**1. Introduction**

In recent years, the supply chain of prefabricated buildings has been widely considered by society in shortening the construction period, saving labour, and improving the quality of construction enterprises. At the same time, states have successively issued relevant policies to encourage and support the development of the supply chain of prefabricated buildings, which has brought prefabricated construction industrialization to a new level [1]. According to the statistics of the prefabricated building industry, the market size of China's prefabricated buildings in 2011 was about CNY 4.3 billion, reached CNY 262.3 billion in 2017, and soared to CNY 1.02 trillion by 2022, with a compound growth rate of more than 100%, and the national prefabricated buildings accounted for more than 22%. Supply chain management is chosen as a management mode for prefabricated buildings to enhance their strength, integrating the advantages of various related enterprises, connecting all the enterprise nodes in the supply chain, and forming a chain network with the owner's demand as the guide and the general contractor as the core. Through the commercial interaction among the enterprises of each node, information flow, resource flow, cash flow, and logistics run through the design and construction units, material purchasing units, logistics and transportation units, assembly and construction units, and structural design units, thus promoting the functional operation of the overall construction chain [2]. However, with the continuous improvements of the technology of prefabricated building

**Citation:** Wang, Y.; Sun, R.; Ren, L.; Geng, X.; Wang, X.; Lv, L. Risk Propagation Model and Simulation of an Assembled Building Supply Chain Network. *Buildings* **2023**, *13*, 981. https://doi.org/10.3390/ buildings13040981

Academic Editor: Tarek Zayed

Received: 10 March 2023 Revised: 26 March 2023 Accepted: 3 April 2023 Published: 7 April 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

supply chains, the number of node enterprises is increasing, and the types of nodes are increasing from the initial single type to the present full coverage. The prefabricated links involved are more complicated, and the suppliers are more diverse, which leads to increasing risks in the supply chain. External environmental factors, such as natural environmental changes, policies and regulations, and internal factors, such as the planning stage, purchasing stage, manufacturing stage, transportation stage, and assembly stage, all affect the change in risks [3]. While the risks in the supply chain increase, the stability of the supply chain worsens, and the changes in these risks affect the normal construction and transportation of the node enterprises in the supply chain [4]. On the other hand, after being affected by the risks, the node enterprises will further spread the risks through the propagation path of the supply chain, which will have an impact on the normal operation of the whole supply chain. Therefore, the risk problem in the supply chain of prefabricated buildings has become an urgent problem in supply chain management.

Risk research on supply chains was first influenced by enterprise risk, and the connection between enterprises increased, so risk arose. A supply chain is a network composed of many enterprises to meet the needs of customers, and the risks among enterprises are bound to affect the risks of the whole supply chain. However, the prefabricated building supply chain has many participants, complicated strategic relationships, and opaque information among enterprises, so it is difficult for the traditional supply chain to meet the development of the prefabricated building supply chain. Therefore, research on the risk of prefabricated building supply chains has gradually attracted people's attention, and many scholars have explored and studied the risks of prefabricated building supply chains from different perspectives. Considering the risks in all stages of the whole life cycle of prefabricated building projects, in Zhang and Qiao [5] an index system was constructed, a risk evaluation model is formed, and relevant suggestions are put forward for the risk management of the prefabricated building supply chain. Chan et al. used a literature review, structured discussion, and multi-attribute group decision models to explore the benefits and challenges of modular integrated architecture [6]. Wang et al.'s factor analysis was used to identify the key points of risk control in the supply chain of prefabricated buildings, and the relationship and weight of risk factors in each stage are quantitatively analysed. It is concluded that the risk mainly exists in the manufacturing stage, and the risk in the delivery stage is the lowest [7]. Zhang refers to the information dissemination model, identifies the relationship among 19 risk dissemination factors, and introduces blockchain technology to construct the information flow model of the assembled supply chain [8]; Gu identified and analysed the risk factors for the supply chain based on the Supply Chain Operations Reference model (abbr. SCOR) theory, reduced the dimension of risk factors by principal component analysis, and simulated and analysed it by system dynamics [9]; Zhang evaluated the risk of a prefabricated concrete supply chain by using a cloud model under an engineering procurement construction (abbr. EPC) general contracting project [10]; An et al. established a Structural Equation Model (abbr. SEM) by multivariate data analysis and obtained the order of influencing factors for the research objective of supply chain integration [11]; Wang et al. introduced the idea of interface management, analysed the risk mechanism of supply chain integration interface, and used a Combination Ordered Weighted Averaging (abbr. C-OWA) operator weighting and grey clustering evaluation method to build a risk evaluation model [12]; Al-Hussein M. H. Al-Aidrous analysed the relationship between the influencing factors of ground-floor housing in prefabricated buildings, and proposed relevant measures to promote the development of prefabricated buildings in Malaysia [13]; Kristopher Orlowski analysed the manufacturing principles of special weatherproof seals for prefabricated buildings and evaluated their influencing factors [14]; Luo et al. analysed the supply chain risk network of stakeholders by using social network, to explore the supply chain network risks of prefabricated houses in Hong Kong and help employees to deal with these risks more effectively [15]; Ibrahim Yahaya Wuni et al. identified the risk factors of modular integrated buildings by using the fuzzy comprehensive evaluation method [16]; Syed Saad examined the key factors for success

in Malaysia's construction industry from five areas: stakeholder understanding, resource availability, process management, issues and perceptions, and future needs [17]. Most of these studies adopted quantitative research methods and explored the risk generation mechanism in different ways, defined the risk propagation angle, and constructed a risk propagation model.

The infectious disease model is a typical transmission dynamics model in mathematical engineering, and the research on the infectious disease dynamics model has great practical significance, mainly including the following three aspects: establishing a mathematical model to simulate the process of infectious disease transmission, analysing the spread trend of the infectious disease model, and studying targeted prevention and control strategies. Typical infectious disease models include the Susceptible–Infected (abbr. SI) model, the Susceptible–Infected–Susceptible (abbr. SIS) model, the Susceptible–Infected–Recovered (abbr. SIR) model [18], and the Susceptible–Exposed–Infected–Recovered (abbr. SEIR) model [19]. Since then, many scholars have improved on the basis of the above classical models based on different considerations: Yiping Tan built a stochastic Susceptible–Infected– Susceptible (abbr. SIS) dynamics model from the perspective of media coverage and used dynamics for research [20]; Ayse Peker Dobie studied a Susceptible–Infectious–Susceptible (SIS) model with virus mutation in a variable population size, calculating equilibrium points [21]; Wonhyung Choi studied a spatial Susceptible—nfected–Susceptible epidemic model with a free boundary [22]; Jianhua Chen considered that the possibility of infected people becoming removed after treatment is very small, and improved the Susceptible– Infected–Recovered (abbr. SIR) model to analyse the emergency supply chain transmission mechanism [23]; Di Liang studied supply chain risk propagation based on the Susceptible– Infected–Recovered (abbr. SIR) epidemic model [24]. Based on the Susceptible–Exposed– Infected–Recovered (abbr. SEIR) model, Guanhua Ni proposed a Susceptible–Contacts– Exposed–Infected–Recovered (abbr. SCEIR) model that incorporates close contacts (C) and self-protectors (P) into a Parameters Sensitivity Analysis of COVID-19 [25]; Isa Abdullahi Baba studied the transmission dynamics of the disease by incorporating the saturated incidence rate into the model, and the Caputo sense was constructed for studying the risk balance point [26]; Mauro Aliano proposed a time delay differential system describing risk diffusion among companies inside an economic sector by means of a Susceptible–Infected– Recovered (abbr. SIR) dynamics [27]; based on the Bo Li epidemiological model, the stability of discrete time and local bifurcation were considered, and the dynamic behaviour of the infectious disease model was analysed [28]. Tchavdar T. Marinov proposed an adaptive susceptible infection-removal (abbr. A-SIR) epidemic model with time-dependent transmission and clearance rates and applied it to address COVID-19 in Latin America [29].

At present, research on prefabricated building supply chains mainly focuses on risk evaluation and risk generation mechanisms. Most of these studies regard risk factors as independent individuals and identify and evaluate related risk factors by using factor analysis and grey cluster evaluation methods, but there are few studies on the spread of risk factors in the supply chain. At the same time, the wide spread of risks in the supply chain can easily break the safe operation of the supply chain, making it unable to achieve the expected goal of supply chain management, resulting in a decrease in supply chain efficiency and an increase in the cost of each subject in the supply chain. Therefore, it is crucial to study risk transmission from the perspective of the prefabricated building supply chain. Due to the similarities between virus transmission and risk transmission in terms of spread object, spread process, and spread environment, scholars have applied basic virus transmission models such as SI, SIS, SIR, SEIR, and so on to the field of risk transmission. However, the traditional virus model fails to take into account the symptoms of enterprises in the supply chain after being eroded by risk, and asymptomatic infected enterprises have no obvious characteristics after risk erosion and are easy to ignore, thereby underestimating the harm brought by risk and making it difficult to comprehensively solve the corresponding problems caused by risk transmission. At the same time, the performance of diseases caused by virus transmission after enterprises in the supply chain

are eroded by risk is similar, and some disease problems cannot be completely solved at one time and recur after treatment; enterprises in the supply chain may not be able to completely solve the risk problem once after being eroded by risk, and may reappear after a period of time, forming a secondary diffusion. Focusing on the problem of risk propagation in the prefabricated building supply chain, this paper uses the Vensim PLE software of system dynamics to identify the causes and consequences of the prefabricated building supply chain risk, determine the key risk factors, innovatively introduce the SEAIR model in the complex network infectious disease transmission dynamics model to establish the risk transmission model, and study the risk transmission process. Table 1 shows the relationship between the existing literature on risk in the prefabricated building supply chain and the virus model improvement perspective and this study.

**Table 1.** Shows a summary of the existing research on risk in the prefabricated building supply chain and virus model improvement perspectives.



#### **Table 1.** *Cont.*

The contributions of this article are as follows:


The remainder of this paper is organized as follows: Section 2 identifies risk factors associated with the prefabricated building supply chain. Section 3 establishes a SEAIR model that considers relapse. In Section 4, based on the recurrent SEAIR model, a supply chain risk propagation model was established and simulated for research. Finally, in Section 5, the relevant conclusions provide some directions for further research.
