Research Article  Open Access
A Periodic Collaboration and Coexistence Management Model with the Oscillation Effect for Complex Mega Infrastructure Project under the Risk of Infection
Abstract
In view of the intergroup structural relationships involved with the complex mega infrastructure projects (MIP), we develop a periodic collaboration and coexistence model that can efficiently curb the risk of infection and ensure normal, orderly progress. We conduct a systematic analysis of the periodic collaboration and the coexistence process for mega infrastructure projects. The results suggest that when a complex major engineering project develops the risk of infection and one group keeps expanding, assimilation or disappearance of the other group may occur, impeding the normal project construction process. If a disruption occurs, it is difficult to resume construction, and substantial economic losses can result. Furthermore, the periodic collaboration and coexistence management model with the oscillation effect can effectively prevent the risk of problems from spreading among groups. By doing so, the model stabilizes the intergroup structural balance in a complex major engineering system. In summary, our model and analysis help, to some extent, to elucidate the critical mechanisms influencing the risk of infection in engineering collaborative management and explain the collaboration and coexistence rules for a complex mega infrastructure project that may risk spreading infection to other parts of the project. The research findings, while further verifying and supplementing relevant theories, also provide beneficial references for collaborative management practices.
1. Introduction
Major infrastructure construction engineering is a general term for governmentoriented largescale complex engineering projects with a vital influence on economic and social development [1]. Driven by growing market demand, deepening global cooperation, government support, and huge capital input, complex engineering systems have not only profoundly changed production and lifestyle but also brought new challenges and opportunities to the management of mega infrastructure projects [2]. Most scholars focus on how to reduce construction engineering costs, accelerate the progress of construction engineering projects, and guarantee smooth construction engineering [3, 4]. In addition, attention has gradually been paid to the risk of infection via the “butterfly effect,” and efforts are being made to find an innovative theoretical method to assist in managing complex engineering projects (the risk of infection refers to the risk incurred by personality, emotion, and awareness factors, such as counterproductive work behavior, antisocial behavior, altruistic behavior, and conflicts among groups or individuals, which can influence the overall engineering interests; the risk of infection can induce other risks in a complex major engineering system or even trigger risks with more expansive and continuous influences).
1.1. Complexity of Engineering Management
Research into the complexity of engineering management originates from the complexity of engineering itself. Along with greater international cooperation, the increases in engineering quantity and complexity present huge challenges to decisionmakers to improve the effectiveness of their decisions [5]. In response to this dilemma, theoreticians have started studying the complexity of mega infrastructure projects and have proposed new and systematic scientific methods to address the related issues [6]. Academic and engineering circles have gradually realized that the previous engineering management theories cannot effectively address the challenges brought by the complexity of mega infrastructure projects. A pioneering scholar, Ottino [7] found that engineering management personnel should be able to comprehend and tackle this complexity but that the ability to do so relies on the examination of many complex phenomena not within the sphere of engineering projects. Sauser and Boardman [8] also noted that knowledge management, an aspect of engineering management, has failed to catch up with the increasing complexity and integration of projects.
Humans drive the operation of engineering management systems. Therefore, research into the operation rules of these systems cannot be separated from the analysis of individuals and groups participating in engineering management [9]. To this end, Du and ElGafy [10] examined the influence of the interplay between organizational and human factors using agentbased modelling (ABM). As one of the critical factors influencing engineering management, agent behavior arises from the interplay among certain psychological processes, psychological status, and personality psychology and is also subject to external environmental factors, such as culture and society [11]. Thus far, the behaviors of engineering management agents have been mainly investigated at the group and individual levels. The major influencing factors can be grouped into three categories, namely, psychological, behavioral, and cultural (see Table 1).

1.2. Risk of Infection in Mega Infrastructure Projects
Research into the risk of infection dates back to the 19th century, when Thornton [17] described the concept of the “risk spillover effect.” Since then, many scholars have applied the idea to different research fields, including finance, insurance, information networks, and biomedicine. These works have indirectly enriched the research into the risk of infection, making it an emerging interdisciplinary research field. Unfortunately, there has not yet been any specific research on infection risk management for mega infrastructure projects.
Almeida et al. [18] noted that managing risk in mega infrastructure projects has become an increasing concern to stakeholders. “Man,” “technology,” and “external environment” constitute three critical factors influencing major engineering risks. To start with, the agent behaviors of “man” are influenced by personality psychology, psychological processes, and psychological status. Human factors influencing engineering risks include personality, emotion, motive, and consciousness [19]. Second, technological factors also influence the risk factors. Katalin et al. [20] theorized that engineering risk management and decisionmaking rely, to a large extent, on the strategy adopted to reduce risks. Cheng and Ding [21] described how to coordinate complexity through the establishment of clear boundaries. Finally, among external environmental factors influencing the risk of infection in engineering projects, the realization of external collaboration rests on the interaction between relevant agents, and intergroup collaboration is considered an effective approach to improving efficiency [22]. Horst [23] argued that homogeneity can contribute to the stability of the correlation system, while differentiation is attributable to the risk of infection. The infection effect first affects a minority and then spreads to the whole system through the interconnected relationship chain.
1.3. Collaborative Management of Complex Engineering Projects
The engineering system for a major project is a complex system with multiple departments involved in contrast to a simple physical facility construction [24]. Successful collaborative management is mainly determined by the interactions between the organizations and other relevant agents [25]. Engineering collaborative management can be defined as a crossorganizational collaboration process integrating multiple resource elements. In engineering collaborative management, efficient cooperative relationships among different groups and methods that can promote engineering efficiency and innovation are of equal importance. Some scholars have developed methods to improve organizational performance through structured system analysis and interactive management [26]. Therefore, an essential approach to improving complex engineering management is to improve the efficiency of the intergroup collaborative management.
Understanding the intergroup interactive complexity during the engineering collaborative management process, especially research on collaborative management mechanisms, such as intergroup interface and boundaries, is important [27]. In recent years, a few scholars have made preliminary investigations. For example, Antoniadis et al. [27] deemed that the key to successful implementation of an organizational engineering project lies in establishing a favorable communication and coordination mechanism across functional departments and creating a cohesive partnership. Senescu et al. [28] deepened the understanding of the correlation between project complexity and communication through indepth analysis, and his work laid the foundation for the development of communication tools for more effective project management. Pauget and Wald [29] observed, after studying the relationship among members of research projects, that cooperative relationships between different groups or different individuals can influence the organizational network stability of mega infrastructure projects.
In summary, with the risk of infection, collaborative management of complex mega infrastructure projects is a top priority for engineering management research. However, the existing research mainly focuses on expounding on the importance and correlation of engineering management research. Based on previous engineering collaborative management research findings [30], and on life cycle theory and oscillation and dynamic behavior, this study is a pioneering work that, for the first time, defines the risk of infection in engineering projects, carries out indepth analytical research into group collaboration within a complex engineering system under the influence of the risk of infection, and provides valuable insights.
We make the following major research contributions in this study: We (1) observe the periodic characteristics and group member characteristics of complex engineering projects through experiments and define relevant concepts concerning the risk of infection in complex engineering projects, (2) establish a periodic collaboration and coexistence model with the vibration effect and conduct empirical analysis of the model based on the theory of oscillation and using mathematical derivation, (3) reveal that when a complex major engineering project develops the risk of infection and one group keeps expanding, it might result in assimilation or disappearance of the other group or impede the major engineering project from normal construction, after which it is difficult to resume production, causing immeasurable economic losses, and (4) ascertain the conditions for the outbreak of infection and observe that the periodic collaboration and coexistence management model with the oscillation effect can effectively avoid the spread of infection between groups and guarantee a stable structure between groups in a complex major engineering system.
2. Hypotheses
Based on the characteristics of complex engineering systems, combining the theories of dynamics, oscillation, and life cycle, and in an environment of impending risk of infection, we observe that the engineering personnel density in the initial period of a complex major engineering project is relatively low, and there are structurally balanced resources for allocation. The numbers of personnel in different types of groups grow steadily along with the progress of the project construction period, but all these groups are still developing in independent environments [31].
As the initial period gradually moves into the maturity period, the engineering personnel density rises and there are fewer structurally balanced resources for allocation. In this period, different groups within the complex major engineering system might compete for their interests and attempt to increase their influence to occupy more resources. Consequently, the intergroup personnel relative density changes. Onsite data of mega infrastructure projects suggests that “natural, small teams” exist in most major engineering construction groups. The “teammates” are either from the same area, have worked together for years, or are relatives. If adverse language and behavior occur in a teammate or several teammates, the connection between the personnel might cause conflict. In addition, if a group carrying or possessing the risk of infection is in a dominant position, then the infection influencing forces will grow and eventually assimilate or eliminate personnel in the other groups, thus decreasing the latter’s personnel density (see Figure 1).
When the risk of infection is spread to a subsidiary engineering project, a construction period, or a department, the risk cannot be immediately classified when effective segregation measures are adopted. Under this condition, highpressure strategies are often adopted, which can damage the interests of irrelevant groups or force them to suspend work or even leave. In addition, highpressure strategies can destroy the structural balance between the groups within an organization. Due to structural defects or the imbalance of major engineering personnel, the engineering project might come to a halt partially or completely, different engineering links will have difficulty in integrating with one another, and production cannot be resumed, resulting in serious economic losses and public security threats.
Therefore, regarding the periodic characteristics of major engineering systems, as well as the occurrence of traditional risks, we adopt an intergroup collaboration and coexistence mechanism through the risk infection medium (major engineering personnel group). Assuming and building a more practical periodic collaboration and coexistence model with the oscillation effect, we attempt to demonstrate and expound upon the role of the model in preventing the risk of infection from spreading among the groups involved in complex mega infrastructure projects.
3. Modelling and Demonstration
The structural interaction of different groups in a complex engineering system is usually subject to environmental influences, both internally and externally. When the risk of infection occurs, or is about to occur, some personnel in the group associated with the risk are, in the traditional approach, separated from the rest or directly expelled. However, based on the above discussion and existing theories, we adopt the periodic collaboration and coexistence management model to not only avoid triggering the risk of infection but also free the nonaffected groups from the influence of the risk, ensuring the smooth progress of the engineering project. To this end, we build the basic model, denoted as Model (1), as follows: where denotes the influence of oscillatory collaborative management on the group and denotes the influence of oscillatory collaborative management on the group . Model (1) assumes that the collaborative management behavior is continuous. However, in reality, the occurrence of the risk of infection is periodic in major engineering management practices. This condition results in discreteness of infection risk prevention and control. In addition, considering the influence of the enrollment rates and turnover rates of different groups within the life cycle of mega infrastructure projects, as well as periodic changes, the traditional continuous, largescale management methods to curb the risk of infection might be effectively replaced by collaborative management featuring discrete oscillation.
Based on the above argument, we develop the period collaboration and coexistence model with the oscillation effect as follows: where and constitute a continuous period function and meet the condition , , and . Assume that and are constants, and there exists an integer such that , , and . In view of real practice, there are natural restrictions that and if . denotes the period when oscillation collaboration is implemented. In other words, the moment of collaboration occurs at the discrete point ; in an oscillating style, the time can be written as .
Model (2) attempts to prevent and control the risk of infection and considers the influence of collaborative management adopted by internal management personnel on the number of group members. In a complex major engineering system, there are different types of groups. In this study, we use Model (2) only to describe collaborative management between two group structures. and , where and , represent the oscillation collaboration degree between the engineering personnel group and the engineering personnel group , respectively.
From the life cycle perspective, the sustainable use of one resource (such as human resources) means the resource is never exhausted. In particular, the continuity of project construction usually indicates the resistance of a complex major engineering system to the risk of infection or an ability to quickly resume construction.
3.1. Continuous Existence of the System
Model (2) at must have a semitrivial periodic solution within the value , which can be simplified as (, 0). Similarly, Model (2) at must have a semitrivial periodic solution within the value , which can be simplified as . Furthermore, the necessary and sufficient conditions can be expressed as and , respectively. Through deduction, we obtain the following equations:
In the following, we examine the situation when Model (2) does not allow coexistence whereby one side is fully assimilated or disappears. In other words, we seek to determine the conditions under which asymptotic steadiness of the trivial periodic solution and the semitrivial periodic solutions and are achieved. We state and prove the following hypothesis.
Hypothesis 1. (i)The trivial solution is linear and unsteady when Model (2) has a semitrivial periodic solution, that is, if and only if (ii)If , then is linear and unsteady when and is linearly steady when .(iii)If , then is linear and unsteady when , and is linearly steady when .
Proof 1. The equation can be used to study the asymptotic steadiness of Model (2)’s periodic solution . After conversion by , we have
where meets the condition
and is the unit matrix.
is given by the oscillation condition of Model (2).
Hence, if the module of the two eigenvalues of the following monodromy matrix
is smaller than one, then the periodic solution is locally steady.
(i)The asymptotic steadiness of the trivial solution .The fundamental solution matrix of the trivial solution is
so its monodromy matrix is
This proves that the trivial solution has two multipliers at the point : and . Therefore, the conclusion stated in (i) is correct.
(ii)The linear steadiness of the semitrivial periodic solution.Corresponding to every semitrivial solution is
Therefore, the fundamental solution matrix of the semitrivial solution is as follows:
Since the following computing process does not rely on (), there is no need to provide its exact formula. The steadiness of the semitrivial solution is confirmed by the two eigenvalues of the following monodromy matrix
where
and
If , then and the linear steadiness of the semitrivial solution is confirmed by . If , then and the unsteady manifold of the semitrivial solution is onedimensional. If , then features neutral steadiness. If , then and is linearly steady.
(iii)The linear steadiness of the semitrivial periodic solution.Similarly, the linear steadiness of the semitrivial periodic solution is confirmed by .
The steadiness of the semitrivial periodic solutions and suggests that when the risk of infection occurs, one personnel group in a major engineering system might be fully assimilated or disappear and the other personnel group might expand. Therefore, to guarantee equilibrium steadiness of the groups in a complex major engineering system, a necessary condition for coexistence of the two kinds of personnel is that the following two equations exist. If one group is fully infected and thus becomes fully assimilated or disappears, the engineering construction might be unable to carry on normally and it is difficult to resume construction, resulting in tremendous economic losses. In other words, all the trivial periodic solutions and semitrivial periodic solutions are linearly unsteady, that is,
In the following section, we examine how to achieve sustainable coexistence and avoid the occurrence of the risk of infection when the personnel groups are structurally stable in the major engineering system. We write Model (2) under the condition of coexistence as follows:
Assume that there exist a constant () and a finite time , which are independent of the system’s initiative values, and make all the initiative values of Model (2) as the solution of . When , then and Model (2) is coexisting. Here, depends on the initial value .
Hypothesis 2. If the following inequalities are valid, then Model (2) suggests that the two kinds of engineering personnel groups coexist and collaborate with each other in a complex major engineering system. Based on Model (2), when is sufficiently large, we have where . Similarly, when is sufficiently large, we have . Then, there exists a constant such that when , then and . Thus, when , we have and, when is sufficiently large, we have . Similarly, when is sufficiently large, we have . Hence, there exists a constant such that when , then and . Assume that Then, when , we have and .
3.2. Bifurcation of the Solution
In this study, we use bifurcation theory to discuss the bifurcation of Model (2) near the semitrivial periodic solutions and . To validate the analytical findings of this study, first examine the two results of the linear oscillation periodic equation.
Hypothesis 3. Assume that . (i)If and , then the periodic linear homogeneous oscillation equation can be written as follows:On , there is no nontrivial periodic solution. Under this condition, if , we write the nonhomogeneous oscillation equation as follows: The periodic solution is exclusive and . Furthermore, the operator , defined by , is linear and relatively compact. (ii)If and are valid, then (22) has only one periodic solution on .
Proof 1. (i)Since suggests unless . Similarly, since suggests unless . Under the condition there is only one periodic solution, and is defined by and is linear and relatively compact. Furthermore, when , the equation has only one periodic solution, and defines a linear and relatively compact operator . Therefore, (22) has only one periodic solution determined by and is the only solution on , where (ii)Under the hypothesis, the solution adopting a random as the initial value is a periodic solution and is given by (23). If , then does not have a nontrivial periodic solution on . Then, we have On , the periodic solution is exclusive. The hypothesis is true.
Hypothesis 4. Assume that and , then The necessary and sufficient condition for the periodic solution is as follows:
Proof 1. The matrix of the linear homogeneous oscillation equation is as follows:
The matrix is , . Then, (31) is the exclusive multiplier
Hence, all the solutions of (31) are periodic solutions and all the solutions of the adjoining equation
are also periodic. In particular, is also a periodic solution of (27), where . The necessary and sufficient condition for (29) to have only one periodic solution is , that is,
Assume and regard as the bifurcation referential value. This condition suggests that is the bifurcation value of the positive periodic solution bifurcated from the semitrivial periodic solution . Assume that , where . Then,
Assume to be the only periodic solution of the following linear oscillation differential equation:
Then
where
and .
Note that exists if . Since
Then, for any , .
Define . The main conclusions are stated in the following hypothesis.
Hypothesis 5. Assume . is valid in terms of and . For any , there is . Then, (i)If , then there exists a constant , which is sufficiently small to ensure that, in terms of every existing under the restriction of , Model (2) has a periodic solution , which can guarantee and as long as .(ii)If , there exists a constant , which is sufficiently small to ensure that, in terms of every existing under the restriction of , Model (2) has a periodic solution , which can guarantee and as long as .
Proof 1. In Model (2), after conversion, we have . Then, where and .
Define , where . Then, and . Then, we write (40) as follows:
Its linear homogeneous function can be written as follows:
Noting that , we have and
We apply (41) to obtain the relatively compact linear operator . Based on , we write (41) as the following equivalent operator function where and
Here, is linear and relatively compact and is quasiequicontinuous and relatively compact and meets the condition when .
Regard in (45) as a real value parameter. As long as , any nontrivial periodic solution of (45) on can generate a periodic solution in the collaboration and coexistence model, where . Hence, the issue of whether the periodic solution of Model (2) exists reduces to the issue of bifurcation of (45). Applying local bifurcation theory to (45), we see that bifurcation occurs only when the linear equation has a nontrivial solution.
Assume that if , is a periodic solution of (48). By the definition of , is the solution to the following equation and vice versa. Based on Hypothesis 3(i), if , where (49) indicates that (45) has the only nontrivial periodic solution on . In view of Hypothesis 3(ii), (41) has the only periodic solution on and bifurcation occurs only under the simple eigenvalue . Then, bifurcation of the sectionally continuous periodic solution of (45) can be obtained. Apart from , all the other solutions are nontrivial.
The expanded form of in the neighbourhood of suggests that the periodic solution of (45) is equivalent to the positive periodic solution of Model (2). In fact, when is sufficiently small and , then
Upon substituting the above equation into (41), and if the coefficients and on both ends are the same, then
Hence, is the solution of (48). A special solution meeting the condition of is
Hence, in terms of , is valid. Applying Hypothesis 2 to the second equation in (53), we have .
If , then there exists a bifurcation , where , and as long as , then and . When becomes sufficiently close to , it is equivalent to the positive periodic solution of Model (2). Therefore, there exists a that is sufficiently small to ensure that the bifurcation can be solved and meets the condition, such that when , if , then and . Since is equivalent to , Hypothesis 5(i) is valid. If , it is valid similarly that there exists a constant that is sufficiently small to ensure that the solution of Model (2) meets the condition such that when , if