A Systematic Study of Mathematical Modeling for Sustainable Community-Based Disaster Risk Management

Abstract

This study aimed to evaluate the application of mathematical modeling in sustainable community-based disaster risk management (CBDRM), paying particular attention to the incorporation of financial risk mitigation mechanisms such as insurance and community-based risk pooling. A structured literature search was conducted in the Scopus and ScienceDirect databases, followed by bibliometric and qualitative analysis of relevant studies in mathematics, economics, and disaster management. During the analysis, 17 peer-reviewed journal articles met the inclusion criteria and were examined based on publication trends, geographical distribution, modeling methods, and the extent to which financial protection mechanisms were incorporated into quantitative frameworks. The findings indicated growing academic interest in recent years and showed considerable methodological diversity, including stochastic optimization, vulnerability indices, agent-based simulations, and econometric models. Despite these advancements, major financial risk mitigation elements, such as premium design, fund management, and payout procedures, remained inadequately incorporated into existing modeling structures and were frequently addressed as separate analytical components. The focus on studies in high-income countries raised concerns about contextual applicability in climate-vulnerable and low-income regions. The review showed the need for more operationally incorporated modeling frameworks that connect quantitative risk assessment with community-level financial resilience strategies to support sustainable CBDRM.

1. Introduction

The increasing frequency and severity of natural disasters are showing the urgent need for more resilient and inclusive disaster risk management strategies. As large-scale government interventions remain essential, there is a growing recognition that effective disaster preparedness and response should be rooted at the community level. This situation has given rise to community-based disaster risk management (CBDRM), a method that empowers local populations to assess risks, plan mitigation actions, and strengthen social and economic resilience from the ground up [1,2]. Disaster risk management is understood as a structured system including hazard identification, probabilistic risk assessment, implementation of preventive as well as mitigative controls, and management of residual risk from a safety engineering perspective [3]. In this study, CBDRM is interpreted as the community-level operationalization of the safety engineering cycle. Community risk mapping corresponds to hazard identification, vulnerability assessment reflects probabilistic risk estimation, local mitigation planning represents risk control implementation, and post-disaster recovery arrangements address residual risk management. Therefore, the safety engineering framework does not replace CBDRM; it provides the analytical and structural backbone by incorporating community actions in a systematic risk management architecture.

CBDRM initiatives have proven effective in increasing community awareness, improving early warning systems, and promoting local leadership in disaster-prone areas. However, managing disaster risk is a technical or logistical challenge, as it also includes addressing the significant financial impacts that disasters impose on vulnerable populations. For many communities, financial losses from disasters can be catastrophic for those in low-income or high-risk regions, leading to long-term socio-economic setbacks [4]. In a safety engineering framework, these financial losses are categorized as residual risk, losses that remain after preventive and mitigative controls are implemented [5]. This classification clarifies that financial resilience mechanisms in CBDRM are not secondary social measures, but integral components of systemic risk management.

A promising avenue to bridge the gap between disaster risk management and financial resilience is the application of insurance mechanisms and other financial risk mitigation tools in CBDRM frameworks. Community-based insurance schemes, risk pooling, and contingency funds are increasingly explored as mechanisms to provide rapid financial support to affected populations, reduce reliance on external aid, and promote recovery [6]. In safety engineering theory, these instruments are categorized as residual risk treatment mechanisms, applied when physical risk reduction cannot fully eliminate expected losses [5]. Insurance and risk pooling are not external financial add-ons, but formal components of systemic risk control. In practical terms, this signifies that the quantitative outputs of CBDRM activities, such as hazard probability, exposure levels, and vulnerability indices, should inform the design of financial instruments in the same safety management cycle. However, the design and implementation of the schemes often require rigorous assessment of risk patterns, premium structures, and potential loss distributions, which are areas where mathematical modeling plays a crucial role.

Mathematical models have long been used in the broader field of disaster risk assessment, including hazard prediction, vulnerability mapping, and probabilistic loss estimation [7]. In safety engineering, these quantitative tools function as the analytical core of risk evaluation, supporting decision-making concerning acceptable risk levels and allocation of mitigation resources [8]. Mathematical modelling enables community-level risk quantification to be associated with formal safety engineering principles in the context of CBDRM, ensuring that mitigation planning and financial preparedness are based on measurable risk indicators rather than ad hoc judgment. The specific application in CBDRM, particularly concerning insurance design and financial risk mitigation, remains underexplored. Some studies have started to incorporate stochastic models, cost–benefit analyses, and risk indices into community-level planning, but the extent to which these tools inform or improve financial protection mechanisms is not well understood.

Following the discussion above, practical examples that combine both concepts supported by quantitative models are still limited despite the conceptual arrangement between CBDRM and community-based insurance. Existing initiatives often face barriers such as data scarcity, limited technical capacity at the community level, and institutional fragmentation [9]. Therefore, this study adopts a safety engineering framework defined as an incorporated cycle of (i) hazard and vulnerability modeling, (ii) quantitative risk estimation, (iii) mitigation and preparedness implementation, and (iv) structured residual risk financing. This definition follows contemporary risk management theory that treats risk analysis and risk treatment as interdependent components of a single system [3]. By positioning CBDRM in the structured cycle, the study explicitly conceptualizes CBDRM as the localized execution of safety engineering principles, rather than as a parallel or independent method. Consequently, there is a pressing need to systematically examine how mathematical models have been used to support CBDRM initiatives, with a particular focus on their role in insurance applications and financial risk mitigation. This systematic review addresses the following study questions:

a.

What types of mathematical models have been applied to support CBDRM?

b.

Do these studies include financial risk mitigation partially, conceptually, or through practical implementation? Additionally, who are the major organizations or stakeholders engaged in the CBDRM efforts, irrespective of the connection to financial mechanisms?

c.

What essential components have been identified in existing studies for the design and implementation of financial risk mitigation mechanisms in CBDRM?

Through answering these questions, the review helps eliminate the study gap. It synthesizes existing literature on mathematical modeling in the context of CBDRM, with a focus on applications in community-based insurance schemes, risk pooling mechanisms, and other financial resilience strategies. Through this process, the review aims to identify methodological trends, practical challenges, and potential innovations that can strengthen the financial dimension of community disaster risk management.

The structure of this study is organized in the following way. Section 2 describes the study method, detailing the search strategy, inclusion criteria, and analytical method applied in the review process. Section 3 presents the results, including an overview of the identified mathematical models and a categorization of their applications in CBDRM. Section 4 provides a critical discussion of study gaps and shows opportunities for incorporating mathematical modeling with insurance and financial risk mitigation for future study directions. Finally, Section 5 concludes the study by reviewing the main perceptions concerning the process.

2. Methods

The systematic literature review (SLR) was conducted following the PRISMA 2020 (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) guidelines to ensure transparency, reproducibility, and scientific rigor throughout the process [10]. The PRISMA checklist is provided in the Supplementary Materials. The main objective of this study was to systematically explore the role of mathematical modeling in CBDRM, with a particular focus on the applications in financial risk mitigation strategies. As CBDRM initiatives increasingly incorporated financial instruments such as microinsurance or community risk pooling, the combination of robust mathematical models became essential to ensure the schemes were actuarially sound, sustainable, and equitable.

The literature search in this study was conducted using two well-established databases, namely, Scopus and ScienceDirect. The selection of Scopus was based on the extensive multidisciplinary coverage and the documented suitability for systematic reviews in interdisciplinary fields such as disaster risk management, economics, and applied mathematics. Scopus has been shown to provide broader journal coverage and citation indexing compared to other databases in engineering and social sciences, which improves the comprehensiveness of evidence identification [11,12]. Following the process, ScienceDirect was selected as a complementary source because it provided full-text access to peer-reviewed journals published by Elsevier, particularly in applied mathematics, risk analysis, actuarial science, and safety engineering. The structured subject classification and advanced filtering system of the database facilitated precise retrieval of domain-specific literature relevant to mathematical modeling and disaster risk financing. The combined use of Scopus for comprehensive indexing and ScienceDirect for full-text domain depth followed recommendations in the systematic review method that supported database triangulation to reduce selection bias and improve coverage [13].

After preliminary testing, ScienceDirect was selected as the primary source for the full systematic search due to the broad coverage of scientific literature, particularly in certain subject areas. The filtering capabilities of the database also facilitated access to peer-reviewed, English-language, and open-access publications relevant to this topic. The final database search was conducted on 24 June 2025, using the following Boolean search string: “modelling” AND ((“community-based disaster risk management” OR “CBDRM” OR “community disaster risk reduction”) OR (“insurance” OR “disaster insurance” OR “financial risk mitigation”)). Search results were refined by restricting subject areas to mathematics and economics/econometrics/finance and including only articles, conference papers, study articles, or reviews written in English and available as open-access.

All retrieved records were exported into JabRef reference manager, similar to the method adopted by [14], to facilitate the organization and removal of duplicate entries. After the deduplication procedure was completed, a structured screening process started. Titles and abstracts were initially screened independently by two reviewers to identify potentially relevant studies, followed by full-text assessment based on predefined eligibility criteria. No automation tools, machine learning algorithms, or AI-assisted screening were used at any stage to maintain manual control and ensure contextual understanding of the materials. Studies were included when the following criteria were met. First, the study explicitly addressed community-based methods for disaster risk financing, insurance, or risk pooling, with a clear orientation toward disaster risk reduction. Second, the study showed the application, development, or theoretical examination of mathematical, statistical, stochastic, or actuarial models in the CBDRM context.

The entire selection process is shown in the PRISMA flow diagram in Figure 1, which was created using the online PRISMA diagram generator provided by ESTEch (available at https://estech.shinyapps.io/prisma_flowdiagram/, accessed on 1 July 2025). This diagram systematically presents the stages of identification, screening, eligibility assessment, and inclusion, improving the transparency and replicability of the review process in accordance with the PRISMA 2020 guidelines. As shown in the PRISMA flow diagram (Figure 1), 91 records were initially identified from database searches. A total of 4 duplicate records were removed, leaving 87 for initial screening based on titles and abstracts. From this stage, 55 records were excluded, and 32 were required for full-text retrieval. A total of 9 records could not be retrieved, leaving 23 for full-text eligibility assessment. After applying the inclusion criteria, 9 records were excluded for not meeting the review scope. Eventually, a total of 14 new studies were included in this study. A total of 3 studies from a previous version of the review were incorporated, bringing the total number of reviewed articles to 17. The combined set of articles formed the basis for both bibliometric analysis and a detailed synthesis to answer the questions presented in this study.

Data extraction was conducted using a structured data extraction form developed specifically for this review. The primary outcomes sought from each included study were: (i) the type of mathematical model, (ii) the level of incorporation of financial risk mitigation in CBDRM, and (iii) the structural components. For each outcome domain, all relevant results reported within each study were collected, including theoretical formulations and model outputs. No restrictions were imposed regarding time horizon, analytical method, or performance metrics, provided they were consistent with the objectives of this review. When information was missing or unclear, assumptions were not made beyond what was explicitly stated in the text. In cases of incomplete reporting, the available information was recorded, and the limitation was noted during synthesis. No attempts were made to contact study authors for additional data.

The methodological quality and potential risk of bias of the studies included in this review were assessed using an adapted qualitative appraisal approach tailored to interdisciplinary modeling-based research. Given that most of the reviewed articles consisted of conceptual frameworks, simulation-based analyses, and mathematical model applications, conventional risk-of-bias assessment tools commonly employed in clinical or experimental studies were considered inappropriate for this context. The assessment was conducted by considering several key aspects: (i) the clarity of model assumptions, (ii) the transparency of data sources, (iii) the appropriateness of modelling techniques in relation to the stated research objectives, and (iv) the completeness of reporting of model parameters and outputs. Each study was evaluated independently by two reviewers. Any discrepancies in the assessments were resolved through discussion until consensus was reached. No automation tools or artificial intelligence-assisted software were used in the risk-of-bias evaluation process. This review did not perform a quantitative meta-analysis; therefore, conventional statistical effect measures such as risk ratios, odds ratios, or mean differences were not applied. The synthesis of findings was conducted narratively and descriptively, with a focus on the classification of mathematical modeling approaches, the degree of integration with financial risk mitigation mechanisms, and the structural components identified in each study. Sensitivity analyses in the statistical sense were not conducted, as no pooled quantitative estimates were generated. However, robustness of the synthesis was ensured by cross-verifying thematic classifications and model categorizations between reviewers.

3. Results

3.1. Bibliometric Analysis

A bibliometric analysis was conducted to complement the qualitative synthesis, providing an overview of publication trends and study information in the selected articles. Bibliometric mapping has become an essential tool to identify study hotspots, influential authors, and gaps in scientific literature [15,16]. Despite the relatively small number of included articles, this analysis provided valuable information on the disciplinary distribution and journal outlets, which could guide future study directions.

All records included in this study were published as peer-reviewed journal articles. This indicated that the study on mathematical modeling in community-based disaster insurance (CBDI) and financial risk mitigation had primarily been developed in academic settings, prioritizing methodological rigor and scientific validation. The absence of other publication types, such as conference papers or technical reports, indicated that the topic had matured beyond preliminary discussions and was being addressed through more established scholarly contributions. As the process improved the reliability of the findings, it also showed a potential gap in practical perceptions from industry reports or gray literature, which could complement academic learning in future studies.

Following the discussion above, Figure 2 shows the distribution of publications over time based on the 17 articles included in this study. The earliest study contributing to this topic appeared in 2011 and 2012, each recording one publication. However, from 2013 to 2015, no relevant studies were identified, reflecting a period of limited academic focus on mathematical modeling in CBDRM and disaster insurance schemes. The field started to receive modest attention in 2016, with a steady and low output of one publication per year until 2021.

A visible shift occurred in 2022, marking the first significant increase in study activity, with three studies published in the same year. Although there was a slight decline to two publications in 2023, the upward trajectory resumed in 2024, reaching three publications, which was the highest annual output observed. The flow indicated increasing academic attention to the role of mathematical modeling in strengthening financial risk transfer instruments at the community level. In this study, CBDI referred specifically to locally organized or community-participatory insurance and risk pooling mechanisms designed to provide financial compensation for disaster-related losses. The mechanism focused on collective premium contributions, risk sharing among members, and simplified or index-based payout structures to improve affordability as well as accessibility, including community-based and microinsurance models discussed in disaster risk contexts. Conceptually, CBDI was not equivalent to CBDRM, which was a broader governance and risk reduction framework, including hazard assessment, preparedness, mitigation, response, and recovery activities led by communities. Consequently, CBDI represented a financial risk transfer component capable of operating in the broader CBDRM framework to address residual financial risk after mitigation efforts. This distinction was associated with disaster risk management literature, which separates risk reduction from risk transfer mechanisms. For example, risk layering frameworks in climate and disaster finance distinguished between prevention/mitigation and insurance-based transfer instruments [9]. Microinsurance research similarly prioritized its role in improving financial resilience rather than replacing structural risk reduction [17,18]. Therefore, the observed increase in publications reflected growing recognition of the need to mathematically model community-level insurance schemes as part of incorporated disaster risk financing strategies, particularly under escalating climate-related hazards [16,19].

By mid-2025, two publications had already been identified, further confirming that study in this area was active and continued to grow. The recent upward trend reflected an increasing demand for evidence-based methods and innovative risk financing tools to support disaster-prone communities, associated with broader global agendas on disaster resilience and financial inclusion. This progression indicated that CBDRM modeling was no longer a peripheral topic but was gradually becoming part of mainstream academic discourse, opening avenues for deeper methodological exploration and interdisciplinary collaboration.

The selected studies were published across a diverse range of reputable journals, reflecting the multidisciplinary nature of the study of CBDRM and its intersection with mathematical modeling. In total, the 17 articles were distributed across 12 different journals, as shown in Figure 3. Progress in Disaster Science contributed the highest number of publications, with three articles. This finding was consistent with the focus of the journal on advancing disaster science, practical risk management solutions, and resilience-building efforts, core elements that were closely associated with CBDRM-related studies.

Based on the discussion above, the journals Natural Hazards and the International Journal of Disaster Risk Reduction each accounted for two articles. This further prioritized the relevance of disaster-focused outlets in disseminating studies that incorporated risk modeling, insurance mechanisms, and community resilience. Natural Hazards, in particular, had long been recognized for publishing work on hazard assessment, vulnerability analysis, and risk management strategies, making it a natural outlet for studies exploring mathematical tools in CBDRM [20].

The remaining articles were distributed across various interdisciplinary journals, including World Development, Ecological Indicators, the Journal of Geographical Sciences, and Socio-Economic Planning Sciences. This broad journal landscape showed that CBDRM and related mathematical models were not confined to disaster risk research alone but increasingly intersected with socio-economic development, urban planning, environmental sustainability, and public health domains. For instance, outlets such as Resilient Cities and Structure or The Journal of Climate Change and Health signified the growing academic recognition of CBDRM in the broader context of urban resilience and climate adaptation challenges.

The presence of journals such as Cogent Economics and Finance and Energy Procedia showed that the role of quantitative modeling and economic analysis in disaster risk management was receiving heightened scholarly attention. These findings reinforced the changing, interdisciplinary character of CBDRM research, where concepts from disaster science, economics, engineering, and environmental management converge to address the complex risks faced by vulnerable communities [19,21]. The diversity of publication sources reflected the growing academic momentum to position CBDRM and its mathematical foundations as an integral part of disaster risk reduction efforts, financial innovation, and sustainable development agendas.

The distribution of publications from a geographical perspective reflected both academic focus in developed countries and a growing study footprint in disaster-prone regions globally. As shown in Figure 4, the United States led with three articles, which were consistent with the well-established academic infrastructure of the country in disaster risk management, mathematical modeling, and community resilience studies [22].

Following the USA, significant contributions originated from New Zealand, Austria, and China, each with two articles. The presence of New Zealand was unsurprising given the extensive exposure of the country to natural hazards such as earthquakes and its strong policy importance on community-based disaster preparedness. Similarly, the active participation of China reflected both the large-scale disaster experiences and increasing investment in studies related to resilience, insurance, and risk modeling.

The remaining studies were distributed across a diverse set of countries, including Germany, Spain, Latvia, Greece, Ethiopia, South Africa, the United Kingdom, and Bangladesh, each represented by one publication. The inclusion of countries such as Bangladesh, Ethiopia, and South Africa was particularly significant, as these nations often encounter high disaster risk but limited access to traditional financial risk mitigation tools [23]. The appearance of the countries in the literature showed a positive shift toward more inclusive global study efforts aimed at modifying mathematical models and insurance schemes to the realities of vulnerable low- and middle-income settings.

Based on the discussion above, the world map in Figure 4 shows the global spread, signifying intensities of study activity across North America, Europe, Oceania, Asia, and parts of Africa. While the majority of studies still originated from high-income or upper-middle-income countries, the developing participation from the Global South showed the growing recognition that CBDRM required context-specific, localized modeling methods. This was particularly relevant, as mathematical models and insurance-based interventions should be adapted to diverse socio-economic, institutional, and hazard landscapes to effectively improve disaster resilience at the community level [16].

As the current study landscape was slightly dominated by developed countries, the geographic diversity observed in these 17 articles showed increasing momentum toward global knowledge exchange and capacity building in CBDRM, particularly for countries most exposed to disaster risks. In terms of scholarly impact, the citation analysis presented in Figure 5 shows a small but developing group of authors shaping the academic conversation around mathematical modeling in CBDRM. Among the reviewed studies, Yin et al. [24] was the most cited work, with 114 citations recorded in Scopus as of June 2025. This reflected the academic relevance of the contribution and also the growing recognition of CBDRM as a critical area in disaster risk research, particularly in connection to quantitative modeling and community resilience.

The second most cited publication was from Holcombe et al. [25], with 38 citations, followed by Fakhruddin et al. [26] with 27. Other authors, including Tadesse and Zeleke [27], Kalogiannidis et al. [28], and Hochrainer-Stigler et al. [29], also received moderate scholarly attention, each accumulating between 10 and 20 citations. As the number showed growing engagement, it reflected the relative novelty and fragmented nature of the study, incorporating mathematical models with community-based methods for disaster risk management.

Several authors, such as Khan et al. [30], Harrison et al. [31], and Han and Koliou [32], recorded fewer than five citations, which was expected since the studies were only recently published. Relating to this discussion, citation counts might not fully capture the practical relevance of CBDRM studies, particularly when considering real-world applications in community settings often operated beyond traditional academic dissemination channels [41]. The citation distribution showed that while a few key publications were starting to serve as reference points, the field remained in the developmental phase. Future studies can benefit from promoting stronger collaborative networks, cross-regional knowledge exchange, and greater incorporation of academic perceptions into practical CBDRM initiatives, ensuring both scientific advancement and community-level impact.

A co-occurrence analysis of terms from abstracts and keywords was conducted using VOSviewer version 1.6.20 to explore thematic patterns and conceptual linkages in the selected studies. The analysis applied a minimum occurrence threshold of 2, ensuring only relevant and frequently discussed terms were visualized. The resulting network map, as shown in Figure 6, presented several interconnected thematic clusters that characterized the study landscape of mathematical modeling in CBDRM.

The importance of the term “community” showed the foundational role of community-level engagement in the discourse, consistent with the core principles of CBDRM that prioritize local participation, risk ownership, and resilience building. Closely connected to “community” were terms including “risk,” “resilience,” “population,” and “insurance”. This indicated that many studies focused on how mathematical or conceptual models could be applied to assess risk, improve resilience, and structure insurance mechanisms in addressing community vulnerabilities.

Another prominent cluster rotated around terms such as “model,” “method,” “analysis,” “understanding,” and “vulnerability,” reflecting the methodological importance of the body of literature. These terms implied that a significant portion of studies focused on empirical assessments and also on developing analytical frameworks to better understand disaster dynamics, socio-economic impacts, and risk mitigation strategies [16].

A distinct grouping of terms such as “disaster,” “climate change,” “disaster risk reduction,” and “development” showed the growing intersection between CBDRM studies and broader global agendas, particularly in the context of climate adaptation and sustainable development. This was associated with the increasing recognition that community resilience could not be dissociated from systemic challenges such as climate change, socio-economic inequalities, and governance gaps.

The map showed technical and policy-oriented terms, including “insurance company,” “policy,” “participation,” and “decision”. This signified that beyond purely academic modeling, there was a developing dialogue around the operational and institutional mechanisms required to translate risk assessments into actionable community-based interventions. Furthermore, terms related to physical infrastructure and hazard exposure, such as “household,” “residential building,” “hazard,” and “damage,” reflected ongoing concerns about disaster impacts on the built environment, consistent with the applied focus of CBDRM.

The dense network of connections across clusters showed the highly interdisciplinary nature of the field, where concepts from disaster science, economics, behavioral studies, and environmental management intersected. This complexity reinforced the need for incorporating methods when applying mathematical models to support CBDRM, ensuring that technical solutions were grounded in local realities and social contexts. The keyword co-occurrence analysis provided a comprehensive overview of current study directions, showing the importance of both theoretical innovation and practical application in advancing community-based methods for disaster risk management.

3.2. Main Review Results

Reviewed studies were organized according to three dimensions to improve analytical clarity. These dimensions included (1) model type, (2) degree of incorporation of financial risk mitigation mechanisms, and (3) major structural components supporting community-based financial resilience. Mathematical modeling has become an increasingly important tool in supporting CBDRM, offering structured methods to quantify hazards, vulnerability, and resilience. These tools enable stakeholders to translate complex disaster scenarios into measurable understandings, supporting more targeted interventions at the community scale [25,37].

3.2.1. Classification by Model Type

The studies reviewed during the process of this analysis showed considerable methodological diversity. Based on each analytical structure, the identified models were grouped into four main categories, namely, (1) optimization and decision-support, (2) vulnerability and resilience index, (3) simulation-based and recovery, and (4) econometric and statistical models. Optimization and decision-support models, such as mixed-integer linear programming and mean-risk formulations [33], focused on allocating resources under uncertainty as well as evaluating trade-offs between cost and resilience outcomes. Moreover, vulnerability and resilience index models used weighted aggregation and regression-based methods to quantify multidimensional exposure as well as adaptive capacity at the community level [34]. Simulation-based methods, including agent-based models and staged recovery frameworks, represented dynamic disaster impacts and post-event reconstruction processes over time [24,37]. Meanwhile, econometric models and discrete selection experiments analyzed behavioral responses, institutional factors, and insurance preferences.

According to the discussion above,

Table 1. Mathematical models, specifying functions and contributions to CBDRM across the reviewed studies.

No	Author(s)	Mathematical Model		Model Function
1	Faiz and Harrison [33]	Mean Risk Model as Mixed-Integer Linear Programing	$\underset{\mathit{x}\in \mathcal{X};\mathit{y},\mathit{u}ϵ\mathcal{Y};\mathit{vϵ}\mathbb{R}}{\min }\left\{\begin{array}{c}\left(1+\gamma \right){\overline{\mathbf{c}}}^{\mathrm{T}}x+\left({\displaystyle \frac{1}{\rho }}\right){\displaystyle \sum _{\omega ϵ\mathsf{\Omega }}}{\lambda }_{\omega }{\overline{\mathbf{q}}}_{\omega }^{\mathrm{T}}{\mathit{y}}_{\omega }+\left({\displaystyle \frac{\gamma }{\rho }}\right)\\ \left(v+{\displaystyle \frac{1}{1-\alpha }}{\displaystyle \sum _{\omega ϵ\Omega }}{\lambda }_{\omega }{\mathit{u}}_{\omega }\right)\end{array}\right\}$ The definitions of sets, parameters, and the decision variables are as shown in [33].	Incorporating hazard uncertainty alongside low-probability, high-impact events. Combining fragility functions to model building vulnerability, with population dislocation models based on demographic data. Analyzing community resilience through two major metrics, namely, the number of households displaced and the time to restore essential services.
2	Eze and Siegmund [35]	-	-	-
3	Fakhruddin et al. [26]	Additive Weight-Based Aggregation	$V_m={\displaystyle \sum _{i=1}^n}\left(W_i\times I_{mi}\right)$ $\mathrm{where} V_m$$\mathrm{is} \mathrm{the} \mathrm{vulnerability} \mathrm{of} \mathrm{a} \mathrm{given} \mathrm{infrastructure} \mathrm{system} (m$$), n$$\mathrm{represents} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{indicators} \mathrm{of} \mathrm{the} \mathrm{vulnerability} \mathrm{model}, W_i$$\mathrm{is} \mathrm{the} \mathrm{weighted} \mathrm{importance} \mathrm{of} \mathrm{a} \mathrm{given} \mathrm{indicator} (i$$), \mathrm{and} I_{mi}$ signifies the normalized indicator.	Assessing the vulnerability of urban infrastructure systems to disasters and climate change.
4	Aroca- Jiménez et al. [34]	Incorporated Multidimensional Vulnerability Index and Incorporated Multidimensional Resilience Index	$Index={\displaystyle \sum _{i=1}^n}{W}_i\times F_i$ with statistical models, Spearman correlation, ordinary least squares, and geographically weighted regression.	Measuring multi-dimensional vulnerability (social, economic, physical, ecosystem, institutional, cultural).
5	Chen et al. [36]	Community Risk Management Capacity for Debris Flow	$F={\displaystyle \sum _{j=1}^n}{\omega }_j\times x_i^{\left(j\right)} \mathrm{a}\mathrm{n}\mathrm{d} C_j={\displaystyle \frac{{\omega }_j\times x_i^{\left(j\right)}}{{\displaystyle {\sum }_{j=1}^n}{\omega }_j\times x_i^{\left(j\right)}}}\times 100\%$ $\mathrm{where} {\omega }_j$$\mathrm{is} \mathrm{the} \mathrm{entropy} \mathrm{weight} \mathrm{of} \mathrm{indicator} j$$, x_i^{\left(j\right)}$$\mathrm{signifies} \mathrm{the} \mathrm{dimensionless} \mathrm{vector} \mathrm{of} \mathrm{sample} \mathrm{values} \mathrm{for} \mathrm{the} j$$-\mathrm{th} \mathrm{evaluation} \mathrm{indicator} \mathrm{in} \mathrm{category} i$$, \mathrm{and} x_i$$\mathrm{is} \mathrm{the} \mathrm{sample} \mathrm{average} \mathrm{value} \mathrm{vector} \mathrm{of} \mathrm{the} i$$-\mathrm{th} \mathrm{category} \mathrm{index}. C$$\mathrm{is} \mathrm{the} \mathrm{contribution} \mathrm{degree} \mathrm{toward} \mathrm{community} \mathrm{risk} \mathrm{management} \mathrm{capacity} \mathrm{for} \mathrm{debris} \mathrm{of} \mathrm{the} \mathrm{three} \mathrm{evaluation} \mathrm{criteria} (\mathrm{pre}-\mathrm{disaster} \mathrm{prevention} \mathrm{and} \mathrm{regulation} \mathrm{capacity} (\mathrm{PPRC}), \mathrm{disaster} \mathrm{emergency} \mathrm{rescue} \mathrm{capacity} (\mathrm{DERC}), \mathrm{and} \mathrm{post}-\mathrm{disaster} \mathrm{recovery} \mathrm{and} \mathrm{assistance} \mathrm{capacity} (\mathrm{PRAC}). \mathrm{In} \mathrm{addition}, F$ signifies the capabilities of each criterion layer, with the statistic model R hierarchical clustering for the category of indicator, coefficient of variation for indicator selection, entropy-weighted gray correlation analysis for entropy weight and correlation, linear weighted synthesis for score aggregate, and contribution degree model for three different management types (prevention-oriented, emergency-oriented, recovery-oriented).	Developing a risk management evaluation system for multi-ethnic communities that were vulnerable to disasters and had socio-economic challenges, with three categories.
6	Naqvi [37]	Spatially Explicit Agent-Based Model	$\mathrm{The} \mathrm{index} \mathrm{of} \mathrm{agents} \mathrm{and} \mathrm{locations} \mathrm{was} i=\mathrm{1,2},\dots ,n$$\mathrm{and} j=\mathrm{0,1},\dots ,m,$$\mathrm{where} 0$$\mathrm{was} \mathrm{the} \mathrm{current} \mathrm{location} \mathrm{of} \mathrm{an} \mathrm{agent}. \mathrm{These} \mathrm{were} \mathrm{procedures} \mathrm{of} \mathrm{the} \mathrm{model}. \mathrm{Production} \mathrm{was} \mathrm{defined} \mathrm{as} \mathrm{agricultural} \mathrm{output}, \mathrm{referred} \mathrm{to} \mathrm{as} “\mathrm{food}” \mathrm{in} \mathrm{villages} \mathrm{and} \mathrm{a} \mathrm{tradeable} “\mathrm{good}” \mathrm{output} \mathrm{in} \mathrm{cities}. \mathrm{Owners} \mathrm{of} \mathrm{productive} \mathrm{capital} \mathrm{produced} \mathrm{an} \mathrm{amount} u_j\le X_{jt}\le X_j^{max}$$\mathrm{of} \mathrm{the} \mathrm{total} \mathrm{output} \mathrm{using} \mathrm{existing} \mathrm{technologies}. \mathrm{Total} \mathrm{output} \mathrm{was} \mathrm{determined} \mathrm{at} \mathrm{the} \mathrm{location} \mathrm{level}, \mathrm{where} \mathrm{output} \mathrm{per} \mathrm{worker} \mathrm{for} \mathrm{each} \mathrm{location} j$ was defined as ${\lambda }_{ijt}={\displaystyle \frac{X_{jt}-u_j}{n_{jt}}}$ $\mathrm{where} n_{jt}$$\mathrm{signifies} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{workers} \mathrm{used} \mathrm{at} \mathrm{location} j$$\mathrm{at} \mathrm{time} t$$. \mathrm{The} \mathrm{total} \mathrm{wage} \mathrm{bill} \mathrm{was} \mathrm{determined} \mathrm{by} \mathrm{a} \mathrm{fixed} \mathrm{rate} w$ per unit of output times the total output produced by workers: ${WB}_{jt}=w(X_{jt}-u_{jt})$ $\mathrm{Wage} \mathrm{earned} \mathrm{per} \mathrm{worker} i$$\mathrm{in} \mathrm{location} j$$\mathrm{at} \mathrm{time} t,$ or wage rate times worker productivity siginfied by $W_{ijt}={\displaystyle \frac{{WB}_{jt}}{n_{jt}}}=w{\lambda }_{ijt}$ $\mathrm{The} \mathrm{amount} \mathrm{of} \mathrm{goods} \mathrm{purchased} \mathrm{was} \mathrm{defined} \mathrm{by} \mathrm{two} \mathrm{parameters}: \mathrm{a} \mathrm{preference} \mathrm{to} \mathrm{consume} \mathrm{at} \mathrm{least} \mathrm{a} \mathrm{minimum} \mathrm{level} \mathrm{of} \mathrm{subsistence} C^{min}$$, \mathrm{evaluated} \mathrm{at} \mathrm{current} \mathrm{market} \mathrm{prices} p_{jt}$$, \mathrm{and} \mathrm{a} \mathrm{preference} \mathrm{toward} \mathrm{holding} \mathrm{inventories} \mathrm{of} \mathrm{food} \mathrm{for} \mathrm{a} \mathrm{certain} \mathrm{time} (\delta$$\mathrm{days}) \mathrm{to} \mathrm{allow} \mathrm{minor} \mathrm{consumption} \mathrm{smoothing}. \mathrm{The} \mathrm{amount} \mathrm{of} \mathrm{goods} \mathrm{purchased} B_{ijt}$$\mathrm{by} \mathrm{a} \mathrm{worker} i$ was defined as $B_{ijt}=Max\left[p_{jt}{C}^{min}, c_{1it}{W}_{ijt}+c_{2it}{m}_{ij,t-1}\right]$ $\mathrm{where}, c_1\le c_{1it}\le 1$$\mathrm{signifies} \mathrm{the} \mathrm{marginal} \mathrm{propensity} \mathrm{to} \mathrm{consume} \mathrm{out} \mathrm{of} \mathrm{income}, \mathrm{and} c_{2it}$$\mathrm{is} \mathrm{the} \mathrm{marginal} \mathrm{propensity} \mathrm{to} \mathrm{consume} \mathrm{out} \mathrm{of} \mathrm{savings} <m_{ijt}$$. \mathrm{Agents} \mathrm{held} \mathrm{food} \mathrm{inventories} F_{ijt}$$, \mathrm{out} \mathrm{of} \mathrm{which} \mathrm{users} \mathrm{consumed} \mathrm{a} \mathrm{fraction} \delta$ every time, defined as $C_{ijt}=Max\left[C^{min}, \delta F_{ij,t-1}\right]$ $\mathrm{Network} \mathrm{distances} \mathrm{were} \mathrm{normalized} \mathrm{between} 0$$\mathrm{and} 1$$\mathrm{to} \mathrm{ensure} \mathrm{consistency} \mathrm{in} \mathrm{calibrating} \mathrm{the} \mathrm{decision}-\mathrm{making} \mathrm{processes}. \mathrm{The} \mathrm{normalized} \mathrm{network} \mathrm{distance} {\mathcal{X}}_j$$\mathrm{to} \mathrm{a} \mathrm{location} j$$\mathrm{took} \mathrm{a} \mathrm{value} \mathrm{of} 0$$\mathrm{when} \mathrm{it} \mathrm{was} \mathrm{the} \mathrm{distance} \mathrm{to} \mathrm{self}, \mathrm{otherwise} {\mathcal{X}}_j>0$. Locations sold the goods being produced in different locations in the region or exported the goods outside the region. Unit costs were determined as $r_{jt}={\displaystyle \frac{{WB}_{jt}}{X_{jt}}}+{\mathcal{X}}_j$ $\mathrm{The} \mathrm{probability} \mathrm{of} \mathrm{migrating} \mathrm{to} \mathrm{a} \mathrm{location} j$$\mathrm{was} \mathrm{based} \mathrm{on} \mathrm{a} \mathrm{joint} \mathrm{probability} \mathrm{distribution} {\mathsf{\Pi }}_{jt}$, defined as ${\mathsf{\Pi }}_{jt}={\mathsf{\Pi }}_t^{{\mathcal{X}}_j}+{\mathsf{\Pi }}_t^{{\widehat{w}}_{jt}}$ $\mathrm{where} {\mathsf{\Pi }}_t^{{\mathcal{X}}_j}$$\mathrm{is} \mathrm{the} \mathrm{probability} \mathrm{of} \mathrm{migration} \mathrm{based} \mathrm{on} \mathrm{network} \mathrm{distances}, \mathrm{and} {\mathsf{\Pi }}_t^{{\widehat{w}}_{jt}}$ signifies the probability of migrating based on the real income ratio of the target location to the current location.	Mapping the impact of major earthquakes.
7	Pagano et al. [38]	-	-	-
8	Kalogiannidis et al. [28]	Multiple Linear Regression	$Y={\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\epsilon$ $\mathrm{where} Y$$\mathrm{is} \mathrm{disaster} \mathrm{risk} \mathrm{management}, {\beta }_0$$\mathrm{signifies} \mathrm{a} \mathrm{constant} (\mathrm{coefficient} \mathrm{of} \mathrm{intercept}), \mathrm{and} X_1$$\mathrm{is} \mathrm{the} \mathrm{school} \mathrm{curriculum}. X_2$$\mathrm{represents} \mathrm{knowledge} \mathrm{acquisition}, \epsilon$$\mathrm{is} \mathrm{the} \mathrm{error} \mathrm{term} \mathrm{in} \mathrm{the} \mathrm{multiple} \mathrm{regression} \mathrm{model}, \mathrm{and} {\beta }_1$$\mathrm{and} {\beta }_2$ are the regression coefficients of the two independent variables.	Testing the relationship between school systems and increased disaster risk reduction capacity.
9	Tadesse and Zeleke [27]	Logistics Distribution Function	$P_i=E\left(Y=1|X\right)={\displaystyle \frac{e^{Z_i}}{1+e^{Z_i}}}$ $\mathrm{where} Z_i={\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\dots +{\beta }_k{X}_k+U_i$$, {\beta }_0$$\mathrm{is} \mathrm{the} \mathrm{intercept}, \mathrm{and} {\beta }_i$$\mathrm{signifies} \mathrm{the} \mathrm{regression} \mathrm{coefficient} \mathrm{to} \mathrm{be} \mathrm{estimated}. X_i$$\mathrm{is} \mathrm{the} \mathrm{pre}-\mathrm{intervention} \mathrm{characteristics}, \mathrm{and} U_i$ represents the disturbance term.	Controlling selection bias and evaluating the impact of the productive safety net program on daily calorie consumption, annual consumption expenditure, household income, livestock ownership, and housing conditions.
10	Hochrainer-Stigler et al. [29]	General Linear Modeling	Statistical analysis using $Y={\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\dots +{\beta }_k{X}_k+\epsilon$	Measuring community flood resilience.
11	Wang and Lindt [39]	Two-Step Residential Recovery Model	$\mathrm{The} \mathrm{recovery} \mathrm{process} \mathrm{of} \mathrm{residential} \mathrm{buildings} \mathrm{was} \mathrm{modeled} \mathrm{in} \mathrm{two} \mathrm{sequential} \mathrm{stages}. \mathrm{The} \mathrm{delay} \mathrm{phase} T_{Delay, ij}$ captured administrative and resource-related delays. The repair phase captured physical construction time based on damage state, using fragility and restoration data. The total time required for each building to be fully restored could be determined by combining the delay time and repair time. For each time step, there was a probability that each building was fully recovered. Therefore, the percentage of residential buildings that were fully restored could be calculated. $B_{rj}^i\left(t\right)=\{\begin{array}{c}1 R_r\left(t\right)\ge R_{j,th}^i\\ 0{ R}_r\left(t\right)<R_{j,th}^i\end{array}, i\in \mathrm{1,2},\dots ,n$ and ${RP}_{rj}\left(t\right)={\displaystyle \frac{{\sum }_{i=1}^n{B}_{rj}^i\left(t\right)}{n}}, i\in \mathrm{1,2},\dots ,n$ $\mathrm{Resilience} \mathrm{was} \mathrm{defined} \mathrm{as} \mathrm{the} \mathrm{area} \mathrm{under} \mathrm{the} \mathrm{functionality} \mathrm{curve} \mathrm{from} \mathrm{the} \mathrm{time} \mathrm{of} \mathrm{the} \mathrm{disaster} t_0$$\mathrm{to} \mathrm{full} \mathrm{recovery} \mathrm{time} t_0+t_r$ $R_i={\int }_{t_0}^{t_0+t_r}{RP}_{r\_mean}\left(t\right) dt$ where ${RP}_{r\_mean}\left(t\right)={\displaystyle \frac{{\sum }_{j=1}^N{RP}_{rj}\left(t\right)}{N}}$	Modeling two stages of recovery, namely, delay time and repair time, and considering the distribution of household income in determining the source of repair financing.
12	Harrison et al. [31]	-	-	-
13	Lefutso et al. [40]	Discrete Choice Experiment and Mixed Logic Model	The utility that individuals derive from selecting alternative insurance options $U_{ij}={\beta }_i^T{X}_{ij}+{ϵ}_{ij}$ $\mathrm{where} X_{ij}$$\mathrm{is} \mathrm{a} \mathrm{vector} \mathrm{of} \mathrm{unobservable} \mathrm{attributes}, {\beta }_i$$\mathrm{signifies} \mathrm{vectors} \mathrm{of} \mathrm{individual} \mathrm{specific} \mathrm{coefficients} \mathrm{representing} \mathrm{the} \mathrm{preferences} \mathrm{of} \mathrm{an} \mathrm{individual} \mathrm{for} \mathrm{the} \mathrm{attributes}, \mathrm{and} {ϵ}_{ij}$ is the random error term that captures unobserved factors assumed to follow a Gumbel distribution. The econometric model that was finally applied to derive the WTP of individuals was $U_{ij}={\beta }_0+{\beta }_{1}HContent+{\beta }_2{Buliding}_{Cover}$ $+{\beta }_{3}Flood Insurance \mathrm{O}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}1$ $+{\beta }_{4}Flood Insurance \mathrm{O}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}2+{ϵ}_{ij}$ $\mathrm{The} \mathrm{probability} \mathrm{that} \mathrm{individual} n$$\mathrm{selected} \mathrm{alternative} j$ in selected task t was $P_{ij}=\int {\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left({\beta }_i^T{X}_{ij}\right)}{\sum \mathrm{e}\mathrm{x}\mathrm{p}\left({\beta }_i^T{X}_{ik}\right)}}f\left(\beta |\theta \right)d\beta$ $\mathrm{where} f\left(\beta |\theta \right)$$\mathrm{signifies} \mathrm{the} \mathrm{density} \mathrm{function} \mathrm{of} \beta$$, \mathrm{often} \mathrm{assumed} \mathrm{to} \mathrm{be} \mathrm{normal} \mathrm{or} \log -\mathrm{normal}, \mathrm{and} \theta$ is the parameter. The willingness to pay of individuals was computed by $WTP={\displaystyle \frac{{\beta }_{Attribute}}{{\beta }_{Premium}}}$ $\mathrm{The} \mathrm{coefficients} {\beta }_1$$, {\beta }_2$$, {\beta }_3$$, \mathrm{and} {\beta }_4$ represent the marginal utility of each attribute.	Exploring the preferences of poor individuals toward flood insurance attributes (coverage size, premiums, and excess costs) and considering the heterogeneity of preferences between individuals.
14	Han and Koliou [32]	Resilience Index, Community Resilience Index, and Model Fragility Functions	The resilience index for an individual system was $R={\displaystyle \frac{1}{T_{LC}}}{\int }_0^{T_{LC}}Q\left(t\right)dt$ $\mathrm{where} R$$\mathrm{signifies} \mathrm{the} \mathrm{resilience} \mathrm{index}, T_{LC}$$\mathrm{is} \mathrm{the} \mathrm{control} \mathrm{time}, \mathrm{and} Q\left(t\right)$ represents the time variant functionality of the system. The community resilience was calculated by $R_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{y}}={\left({\displaystyle \prod _{i=1}^{N_s}}{R}_i^{w_i}\right)}^{{\displaystyle \frac{1}{{\sum }_{i=1}^{N_s}{w}_i}}}$ $\mathrm{where} N_s$$\mathrm{is} \mathrm{the} \mathrm{total} \mathrm{number} \mathrm{of} \mathrm{systems} \mathrm{included}, R_i$$\mathrm{represents} \mathrm{the} \mathrm{resilience} \mathrm{index} \mathrm{of} \mathrm{the} i$$\mathrm{th} \mathrm{system}, \mathrm{and} w_i$$\mathrm{is} \mathrm{the} \mathrm{weighting} \mathrm{factor} \mathrm{of} \mathrm{the} i$th system. This mitigation strategy focused on business resilience after a tornado. Following damage assessment, contractor mobilization and financing proceeded in parallel, and insured businesses filed claims, while others took loans. Each step, including permit acquisition, introduced delays that could be modeled using fragility functions. $F_{rt}\left(T_{del}\right)=\mathsf{\Phi }\left[{\displaystyle \frac{\ln \left(T_{del}\right)-\ln \left(\theta \right)}{\beta }}\right]$ $\mathrm{where} F_{rt}\left(·\right)$$\mathrm{is} \mathrm{the} \mathrm{associated} \mathrm{fragility} \mathrm{function}, T_{del}$$\mathrm{represents} \mathrm{the} \mathrm{delay} \mathrm{time} \mathrm{of} \mathrm{a} \mathrm{process}, \mathrm{and} \theta$$\mathrm{and} \beta$ are the median and dispersion of the fragility function, respectively.	Modeling community resilience to tornadoes for structural damage and recovery delays.
15	Holcombe et al. [25]	Present Value Calculation of Landslide Costs	$\mathrm{The} \mathrm{present} \mathrm{value} \mathrm{of} \mathrm{expected} \cos \mathrm{ts} \mathrm{from} \mathrm{landslides} \mathrm{of} \mathrm{type} A$ was ${\displaystyle \frac{q_{A,I}{c}_A\left(1-{\left[\frac{\left(1-q_{A,I}\right)}{\left(1+r\right)}\right]}^N\right)}{1-\left[\frac{\left(1-q_{A,I}\right)}{\left(1+r\right)}\right]}}+{\displaystyle \frac{q_A{c}_A{\left[\frac{\left(1-q_{A,I}\right)}{\left(1+r\right)}\right]}^N}{1-\left[\frac{\left(1-q_A\right)}{\left(1+r\right)}\right]}}.$ $\mathrm{The} \mathrm{present} \mathrm{value} \mathrm{of} \mathrm{expected} \cos \mathrm{ts} \mathrm{from} \mathrm{landslides} \mathrm{of} \mathrm{type} B$$\mathrm{in} \mathrm{location} L$ was ${\displaystyle \frac{q_{B,I}{c}_{B,L}\left(1-q_{A,I}\right)\left(1-{\left[\frac{\left(1-q_{A,I}\right)\left(1-q_{B,I}\right)}{\left(1+r\right)}\right]}^N\right)}{1-\left[\frac{\left(1-q_{A,I}\right)\left(1-q_{B,I}\right)}{\left(1+r\right)}\right]}}$ $+{\displaystyle \frac{q_B{c}_{B,L}\left(1-q_A\right)\left(1-{\left[\frac{\left(1-q_{A,I}\right)\left(1-q_{B,I}\right)}{\left(1+r\right)}\right]}^N\right)}{1-\left[\frac{\left(1-q_A\right)\left(1-q_B\right)}{\left(1+r\right)}\right]}}.$ The definitions of variables and assumption are shown in [25].	Calculating the probability of a landslide event multiplied by the exposure and vulnerability of elements at the location.
16	Khan et al. [30]	Livelihood Vulnerability Index and Ordered Logistic Regression Model	Various factors such as socio-demographics, livelihood strategies, social networks, health, food, water, and climate-related disasters were considered to assess livelihood vulnerability at the union level. Each sub-component was first standardized to a 0–1 scale using this equation. ${Index}_{S_i}={\displaystyle \frac{S_i-S_{min}}{S_{max}-S_{min}}}$ $\mathrm{where} {Index}_{S_i}$$\mathrm{is} \mathrm{the} \mathrm{standardized} \mathrm{value} \mathrm{of} \mathrm{an} \mathrm{indicator} \mathrm{for} \mathrm{a} \mathrm{household}, S_i$$\mathrm{represents} \mathrm{the} \mathrm{actual} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{same} \mathrm{indicator}, \mathrm{and} S_{min}$$\mathrm{and} S_{max}$ are the minimum and maximum values, respectively, of the same indicator. Calculation of the average of the principal components: $M_u={\displaystyle \frac{{\sum }_{i=1}^n{Index}_{S_{u^i}}}{n}}$ $\mathrm{where} M_u$$\mathrm{ss} \mathrm{a} \mathrm{major} \mathrm{component} \mathrm{for} \mathrm{union} u$$[\mathrm{socio}-\mathrm{demographic} \mathrm{profile} (\mathrm{SDP}), \mathrm{livelihood} \mathrm{strategies} (\mathrm{LS}), \mathrm{social} \mathrm{networks} (\mathrm{SN}), \mathrm{health} (\mathrm{H}), \mathrm{food} (\mathrm{F}), \mathrm{water} (\mathrm{W}), \mathrm{or} \mathrm{climatic} \mathrm{variability} \mathrm{and} \mathrm{disasters} (\mathrm{CVD}\left)\right], {Index}_{S_{u^i}}$$\mathrm{represents} \mathrm{the} \mathrm{sub}-\mathrm{components} \mathrm{indexed} \mathrm{by} i$$\mathrm{that} \mathrm{made} \mathrm{up} \mathrm{each} \mathrm{major} \mathrm{component}, \mathrm{and} n$ is the number of sub-components in each major component. Calculation of the weighted value of each dimension of LVI (IPCC): ${CF}_u={\displaystyle \frac{{\sum }_{i=1}^n{W}_{M_i}{M}_{ui}}{{\sum }_{i=1}^n{W}_{M_i}}}$ $\mathrm{where} {CF}_u$$\mathrm{represents} \mathrm{the} \mathrm{IPCC}-\mathrm{defined} \mathrm{dimensions} (\mathrm{exposure}, \mathrm{sensitivity}, \mathrm{or} \mathrm{adaptive} \mathrm{capacity}) \mathrm{for} \mathrm{union} u$$, M_{ui}$$\mathrm{signifies} \mathrm{the} \mathrm{weighted} \mathrm{value} \mathrm{of} \mathrm{each} \mathrm{major} \mathrm{component} \mathrm{for} \mathrm{union} u$$\mathrm{indexed} \mathrm{by} i$$, \mathrm{and} W_{M_i}$ is the number of major components under each dimension. The three dimensions were finally combined to generate the LVI (IPCC) after computing the exposure, sensitivity, and adaptive capacity: ${LVI\left(IPCC\right)}_u={\displaystyle \frac{E_u+S_u+\left(1-{AC}_u\right)}{3}}$ $\mathrm{where} {LVI\left(IPCC\right)}_u$$\mathrm{is} \mathrm{the} \mathrm{livelihood} \mathrm{vulnerability} \mathrm{index} \mathrm{for} \mathrm{union} u$$, \mathrm{and} E_u$$\mathrm{signifies} \mathrm{the} \mathrm{measured} \mathrm{exposure} \mathrm{score} \mathrm{of} \mathrm{the} \mathrm{union}u$$(\mathrm{corresponding} \mathrm{to} \mathrm{the} \mathrm{major} \mathrm{component}, \mathrm{namely}, \mathrm{climate} \mathrm{variability} \mathrm{and} \mathrm{disasters}). {AC}_u$$\mathrm{represents} \mathrm{the} \mathrm{measured} \mathrm{adaptive} \mathrm{capacity} \mathrm{score} \mathrm{of} \mathrm{union} u$$(\mathrm{the} \mathrm{weighted} \mathrm{mean} \mathrm{of} \mathrm{the} \mathrm{major} \mathrm{components}, \mathrm{namely}, \mathrm{socio}-\mathrm{demographic}, \mathrm{livelihood} \mathrm{strategies}, \mathrm{and} \mathrm{social} \mathrm{networks}). S_u$$\mathrm{is} \mathrm{the} \mathrm{calculated} \mathrm{sensitivity} \mathrm{score} \mathrm{for} \mathrm{union} u$ (the weighted mean of the major components, namely, health, food, and water). The Kruskal–Wallis H Test was used to differentiate vulnerability between groups, and an ordered logistic regression model was used to identify significant indicators that influenced vulnerability.	Measuring household vulnerability to climate disasters and identifying significant indicators that influence vulnerability.
17	Yin et al. [24]	Disaster Risk Formula, Urban Rainfall Intensity Formula, Surface Runoff Model using SCS Curve Number, Waterlogging Estimation Model, Loss Estimation Based on Stage-Damage Curve, and Annual Risk Model	The possibility of expected losses: $R=H\times V\times E$ $\mathrm{where} R$$\mathrm{is} \mathrm{the} \mathrm{risk}, H$$\mathrm{represents} \mathrm{the} \mathrm{hazard}, \mathrm{the} \mathrm{probability} \mathrm{of} \mathrm{a} \mathrm{potentially} \mathrm{damaging} \mathrm{physical} \mathrm{event}, \mathrm{and} V$$\mathrm{is} \mathrm{vulnerability}, \mathrm{the} \mathrm{degree} \mathrm{of} \mathrm{loss} \mathrm{to} \mathrm{a} \mathrm{given} \mathrm{element} \mathrm{of} \mathrm{risk} \mathrm{or} \mathrm{a} \mathrm{set} \mathrm{of} \mathrm{elements} \mathrm{at} \mathrm{risk} \mathrm{from} \mathrm{the} \mathrm{occurrence} \mathrm{of} \mathrm{a} \mathrm{natural} \mathrm{phenomenon} \mathrm{of} \mathrm{a} \mathrm{given} \mathrm{magnitude}. E$ signifies exposure to the element of risk, including buildings, population, property, or other human activities. Calculation of different intensities of rainfall: $q={\displaystyle \frac{1995.84\left(P^{0.3}-0.42\right)}{{\left(t+10+7\log P\right)}^{0.82+0.07\log P}}}$ $\mathrm{where} q$$\mathrm{is} \mathrm{the} \mathrm{rainfall} \mathrm{intensity}, P$$\mathrm{signifies} \mathrm{the} \mathrm{return} \mathrm{period} \mathrm{of} \mathrm{rainfall}, \mathrm{and} t$ representes the duration of rainfall. Surface runoff model using the SCS curve number: $Q={\displaystyle \frac{{\left(T-la\right)}^2}{\left(T+S-la\right)}}$ $\mathrm{where} S={\displaystyle \frac{25400}{CN-254}}.$$\mathrm{In} \mathrm{detail}, Q$$\mathrm{is} \mathrm{the} \mathrm{direct} \mathrm{runoff}, T$$\mathrm{signifies} \mathrm{the} \mathrm{total} \mathrm{rainfall}, \mathrm{and} la$$\mathrm{is} \mathrm{the} \mathrm{initial} \mathrm{abstraction}. S$$\mathrm{represents} \mathrm{the} \mathrm{potential} \mathrm{maximum} \mathrm{retention}, \mathrm{and} CN$ is curve number. Waterlogging estimation model: $W=\left(Q-D\right)\times S^{\mathrm{*}}$ $\mathrm{where} W$$\mathrm{is} \mathrm{the} \mathrm{amount} \mathrm{of} \mathrm{waterlogging}, D$$\mathrm{represents} \mathrm{the} \mathrm{amount} \mathrm{of} \mathrm{drainage}, \mathrm{and} S^{\mathrm{*}}$$\mathrm{is} \mathrm{the} \mathrm{catchment} \mathrm{area}. \mathrm{The} \mathrm{average} \mathrm{annual} \mathrm{waterlogging} \mathrm{loss} (AWWL$) was defined as the area under the risk curve: $AWWL=\int xf\left(x\right)dx$ $\mathrm{where} x$$\mathrm{is} \mathrm{the} \mathrm{occurrence} \mathrm{probability} \mathrm{of} \mathrm{a} \mathrm{waterlogging} \mathrm{event}, \mathrm{and} f\left(x\right)$$\mathrm{represents} \mathrm{the} \mathrm{loss} \mathrm{of} x$.	Spatial mapping of inundation, vulnerability, losses, and risk evaluation.

shows the modeling methods, reviewing each specific function, contributions to CBDRM, and addressing the study question (a): What types of mathematical models have been applied to support CBDRM? Among the 17 reviewed articles, most explicitly used quantitative modeling methods to support disaster risk assessment or financial risk mitigation strategies, including stochastic optimization [33], vulnerability indices [34], and disaster impact simulations [25]. These methods contributed to risk quantification, resilience measurement, and evidence-based policy formulation. A small subset of studies adopted conceptual or participatory frameworks without implementing formal quantitative simulations. Although these did not develop predictive mathematical models, the studies provided valuable governance perceptions and stakeholder perspectives that served as foundations for future quantitative incorporation. The increasing use of mathematical models in CBDRM reflected a growing emphasis on structured, data-informed strategies to improve disaster preparedness, response, and recovery [24,33].

3.2.2. Level of Incorporation of Financial Risk Mitigation

Outside the application of technical modeling methods, an important question was the degree to which CBDRM frameworks incorporated financial risk mitigation tools. These instruments, including insurance schemes, community funds, or risk pooling mechanisms, served as safety nets in the aftermath of disasters [38,40]. This section addresses study question (b): Do these studies include financial risk mitigation partially, conceptually, or through practical implementation? Additionally, who are the major organizations or stakeholders engaged in these CBDRM efforts, irrespective of the connection to financial mechanisms?

Table 2. Incorporation of financial risk mitigation and stakeholder participation in CBDRM across the reviewed studies.

Author(s)	Financial Risk Mitigation	Local Government/Government	Community	Institute/ Company/ NGO
Faiz and Harrison [33]	✔	✔	✔	✔
Eze and Siegmund [35]	✔	✘	✘	✔
Fakhruddin et al. [26]	✔	✔	✔	✔
Aroca- Jiménez et al. [34]	✘	✔	✔	✔
Chen et al. [36]	✘	✔	✔	✔
Naqvi [37]	✔	✔	✔	✔
Pagano et al. [38]	✔	✔	✔	✔
Kalogiannidis et al. [28]	✘	✔	✔	✔
Tadesse and Zeleke [27]	✔	✔	✔	✔
Hochrainer-Stigler et al. [29]	✔	✘	✘	✔
Wang and Lindt [39]	✔	✔	✔	✔
Harrison et al. [31]	✘	✔	✔	✔
Lefutson et al. [40]	✔	✔	✔	✔
Han and Koliou [32]	✔	✔	✔	✔
Holcombe et al. [25]	✔	✔	✔	✘
Khan et al. [30]	✘	✔	✔	✔
Yin et al. [24]	✘	✔	✔	✔

shows the presence or absence of financial mechanisms and the actors involved in each CBDRM context.

Based on the analytical treatment of financial mechanisms, the reviewed studies were grouped into three levels of incorporation.

• Level 1—No explicit financial incorporation

The studies focused on hazard assessment, vulnerability measurement, or resilience evaluation without incorporating insurance schemes or structured financial instruments into the modeling framework. Examples included studies prioritizing community vulnerability or disaster risk mapping without reference to post-disaster financial tools.

• Level 2—Conceptual or partial financial reference

In this category, financial risk mitigation mechanisms were recognized but not formally incorporated into the mathematical structure of the model. Insurance, catastrophe modeling or recovery financing were discussed as policy recommendations or contextual elements rather than operational components.

• Level 3—Explicit financial incorporation

These studies incorporated financial mechanisms directly into the analytical framework. Examples included modeling recovery financing, evaluating willingness to pay for insurance, or optimizing risk-sharing strategies under uncertainty [26,38,39]. In these cases, financial risk mitigation was treated as a structural element rather than an external recommendation.

Among 17 reviewed articles, 11 referenced insurance or financial risk mitigation in some form. However, only a subset incorporated these mechanisms quantitatively in the modeling structure. This distribution indicated that as financial resilience was increasingly recognized in CBDRM studies, formal incorporation into mathematical models remained uneven. Concerning stakeholder participation, most studies reported participation from local governments, communities, and NGOs. On the other hand, the presence of financial mechanisms did not often correspond with structured stakeholder incorporation. For example, some studies discussed insurance-related issues without direct engagement of public institutions or community actors. Meanwhile, others prioritized strong institutional (community) collaboration and did not address insurance or financial tools. The findings implied that financial risk mitigation and stakeholder participation were both present in CBDRM literature, and each arrangement in coherent modeling frameworks varied significantly across studies.

3.2.3. Structural Components

Building on the previous classification, study question (c) was formulated as follows: What essential components have been identified in existing studies for the design and implementation of insurance or financial risk mitigation mechanisms in CBDRM?

Table 3. Key components of financial risk mitigation in CBDRM and examples across reviewed studies.

No.	Component	Description	Example from Reviewed Studies
1	Risk Pooling Mechanism	A collective fund to spread disaster risk in the community or across regions	Spatial agent-based model explored impact zones and potential to incorporate risk pooling geographically [37]
2	Premium Structure	Contribution scheme paid by community members, including subsidies for vulnerable groups	Examined community preferences for premium size, coverage, and cost-sharing [40]
3	Trigger Mechanism	Conditions for payout based on objective indices or verified damages	Showed the role of index-based insurance activated by weather parameters [38]
4	Fund Holder/Financial Institution	Entity responsible for managing funds (government, NGO, private sector, or hybrid)	Prioritized multi-stakeholder coordination in risk-informed financial management [26]
5	Payout and Disbursement Process	Mechanism to deliver funds or assistance quickly post-disaster	Considered household income in recovery funding and timing of repairs [39]
6	Community Engagement and Capacity Building	Participation of local communities in design, monitoring, and education about schemes	Participatory modeling built awareness and strengthened ownership of risk solutions [31]
7	Regulatory and Policy Support	Enabling laws and policies to facilitate insurance, subsidies, or public–private partnerships	Discussed governance challenges and policy gaps in disaster financial protection [35]

shows these components and provides examples drawn from the reviewed literature.

Across the selected studies, seven recurring structural components were identified. First, risk pooling mechanisms were shown as collective arrangements to distribute disaster losses in or across communities. For example, spatial modeling methods presented the potential for geographically distributed risk-sharing structures [40]. Second, premium structures determined how financial contributions were collected from community members, including considerations of affordability and cost-sharing. Discrete selection experiments showed how communities evaluated premium size, coverage levels, and excess costs [40]. Third, trigger mechanisms defined the conditions under which payouts were activated. Index-based insurance schemes connected to measurable parameters, such as rainfall intensity or weather thresholds, were frequently discussed in this context [39]. Fourth, fund governance and financial management arrangements specified which entities—government agencies, NGOs, private insurers, or hybrid partnerships—were responsible for managing pooled resources. Multi-stakeholder coordination frameworks prioritized the importance of institutional roles in financial inaccuracy. Fifth, payout and disbursement processes addressed how and when financial support was delivered following a disaster. Recovery models that incorporated income distribution and repair financing showed the operational implications of delayed or phased disbursement [39]. Sixth, community engagement and capacity building were prioritized as critical for ensuring transparency, trust, and long-term participation. Participatory modeling methods showed how community participation strengthened ownership of risk mitigation initiatives [31]. The regulatory and policy support provided the enabling environment for insurance mechanisms, subsidies, and public–private partnerships. Governance-oriented studies identified institutional gaps and policy challenges in disaster financial protection systems [35]. Although individual studies addressed specific components, few incorporated all elements in a single comprehensive framework. As a result, the reviewed literature presented these components largely in fragmented form rather than as a fully operationalized financial risk mitigation system.

4. Discussion

4.1. Study Gaps

The bibliometric analysis conducted in this study offered several major perceptions into the current state of study on mathematical modeling in CBDRM, particularly in relation to insurance and financial risk mitigation. All 17 articles included in the review were published as peer-reviewed journal articles, reflecting strong academic importance on methodological rigor while also indicating that discussions on CBDRM modeling remain largely focused on scholarly circles. In terms of publication trends, study output was relatively limited before 2016, with only sporadic contributions. The limited number of eligible studies further indicated that the intersection between mathematical modeling, CBDRM, and financial risk mitigation remained underexplored in the global literature. This signified a structural study gap rather than a lack of practical relevance.

The years 2022–2024 achieved a visible increase in publication frequency, signifying growing academic attention toward incorporating mathematical models into community-based disaster risk strategies [16]. This momentum was associated with broader global concerns about disaster resilience and the need for innovative financial mechanisms to protect vulnerable communities. The distribution of publications across 12 different journals, including Progress in Disaster Science, Natural Hazards, and the International Journal of Disaster Risk Reduction, showed the interdisciplinary character of the field. Following the discussion, this study was rooted in disaster science and also extended into socio-economic planning, environmental management, and development analysis. This showed the need for incorporated methods in CBDRM study and practice.

Most studies geographically originated from countries with advanced disaster risk management infrastructures, such as the USA, New Zealand, China, and Austria. The participation of studies from Bangladesh, South Africa, Ethiopia, and Greece reflected a promising, but still limited, expansion of CBDRM modeling analyses into more vulnerable and disaster-prone regions of the Global South [22]. This distribution raised concerns regarding contextual transferability, particularly for low-income and climate-vulnerable regions where community-based financial protection mechanisms were most urgently needed. The keyword mapping based on abstracts and keywords showed strong thematic clusters around concepts such as “community,” “risk,” “insurance,” “model,” and “resilience.” Moreover, the co-occurrence of terms including “climate change,” “vulnerability,” and “disaster risk reduction” further indicated that mathematical modeling in CBDRM was increasingly situated in global agendas on climate adaptation as well as sustainable development [21].

This study identified several major gaps that limited the practical incorporation of technical modeling with disaster risk financing at the community level despite the increasing use of mathematical models to support CBDRM. First, as models including stochastic optimization, vulnerability indices, and agent-based simulations were widely used to quantify disaster risks, each connection to financial risk mitigation mechanisms, namely, insurance, risk pooling, or community funds, remained largely underdeveloped. In many cases, financial protection was discussed only as a theoretical recommendation, without being incorporated into the mathematical structure of each model. Second, studies that explicitly addressed insurance or financial risk mitigation tended to focus on individual aspects, such as community preferences for insurance attributes [40] or conceptual proposals for index-based products [38]. However, these analyses rarely presented incorporated operational frameworks grounded in quantitative modelling. This fragmentation created a disconnect between technical risk assessment and real-world financial protection systems, leaving communities vulnerable in terms of both knowledge and resources.

Following the discussion above, the review showed that critical components essential for functioning risk pooling mechanisms, such as premium structures, payout processes, fund management, and regulatory support, were often addressed in isolation and lacked coherent incorporation into existing CBDRM initiatives. Few studies provided guidance on how these elements could work together in a community-driven financial protection scheme supported by robust, data-informed modelling. As stakeholder participation was frequently mentioned in CBDRM, the participation of governments, financial institutions, and communities in designing, managing, and implementing financial risk mitigation remained very limited and inconsistently described across the literature. This showed the need for future studies that bridged technical, institutional, and financial dimensions through incorporated modeling methods and participatory processes.

Beyond identifying the gaps, considering the basic causes of the limited incorporation between mathematical modeling and financial risk mitigation in CBDRM was essential. A contributing factor was disciplinary fragmentation, where disaster risk modeling was primarily developed in engineering and environmental sciences. Meanwhile, insurance design and risk financing mechanisms were often addressed in economics or actuarial disciplines. The separation restricted the development of unified analytical frameworks. In addition, the limited availability of community-level claims data and institutional constraints hindered the operational incorporation of insurance mechanisms into quantitative models.

Future studies should move beyond conceptual discussions and explore more operational incorporation pathways to advance the field. For instance, vulnerability and hazard assessment models can incorporate premium calculation modules and payout trigger mechanisms. Agent-based simulations may be extended to represent household participation in community risk pooling schemes, while recovery models can incorporate claim settlement timing and liquidity constraints. Optimization methods may also support the design of subsidy structures that balance affordability with long-term sustainability. Therefore, an operational incorporation framework can combine (1) quantitative risk assessment, (2) financial structuring mechanisms such as pooling and premium design, (3) institutional governance arrangements, and (4) implementation strategies at the community level. This method should provide a more coherent bridge between technical modeling outputs and practical financial resilience mechanisms in CBDRM.

4.2. Future Study

This study showed a significant study gap in the practical design of community-based risk pooling mechanisms grounded in mathematical models, despite the growing recognition of financial risk mitigation as part of CBDRM. Several studies prioritized the need for insurance schemes or disaster funds, but few provided operational frameworks that followed the financial realities and institutional capacities of disaster-prone communities [38,40].

Building on the gaps, this study proposed a conceptual structure for a community-based risk pooling mechanism, shown in Figure 7. At the core of this scheme was the principle of collective risk sharing, supported by essential components such as affordable premium structures, transparent fund management, clear payout procedures, regulatory alignment, and strong community engagement. The incorporation of these components strengthened financial resilience and also ensured that risk pooling remained accessible, trusted, and responsive to community needs. Future studies should focus on translating this conceptual model into quantitative designs using stochastic modeling, optimization, or hybrid methods that can balance equity, affordability, and operational sustainability in CBDRM frameworks.

4.3. Limitations of the Review

Several limitations were recognized, even though the systematic review followed a rigorous screening and selection process. First, the final sample consisted of 17 peer-reviewed journal articles. As this reflected the relatively narrow intersection between mathematical modeling, CBDRM, and financial risk mitigation in existing literature, the limited number of eligible studies constrained the scope of generalizable assumptions. Second, the geographical distribution of the selected studies indicated a focus on high-income countries with relatively advanced disaster risk governance systems. Differences in institutional capacity, insurance penetration, regulatory environments, and financial inclusion between high-income and low- and middle-income countries influenced the applicability of the identified modeling methods across diverse contexts.

Third, the reliance on major academic databases and English-language publications excluded relevant studies from regional journals or policy reports, particularly those documenting community-level financial practices in developing countries. These limitations showed the need for future systematic reviews to expand database coverage, incorporate gray literature where methodologically appropriate, and explore multilingual sources to capture a broader spectrum of CBDRM financial innovations globally. The restrictions did not damage the analytical contribution of this study but rather delineate the boundaries within which the conclusions should be interpreted.

5. Conclusions

In conclusion, this systematic review examined the application of mathematical modeling in CBDRM, paying particular attention to the incorporation of insurance and financial risk mitigation mechanisms. The results showed substantial methodological diversity, including optimization and simulation models, vulnerability indices, and econometric methods. However, as quantitative modeling was increasingly central to CBDRM studies, the incorporation of financial risk mitigation mechanisms remained uneven and often fragmented. This study contributed by synthesizing existing modeling methods and performing classification according to model type, level of financial incorporation, and major structural components, following an academic perspective. Moreover, the analysis showed a structural gap between technical disaster risk modeling and operational financial protection design, identifying the need for interdisciplinary incorporation between engineering, environmental sciences, and actuarial or financial disciplines.

The study practically showed the importance of associating quantitative risk assessment with insurance design, risk pooling structures, governance arrangements, and community engagement mechanisms. Strengthening this association would be essential for improving community-level financial resilience, particularly in disaster-prone and climate-vulnerable regions. On the other hand, this study was subject to certain limitations. The number of eligible studies remained very limited, and the majority originated from high-income countries, potentially affecting contextual generalizability. The reliance on peer-reviewed English-language publications excluded relevant regional or policy-oriented contributions. As a result, future studies should focus on developing operational incorporation frameworks that embed insurance mechanisms in quantitative modeling structures. Expanding geographical representation, incorporating community-level financial data, and fostering interdisciplinary collaboration will also be critical to advance the practical applicability of CBDRM modeling.