# Supporting Post-disaster Recovery with Agent-based
Modeling in Multilayer Socio-physical Networks

Jiawei Xue  Sangung Park  Washim Uddin Mondal  Sandro Martinelli Reia  Tong
Yao  Satish V. Ukkusuri  Lyles School of Civil Engineering, Purdue University,
West Lafayette, IN, USA.  School of Electrical and Computer Engineering,
Purdue University, West Lafayette, IN, USA.  Corresponding Author:
<EMAIL_ADDRESS>
###### Abstract

The examination of post-disaster recovery (PDR) in a socio-physical system
enables us to elucidate the complex relationships between humans and
infrastructures. Although existing studies have identified many patterns in
the PDR process, they fall short of describing how individual recoveries
contribute to the overall recovery of the system. To enhance the understanding
of individual return behavior and the recovery of point-of-interests (POIs),
we propose an agent-based model (ABM), called PostDisasterSim. We apply the
model to analyze the recovery of five counties in Texas following Hurricane
Harvey in 2017. Specifically, we construct a three-layer network comprising
the human layer, the social infrastructure layer, and the physical
infrastructure layer, using mobile phone location data and POI data. Based on
prior studies and a household survey, we develop the ABM to simulate how
evacuated individuals return to their homes, and social and physical
infrastructures recover. By implementing the ABM, we unveil the heterogeneity
in recovery dynamics in terms of agent types, housing types, household income
levels, and geographical locations. Moreover, simulation results across nine
scenarios quantitatively demonstrate the positive effects of social and
physical infrastructure improvement plans. This study can assist disaster
scientists in uncovering nuanced recovery patterns and policymakers in
translating policies like resource allocation into practice.

###### keywords:

Post-disaster recovery, Agent-based model, Multilayer network, Socio-physical
system, Hurricane Harvey

††journal: X

##  1 Introduction

Natural hazards such as hurricanes, typhoons, tsunamis, earthquakes, and fires
cause catastrophic damage to the human infrastructure system worldwide [76].
Recent studies have indicated that climate change has increased the intensity
and frequency of natural hazards, thus augmenting economic losses caused by
damages over time [67, 55, 13]. After a disaster, fast restoration of social
and economic activities helps mitigate the vulnerability of the affected
population. We define post-disaster recovery (PDR) as the process of
rebuilding human communities and infrastructures from dysfunctional states to
normal functional states [27, 52]. The PDR reduces economic and social losses
through the resumption of terminated activities in production, service, and
transport [29, 79].

Understanding PDR is challenging for at least four reasons. First, data that
inform the state of the system are scarce given the failure of sensing
infrastructures (e.g., surveillance cameras, loop detectors) caused by the
disaster [46]. Second, human behavior during PDR is complex given the
affection of many factors, including a limited budget and neighborhoods [69].
For a coastal city in Indonesia after the 2004 Indian Ocean Tsunami, many
residents hoped to immigrate to inland areas, but finally stayed in coastal
regions, due to the rising prices of inland housing [45]. Third, recovered
infrastructure statuses intricately shape human perception and behavior in the
environment [75]. For example, in the New York metropolitan area, commuters
changed their travel decisions about the recovery of the subway system, power
supply, daycare centers, and schools after Hurricane Sandy in 2012 [31].
Fourth, the functional states of the socio-physical system evolve in an
uncertain manner influenced by unforeseen events, making the state of the
system difficult to anticipate [58]. For example, Sovacool et al. found that
the social, economic, and political factors influence the PDR of hurricanes,
tsunamis, and earthquakes in the United States, New Zealand, Thailand, and the
Philippines [60].

Figure 1: A schematic diagram of background, source data, multilayer socio-
physical network, and the ABM for the PDR. Background maps © the National
Hurricane Center [10].

Past studies characterized the PDR from different angles. To perceive the PDR,
researchers utilized various types of data like mobile phone location data
[75], household survey data [31, 32], and satellite images before and after
the disaster [27]. By analyzing these data, researchers have unraveled
numerous PDR patterns. Kates et al. concluded that the recovery of New Orleans
after Hurricane Katrina in 2005 underwent three phases in its temporal
evolution [35]. Oloruntoba et al. established a framework to describe the
disaster preparedness and recovery process of a bushfire and a cyclone in
Australia [53]. For related factors in the PDR, Yabe et al. revealed that
human community recovery after hurricanes, earthquakes, and tsunamis in Japan
and Puerto Rico from 2015 to 2017 was associated with socio-demographic
factors such as population and median household income [73]. Analogously, Lee
et al. reported that age, income levels, and social support unitedly shaped
the PDR in New Jersey after Hurricane Sandy in 2012 [41]. Although these
aggregated studies provide valuable characterizations of the PDR, they are
insufficient to capture the specific recovery dynamic of an individual and
each infrastructure facility, which play crucial roles in target humanitarian
aids [1] or tangible infrastructure repair plans [26].

Agent-based models (ABMs) offer the opportunity to simulate a dynamic and more
detailed picture of the PDR with a high temporal and spatial resolution
compared to aggregated studies. ABM defines each agent as an individual entity
(e.g., a person, an infrastructure facility) and simulates the nonlinear
collective behavior of the system using simple interaction rules among
heterogeneous agents [11]. It is especially pragmatic when interactions
between agents depict complex human behavior such as learning and adaptation
[57]. Given these peculiarities, ABM has been used in many social and urban
science problems, such as modeling a transit network [61], vehicle movement
simulations [20, 25, 42], policy-making for natural resources [54], disease
spreading intervention [77], and infrastructure resilience assessment [19]. In
the field of natural hazards, existing studies have built multiple ABMs to
describe when, where, and how people evacuated during different types of
natural disasters, such as hurricanes [25, 78], tsunamis [70, 36], and wild
fires [59]. A limitation of these ABM studies is that they do not adequately
describe the interaction between humans and diverse types of infrastructure,
such as service business and water supply systems, which can result in an
incomplete understanding of the system.

To overcome the shortage of existing ABM studies, we refer to the multilayer
network, which is a set of layers of networks with intralayer and interlayer
connections [37, 7]. There are two advantages to establishing the ABM on the
multilayer network. First, node and edge entities in the network naturally
mimic agents and their interactions in the ABM, allowing us to specify details
in the complex socio-physical systems during the PDR. Second, the multilayer
structure distinguishes different types of agents by assigning them to
different layers, making the modeling procedure intuitive. This study proposes
an ABM with multilayer socio-physical networks to model the recovery of
individuals and infrastructures after Hurricane Harvey hit Texas in August
2017 (Fig. 1, left). Our ABM provides high-resolution spatial and temporal
system status information, which can be used as the feedback of environment in
subsequent disaster management such as resource allocation. First, we define
the three-layer network $G$ with a human layer $G_{h}$, a social
infrastructure layer $G_{s}$, a physical infrastructure layer $G_{p}$, and
interlayer edges $E_{h,s}$, $E_{h,p}$, and $E_{s,p}$. The network is
constructed using real-world mobile phone location data and point-of-interests
(POIs) data (Fig. 1, middle). Second, we characterize the dynamics in $G$ by
modeling human return behavior via a binary logit model and then delineating
the recovery process of the infrastructures through differential equations.
Models are calibrated based on a survey conducted in Texas after Hurricane
Harvey (Fig. 1, middle). Third, we build the ABM that covers the recovery
process through the design of a recovery flowchart (Fig. 1, right). Based on
the ABM simulation results, we present the recovery curves of the three layers
under nine scenarios.

The remaining paper is organized as follows. We review related work about
ABMs, multilayer networks, and policy interventions in Section 2. Section 3
defines the three-layer socio-physical network. Section 4 describes the
details of various agent interactions. Section 5 shows the ABM simulation
process. Section 6 performs the simulation under nine recovery scenarios.
Sections 7 and 8 summarize our ABM.

##  2 Literature Review

###  2.1 Agent-based modeling in post-disaster recovery

ABMs play vital roles in simulating the PDR. We review relevant studies about
applications in various disasters such as hurricanes [31, 47], typhoon [28],
flood [63], and earthquake [15, 4] in Table 1. Residents made decisions to
repair their destroyed houses considering multiple factors such as house
conditions, insurance, and statuses of neighborhood houses [15, 49]. In
addition to the decision of physical houses, residents adapted their daily
behavior such as travel mode choices after the disaster [31]. For the PDR of
the infrastructure, the studies modeled the temporal restoration of
functionalities of infrastructures such as houses [49], urban transport [31],
and power system [63]. In summary, these ABM studies built the system recovery
dynamic via the high-resolution agent-agent interactions and provided useful
information to downstream applications about policy recommendations.
Furthermore, they emphasized the importance of the recovery dynamics of the
human system and the infrastructure system.

Study | Disaster | Human-social | Human-physical | Language
---|---|---|---|---
infrastructure | infrastructure
Nejat and Dam. [49] | N/A | $\checkmark$ |  | NetLogo
Huling and Miles [34] | General | $\checkmark$ |  | Python
Hajhashemi et al. [31] | Hurricane |  | $\checkmark$ | MATLAB
Moradi et al. [47] | Hurricane | $\checkmark$ |  | NetLogo
Sun and Zhang [63] | Flood |  | $\checkmark$ | Python
Costa et al. [15] | Earthquake |  | $\checkmark$ | C++
Ghaffarian et al. [28] | Typhoon | $\checkmark$ |  | Python
Alisjahbana et al. [4] | Earthquake | $\checkmark$ |  | Python
This study | Hurricane | $\checkmark$ | $\checkmark$ | Python
Table 1:  Comparison between existing ABMs and this study. Existing studies
have focused either on the interactions between humans and social
infrastructures, or the interactions between humans and physical
infrastructures. In contrast, this study considers both the two types of
interactions.

However, existing studies fail to adequately address the four challenges in
PDR modeling (i.e., scarce data, complicated behavior, intricate influences,
and dynamics uncertainty) mentioned in Section 2. This is primarily due to
their incomplete capture of agent-agent interactions between the human and
infrastructure system, which may ignore crucial factors like water/sewer
infrastructure. Specifically, some studies have concentrated on house repairs
while neglecting the impact of neighboring economic activities on human
movement [31, 63, 15]. Moreover, restoring physical infrastructure (e.g.,
power, roads, water/sewer) is a major driver of socio-physical system
recovery. Specifically, it took more than 600 million dollars in Texas after
Hurricane Harvey [62]. In contrast, it has been overlooked in several agent-
based studies [47, 28, 4, 49, 34]. To close these gaps, this study develops an
ABM that simulates multiple types of agents and their interactions (i.e.,
human-social infrastructures, human-physical infrastructures) by integrating
various data sources, including mobile phone location data, POI data, and
survey data.

###  2.2 Multilayer network

Modeling a complex system as a multilayer network allows us to characterize
various network dynamics, such as cascading failures between layers [24]).
Danziger and Barabási (2022) investigated the recovery coupling phenomenon in
multilayer networks [17]. They established the theoretical connection between
the fraction of failed nodes, the damage rate, the repair rate, and a network
topology measure [17]. Alternatively, multilayer networks have been used to
model various systems, including infrastructure networks [5, 2], disease
spreading [3, 56], and socio-physical system resilience [75, 21]. For example,
Li et al. (2019) created a five-layer network representing interdependent
infrastructure systems (i.e., transportation, community development,
environmental conservation, emergency response, and flood control) to simulate
network dynamics and guide resilience planning [43]. Dong et al. (2020)
combined road network data and channel network data in Texas to investigate
how the overflow in the channel network cascaded to the road network [18]. In
the context of multimodal transportation in the urban area, Bellocchi et al.
proposed a multilayer network, representing the motorways, walking roads, the
bus system, and the metro [7]. While these studies examined infrastructures,
Fan et al. (2022) introduced the human layer and optimized human accessibility
to urban facilities by minimizing total travel cost [21]. Given the merits of
multilayer networks in modeling a system with interactions between its
entities, we construct a multilayer network comprising the human, social
infrastructure, and physical infrastructure layers to simulate the PDR
process.

###  2.3 Policy interventions in post-disaster recovery

After the disaster, the federal and state governments approve relief packages
consisting of tangible policy interventions to assist the victims to rebuild
their communities from the following two aspects.

  * 1.

Individuals, business. To assist individuals, agencies offer medical
assistance to injured persons to facilitate their recuperation and provide
insurance support to encourage worker retention within industries [52]. As for
businesses, the recovery of production, warehousing, and logistics stimulates
their overall recovery [52]. Collectively speaking, tax relief serves as a
useful strategy to support both individuals and businesses. For instance, the
U.S. government enacted the Disaster Tax Relief in Sept. 2017 to grant tax
benefits to affected Texans after Hurricane Harvey. Individuals could be
eligible for a tax deduction of $29,500 if they experienced a disaster loss
exceeding $30,000 [14].

  * 2.

Physical infrastructures. Another aspect of the PDR policies involves the
restoration of physical infrastructures including streets, roadways, bridges,
water/sewer systems, electricity facilities, drainage, and wetlands. After
Hurricane Harvey, over 300 recovery budgets at the county and state levels
were created to repair and enhance physical infrastructures. Notably, the
Texas government put forward a funding proposition of $80 million for
constructing a wastewater treatment plant [62].

The recovery data display a mixed outcome that is shaped by both the inherent
recovery of the system and the interventions implemented through policies. We
utilize empirical data to establish the baseline scenario. In order to
evaluate the effects of further policy involvement on the entire system, we
formulate eight different infrastructure recovery scenarios and simulate the
recovery process within each of these scenarios.

##  3 Data-driven Three-layer Socio-physical Network

###  3.1 Study area

Figure 2: Study area and used data. (a) Locations of the five selected
counties in Texas; (b) The five selected counties; (c) Estimated home
locations based on mobile phone location data. The numbers denote counts of
homes within each county; (d) Locations of POIs. The numbers denote counts of
POIs within each county.

We build a three-layer network model for Hurricane Harvey, a Category four
hurricane that occurred in August 2017 [10, 30]. It made two landfalls,
resulting in historical amounts of rainfall in Houston and surrounding areas,
with over 1,300 mm recorded between August 25 and 30 [80]. The impact of the
hurricane included at least 68 fatalities in Texas and the highest economic
losses among all hurricanes in the United States from 2005 (when Hurricane
Katrina occurred) to 2017 [10]. The PDR process lasted for several weeks,
involving large-scale human resettlement, house reconstruction, and
infrastructure repair. In this study, we focus on modeling the PDR process in
five Texas counties: Harris, Fort Bend, Brazoria, Galveston, and Jefferson
(Fig. 2a and Fig. 2b) for two reasons. First, these counties are near the Gulf
of Mexico where the hurricane made landfall [10]. Second, they are diverse
counties with varying populations (Table 2), which enables the investigation
of spatial heterogeneity of the PDR. Besides, we set the modeling period as
from August 30 to October 30, encompassing the predominant duration of the
recovery endeavors.

County | Harris | Fort Bend | Brazoria | Galveston | Jefferson
---|---|---|---|---|---
2019 Population [12] | 4.71 million | 812 thousand | 374 thousand | 342 thousand | 252 thousand
Land area (square miles) [51] | 1,729 | 875 | 1,387 | 399 | 904
Direct death [10] |  $\geq$36 |  $\geq$3 |  $\geq$0 |  $\geq$3 |  $\geq$5
Table 2:  Population, area, and fatalities associated with Harvey in the five
counties.

###  3.2 Used data

We use mobile phone location data and foot traffic data to construct the
socio-physical network.

Mobile phone location data. The mobile phone location data are utilized to
quantify human recovery behavior. They are owned by
Quadrant111https://www.quadrant.io/mobile-location-data and cover the entire
Texas state. They consist of the longitude and latitude coordinates of mobile
phone users, with an average sampling interval of 1.93 hours. On average, each
user generates 12.44 mobile phone location data points per day. The collection
of these data is accomplished through various sources, including the Global
Positioning System (GPS) and software development kit (SDK)222
https://www.quadrant.io/resources/location-
data?hsCtaTracking=d0ad6bf2-1422-4749-bc50-a13e33d3ed0a%7Ce2784482-0f78-4f4b-9651-dfef24ce65ce.
For GPS data, user devices receive messages from GPS satellites to compute
their longitude and latitude coordinates. SDK data, on the other hand, allows
an application to record the device’s location using GPS when the application
is in operation. The location data is precise up to six decimals (e.g.,
29.658068∘N, 95.158485∘W), indicating a resolution of approximately 0.1
meters. Due to their detailed information, the location data find wide
applications in computational social science studies [74, 72, 71].

Mobile phone data enable us to monitor the evacuation and return patterns of a
large group of individuals. To accomplish this, we first extract location data
points of each user during nighttime hours (specifically, from 9:00 PM to 6:00
AM) before the hurricane (from August 1 to 16), and then estimate the user’s
home location as the centroid of these nighttime points (Fig. 2c). These home
location estimations serve as benchmarks to track whether and when the
individual evacuated and returned. In estimating home locations and comparing
them with user trajectories across different days, we are able to measure the
return-home behavior of individuals.

Foot traffic data. We employ the foot traffic data in the entire Texas state
collected by SafeGraph333https://www.safegraph.com/ (Fig. 2d) to monitor the
statuses of POIs. SafeGraph combines location data from various mobile phone
applications and generates foot traffic data on numerous types of POIs (e.g.,
clothing stores, automobile dealers, gasoline stations, and postal services).
Note that the foot traffic data reveal only aggregated measures and do not
convey any personal socio-demographic or trajectory information. Especially,
each POI is associated with a unique place ID (i.e., safegraph_place_id) and
its polygon geolocation. For each POI (e.g., a restaurant), the data include
the number of hourly visits to this POI on different days. We focus on the
foot traffic data of POIs whose locations are within our selected five
counties.

Based on the phone data and foot traffic data, we finally identify 43,147 user
homes and 90,513 POIs in the five counties.

###  3.3 Three-layer network

To describe the PDR dynamic, we model the socio-physical system as a three-
layer network $G=\\{G_{h},G_{s},G_{p},E_{h,s},E_{h,p},E_{s,p}\\}$. Here,
$G_{h}=\\{V_{h},E_{h}\\}$, $G_{s}=\\{V_{s},E_{s}\\}$, and
$G_{p}=\\{V_{p},E_{p}\\}$ denote the human layer, the social infrastructure
layer, and the physical infrastructure layer, respectively. $E_{h,s}$,
$E_{h,p}$, and $E_{s,p}$ represent inter-layer connections (Fig. 3a). We
present definitions of nodes in $G$.

  * 1.

Nodes in the human layer. In $G_{h}$, a node $v_{h}\in V_{h}$ represents the
identified home of an individual. For example, if two persons $\mathcal{A}$
and $\mathcal{B}$ belong to the same household and reside together in a house
before the hurricane, then we introduce one node for $\mathcal{A}$ and the
other node for $\mathcal{B}$. Constructing distinct nodes for different
members in one household has two merits: (1) to provide a more nuanced and
accurate description of human return-home behavior at the high granularity
level (i.e., individual level); (2) to simulate the situation that different
members in one household do not return to the home simultaneously. The edge
set (i.e., $E_{h}$) will be defined in later paragraphs.

  * 2.

Nodes in the social infrastructure layer. In $G_{s}$, a node $v_{s}\in V_{s}$
denotes a POI. We extract the customer visit data on different days from
SafeGraph and set it as the feature of the POI node. The edge set (i.e.,
$E_{s}$) will be discussed in later paragraphs.

  * 3.

Nodes in the physical infrastructure layer. In $G_{p}$, we define five nodes
(i.e., $v_{p}\in V_{p}$) as the water/sewer system in the five counties,
respectively. We associate each node with its recovery level of water/sewer
system, which is a value between 0 and 1 (0: full damage; 1: full recovery)
and will be rigorously defined as $r_{a}(t)$ in Subsection 4.3. While the
mobile phone location data and foot traffic data provide high spatial
resolution information about nodes in $G_{h}$ and $G_{s}$, we lack such high-
resolution data on the water/sewer system. Hence, we resort to the coarse
five-node setting based on a survey (Subsection 4.6). We acknowledge this
limitation and recommend readers adopt a more nuanced definition if accurate
water/sewer data is available.

Note that each node has its unique geolocation ($v_{h}$: the location of the
estimated home; $v_{s}$: the location of a POI; $v_{p}$: the centroid of the
county). In summary, the numbers of nodes in $G_{h}$, $G_{s}$, and $G_{p}$ are
43,147, 90,513, and 5. The intra-layer and inter-layer edges are defined as
follows.

  * 1.

Intra-layer edges (i.e., $E_{h}$, $E_{s}$, and $E_{p}$). Recall that a node
$v$ in $G_{h}$ (or $G_{s}$) has its specific geolocation. Within $G_{h}$ (or
$G_{s}$), we assume the edge between nodes $v_{h1}$ and $v_{h2}$ (or $v_{s1}$
and $v_{s2}$) exists if and only if $d(v_{h1},v_{h2})$ (or $d(v_{s1},v_{s2})$)
(i.e., the spatial distance between them) is not larger than a predefined
threshold $\delta$. The underlying rationality lies in the theory of Tobler’s
First Law of Geography, which states that the statuses of many spatial
entities (e.g., houses) are affected by their adjacent spatial entities [66,
65]. We now explain the real-world implication of the edges in $E_{h}$ and
$E_{s}$. Two endpoints of an edge (i.e., two humans, or two POIs) could affect
each other during the PDR by reducing the skepticism about normal activities
and encouraging the recovery of humans (i.e., residents in the human layer,
and business owners of POIs in the social infrastructure layer). Based on this
definition, the intra-layer edges are bidirectional. In our model, we set
$\delta$ as 1 km, resulting in the numbers of edges in $G_{h}$, $G_{s}$ as
2,308,629 and 5,698,425, respectively. Finally, since the recovery level of
physical nodes in each county can be uniquely captured by the survey
(Subsection 4.6), we set $E_{p}$ as the empty set. In this way, we model the
recovery dynamics of nodes in $G_{p}$ independently.

  * 2.

Inter-layer edges (i.e., $E_{h,s}$, $E_{h,p}$, and $E_{s,p}$). The edge
between the node $v_{h}$ in $G_{h}$ and the node $v_{s}$ in $G_{s}$ is
constructed when their spatial distance is not larger than $\delta$. In
reality, human returning home decisions depend on the activities of
neighborhood POIs (which will be demonstrated in Subsection 4.2). In this way,
the inter-layer edges between $G_{h}$ and $G_{s}$ are from nodes in $G_{s}$ to
nodes in $G_{h}$. The number of edges in $E_{h,s}$ is 8,845,359. Finally, we
define the directional edges from each node in the $G_{p}$ (i.e., the
water/sewer system) to each node in $G_{h}$ and $G_{s}$ within the same
county, because the water/sewer system affects both the human returning home
and POI recovery.

To unearth the property of the built three-layer network, we display the
degree distribution of nodes in the three layers in Fig. 3b and Fig. 3c. In a
network, the degree of a node is defined as the number of its adjacent nodes
[6]. The degree distribution of all nodes in the network is one of the most
predominantly used measures of network topology. Fig. 3b shows the
distribution of node degree in the human layer. It appears that Harris County
has a larger average degree than the other four counties, which aligns with
the fact that Harris County includes many densely populated areas where each
home has a large number of nearby neighborhoods. Furthermore, the points ($x$:
degree, $y$: logarithmic count) can be roughly approximated by five straight
lines, implying the node degrees in the human layer in the five counties are
close to the exponential distributions. Fig. 3c also exhibits the straight
line fitting and reveals exponential node degree distributions of the social
infrastructure layer. Compared to the human layer, the discrepancy of degree
distributions among the five counties is not such obvious in the social
infrastructure layer, which is inferred from the close distance between points
representing Harris County and the other four counties in Fig. 3c. In Texas,
many POIs (such as automobile dealers, large supermarkets) have wide coverage
in suburban and rural areas, leading to an overall more homogeneous
configuration of POIs in urban, suburban, and rural sectors. In this way, the
degree distribution of nodes in the social infrastructure layer (Fig. 3c) in
Harris County (mostly urban areas) and the other four counties (mostly
suburban and rural areas) have a smaller difference than those in the human
layer (Fig. 3b).

Figure 3: (a) Three-layer socio-physical network. Nodes in $G_{h}$ (i.e., the
human layer), $G_{s}$ (i.e., the social infrastructure layer), and $G_{p}$
(i.e., the physical infrastructure layer) denote homes, POIs, and the
water/sewer system in the five counties; (b) Degree distribution of nodes in
$G_{h}$; (c) Degree distribution of nodes in $G_{s}$.

##  4 Interactions between Agents

After defining the three-layer network $G$, we now specify how humans, social
infrastructures, and physical infrastructures adapt to the changes in the
environment during the PDR. The relationship between either (1) multiple
humans or (2) humans and infrastructures involve complex human behavior, that
is, how individuals decide to return to their original homes considering the
recovery levels of their neighbors, nearby POIs, and water/sewer systems. To
capture the human behavior, we extract information from a survey conducted on
Texas residents, and develop a post-disaster behavioral model (PD-BM).
Furthermore, the interactions between social infrastructures and physical
infrastructures also exhibit unique dependencies. We study these dependencies
using a socio-physical system dynamic model (SP-DM) [75].

###  4.1 Using survey data to obtain the PD-BM in the human layer

The accurate description of human behavior plays a crucial role in the ABM of
the PDR by determining the changes in recovery levels. While human evacuation
and returning behavior are difficult to describe due to multiple factors
(e.g., demographics), we use a post-disaster survey to approximate such
behavior and feed the results into the ABM. Our ABM applies to other natural
hazards (e.g., earthquakes and wildfires) if we change the human behavior
module based on the case-specific behavior data. We now measure human behavior
in the three-layer network for Hurricane Harvey as an example.

We build the PD-BM based on results from the survey, namely, Texas Household
Recovery Survey (THRS). This survey was conducted on residents living in the
five counties in Texas from Sept. 17, 2021, to Feb. 23, 2022, and consists of
776 respondents. In particular, THRS queries questions regarding demographic
information (e.g., age, sex), hurricane-related individual behavior (e.g.,
evacuation, returning home), returning behavior of neighborhoods, the
reopening of POIs (e.g., school, business), and statuses of infrastructure
(e.g., power, roads). For example, one question is “To what extent has your
home and community experienced damage/recovery from Hurricane Harvey?”.
Another question is “To what extent have the following people and
organizations returned back to your community or reopened?”. Based on these
questions, we define variables (i.e., $q_{age},...,y_{return}$) to record the
answers of respondents. Please find the summary of survey questions,
variables, and their values in Table 5.

The data pre-processing incorporates two steps: data filtering and missing
data imputation. First, note that in the survey, only an evacuated person
could answer the question about returning home. Hence, there are three types
of persons: (1) not evacuate; (2) evacuate but not return permanently; (3)
evacuate and return permanently. Recall that the objective of this study is to
simulate the recovery of the socio-physical system. Therefore, we focus on the
collected information of the evacuated group (i.e., types (2) and (3)). After
data filtering, we get 170 response samples for 170 residents in the five
counties. Second, we fill in the missing data. For each response sample, we
compute its missing data ratio as the number of vacant answers over the total
number of answers in the survey. Among these 170 response samples, the average
missing data ratio is 5.2%, and the median missing data ratio is 0.0%, because
more than half of the response samples do not have vacant answers. In
consideration of the low missing data ratio and the relatively small dataset
(i.e., 170 response samples), we adopt the mean substitution method, which
replaces the missing value under one variable with the mean of existing values
under the same variable. While the mean substitution method retains the mean
value of each variable unchanged, it distorts the statistical inference by
enlarging the valid dataset. Here, we recommend readers directly filter out
survey records with any missing data entries, if the number of full
observations in the post-disaster survey is large (e.g., 839 in Bian et al.
[8]).

Until now, we have obtained complete information about demographics,
neighborhood returning, nearby POIs, and infrastructure for individuals who
evacuated during the hurricane. Here, we construct the PD-BM to investigate
the underlying relationship between different variables (i.e., $q_{age}$, …,
$q_{physical,e}$) and returning home decisions (i.e., $y_{return}$). Since we
consider two decisions (i.e., evacuate but not return, evacuate and return) in
our study, we exploit the binary logit (BL) model to predict whether the
evacuated individual will return. For other types of post-disaster surveys,
readers can refer to the multinomial logit (MNL) model [31] if there are at
least three decision options, the nested logit (NL) model [8] if the different
types of recovery decisions (e.g., return time, mode choices) are dependent
with each other.

Our BL model fixes $y_{return}$ as the dependent variable and other variables
except for $y_{evacuate}$ in Table 5 as candidate independent variables. Here,
$q_{human,*}$ serves as an independent variable, enabling our BL model to
capture the impact of neighborhood individuals on the return of an individual.
This setting can reflect real-world situations where different adjacent human
agents mutually affect each other if we apply the BL model to every human
agent.

It is worth noting that the PD-BM includes one variable among $q_{human,a-b}$,
one variable among $q_{social,a-e}$, and one variable among $q_{physical,a-e}$
for two reasons. First, we discern that the variables within the same category
have strong positive correlations and thus select only one variable in each
category to avoid multicollinearity. Second, since the statistical model
characterizes inter-layer influence on human returning home behavior for the
ABM, employing more than one variable for each agent leads to burdensome agent
interactions. Finally, in the raw survey data, $q_{income}$ belongs to one of
the nine intervals (Table 5). We replace intervals [0, 15K) and (120K,
$+\infty$) with 15K and 120K, respectively, in the BL model. For the other
seven intervals (e.g., [60K, 75K]), we put the average income value of the
interval (e.g., $67.5$K) into our BL model. Under these settings, we determine
the final model with the largest $R^{2}$ values.

###  4.2 The PD-BM-Harris and the PD-BM-Other

Given that Harris County has an apparently distinct population density (Table
2) and network properties (Fig. 3b and Fig. 3c) from the other four counties,
their decisive factors of human returning behavior may not be identical.
Hence, we build two separate models for Harris County and the other four
counties, respectively. We present the final models (i.e., the PD-BM-Harris
and the PD-BM-Other) in Table 3.

PD-BM-Harris | PD-BM-Other
---|---
Variable | Coefficient | Standard Error |  $p$-value | Variable | Coefficient | Standard Error |  $p$-value
Intercept | -1.904 | 0.419 | <$0.001^{**}$ | Intercept | -2.379 | 0.682 | <$0.001^{**}$
$q_{house}$ | 1.520 | 0.508 | $0.003^{**}$ | $q_{income}$ | 2.26$\times$ $10^{-5}$ | 8.82$\times$ $10^{-6}$ | $0.010^{*}$
$q_{human,a}$ | 1.638 | 0.776 | $0.035^{*}$ | $q_{human,b}$ | 3.298 | 1.426 | $0.021^{*}$
$q_{social,e}$ | -1.756 | 0.565 | $0.002^{**}$ | $q_{social,c}$ | -4.845 | 1.850 | $0.009^{**}$
$q_{physical,b}$ | 1.171 | 0.490 | $0.017^{*}$ | $q_{physical,b}$ | 1.675 | 0.567 | $0.003^{**}$
$n$ | 99 |  |  | $n$ | 71 |  |
$R^{2}$ | 0.416 |  |  | $R^{2}$ | 0.398 |  |
Adjusted $R^{2}$ | 0.342 |  |  | Adjusted $R^{2}$ | 0.297 |  |

Table 3:  Binary logit models: the PD-BM-Harris, the PD-BM-Other (**:
$p$<0.01; *: $p$<0.05). $n$: the number of samples. Here, $R^{2}$ and adjusted
$R^{2}$ values for PD-BM-Harris and PD-BM-Other are not large due to the
complex nature of human behavior during the PDR. In fact, the two models have
comparable adjusted $R^{2}$ values (i.e., 0.342, 0.297) with existing studies
such as [31] (adjusted $R^{2}$ values: 0.337, 0.315).

From the PD-BM-Harris (Table 3, Left), we recognize that the selected
variables from the human layer, the social infrastructure layer, and the
physical infrastructure layer are respectively $q_{human,a}$ (neighborhoods),
$q_{social,e}$ (other community), and $q_{physical,b}$ (water/sewer).
Individuals owning the house (i.e., $q_{house}=1$) are more likely to return
to their homes, which could be inferred from the positive coefficient (i.e.,
$\beta$=1.520). Compared with people who rent the house (i.e., $q_{house}$=0),
evacuated house owners take more responsibility during the house repair if the
house is damaged, so they have stronger motivation to return [49].

Furthermore, we conclude a negative relationship between $y_{return}$ and
$q_{social,e}$ from the coefficient -1.756. It seems to be counterintuitive
because the recovery of social infrastructures may accelerate human returning
home. Nevertheless, it makes sense for the following reason. Humans have
multiple choices of evacuation destinations (e.g., home of friends/relatives,
public shelters, hotels or motels, vehicles, etc.). It may happen that humans
stay at evacuated locations and visit nearby POIs (which leads to high
recovery levels of the social infrastructure layer), but do not return to
their original homes (which leads to low recovery levels of the human layer)
because of issues such as house damages for a long period.

Finally, water/sewer (i.e., $q_{physical,b}$) is found to be significantly
associated with human return behavior. While returned individuals may employ
power generators to temporarily produce electricity for their daily usage,
they have a stronger dependency on public-supplied water/sewer facilities,
because water/sewer services can not be accomplished by themselves.
Additionally, existing studies also argue that the water service deficit could
properly approximate the recovery level of the physical infrastructure system
based on empirical data [75, 39].

Analogous to the PD-BM-Harris, water/sewer service is incorporated in the PD-
BM-Other (Table 3, Right) to associate human returning decision as a
representative variable of the physical infrastructure system.

Moreover, the positive parameter on the variable $q_{income}$ (i.e.,
$2.26\times 10^{-5}$) reveals that evacuated people in high-income families
are more prone to return to their homes than those from low-income families.
Note that we use the original values of $q_{income}$ outlined in Table 5 in
PD-BM-Harris and PD-BM-Other so that the parameter has a small value. We
interpret the finding from two aspects. First, wealthy people have high levels
of mobility (e.g., vehicle ownership, support from friends, and accessibility
to gas stations), allowing them to easily return to their original communities
[23]. Second, their pre-disaster living houses are sturdy since the building
materials are of high quality, resulting in less housing damage than the
temporal houses of low-income groups. In this way, high-income people could
recover more promptly than low-income people as the mild housing damage is
repaired more efficiently. This observation is commensurate with the finding
from the existing study which demonstrates that the PDR rate after natural
hazards is positively related to income levels [73]. Furthermore, the impact
of income on human recovery is more apparent in the four counties (with a
large ratio of rural areas) than those in Harris County, because rural
inhabitants reside more sparsely and are therefore more self-dependent [23].

Similar to the PD-BM-Harris, the negative relation between $y_{return}$ and
$q_{social,c}$ in the PD-BM-Other (i.e., parameter -4.845) results from the
events that many evacuated individuals do not return back to their homes but
still visit surrounding POIs (e.g., business entities).

Recall that the objective of our study is to establish a rigorous ABM to
simulate the PDR where human behavior plays a crucial role. The goal of this
subsection is to quantify human recovery behavior via an example exploiting
survey data and the BL model. Here we acknowledge the limitation of using the
small-size survey data (i.e., 99 and 71) to characterize the complex human
return behavior in the five counties with more than 6 million population. We
suggest readers deliver a more complete and representative investigation to
gauge human behavior and then encode the behavior model in the ABM for other
disasters.

###  4.3 Defining node attributes as their recovery levels

On the day $t$, each node (i.e., a node $v_{h}\in V_{h}$, or a node $v_{s}\in
V_{s}$, or a node $v_{p}\in V_{p}$) is associated with a recovery level
$r_{a}(t)$. It is a non-negative value, encodes its recovery status, and will
evolve with $t$. Specially, $r_{a}(t)$ for agents in the three layers are
respectively:

  * 1.

Human layer. $r_{a}(t)=\left\\{\begin{aligned}
1&,&the\;\;resident\;\;stays\;\;at\;\;his/her\;\;home\;\;during\;\;night\;\;on\;\;the\;\;day\;\;t,\\\
0&,&the\;\;resident\;\;does\;\;not\;\;stay\;\;at\;\;his/her\;\;home\;\;during\;\;night\;\;on\;\;the\;\;day\;\;t.\\\
\end{aligned}\right.$

  * 2.

Social infrastructure layer.
$r_{a}(t)=\frac{the\;\;number\;\;of\;\;daily\;\;visits\;\;to\;\;the\;\;POI\;\;on\;\;the\;\;day\;\;t}{the\;\;average\;\;daily\;\;visits\;\;to\;\;the\;\;POI\;\;from\;\;Aug.\;1\;\;to\;\;Aug.\;16.}$.

  * 3.

Physical infrastructure layer. $r_{a}(t)$ = the functionality of physical
infrastructures. $r_{a}(t)=0$: completely damage; $r_{a}(t)=1$: full function.

###  4.4 Using survey data to obtain the SP-DM in the social infrastructure
layer

While the PD-BM-Harris and the PD-BM-Other describe the human return-home
behavior in $G_{h}$, we still lack agent interaction rules specifying the
recovery dynamics of nodes in $G_{s}$. In this subsection, we measure the
recovery of nodes in the social infrastructure layer as $r_{a}(t)$ defined in
Subsection 4.3. Distinct from the BL models (i.e., the PD-BM-Harris, the PD-
BM-Other), the dynamic models (i.e., the SP-DM-Harris, the SP-DM-Other) are
exploited to capture recovery dynamics in $G_{s}$ in Harris County and the
other four counties. We first illustrate the meaning of the dynamic model
using an example (i.e., Eq. 1) and then propose the dynamic model for our ABM
(i.e., Eq. 2).

####  4.4.1 An example of the dynamic model

A dynamic model on a variable $\Phi(t)$ refers to a differential equation as
follows:

$\frac{d\Phi(t)}{dt}=f_{\theta}(\Phi(t)),$ (1)

which is parameterized by $\theta$ and characterizes the temporal variations
of $\Phi(t)$. Note that $\Phi(t)$, $\theta$, and $f_{\theta}(.)$ are used in
this example but not in our ABM. Besides, $\theta$ is estimated from the
observable data points at different times (i.e.,
$\\{(t_{i},\Phi(t_{i}))\;|\;t_{i}\in\Gamma\\}$, where $\Gamma$ is the set of
time points). Building upon dynamic models such as the exponential growth
model (i.e., $\Phi(t)=\Phi(0)e^{Bt}$, where $\Phi(0)$ is the value of
$\Phi(t)$ at time $t=0$, and $B$ is the parameter) and the logistic growth
model (i.e., $\Phi(t)=\frac{A}{1+Be^{-Ct}}$, where $A$, $B$, and $C$ are model
parameters that are estimated from the data), researchers propose various
differential equations to model the dynamics of the socio-physical systems
[75, 38].

####  4.4.2 The dynamic model in our ABM

We now build the SP-DM-Harris and the SP-DM-Other for our ABM. For one county,
we use $\overline{r}_{s}(t)$ and $\overline{r}_{p}(t)$ to denote the average
recovery levels of its nodes in $G_{s}$ and $G_{p}$ on the day $t$,
respectively. Based on these definitions, we design the dynamic model
describing the changes of $\overline{r}_{s}(t)$ as follows:

$\frac{d\overline{r}_{s}(t)}{dt}=0.001\beta_{s}\overline{N}\overline{r}_{s}(t)(1-\frac{\overline{r}_{s}(t)}{K_{s}})+0.1\beta_{p}\overline{r}_{p}(t)(1-\frac{\overline{r}_{p}(t)}{K_{p}}),$
(2)

where $\beta_{s}$, $K_{s}$, $\beta_{p}$, and $K_{p}$ are parameters and will
be discussed in Subsection 4.4.3. We use $\overline{N}$ to represent the
average number of intra-layer neighborhoods for nodes in $G_{s}$. The values
of $\overline{N}$ for nodes in $G_{s}$ are respectively: 139.1 (Harris
County), 107.7 (Fort Bend County), 79.9 (Brazoria County), 78.5 (Galveston
County), and 70.2 (Jefferson County). In this way, we build five dynamic
equations (i.e., Eq. 2) for the five counties, respectively. Note that the
four dynamic equations associated with the four counties (i.e., Fort Bend
County, Brazoria County, Galveston County, and Jefferson County) share the
same parameters of $\beta_{s}$ and $\beta_{p}$ given their similar socio-
demographic properties (Table 2), resulting in the model SP-DM-Other. Finally.
Eq. 2 serves as the prototype of the updating rules in $G_{s}$ developed later
(in Subsection 5.4.2).

####  4.4.3 The interpretation of Equation 2

In Eq. 2, the first term on the right-hand side captures the internal
influence from the neighborhood social infrastructure nodes, while the second
term models how the physical infrastructure nodes (i.e., water/sewer
facilities) affect the social infrastructure nodes (i.e., POIs). In
particular, we include
$\overline{r}_{s}(t)(1-\frac{\overline{r}_{s}(t)}{K_{s}})$ and
$\overline{r}_{p}(t)(1-\frac{\overline{r}_{p}(t)}{K_{p}})$ in the model to
describe the underlying logistic growth patterns that have been discovered in
the existing study [73]. Note that for a the logistic model (i.e.,
$\Phi(t)=\frac{A}{1+Be^{-Ct}}$), we have

$\frac{d\Phi(t)}{dt}=C\;\Phi(t)(1-\frac{\Phi(t)}{A}).$ (3)

In Eq. 2, $K_{s}$ and $K_{p}$ play the same role as $A$ in Eq. 3 to capture
the ultimate recovery level within one period, $\beta_{s}$ and $\beta_{p}$
imitate $C$ in Eq. 3. Besides, we introduce $\overline{N}$ (i.e., the average
number of spatial neighborhoods) in the first term of Eq. 2 to consider the
impact of the neighborhood at the aggregated level. In addition, we
incorporate two constants 0.001 and 0.1 in Eq. 2 for better convergence of the
estimated probability distribution of parameters from the experiments.

Here, we acknowledge that the individual-level model (which uses the exact
number of neighborhood nodes $N_{i}$ for node $i$) leads to more accurate
modeling than the aggregated model (which uses $\overline{N}$). The barrier of
such individual-level modeling is that it significantly increases the number
of dynamic equations (i.e., Eq. 2) to 90,513 (i.e., $|V_{s}|$), making the
parameter estimation process computationally intractable. Given the
computational constraint, we resort to the aggregate-level model in estimating
parameters $\beta_{s}$ and $\beta_{p}$. However, our attribute updates in the
ABM are still designed as individual-level updates in order to preserve the
high-resolution advantage of the ABM.

###  4.5 The SP-DM-Harris and the SP-DM-Other

Now we conduct the parameter estimation for Eq. 2 to obtain the SP-DM-Harris
and the SP-DM-Other. We utilize the maximum a posterior (MAP) method to
estimate $\beta_{s},K_{s},\beta_{p}$, and $K_{p}$ based on the observable
survey data (i.e., $D=\\{(t_{i},r_{s}(t_{i}))\;|\;t_{i}\in\Gamma\\}$, where
$\Gamma$ is the set of time points) and assumed prior distribution. Here, the
estimation is a joint estimation where the four parameters are updated
simultaneously. For $\boldsymbol{\beta}=[\beta_{s},K_{s},\beta_{p},K_{p}]$,
the MAP estimates $\boldsymbol{\beta}$ as:

$\hat{\boldsymbol{\beta}}_{MAP}=argmax_{\boldsymbol{\beta}}P(D\;|\;\boldsymbol{\beta})P(\boldsymbol{\beta}),$
(4)

where $P(\boldsymbol{\beta})$ is the density function of the prior
distribution of $\boldsymbol{\beta}$, and $P(D\;|\;\boldsymbol{\beta})$ is the
conditional probability. Specially, we initialize the prior distribution of
$\beta_{s},K_{s},\beta_{p}$, and $K_{p}$ as follows:

$\beta_{s},\beta_{p}\sim HalfCauchy(1);K_{s},K_{p}\sim Uniform(0.5,1.0).$ (5)

The prior distributions are chosen based on the existing study [75]. Next, we
apply the Hamiltonian Monte Carlo (HMC) sampling method to draw the model
parameter samples given the distribution using the PyMC3 package444
https://docs.pymc.io/en/v3/index.html. HMC works well for large data sets by
introducing the momentum term of the posterior distribution and the energy
term [33]. We check the convergence of estimated parameters via the acceptance
rate in the HMC. Finally, we estimate $\beta_{s},K_{s},\beta_{p}$, and $K_{p}$
as the peak points of the density function of the posterior distribution of
$\beta_{s},K_{s},\beta_{p}$, and $K_{p}$ based on the distribution of
$\boldsymbol{\beta}$. The final estimated parameters are shown in Table 4.

| SP-DM-Harris | SP-DM-Other
---|---|---
Variable | $\beta_{s}$ | $K_{s}$ | $\beta_{p}$ | $K_{p}$ | $\beta_{s}$ | $K_{s}$ | $\beta_{p}$ | $K_{p}$
Estimated value | 0.026 | 0.671 | 1.432 | 0.901 | 0.093 | 0.736 | 1.114 | 0.935

Table 4:  The parameters in the dynamic models: the SP-DM-Harris, the SP-DM-
Other.

###  4.6 Using survey data to obtain recovery dynamics of the physical
infrastructure layer

Figure 4: The recovery level of the physical infrastructures (i.e.,
water/sewer facilities) in the five counties fitted by five generalized
logistic curves. (a, b) Harris County, Fort Bend County, and Brazoria County;
(c, d) Galveston County, Jefferson County. Note that (b, d) are enlarged parts
of (a, c).

In the proposed three-layer network $G$, the physical infrastructure layer
contains five nodes representing the water/sewer systems in the five counties.
For a node $v_{p}$ in the physical infrastructure layer, $r_{v_{p}}(t)$
represents the recovery level of this node on the day $t$ (please find
$r_{a}(t)$ in Subsection 4.3). We attain $r_{v_{p}}(t)$ from the survey. In
particular, Question 9 in the THRS includes a subquestion consulting about the
water/sewer recovery level since Aug. 30, 2017. For each county on the day
$t$, we compute the average score provided by respondents living in the county
and conclude the results as the scatter points displayed in Fig. 4. To
evaluate the recovery level on the days that are not covered in the survey
(e.g., 20 days or 40 days since Aug. 30, 2017), we apply five generalized
logistic functions [50]:

$r_{v_{p}}(t)=\frac{A_{v_{p}}}{1+e^{-C_{v_{p}}(t-D_{v_{p}})}}+B_{v_{p}},$ (6)

to fit the scatter dots for the five counties. Here, $A_{v_{p}}$, $B_{v_{p}}$,
$C_{v_{p}}$, and $D_{v_{p}}$ are parameters that are estimated based on the
data. Through the full-period curves exhibited in Fig. 4a and Fig. 4c and the
first-two-month curves exhibited in Fig. 4b and Fig. 4d, we notice that the
scatter points can be seamlessly fitted by the generalized logistic curves.
Additionally, we present the numerical evidence as the Pearson correlations
(i.e., $\rho$) between the real recovery level time series and the fitted
recovery level time series. We report that all five Pearson correlations are
greater than 0.950, entailing a consistent tendency between real data and
fitted data. Eq. 6 will be used to generate the daily recovery level of
physical infrastructures in the five counties in our ABM.

##  5 Long-term ABM Simulation

Figure 5: The long-term PDR processes. (a) The dynamics of recovery levels for
agents representing homes and POIs; (b) The flowchart summarizing the
procedure of agent attribute updates during the PDR process.

After building the three-layer network $G$ (i.e., Subsection 3.3), specifying
human behavior (i.e., Subsection 4.2), recovery dynamics of social
infrastructures (i.e., Subsection 4.5) and physical infrastructures (i.e.,
Subsection 4.6), we now develop the ABM of the PDR process. Our goal is to
simulate how individuals return to their original homes, infrastructures
resume themselves to pre-disaster activity levels, and to further investigate
the impact of physical infrastructures (i.e., water/sewer facilities) on
humans and social infrastructures (i.e., POIs).

###  5.1 Agent definitions

The ABM comprises three types of agents: homes of residents, POIs, and county-
level water/sewer systems whose dependencies constitute recovery channels in
the system. From Subsection 4.2, we know an evacuated individual decides to
return to his/her home by jointly considering the status of physical
infrastructures, social infrastructures, and whether neighborhood households
have returned. Specifically, we define agents in the ABM as home nodes, social
infrastructure nodes, and physical infrastructure nodes in $G$.

###  5.2 Agent attributes

For each agent in the ABM, we define its attribute on the day $t$ as its
recovery level $r_{a}(t)$, which is defined in Subsection 4.3. The $r_{a}(t)$
is updated with the temporal resolution of one day, which was adopted in
existing recovery studies such as Hajhashemi et al. [31]. On each day, the ABM
updates $r_{a}(t)$ for all agents $a$ based on their previous recovery levels
and the recovery level of their neighborhoods in $G$. The outcomes of the ABM
are the dynamics of $r_{a}(t)$ under different scenarios.

###  5.3 ABM initialization

We now customize the initialization of the ABM. Since our ABM simulates the
dynamics of $r_{a}(t)$ for different agents $a$ with evolving time $t$, we
initialize the system by specifying the starting time $t_{0}$ and
$r_{a}(t_{0})$.

Two examples of the changes of $r_{a}(t)$ from the pre-hurricane to the post-
hurricane periods for human home nodes and POI nodes are visualized in Fig. 5a
and Fig. 5b, respectively. During the pre-hurricane period shown in Fig. 5a,
since we define home locations as the centroids of night-hour mobile phone
location points, then $r_{v_{h}}(t)$ are equal to 1 for most home agents
$v_{h}\in V_{h}$, implying that the homes are occupied by residents. When the
hurricane approaches and leaves (i.e., from Aug. 17 to Aug. 30),
$r_{v_{h}}(t)$ for many home agents decreases from 1 to 0, indicating that the
residents evacuate to other places. Afterward, the PDR commences after the
hurricane dies out, and several $r_{v_{h}}(t)$ resume to 1, reflecting that
people return to their homes. The difference between $r_{a}(t)$ shown in Fig.
5b and those in Fig. 5a is that $r_{a}(t)$ are decimal (e.g., 0.97) for POI
nodes and binary (i.e., 0 or 1) for home nodes. It is because the recovery
level of a POI is defined as the ratio of the current number of visits and the
average number of historical visits, while a home node is either occupied or
not occupied. Based on these analyses, we define $t_{0}$ as the day when
Hurricane Harvey left Texas, which was Aug. 30, 2017.

###  5.4 Updating rules

After initialization, the ABM updates the attributes of agents (i.e., homes,
social infrastructure nodes, and physical infrastructure nodes) in an
iterative manner, which is illustrated in Fig. 5c. The simulation covers the
period from Aug. 30 to Oct. 28, so the number of total iterations is $M=60$.
At each iteration $t\in\\{t_{0}+1,t_{0}+2,...,t_{0}+59\\}$, we renew
$r_{a}(t)$ for agents $v_{p}\in V_{p},v_{s}\in V_{s}$, and $v_{h}\in V_{h}$
sequentially.

####  5.4.1 Physical infrastructure layer

For the physical infrastructure agent $v_{p}\in V_{p}$, we use the generalized
logistic curves shown in Fig. 4 and Eq. 6 to evaluate the recovery level of
physical infrastructures on different days (i.e., $r_{v_{p}}(t),v_{p}\in
V_{p},t\in\\{t_{0}+1,t_{0}+2,...,t_{0}+59\\}$). The underlying reason is that
these curves perfectly fit the real data from the survey and provide high-
resolution temporal recovery levels.

####  5.4.2 Social infrastructure layer

Next, we perform the attribute update for nodes $v_{s}\in V_{s}$. Recall that
the SP-DM-Harris and the SP-DM-Other (i.e., Eq. 2 with estimated parameters in
Table 4) inform us of the relationship between the recovery rate of social
infrastructure nodes (i.e., $\frac{d\overline{r}_{s}(t)}{dt}$) and the
recovery level of physical infrastructure nodes (i.e., $\overline{r}_{p}(t)$).
For the node $v_{s}\in V_{s}$ and $t\in\\{t_{0}+1,t_{0}+2,...,t_{0}+59\\}$, we
employ this relationship to construct the attribute updating rule as follows:

$r_{v_{s}}(t)=min\\{1.0;\;r_{v_{s}}(t-1)+0.001\beta_{s}\sum_{v_{s^{\prime}}\in
N(v_{s})}r_{v_{s^{\prime}}}(t-1)(1-\frac{r_{v_{s^{\prime}}}(t-1)}{K_{s}})+0.1\beta_{p}r_{p(v_{s})}(t-1)(1-\frac{r_{p(v_{s})}(t-1)}{K_{p}})\\}.$
(7)

In Eq. 7, $N(v_{s})$ is the set of neighborhood nodes in
$G_{s}=\\{V_{s},E_{s}\\}$ of $v_{s}$, and $p(v_{s})$ is the physical
infrastructure node that is connected to $v_{s}$. We specify the values of
$\beta_{s},K_{s},\beta_{p}$, and $K_{p}$ based on Table 4. Eq. 7 shares the
analogous form as Eq. 2, capturing the impact of neighborhood social
infrastructure nodes and physical infrastructure nodes on the recovery of
$v_{s}$. The underlying intuition is two-fold: (1) the business reopening of a
POI can be stimulated by the recovery of geographically adjacent POIs, which
encourages both customers and business stakeholders to restart daily
activities; (2) an increased level of the recovery level of the physical
infrastructures in a county contributes to the recovery of social
infrastructures. It is because the water/sewer facilities play a fundamental
role in economic activities such as restaurants. In addition, we add the upper
bound of 1.0 in Eq. 7 to ensure that the daily visits to each node $v_{s}$ do
not exceed those during the pre-hurricane period.

####  5.4.3 Human layer

The final step within one iteration is to obtain $r_{v_{h}}(t+1)$ for
$v_{h}\in V_{h}$ and $t\in\\{t_{0},t_{0}+1,...,t_{0}+58\\}$. In particular, we
design $r_{v_{h}}(t+1)$ as follows:

$r_{v_{h}}(t+1)=\left\\{\begin{aligned} 1,\;\;&r_{v_{h}}(t)=1,\\\
1,\;\;&r_{v_{h}}(t)=0\;\;and\;\;with\;\;probability\;\;\frac{P_{v_{h}}}{M},\\\
0,\;\;&r_{v_{h}}(t)=0\;\;and\;\;with\;\;probability\;\;(1-\frac{P_{v_{h}}}{M}),\end{aligned}\right.$
(8)

where $P_{v_{h}}$ is calculated based on the binary logistic models (i.e., the
PD-BM-Harris, the PD-BM-Other). Recall that $P_{v_{h}}\in[0,1]$ describes the
probability that an evacuated person returns to his/her home during the long
term. Hence, we design the updating rule with the returning probability as
$\frac{P_{v_{h}}}{M}$, where $M$ is the number of simulated days on one day.
The underlying assumption is that the returning home event of an individual is
evenly distributed among the recovery period (i.e., $M$ days). This assumption
is made given that we have information of an individual returning probability
(i.e., $P_{v_{h}}$) but lack the exact values of returning probability on each
day. We hope to strengthen this assumption using more complete data in the
future.

##  6 Scenarios and Results

###  6.1 Scenarios

Our ABM simulates the recovery levels of each node in $G$ and allows us to
examine the recovery trajectories under different post-disaster policies. We
generate nine scenarios from hypothetical physical infrastructure and social
infrastructure recovery promotion strategies (Fig. 6). Scenario #1 serves as
the baseline scenario representing the real-world recovery level of homes,
social infrastructures (i.e., POIs), and physical infrastructures (i.e.,
water/sewer systems) since Aug. 30, 2017.

Figure 6: Scenarios #1 to #9.

Scenarios #2 and #3 are designed to examine the effect of physical
infrastructure repair enhancement. It was released by the Texas Commission on
Environment Quality that more than 60 public-water systems were not normally
operating because of Hurricane Harvey [40], which illustrates the damaging
effects of such hazards on physical infrastructures. Scenarios #2, #3 model
the situations when the recovery velocity of the water/sewer system (i.e.,
$r^{\prime}_{v_{p}}(t+1)-r^{\prime}_{v_{p}}(t)$) is respectively 2.0 and 4.0
times as those in Scenario #1 (i.e., $r_{v_{p}}(t+1)-r_{v_{p}}(t)$). Here,
$t\in\\{t_{0},t_{0}+1,...,t_{0}+58\\}$. Note that such physical infrastructure
enhancement is time-dependent. The new physical recovery curve still preserves
the first-steep-then-flat pattern shown in Fig. 4. In the real world, physical
recovery enhancement can be implemented by investing more budget (e.g.,
dispatching more repairing workers, and providing more gas and mobility
choices for these workers).

Furthermore, we customize Scenarios #4 and #5 to investigate the consequence
of stimulating the recovery of social infrastructures. In particular, we set
$\beta_{s}^{\prime}=2.0\beta_{s}$, $\beta_{s}^{\prime}=4.0\beta_{s}$ where
$\beta_{s}$ is the original parameter in Eq. 2. These settings describe the
situations where the government conducts POI recovery plans (e.g., the
disaster tax relief discussed in Subsection 2.3).

Finally, we design Scenarios #6, #7, #8, #9 to explore the system recovery
performance under joint physical infrastructure and social infrastructure
policies. Scenarios #6, #7, #8, #9 set
$(\lambda_{p},\lambda_{s})=(\frac{r^{\prime}_{v_{p}}(t+1)-r^{\prime}_{v_{p}}(t)}{r_{v_{p}}(t+1)-r_{v_{p}}(t)}$,$\frac{\beta^{\prime}}{\beta}$)
as (2.0, 2.0), (2.0, 4.0), (4.0, 2.0), and (4.0, 4.0), respectively.

###  6.2 Code structure

We name our simulator as PostDisasterSim (PDS) and implement it using the Mesa
package in Python. Mesa is an agent-based modeling toolkit and the counterpart
to NetLogo [64]. Mesa has been extensively used in human behavior modeling
during natural hazards [27, 63] and can collaborate with downstream decision-
making modules (e.g., optimization, reinforcement learning). The code of the
PDS can be found at https://github.com/JiaweiXue/PostDisasterSim. In the PDS,
we first build the network using mobility and POI data and then simulate the
daily changes in recovery levels of different types of agents. The schematic
diagram of the agent attribute updates is shown in Fig. 10. We run the PDS on
the Ubuntu system, which has a 3.3 GHz w-2155 CPU and 32 GB memory. The
running time for each scenario of 60 days is between 600 and 800 seconds.

###  6.3 Analyzing spatio-temporal recovery under Scenario $\\#$1.

Figure 7: Simulation results under Scenario #1. (a) Recovery curves for the
three layers in Harris County; (b) The average recovery level of home nodes of
people who own or rent houses in Harris County; (c) Recovery curves for the
three layers in the other four counties; (d) The average recovery level of
home nodes with varying income levels in the other four counties.

We present simulation outcomes under Scenario #1 in Fig. 7. Fig. 7a and Fig.
7c display the average recovery level (i.e., $\overline{r}_{a}(t)$) for agents
in $G_{h}$, $G_{s}$, and $G_{p}$ on different days, respectively. Fig. 7b
shows the average recovery level of home nodes representing residents who own
or rent the house. Fig. 7d exhibits the average recovery level of residents
with distinct annual income levels.

First, we notice that the recovery curves of physical infrastructures (i.e.,
yellow curves) and social infrastructures (i.e., blue curves) are concave,
with initially a large slope and then a mild slope. The results agree with the
logistic-shape pattern in Eq. 6 and Eq. 7. Nevertheless, the recovery curves
of the home layer (i.e., red curves) are approximately linear during the two-
month simulation period. It is because our human layer updating rule (i.e.,
Eq. 8) assumes that the returning probability is a constant across different
days.

Second, comparing the recovery curves in Harris County (i.e., Fig. 7a) and the
other four counties (i.e., Fig. 7c), we discern that the recovery levels of
physical infrastructures (i.e., yellow curves) in Harris County are lower than
those in the other four counties on the same day. It is known that commercial
and industrial water demand per capita per day is nearly the same as the
domestic and public needs in the United States [48]. Harris County has more
intensive commercial and industrial activities than the other four rural
counties, resulting in higher water demand and a more complicated water supply
network. Such a complex water network is more difficult to repair, causing a
slow recovery rate. In contrast, the human returning home ratios (i.e., red
curves) in Harris County are higher than the others. In fact, residents in
Harris County could have stronger accessibility to urban facilities such as
food and medical services. It is because Harris County is mostly an urbanized
area with more POIs (i.e., 66,995 from Fig. 2) than the other four counties
(i.e., 8,947, 4,834, 5,330, and 4,407 from Fig. 2). Consequently, they are
more confident about returning home soon after the hurricane, thus engendering
fast human layer recovery.

Third, Fig. 7b reveals that the average recovery levels of residents owning
the house are higher than those of residents renting the house on Sept. 30 and
Oct. 30 in Harris County, even if they are quite similar on Aug. 30. As
mentioned earlier, house owners care more about the statuses of their houses
so that they are more self-motivated to return back.

Fourth, we conclude from Fig. 7d that high-income groups return to their homes
faster than low-income groups in an overall manner. High-income people not
only have higher mobility capabilities and accessibility of recovery sources
[23] but also are more likely to own their houses, thus explaining the return
behavior differences across nine income groups.

Figure 8: Spatio-temporal results in Scenarios #1.

We now scrutinize the spatial heterogeneity of recovery rates in the five
counties since Aug. 30. It is worthwhile to mention that Harris County and
Fort Bend County are inland counties, while Brazoria County, Galveston County,
and Jefferson County are coastal counties and adjacent to the Gulf of Mexico.
The report from the National Hurricane Center [10] informs us of two
discrepancies between the two inland counties and the three coastal counties:
(1) the coastal counties maintained smaller distances to the trajectories of
the centroid of Hurricane Harvey (page 56 in the report); (2) the coastal
counties experienced the storm surge inundation of 2-4 feet (page 59 in the
report).

Fig. 8 summarizes the average recovery levels of agents in the three layers on
Aug. 30, Sept. 30, and Oct. 30. Here we focus on two observations and discuss
the underlying insights.

The first observation is that humans living in inland counties (i.e., Harris
County and Fort Bend County) return to homes more swiftly than the other three
counties, which can be inferred from the three subfigures on the first row of
Fig. 8. For one thing, the two inland counties are more distant from the
hurricane centroid, resulting in less housing damage. In this way, a higher
ratio of residents in the two counties returns back. For another, Interstate
10 and Interstate 45 connect at the inland county, enabling more convenient
roadway-based returning home trips [22].

The second observation is that the physical infrastructure layer (i.e.,
water/sewer system) in Fort Bend County had significantly better functionality
than the other four counties on Aug. 30, which can be found in the first
subfigure on the last row of Fig. 8. On Sept. 30 and Oct. 30, the
functionality of physical infrastructures in Fort Bend County was close to
Harris County, Galveston County, and Jefferson County and was inferior to
Brazoria County. According to Fig. 8 in the NHC report [10], heavy rainfall
occurred near Galveston Bay and Trinity Bay, while the rainfall in Fort Bend
was much milder. Hence, the water/sewer system in Fort Bend maintained a
superior condition than the other four counties that surround Galveston Bay
and Trinity Bay. As time proceeded and water/sewer repairing forces started to
work, the physical infrastructures in all five counties gradually recovered.

###  6.4 Comparing results under Scenarios $\\#$ 1-9.

Figure 9: Comparing baseline (i.e., #1) with physical infrastructure
improvement (i.e., #2, #3), social infrastructure improvement (i.e., #4, #5),
and joint improvement (i.e., #6, #7, #8, #9).

To evaluate the recovery levels of the system under various hypothetical
policies, we run the simulation for each of the nine scenarios shown in Fig.
6. Fig. 9 includes the average recovery levels of social infrastructures in
the five counties on 0, 7, 30, and 60 days since Aug. 30 for Scenarios #1-9.
Recall that Scenarios #2-3 denote physical infrastructure recovery improvement
where $\lambda_{p}$=2.0 and 4.0, Scenarios #4-5 represent social
infrastructure improvement where $\lambda_{s}$=2.0 and 4.0, and Scenarios #6-9
imply the joint improvement.

In Fig. 9a, we observe that the physical infrastructure recovery plan
significantly promotes the average recovery level of POIs from around 0.7 to
around 0.9 in 60 days since Aug. 30, which demonstrates the long-term benefit
of accelerating water/sewer system repair on the recovery of the social
infrastructures. Moreover, substantial physical infrastructure improvement
(i.e., $\lambda_{p}$=4.0, Scenario #3) results in slightly higher recovery
levels of POIs than mild physical infrastructure improvement (i.e.,
$\lambda_{p}$=2.0, Scenario #2) on 30 days and 60 days since Aug. 30. We also
notice at the beginning of the recovery (i.e., 7 days), the physical
infrastructure improvement does not bring positive influence on the POI
recovery. This phenomenon emanates from the nonlinear and non-monotonous
interactions between social infrastructure and physical infrastructure
described in Eq. 7.

We derive similar conclusions from the simulation results under the social
infrastructure improvement shown in Fig. 9b. Strong social infrastructure
improvement (i.e., $\lambda_{s}=4.0$, Scenario #5) leads to faster recovery of
POIs than weak social infrastructure improvement (i.e., $\lambda_{s}=2.0$,
Scenario #4), and further the case without policy intervention (i.e., Scenario
#1). Finally, we implement the simulation for the joint policies (i.e.,
Scenarios #6-9) where both social and physical infrastructure improvement
plans are executed. Results in Fig. 9c manifest that the long-term enhancement
effect from social and physical infrastructure improvement on POIs are
additive, informing the possibility to conduct the two types of improvement
concurrently in the real-world post-disaster practice.

##  7 Discussion

Compared to hurricane-induced evacuation [8, 22, 68, 44, 9], there are fewer
studies focusing on the post-disaster recovery of the socio-physical system
[75, 41]. This study proposes an ABM on the three-layer socio-physical network
to simulate the recovery of individuals, social infrastructures, and physical
infrastructures. The first merit of the ABM is that it broadcasts the recovery
level of each entity (e.g., an individual, a restaurant) at a high temporal
resolution, that is, every day, enabling a nuanced system evaluation and
management. Second, our ABM suitably blends the information from the mobile
phone location data and the survey data by inheriting their advantages.
Indeed, mobile phone location data have wider coverage than the survey data
(i.e., 43,147 and 170 residents in this study) but lack information about
humans’ intentions of returning home. Conversely, survey data provide the
relationship between socio-demographic factors and returning home decisions,
but are limited by the small sample size.

We now disclose two limitations in our ABM and discuss potential solutions.
The first limitation is the unclear match between mobile phone users and
survey respondents. Recall that we employ the trajectories of mobile phone
users to track human evacuation and returning events, and questionnaires to
explore the underlying model describing human return behavior. Both the
trajectory data and human behavioral model apply to nodes in the human layer
in our ABM, thus it is more rigorous to match the mobile phone users and the
survey respondents. The potential solution can be the active data collection:
to use the mobile phone location data to identify users’ home locations, and
then send questionnaires to residents living in these locations. The other
limitation is the unverified assumption that an individual decides to return
his/her home with equal probability across different days (Eq. 8). Although
this assumption does not affect the recovery level of home nodes at the end of
the simulation period (i.e., $\overline{r}_{v_{h}}(60)$) because $P_{v_{h}}$
is accurate, it may lead to inaccurate recovery level of home nodes within the
simulation period. We suggest developing a statistical model describing the
specific time of human return.

We envision two applications of this ABM. First, disaster scientists could
build customized ABMs using our ABM framework and new human return behavior
models (Subsection 4.2) and dynamic models of social infrastructures
(Subsection 4.5) and physical infrastructures (Subsection 4.6) for various
types of natural hazards. Second, the individual-level recovery information
revealed by our ABM may guide the city government or public organizations to
identify the statuses of vulnerable individuals [23] or POI owners and provide
the necessary recovery support [1]. Note that information privacy issues
should also be considered during such target humanitarian aid process.

##  8 Conclusion

This study explores how a socio-physical infrastructure system recovers to its
normal status after a natural hazard, which serves as a basic component in
resilience modeling [16]. We create the ABM to simulate the dynamics of
recovery levels of different entities including homes, social infrastructures,
and physical infrastructures by grasping the interactions between entities.
Our ABM is built upon a three-layer network with the human layer, the social
infrastructure layer, and the physical infrastructure layer, enabling us to
distinguish unique behavior and dynamics associated with different types of
agents. To do this, we fuse multiple types of data (i.e., mobile phone
location data, POI data, and survey data) to define agent attributes, the ABM
initialization, and agent updating rules. To the best of our knowledge, our
ABM is the first ABM to simulate both human returning home, and recovery of
social and physical infrastructures during PDR at the individual level.

We conduct experiments in five counties in Texas after Hurricane Harvey in
August 2017 under nine scenarios. There are three main conclusions derived
from the experiments for the PDR of Hurricane Harvey. First, the recovery
dynamics exhibit heterogeneous patterns in terms of the type of agents (i.e.,
home agents, social infrastructure agents, and physical infrastructure
agents), housing types (i.e., rent or own the house), and annual income
levels. Second, inland counties have higher human return rates than coastal
counties. Third, social and physical infrastructure improvements pose positive
effects on the long-term recovery of social infrastructure and these effects
are addictive. Moreover, from the phase transition analysis, we find that
physical infrastructure improvement may have a more efficient enhancement
effect on the recovery of social infrastructures than social infrastructure
improvements. Our ABM could help disaster scientists to explore the property
of the socio-physical system during PDR with high-resolution recovery
information, and policymakers to design detailed post-disaster support for
individuals and resource allocation strategies.

##  9 Authorship Contribution

Jiawei Xue: Conceptualization, Methodology (ABM framework, three-layer
network, human behavior model, simulation design), Coding, Figure, Writing.
Sangung Park: Conceptualization, Methodology (human behavior model, dynamic
model), Figure, Writing, Editing. Washim Uddin Mondal: Conceptualization,
Methodology, Editing. Sandro Martinelli Reia: Conceptualization, Methodology,
Writing. Tong Yao: Methodology, Editing. Satish V. Ukkusuri:
Conceptualization, Methodology, Discussion, Editing.

##  10 Data and Code Availability

The data and code for the ABM are stored at
https://github.com/JiaweiXue/PostDisasterSim.

##  11 Acknowledgement

The authors are funded by the NSF Grant (No. 1638311): Critical Transitions in
the Resilience and Recovery of Interdependent Social and Physical Networks.
The authors also thank Dr. Bailey Benedict from California State University,
Dr. Seungyoon Lee from Purdue University for their great effort in conducting
the survey.

##  Appendix A Survey table

Survey question | Variable | Value
---|---|---
What is your age? | $q_{age}$ | 0$-$100
What is your sex? | $q_{sex}$ | 0, 1 (0: female; 1: male)
Did you own or rent your house? | $q_{house}$ | 0, 1 (0: rent; 1: own)
Which ethnic groups do you belong to? | $q_{race}$ |  | Seven groups: African American;
---
Asian/Pacific islander; Caucasian; Hispanic;
Native American; Multiple/mixed; Others
What is your yearly household income? | $q_{income}$ |  | Nine intervals: <$15K; $15-30K; $30-45K;
---
$45-60K; $60-75K; $75-90K; $90-105K;
$105-120K; >$120K
| To what extent have the following people returned?
---
a. your neighborhoods
b. others in your community
| $q_{human,\;a}$
---
$q_{human,\;b}$
0$-$1 (0: no return; 1: full return)
| To what extent have the following organizations reopened?
---
a. medical facilities
b. school/childcare
c. businesses
d. your employer
e. other community
| $q_{social,\;a}$
---
$q_{social,\;b}$
$q_{social,\;c}$
$q_{social,\;d}$
$q_{social,\;e}$
0$-$1 (0: no reopen; 1: full reopen)
| To what extent has your community experienced recovery?
---
a. power
b. water/sewer
c. roads/highways
d. bridges
e. buildings/structures
| $q_{physical,\;a}$
---
$q_{physical,\;b}$
$q_{physical,\;c}$
$q_{physical,\;d}$
$q_{physical,\;e}$
0$-$1 (0: complete damage; 1: full function)
Did you evacuate? | $y_{evacuate}$ | 0, 1 (0: no evacuate; 1: evacuate)
Have you returned home permanently? | $y_{return}$ | 0, 1 (0: no return; 1: return)

Table 5:  Description of variables mined from the survey. Note that the social
infrastructure recovery indicators (i.e., $q_{social,*}$) and the physical
infrastructure recovery indicators (i.e., $q_{physical,*}$) describe
respondents’ perception of the neighborhood environment, so that they can be
used to characterize agent interactions between adjacent nodes in $G$
(Subsection 5.4.3).

##  Appendix B Code structure

Figure 10: The flowchart of agent attribute update in the code.

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