual flood-prone villages to reduce flood hazards during the pandemic. At the same time, this does not result in an uncontrolled number of new infections. The proposed approach can generalize to different regions and can provide a framework to stakeholders to manage conflicting objectives in disaster management planning and to handle compound hazards.


Introduction
Flood has been one of the most devastating natural disasters that cause massive loss of lives and property (2,3). Adequate preparedness and disaster management planning are required to minimize these losses and increase recovery speed (4). During floods, evacuation is one of the most critical preparedness measures to minimize the loss of lives, where people from high flood risk areas are shifted to safer areas (5). The objective of evacuation planning is to define a policy for people under high risk to minimize loss of lives and damage to property (6).
Preparedness for floods and cyclones starts by creating safe shelters at strategic locations, not very far from the high hazard areas. The evacuation planning involves shifting the vulnerable populations efficiently to the shelter homes while ensuring the timely distribution of essential commodities (7). Preparing an evacuation strategy well before the flood occurrence is pivotal to avoid last moment chaos that occurs due to the involvement of decision-makers at multiple stages and the need for the necessary arrangements to implement the evacuation in real-time. Informing the evacuees well in advance about the evacuation will ease the process largely and make the evacuee comfortable in following the instructions (8).
Optimal allocation of evacuees to shelters is a key challenge in evacuation planning (4). Under normal circumstances, the only objective is to decrease the number of people under flood risk, so as to maximize the number of people to be evacuated to nearby shelters. These shelter homes are designed to accommodate a very large number of people (for example, the capacity of a shelter on the East coast of India is approximately 2000) during natural disasters. As the shelters are provided for a very short duration (around one week), the per capita area allocated is low (9). This is acceptable during normal scenarios; however, during the pandemic, it is essential to maintain social distancing to control the spread of COVID-19. Hence, during the pandemic scenario, it is not desired to fill the shelters at their full capacity. On the other hand, the evacuation demands for the shifting of maximum people to the shelters from the (possible) flood-affected regions. The two objectives, to reduce the spread of pandemic (COVID-19 here) and increase the number of evacuated people, are in conflict with each other. This poses a challenge to disaster mitigation organizations and policymakers.
People under potential risk are evacuated to safer shelter houses timely and safely (5,10).
Evacuation planning involves a number of decision-makers and disparate individual behavior of evacuees. An effective evacuation planning requires well-defined roles, responsibilities, and communication amongst stakeholders (11). Evacuation planning depends on factors like geographical location, population size, the spatial extent of the event's extremes, duration, the intensity of the event, and uncertainties (12)(13)(14). Understanding the evacuation process and the associated models are necessary for evacuation planning (15). Mathematical modeling and optimization have become helpful tools for evaluating time requirements for evacuation and allocating evacuees in optimal shelters (7,16). Various studies have used optimization models for flood evacuation to minimize losses considering factors like travel time and distance, cost of evacuation, and usage of infrastructure (5,6,14,(17)(18)(19). Most of these studies have considered the objective function as the minimization of the transportation distance and/or time required to reach the shelters.
While the objective of designed evacuation strategies is to minimize the injuries and loss of life during the disaster, the prevalence of contagious diseases, including COVID-19, present conflicting priorities to the stakeholders and policymakers. Flood evacuation strategies are designed to encourage people to take shelters in designated areas. However, violation of social distancing protocols in these shelters could result in a sudden surge in the contractions of infections and mortality rates (20). Besides, immediately following a disaster and throughout the recovery period, healthcare facilities are often disrupted, which results in the reduced capacity of the sector to respond to the primary health consequences of flooding and delivering care to COVID-19 patients (21). Hence, disaster management approaches need to account for the effect of social  (Figure 1(a-b)). We address these multiple objectives using the max-min approach of multi-objective optimization, which has been widely used in areas such as water resources management (23)(24)(25), waste load allocations for water quality management in a stream (26)(27)(28)(29). The model is applied to a flood-prone district on the east coast of India, the Jagatsinghpur district in Odisha. to susceptible, susceptible to exposed, exposed to infected, infected to recovered, infected to fatality state, and recovered to susceptible respectively. Parameters θE and θI are testing rates, whereas ψE and ψI are positivity rate for exposed and infected individuals, respectively.

Case-study and Data
Jagatsinghpur is a coastal (east coast) district in the state Odisha, India (Figure 2  The first step in designing any evacuation strategy is to identify the villages with high flood hazard.
The hazard values associated with 100 years return period were estimated for the Jagatsinghapur district as reported by Mohanty et al. (31). The authors have considered regionalized design rainfall, design discharge, and design storm-tide as primary inputs to a comprehensive 1D-2D coupled MIKE FLOOD model (32) to derive flood hazard values at village level. In the present study, hazard values generated for a flood quantiles corresponding to 100-years return period and 24 hours duration are considered to classify the villages into the category of "high hazard" (31,33

3.1.Optimization model
The optimization model that is needed to be solved for designing evacuation strategies has the Finally, the resulting optimization model is expressed as: = 0 ∀ ∉ Where, , is the capacity of j th shelter, which is considered here to be 2000, as per the information provided by Government agency. is the number of evacuees staying is shelter j.
is the set of shelters that belong to the five closest shelters from village i. The function f is Eq. (7) is an epidemiological model based on the extended Susceptible -Exposed -Infectious -Recovered -Susceptible (SEIRS) model. Eq. (9) takes care of the fact that the people in pucca houses will be evacuated only after the complete evacuation of the population living in the kutcha houses.

3.2.SEIRS Epidemiological Model
To study the effect of social contact network structures on the propagation of the spread of COVID-house, we use extended SEIRS model. In the standard SEIRS model, the entire population is divided into Susceptible (S), Exposed (E), Infectious (I), and Recovered (R) individuals. In the extended SEIRS model, we further divide the population in Detected Exposed (DE) and Detected Infected (DI) by using social tracing and testing parameters. The initial seed is then provided in terms of population in each category. Recent developments in the field of epidemiological modeling further compartmentalize the contagious individuals according to the degree of severity of symptoms. However, given the limited availability of datasets to calibrate the associated parameters, we use a 7-compartment model in this study (Figure 1(c)).
A Susceptible member becomes exposed or infected upon contacting the infected individual during a transmission event. Newly exposed individuals experience a latent period during which they are not contagious (referred to as the Incubation period). Exposed individuals than progress to the infected stage where they can either get tested if they are exhibiting symptoms or they have been selected for testing based on the contact tracing network at the prevalent rate of contact tracing and testing in the society. The infected individual can then progress either to Recovery (R) or succumb to the infection (F).
Since we are interested in decreasing the flood risk in the COVID-19 scenario, we use the deterministic mean-field model implementation of the SEIRS Extended model. Specifically, we assume that despite the underlying interaction social interaction structure that is ubiquitous to any society, the interactions within the shelter homes will primarily be random due to the violation of social distancing norms. Hence, all individuals mix uniformly and have the same rates and parameters in the current implementation of the epidemiological model. We use the SEIRSPLUS package implemented in Python to obtain the number of infected individuals in each shelter filled with the full capacity of 2000 using different values for the initial number of infections. We note that if the underlying social network structure and information on testing and isolation testing protocols are available, stochastic network models are recommended to account for stochasticity, heterogeneity, and deviations from uniform mixing assumptions (34).

3.3.Max-Min Approach
The multi-objective optimization model presented in Eq. (1-11) is solved here with the Max-min approach. The first objective function can have a value between 0 to 1 as per Eq. (4). The second objective function is also standardized by dividing Ij by Ij,max, which is the maximum possible value of Ij, given by, ( , , ,0 ). The max-min approach maximizes the minimum of all the objectives (when objectives are to be minimized, it is considered as the maximization of the negative of the objective function), which will force all the individual objectives to maximize. Following the maxmin approach, the optimization model may be formulated as: The model mentioned above is a non-linear optimization model. We use a search algorithm, known as Probabilistic Global Search Laussane (PGSL) to obtain the feasible optimal solution (35).

3.4.PGSL: Search Algorithms for Optimization Model
PGSL, a global search algorithm, was developed by Raphael and Smith (35) based on the assumption that better results can be obtained by focusing more on the neighborhood of good solutions. In every iteration, the algorithm increases the probability of obtaining a solution from the region of good solutions of the previous iteration. Thus, the search space is narrowed down until it converges to the optimum solution. PGSL is different from other methods as it uses four nested cycles, which helps improve the search, and thus more focus could be given to areas around good solutions (29). The four cycles of PGSL are: Sampling cycle: Samples are generated randomly from the current PDF of each variable.
Each point is evacuated based on the objective functions, and the best point is selected.
Probability updating cycle: Probability of neighborhood of good results increased and bad decreases, and the PDFs of each variable are updated accordingly after each cycle.
Focusing cycle: Search is focused on an interval containing better solutions after a number of probability updating cycles. This is done by dividing the interval containing the best solution for each variable.
Subdomain cycle: The search space keeps narrowing by selecting only a subdomain of the region of good points.

Results and discussion
We apply the developed optimization model (Eq. [12][13][14][15][16][17][18][19][20][21][22][23][24][25] to the case study of Jagatsinghapur District. Due to data non-availability, we have assumed hypothetical values of some of the variables for demonstration purposes. We considered zx to be 0.6, zy to be 0.4 and rf to be 0.8 for all the villages (i). The maximum shelter capacity is considered to be 2000. The stay period in the shelter is considered to be seven days. The values are considered after discussions with planners and management authorities working at different levels of decision making. We applied our optimization model first by considering a uniform initial infection value across the district. The infection value is considered as 1% for demonstration purpose. We first simulated the increase in the number of infections in a shelter assuming different initial infection values (0.1%, 0.25%, 0.5%, 0.75%, and 1%) with shelters at full capacity for seven days. We find, the number of infections in a shelter to be in the range of 7 to 60 at the end of the stay, depending on the initial infection (Supplementary figure S1). Violation of social distancing norms within shelter houses operating at their designated capacity could expose a large number of individuals to the highly contagious diseases, including COVID-19. Once exposed and infected individuals move back to their respective villages, it may result in the widespread outbreak at local scales.
Such a scenario may also become unmanageable, as the medical, as well as other facilities, will be limited after flooding events. Hence, a proper evacuation strategy planning is needed to decrease flood losses and the spread of COVID-19.

Fraction of pucca house people evacuated
To check the applicability of our optimization model, we first used the optimization model for non-pandemic scenarios, which includes the equations from Eq (12) to Eq (25), excluding Eq (14), (19), and (20). The results obtained from the model are presented in figure 3. We find that in most of the villages with high flood hazards, more than 50% of the population is evacuated (Figure   3(a)). We find a good number of shelters (213) remain unused in the Central area (Figure 3(b)), as they are far away from the hazardous villages, and transporting people to those shelters is difficult.
These shelters may not be useful during the flood, but during cyclones, they are extensively used.
In most of the villages considered, more than 75% of the populations in kutcha houses are getting evacuated (Figure 3(c)). For 69 villages, this fraction reaches 100% with evacuations for a few in pucca houses (Figure 3(d)). Such realistic results prove that the model works efficiently under nonpandemic situations.

Summary
Here we address an important compound event problem related to flood evacuation to minimize the loss due to flood hazards, while also demonstrating the potential issues associated with