# A Periodic Collaboration and Coexistence Management Model with the Oscillation Effect for Complex Mega Infrastructure Project under the Risk of Infection

Authors: Na Zhao; T. C. E. Cheng
Journal: Complexity
 (2018)

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## 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.


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## Body

## 1. Introduction

Major infrastructure construction engineering is a general term for government-oriented large-scale 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 decision-makers 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 El-Gafy [10] examined the influence of the interplay between organizational and human factors using agent-based 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).Table 1
Factors influencing individual and group behaviors in mega infrastructure projects.
Individual [12–14]
Group [15, 16]
Psychology
Personality, emotion, motive, and consciousness
Trust, cohesion, and commitment
Behavior
Authorization, leadership, organizational citizen behavior (OCB), counterproductive work behavior, altruistic behavior, and reciprocal behavior
Transactive memory system, shared mental model, conflict, decision-making behavior, cooperation behavior, and learning behavior
Culture
Job satisfaction, motive, and leadership
Learning, trust, learning, and conflict


### 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 decision-making 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 cross-organizational 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 in-depth 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 in-depth 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.


## 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 decision-makers 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 El-Gafy [10] examined the influence of the interplay between organizational and human factors using agent-based 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).Table 1
Factors influencing individual and group behaviors in mega infrastructure projects.
Individual [12–14]
Group [15, 16]
Psychology
Personality, emotion, motive, and consciousness
Trust, cohesion, and commitment
Behavior
Authorization, leadership, organizational citizen behavior (OCB), counterproductive work behavior, altruistic behavior, and reciprocal behavior
Transactive memory system, shared mental model, conflict, decision-making behavior, cooperation behavior, and learning behavior
Culture
Job satisfaction, motive, and leadership
Learning, trust, learning, and conflict


## 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 decision-making 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 cross-organizational 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 in-depth 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 in-depth 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. On-site 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 Figure1).Figure 1
Changes in group personnel densities and resources in different periods of the life cycle.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, high-pressure strategies are often adopted, which can damage the interests of irrelevant groups or force them to suspend work or even leave. In addition, high-pressure 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:
(1)β̇t=βty1t−x11tβt−x12tλt−D1βt,λ̇t=λty2t−x21tβt−x22tλt−D2λt,where D1βt denotes the influence of oscillatory collaborative management on the group βt and D2λt denotes the influence of oscillatory collaborative management on the group λt. 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, large-scale 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 theT-period collaboration and coexistence model with the oscillation effect as follows:
(2)β̇t=βty1t−x11tβt−x12tλt,t≠τk,k∈N,λ̇t=λty2t−x21tβt−x22tλt,t≠τk,k∈N,βτk+=1+hkβτk,t≠τk,k∈N,λτk+=1+gkλτk,t≠τk,k∈N,where yit and xijti,j=1,2 constitute a continuous T-period function and meet the condition xijt≥0i,j=1,2,i≠j, xii>0, and N=1,2,…. Assume that hk and gkk∈N are constants, and there exists an integer q>0 such that hk+q=hk, gk+q=gk, and τk+q=τk+T. In view of real practice, there are natural restrictions that 1+hk>0 and 1+gk>0 if k∈N. τ denotes the period when oscillation collaboration is implemented. In other words, the moment of collaboration occurs at the discrete point t=nτn=0,1,2,…; in an oscillating style, the n time can be written as t=nτ+.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. hk and gk, where hk<0 and gk<0, represent the oscillation collaboration degree between the engineering personnel group β and the engineering personnel group λ, 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 β,0 must have a semitrivial periodic solution within the value y¯1>1/Tln∏k=1q1/1+hk, which can be simplified as (θy1,x11, 0). Similarly, Model (2) at 0,λ must have a semitrivial periodic solution within the value y¯2>1/Tln∏k=1q1/1+gk, which can be simplified as 0,θy2,x22. Furthermore, the necessary and sufficient conditions can be expressed as θy1≔θy1,x11 and θy2≔θy2,x22, respectively. Through deduction, we obtain the following equations:
(3)y¯1=1Tln∏k=1q11+hk+x11θy1¯,y¯2=1Tln∏k=1q11+gk+x22θy2¯.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 0,0 and the semitrivial periodic solutions θy1,0 and 0,θy2 are achieved. We state and prove the following hypothesis.Hypothesis 1.
(i)
The trivial solution0,0 is linear and unsteady when Model (2) has a semitrivial periodic solution, that is, if and only if
(4)y¯1>1Tln∏k=1q11+hkory¯2>1Tln∏k=1q11+gk.
(ii)
Ify¯1>1/Tln∏k=1q1/1+hk, then θy1,0 is linear and unsteady when y¯2>1/Tln∏k=1q1/1+gk+x21θy1¯ and is linearly steady when y¯2<1/Tln∏k=1q1/1+gk+x21θy1¯.
(iii)
Ify¯2>1/Tln∏k=1q1/1+gk, then 0,θy2 is linear and unsteady when y¯1>1/Tln∏k=1q1/1+hk+x12θy2¯, and is linearly steady when y¯1<1/Tln∏k=1q1/1+hk+x12θy2¯.Proof 1.
The equation can be used to study the asymptotic steadiness of Model (2)’s periodic solution β¯t,λ¯t. After conversion by βt,λt=β¯t+at,λ¯t+bt, we have
(5)atbt=Φta0b0,where Φt meets the condition
(6)dΦtdt=y1−2x11β¯−x12λ¯−x12β¯−x21λ¯y2−x21β¯−2x22λ¯Φt,and Φ0=I is the unit matrix.
(7)aτk+bτk+=1+hk001+gkaτkbτk,is given by the oscillation condition of Model (2).
Hence, if the module of the two eigenvalues of the following monodromy matrix(8)M=∏k=10q1+hk00∏k=1q1+gkΦT,is smaller than one, then the T-periodic solution is locally steady.
(i)
The asymptotic steadiness of the trivial solution0,0.
The fundamental solution matrix of the trivial solution0,0 is
(9)dΦtdt=y100y2Φt,so its monodromy matrix is
(10)M=∏k=1q1+hkey¯1T00∏k=1q1+tkey¯2T.
This proves that the trivial solution has two multipliers at the point0,0: β1=∏k=1q1+hkey¯1T and β2=∏k=1q1+gkey¯2T. Therefore, the conclusion stated in (i) is correct.
(ii)
The linear steadiness of the semitrivial periodic solution.
Corresponding to every semitrivial solutionθy1,0 is
(11)dΦtdt=y1−2x11θy1−2x12θy10y2−x21θy1Φt.Therefore, the fundamental solution matrix of the semitrivial solution is as follows:
(12)Φt=exp∫0ty1−2x11θy1ds∗0exp∫0ty2−2x21θy1ds.
Since the following computing process does not rely on (∗), there is no need to provide its exact formula. The steadiness of the semitrivial solution θy1,0 is confirmed by the two eigenvalues of the following monodromy matrix
(13)M=∏k=1q1+hk00∏k=1q1+gkΦT.where
(14)β1=∏k=1q1+hkexp∫0Ty1−2x11θy1ds=∏k=1q1+hkexp∫0Ty1−x11θy1ds−∫0Tx11θy1ds=∏k=1q1+hkexpln∏k=1q11+hkexp−∫0Tx11θy1ds=exp−x11θy1¯T=exp−Ty¯1−1Tln∏k=1q11+hk,and
(15)β2=∏k=1q1+gkexp∫0Ty2−x21θy1ds=∏k=1q1+gkexpy¯2−x21θy1¯T.
Ify¯1>1/Tln∏k=1q1/1+hk, then β1<1 and the linear steadiness of the semitrivial solution θy1,0 is confirmed by y¯2−1/Tln∏k=1q1/1+gk−x21θy1¯. If y¯2>1/Tln∏k=1q1/1+gk+x21θy1¯, then β2>1 and the unsteady manifold of the semitrivial solution θy1,0 is one-dimensional. If y¯2=1/Tln∏k=1q1/1+gk+x21θy1¯, then θy1,0 features neutral steadiness. If y¯2<1/Tln∏k=1q1/1+gk+x21θy1¯, then β2<1 and θy1,0 is linearly steady.
(iii)
The linear steadiness of the semitrivial periodic solution.
Similarly, the linear steadiness of the semitrivial periodic solution0,θy2 is confirmed by y¯1−1/Tln∏k=1q1/1+hk+x12θy2¯.
The steadiness of the semitrivial periodic solutionsθy1,0 and 0,θy2 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,
(16)y¯1>1Tln∏k=1q11+hk+x12θy2¯,y¯2>1Tln∏k=1q11+gk+x21θy1¯.
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 constantm (M>0) and a finite time T0, which are independent of the system’s initiative values, and make all the initiative values of Model (2) as the solution ait of ai0+>0. When t>T0, then m<ait<Mi=1,2 and Model (2) is coexisting. Here, T0 depends on the initial value ai0+.Hypothesis 2.
If the following inequalities are valid,(17)y¯1>1Tln∏k=1q11+hk+x12θy2¯,y¯2>1Tln∏k=1q11+gk+x21θy1¯,
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 t is sufficiently large, we have
(18)β̇t≤βty1t−x11tβt,t≠τk,k∈N,βτk+=1+hkβτk,t=τk,k∈N,where βt≤θy1. Similarly, when t is sufficiently large, we have λt≤θy2. Then, there exists a constant T1>0 such that when t≥T1, then βt≤θy1 and λt≤θy2. Thus, when t>T1, we have
(19)β̇t≥βty1t−x12tθy2−x11tβt,t≠τk,k∈Z+,βτk+=1+hkβτk,t=τk,k∈Z+,and, when t is sufficiently large, we have βt≥θy1−x12θy2,x11>0. Similarly, when t is sufficiently large, we have λt≥θy2−x21θy1,x22>0. Hence, there exists a constant T¯>T1 such that when t≥T¯, then βt≥θy1−x12θy2,x11 and λt≥θy2−x21θy1,x22. Assume that
(20)ρ=infθy1−x12θy2,x11t,θy2−x21θy1,x22t:t∈0,T,0<ρ<∞,σ=supθy1t,θy2t∈0,T,ρ≤σ<∞.
Then, whent≥T¯, we have ρ≤βt≤σ and ρ≤λt≤σ.


### 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 θy1,0 and 0,θy2. To validate the analytical findings of this study, first examine the two results of the linear oscillation periodic equation.Hypothesis 3.
Assume thatcij∈PCT.
(i)
Ifc11¯≠1/Tln∏k=1q1/1+ck and c22¯≠1/Tln∏k=1q1/1+dk, then the periodic linear homogeneous oscillation equation can be written as follows:
(21)ḃ1t=c11b1t+c12b2t,t≠τk,k∈N,ḃ2t=c22b2t,t≠τk,k∈N,b1τk+=1+ckb1τk,t=τk,k∈N,b2τk+=1+dkb2τk,t=τk,k∈N.
OnPCT×PCT, there is no nontrivial periodic solution. Under this condition, if f1,f2∈PCT×PCT, we write the nonhomogeneous oscillation equation as follows:
(22)ż1t=c11z1t+c12z2t+f1,t≠τk,k∈N,ż2t=c22z2t+f2,t≠τk,k∈N,z1τk+=1+ckz1τk,t=τk,k∈N,z2τk+=1+dkz2τk,t=τk,k∈N.
TheT-periodic solution is exclusive and z1,z2∈PCT×PCT. Furthermore, the operator L:PCT×PCT→PCT×PCT, defined by z1,z2=Lf1,f2, is linear and relatively compact.
(ii)
Ifc22¯=1/Tln∏k=1q1/1+dk and c11¯≠1/Tln∏k=1q1/1+ck are valid, then (22) has only one T-periodic solution on PCT×PCT.Proof 1.
(i)
Since(23)b2t=∏0≤τk<t1+dkb20exp∫0Tc22sds,c22¯≠1/Tln∏k=1q1/1+dk suggests b2∉PCT unless b2=0. Similarly, since
(24)b1t=∏0≤τk<t1+ckb10exp∫0Tc11sds,c11¯≠1/Tln∏k=1q1/1+ck suggests b∉PCT unless b1=0. Under the condition
(25)ż2t=c22z2t+f2,t≠τk,k∈N,z2τk+=1+dkz2τk,t=τk,k∈N,there is only one T-periodic solution, and L2:PCT→PCT is defined by z2=L2f2 and is linear and relatively compact. Furthermore, when f3=c12L2f2+f1∈PCT, the equation
(26)ż1t=c11z1t+f3,t≠τk,k∈N,z1τk+=1+ckz2τk,t=τk,k∈N,has only one T-periodic solution, and z1=L1f3 defines a linear and relatively compact operator L1:PCT→PCT. Therefore, (22) has only one T-periodic solution determined by z1,z2=Lf1,f2 and is the only solution on PCT×PCT, where
(27)Lf1,f2=L1c12L2f2+f1,L2f2
(ii)
Under the hypothesis, the solution adopting a randomb20 as the initial value is a T-periodic solution and is given by (23). If c11¯≠1/T∏k=1q1/1+ck, then ḃt=c11bt does not have a nontrivial periodic solution on PCT. Then, we have
(28)ḃ1t=c11b1t+c12∏0≤τk<t1+dkb20exp∫0tc22sds,t≠τk,b1τk+=1+ckb1τk,t=τk,k∈N.
OnPCT, the T-periodic solution is exclusive. The hypothesis is true.Hypothesis 4.
Assume thatx∈PCT and x¯=1/T∏k=1q1/1+ck, then
(29)ȧ=xa+f,t≠τk,k∈N,aτk+=1+ckaτk,t=τk,k∈N.
The necessary and sufficient condition for theT-periodic solution x∈PCT is as follows:
(30)∫0T∏0≤τk<t11+ckexp−∫0txβdβfdt=0.Proof 1.
The matrix of the linear homogeneous oscillation equation is as follows:(31)ȧ=ax,t≠τk,k∈N,aτk+=1+ckaτk,t=τk,k∈N.
The matrix isAt,s=∏s≤τk<t1+ckexp∫stxβdβ, −∞<s≤t<∞. Then, (31) is the exclusive multiplier
(32)μ=AT,0+=∏k=11+ckexp∫0Txsds=1.
Hence, all the solutions of (31) are T-periodic solutions and all the solutions of the adjoining equation
(33)ḃ=−xb,t≠τk,k∈N,bτk+=11+ckbτk,t=τk,k∈N,are also T-periodic. In particular, b=ψt=∏0≤τk<t1/1+ckexp−∫0txβdβ is also a T-periodic solution of (27), where ψT+=ψ0+=1. The necessary and sufficient condition for (29) to have only one T-periodic solution is ∫0Tψtfdt=0, that is,
(34)∫0T∏0≤τk<t11+ckexp−∫0txβdβfdt=0.
Assumey¯1>1/Tln∏k=1q1/1+hk and regard y2 as the bifurcation referential value. This condition suggests that ξ∗=1/Tln∏k=1q1/1+gk+x21θy1¯ is the bifurcation value of the positive periodic solution bifurcated from the semitrivial periodic solution θy1,0. Assume that y2t=ξ+p2t, where p¯2=0. Then,
(35)a21t=∏0≤τk<t1+gkexp∫0tp2+ξ∗−x21θy1ds∈PCT.
Assumea11 to be the only T-periodic solution of the following linear oscillation differential equation:
(36)ȧ11t=y1−2x11θy1a11t−x12tθy1a21t,t≠τk,k∈N,a11τk+=1+hka11τk,t=τk,k∈N.
Then(37)a11t=−∫0tGt,sx12θy1a21sds,where
(38)Gt,s=At1−At−1A−1s,0<s<t≤T,At+T1−AT−1A−1s,0<t≤s≤T,Gt−kT,s−jT,kT<t≤kT+T,jT<s≤jT+T,k∈N,j∈N,and At=∏0≤τk<tt+hkexp∫0ty2−2x11θy1ds.
Note thata21t>0 exists if t>0. Since
(39)At=∏k=1q1+hkexp∫0Ty1−2x11θy1ds=∏k=1q1+hkexp∫0Ty1−x11θy1dsexp∫0T−x11θy1ds=exp−∫0Tx11θy1ds<1,Gt,s>0.
Then, for anyt, a11t<0.
Defineξ1=x21a11t+x22ta21t¯. The main conclusions are stated in the following hypothesis.Hypothesis 5.
Assumeyit. xij∈PCTi,j=1,2 is valid in terms of y¯1>1/Tln∏k=1q1/1+hk and x21θy1>1/Tln∏k=1q1+gk. For any t>0, there is xijt>0. Then,
(i)
Ifξ1>0, then there exists a constant c0>0, which is sufficiently small to ensure that, in terms of every y2ty2t∈PCT existing under the restriction of ξ∗<y¯2<ξ∗+c0, Model (2) has a periodic solution β,λ∈PCT×PCT, which can guarantee βt<θy1 and λt>0 as long as t>0.
(ii)
Ifξ1<0, there exists a constant d0>0, which is sufficiently small to ensure that, in terms of every y2ty2t∈PCT existing under the restriction of ξ∗−d0<y¯2<ξ∗, Model (2) has a periodic solution β,λ∈PCT×PCT, which can guarantee βt<θy1 and λt>0 as long as t>0.Proof 1.
In Model (2), after conversion, we have a1=βt−θy1a2=λt. Then,
(40)ȧ1t=y1−2x11θy1a1−x12θy1a2+g1a1,a2,t≠τk,k∈N,ȧ2t=y2−x21θy1a2+g2a1,a2,t≠τk,k∈N,a1τk+=1+hka1τk,t=τk,k∈N,a2τk+=1+gka2τk,t=τk,k∈N,where g1a1,a2=−x11a12−x12a1a2 and g2a1,a2=−x21a1a2−x22a22.Definep2t=y2t−ξ, where ξ=y¯2. Then, y2=p2+ξ and p¯2=0. Then, we write (40) as follows:
(41)ȧ1t=y1−2x11θy1a1−x12θy1a2+g1a1,a2,t≠τk,k∈N,ȧ2t=p2−x21θy1a2+ξa2+g2a1,a2,t≠τk,k∈N,a1τk+=1+hka1τk,t=τk,k∈N,a2τk+=1+gka2τk,t=τk,k∈N.Its linear homogeneous function can be written as follows:(42)δ̇1t=y1−2x11θy1δ1−x12θy1δ2,t≠τk,k∈N,δ̇2t=p2−x21θy1δ2,t≠τk,k∈N,δ1τk+=1+hkδ1τk,t=τk,k∈N,δ2τk+=1+gkδ2τk,t=τk,k∈N.Noting thatx21θy1¯>1/Tln∏k=1q1+gk, we have
(43)c22¯=p2−x21θy1¯=−x21θy1¯<1Tln∏k=1q11+gk,and
(44)c11¯=y1−2x11θy1¯=y1−x11θy1¯−x11θy1¯=1Tln∏k=1q11+hk−x11θy1¯≠1Tln∏k=1q11+hk.We apply (41) to obtain the relatively compact linear operator L:PCT×PCT→PCT×PCT. Based on L, we write (41) as the following equivalent operator function
(45)a1,a2=ξL∗a1,a2+Ga1,a2,where
(46)L∗a1,a2=L1−x12θy1L2a2,L2a2and
(47)Ga1,a2=L1−x12θy1L2g2a1,a2+g1a1,a2,L2g2a1,a2.Here,L∗:PCT×PCT→PCT×PCT is linear and relatively compact and G:PCT×PCT→PCT×PCT is quasi-equicontinuous and relatively compact and meets the condition G=∘a1,a2PCT when a1,a2PCT→o.Regardξ in (45) as a real value parameter. As long as ξ∈R, any nontrivial periodic solution a1a2≠0,0 of (45) on PCT×PCT can generate a periodic solution β,λ=a1+θy1,a2 in the collaboration and coexistence model, where y¯2=ξ. 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
(48)δ1,δ2=ξL∗δ1,δ2,δ1,δ2≠0,0,ξ∈Rhas a nontrivial solution.Assume that ifξ∈R, δ1,δ2∈PCT×PCT is a T-periodic solution of (48). By the definition of L∗, δ1,δ2 is the solution to the following equation
(49)δ̇1t=y1−2x11θy1δ1−x12θy1δ2,t≠τk,k∈N,δ̇2t=p2+ξ−x21θy1δ2,t≠τk,k∈N,δ1τk+=1+hkδ1τk,t=τk,k∈N,δ2τk+=1+gkδ2τk,t=τk,k∈N,and vice versa. Based on Hypothesis 3(i), if ξ=ξ∗, where
(50)ξ∗=1Tln∏k=1q11+gk+x21θy1¯,(49) indicates that (45) has the only nontrivial T-periodic solution on PCT×PCT. In view of Hypothesis 3(ii), (41) has the only T-periodic solution on PCT×PCT and bifurcation occurs only under the simple eigenvalue ξ∗. Then, bifurcation of the sectionally continuous periodic solution of (45) can be obtained. Apart from ξ∗;0,0, all the other solutions are nontrivial.The expanded form ofξ;a1,a2 in the neighbourhood of ξ∗;0,0 suggests that the periodic solution of (45) is equivalent to the positive periodic solution of Model (2). In fact, when ε is sufficiently small and aij∈PTT, then
(51)ξ=ξ∗+ξ1ε+⋯,ai=ai1ε+ai2ε2+⋯,i=1,2.Upon substituting the above equation into (41), and if the coefficients ε and ε2 on both ends are the same, then
(52)ȧ11=y1−2x11θy1a11−x12θy1a21,t≠τk,k∈N,ȧ21=p2+ξ∗−x21θy1a21,t≠τk,k∈N,a11τk+=1+hka11τk,t=τk,k∈N,a21τk+=1+gka22τk,t=τk,k∈N,(53)ȧ12=y1−2x11θy1a12−x12θy1a22−x11a112−x12a11a21,t≠τk,k∈N,ȧ22=p2+ξ∗−x21θy1a22+ξ1a21−x21a11a21−x22a212,t≠τk,k∈N,a12τk+=1+hka12τk,t=τk,k∈N,a22τk+=1+gka22τk,t=τk,k∈N.Hence,a11,a21∈PCT×PCT is the solution of (48). A special solution meeting the condition of a210=1 is
(54)a21t=∏0≤τk<t1+gkexp∫0tp2+ξ∗−x21θy1ds>0,a11t=−∫0tGt,sx12θy1a21sds.Hence, in terms oft, a11t<0 is valid. Applying Hypothesis 2 to the second equation in (53), we have ξ=x21a11+x22a21¯.Ifξ1>0, then there exists a bifurcation γ+=ξ;a1,a2⊆R×PCT×PCT, where ξ>ξ∗, and as long as t>0, then a1t<0 and a2t>0. When ξ becomes sufficiently close to ξ∗, it is equivalent to the positive periodic solution of Model (2). Therefore, there exists a c0>0 that is sufficiently small to ensure that the bifurcation C+=ξ;β,λ∈R×PCT×PCT can be solved and meets the condition, such that when ξ∗<ξ<ξ∗+c0, if t>0, then βt<θy1 and λt>0. Since ξ∗<ξ<ξ∗+c0 is equivalent to ξ∗<y¯2<ξ∗+c0, Hypothesis 5(i) is valid. If ξ1<0, it is valid similarly that there exists a constant d0>0 that is sufficiently small to ensure that the solution β,λ∈PCT×PCT of Model (2) meets the condition such that when ξ∗−d0<y¯2<ξ∗, if t>0, then βt<θy1 and λt>0. The proof is complete.From Hypothesis5, we note that ξ1 is a measure of the probability that the two personnel groups coexist. ξ∗ is a value of the positive periodic solution bifurcated from the semitrivial periodic solution θy1,0. If ξ1>0, then y¯2>ξ∗. From Hypothesis 1, we see that the unsteady manifold of the semitrivial positive periodic solution θy1,0 increases by one dimension. According to the steadiness exchange principle, when ξ1>0, the periodic solution y¯2;β,λ exhibits linear asymptotic steadiness. If ξ1<0, then the bifurcated periodic solution is unsteady.


## 3.1. Continuous Existence of the System

Model (2) at β,0 must have a semitrivial periodic solution within the value y¯1>1/Tln∏k=1q1/1+hk, which can be simplified as (θy1,x11, 0). Similarly, Model (2) at 0,λ must have a semitrivial periodic solution within the value y¯2>1/Tln∏k=1q1/1+gk, which can be simplified as 0,θy2,x22. Furthermore, the necessary and sufficient conditions can be expressed as θy1≔θy1,x11 and θy2≔θy2,x22, respectively. Through deduction, we obtain the following equations:
(3)y¯1=1Tln∏k=1q11+hk+x11θy1¯,y¯2=1Tln∏k=1q11+gk+x22θy2¯.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 0,0 and the semitrivial periodic solutions θy1,0 and 0,θy2 are achieved. We state and prove the following hypothesis.Hypothesis 1.
(i)
The trivial solution0,0 is linear and unsteady when Model (2) has a semitrivial periodic solution, that is, if and only if
(4)y¯1>1Tln∏k=1q11+hkory¯2>1Tln∏k=1q11+gk.
(ii)
Ify¯1>1/Tln∏k=1q1/1+hk, then θy1,0 is linear and unsteady when y¯2>1/Tln∏k=1q1/1+gk+x21θy1¯ and is linearly steady when y¯2<1/Tln∏k=1q1/1+gk+x21θy1¯.
(iii)
Ify¯2>1/Tln∏k=1q1/1+gk, then 0,θy2 is linear and unsteady when y¯1>1/Tln∏k=1q1/1+hk+x12θy2¯, and is linearly steady when y¯1<1/Tln∏k=1q1/1+hk+x12θy2¯.Proof 1.
The equation can be used to study the asymptotic steadiness of Model (2)’s periodic solution β¯t,λ¯t. After conversion by βt,λt=β¯t+at,λ¯t+bt, we have
(5)atbt=Φta0b0,where Φt meets the condition
(6)dΦtdt=y1−2x11β¯−x12λ¯−x12β¯−x21λ¯y2−x21β¯−2x22λ¯Φt,and Φ0=I is the unit matrix.
(7)aτk+bτk+=1+hk001+gkaτkbτk,is given by the oscillation condition of Model (2).
Hence, if the module of the two eigenvalues of the following monodromy matrix(8)M=∏k=10q1+hk00∏k=1q1+gkΦT,is smaller than one, then the T-periodic solution is locally steady.
(i)
The asymptotic steadiness of the trivial solution0,0.
The fundamental solution matrix of the trivial solution0,0 is
(9)dΦtdt=y100y2Φt,so its monodromy matrix is
(10)M=∏k=1q1+hkey¯1T00∏k=1q1+tkey¯2T.
This proves that the trivial solution has two multipliers at the point0,0: β1=∏k=1q1+hkey¯1T and β2=∏k=1q1+gkey¯2T. Therefore, the conclusion stated in (i) is correct.
(ii)
The linear steadiness of the semitrivial periodic solution.
Corresponding to every semitrivial solutionθy1,0 is
(11)dΦtdt=y1−2x11θy1−2x12θy10y2−x21θy1Φt.Therefore, the fundamental solution matrix of the semitrivial solution is as follows:
(12)Φt=exp∫0ty1−2x11θy1ds∗0exp∫0ty2−2x21θy1ds.
Since the following computing process does not rely on (∗), there is no need to provide its exact formula. The steadiness of the semitrivial solution θy1,0 is confirmed by the two eigenvalues of the following monodromy matrix
(13)M=∏k=1q1+hk00∏k=1q1+gkΦT.where
(14)β1=∏k=1q1+hkexp∫0Ty1−2x11θy1ds=∏k=1q1+hkexp∫0Ty1−x11θy1ds−∫0Tx11θy1ds=∏k=1q1+hkexpln∏k=1q11+hkexp−∫0Tx11θy1ds=exp−x11θy1¯T=exp−Ty¯1−1Tln∏k=1q11+hk,and
(15)β2=∏k=1q1+gkexp∫0Ty2−x21θy1ds=∏k=1q1+gkexpy¯2−x21θy1¯T.
Ify¯1>1/Tln∏k=1q1/1+hk, then β1<1 and the linear steadiness of the semitrivial solution θy1,0 is confirmed by y¯2−1/Tln∏k=1q1/1+gk−x21θy1¯. If y¯2>1/Tln∏k=1q1/1+gk+x21θy1¯, then β2>1 and the unsteady manifold of the semitrivial solution θy1,0 is one-dimensional. If y¯2=1/Tln∏k=1q1/1+gk+x21θy1¯, then θy1,0 features neutral steadiness. If y¯2<1/Tln∏k=1q1/1+gk+x21θy1¯, then β2<1 and θy1,0 is linearly steady.
(iii)
The linear steadiness of the semitrivial periodic solution.
Similarly, the linear steadiness of the semitrivial periodic solution0,θy2 is confirmed by y¯1−1/Tln∏k=1q1/1+hk+x12θy2¯.
The steadiness of the semitrivial periodic solutionsθy1,0 and 0,θy2 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,
(16)y¯1>1Tln∏k=1q11+hk+x12θy2¯,y¯2>1Tln∏k=1q11+gk+x21θy1¯.
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 constantm (M>0) and a finite time T0, which are independent of the system’s initiative values, and make all the initiative values of Model (2) as the solution ait of ai0+>0. When t>T0, then m<ait<Mi=1,2 and Model (2) is coexisting. Here, T0 depends on the initial value ai0+.Hypothesis 2.
If the following inequalities are valid,(17)y¯1>1Tln∏k=1q11+hk+x12θy2¯,y¯2>1Tln∏k=1q11+gk+x21θy1¯,
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 t is sufficiently large, we have
(18)β̇t≤βty1t−x11tβt,t≠τk,k∈N,βτk+=1+hkβτk,t=τk,k∈N,where βt≤θy1. Similarly, when t is sufficiently large, we have λt≤θy2. Then, there exists a constant T1>0 such that when t≥T1, then βt≤θy1 and λt≤θy2. Thus, when t>T1, we have
(19)β̇t≥βty1t−x12tθy2−x11tβt,t≠τk,k∈Z+,βτk+=1+hkβτk,t=τk,k∈Z+,and, when t is sufficiently large, we have βt≥θy1−x12θy2,x11>0. Similarly, when t is sufficiently large, we have λt≥θy2−x21θy1,x22>0. Hence, there exists a constant T¯>T1 such that when t≥T¯, then βt≥θy1−x12θy2,x11 and λt≥θy2−x21θy1,x22. Assume that
(20)ρ=infθy1−x12θy2,x11t,θy2−x21θy1,x22t:t∈0,T,0<ρ<∞,σ=supθy1t,θy2t∈0,T,ρ≤σ<∞.
Then, whent≥T¯, we have ρ≤βt≤σ and ρ≤λt≤σ.


## 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 θy1,0 and 0,θy2. To validate the analytical findings of this study, first examine the two results of the linear oscillation periodic equation.Hypothesis 3.
Assume thatcij∈PCT.
(i)
Ifc11¯≠1/Tln∏k=1q1/1+ck and c22¯≠1/Tln∏k=1q1/1+dk, then the periodic linear homogeneous oscillation equation can be written as follows:
(21)ḃ1t=c11b1t+c12b2t,t≠τk,k∈N,ḃ2t=c22b2t,t≠τk,k∈N,b1τk+=1+ckb1τk,t=τk,k∈N,b2τk+=1+dkb2τk,t=τk,k∈N.
OnPCT×PCT, there is no nontrivial periodic solution. Under this condition, if f1,f2∈PCT×PCT, we write the nonhomogeneous oscillation equation as follows:
(22)ż1t=c11z1t+c12z2t+f1,t≠τk,k∈N,ż2t=c22z2t+f2,t≠τk,k∈N,z1τk+=1+ckz1τk,t=τk,k∈N,z2τk+=1+dkz2τk,t=τk,k∈N.
TheT-periodic solution is exclusive and z1,z2∈PCT×PCT. Furthermore, the operator L:PCT×PCT→PCT×PCT, defined by z1,z2=Lf1,f2, is linear and relatively compact.
(ii)
Ifc22¯=1/Tln∏k=1q1/1+dk and c11¯≠1/Tln∏k=1q1/1+ck are valid, then (22) has only one T-periodic solution on PCT×PCT.Proof 1.
(i)
Since(23)b2t=∏0≤τk<t1+dkb20exp∫0Tc22sds,c22¯≠1/Tln∏k=1q1/1+dk suggests b2∉PCT unless b2=0. Similarly, since
(24)b1t=∏0≤τk<t1+ckb10exp∫0Tc11sds,c11¯≠1/Tln∏k=1q1/1+ck suggests b∉PCT unless b1=0. Under the condition
(25)ż2t=c22z2t+f2,t≠τk,k∈N,z2τk+=1+dkz2τk,t=τk,k∈N,there is only one T-periodic solution, and L2:PCT→PCT is defined by z2=L2f2 and is linear and relatively compact. Furthermore, when f3=c12L2f2+f1∈PCT, the equation
(26)ż1t=c11z1t+f3,t≠τk,k∈N,z1τk+=1+ckz2τk,t=τk,k∈N,has only one T-periodic solution, and z1=L1f3 defines a linear and relatively compact operator L1:PCT→PCT. Therefore, (22) has only one T-periodic solution determined by z1,z2=Lf1,f2 and is the only solution on PCT×PCT, where
(27)Lf1,f2=L1c12L2f2+f1,L2f2
(ii)
Under the hypothesis, the solution adopting a randomb20 as the initial value is a T-periodic solution and is given by (23). If c11¯≠1/T∏k=1q1/1+ck, then ḃt=c11bt does not have a nontrivial periodic solution on PCT. Then, we have
(28)ḃ1t=c11b1t+c12∏0≤τk<t1+dkb20exp∫0tc22sds,t≠τk,b1τk+=1+ckb1τk,t=τk,k∈N.
OnPCT, the T-periodic solution is exclusive. The hypothesis is true.Hypothesis 4.
Assume thatx∈PCT and x¯=1/T∏k=1q1/1+ck, then
(29)ȧ=xa+f,t≠τk,k∈N,aτk+=1+ckaτk,t=τk,k∈N.
The necessary and sufficient condition for theT-periodic solution x∈PCT is as follows:
(30)∫0T∏0≤τk<t11+ckexp−∫0txβdβfdt=0.Proof 1.
The matrix of the linear homogeneous oscillation equation is as follows:(31)ȧ=ax,t≠τk,k∈N,aτk+=1+ckaτk,t=τk,k∈N.
The matrix isAt,s=∏s≤τk<t1+ckexp∫stxβdβ, −∞<s≤t<∞. Then, (31) is the exclusive multiplier
(32)μ=AT,0+=∏k=11+ckexp∫0Txsds=1.
Hence, all the solutions of (31) are T-periodic solutions and all the solutions of the adjoining equation
(33)ḃ=−xb,t≠τk,k∈N,bτk+=11+ckbτk,t=τk,k∈N,are also T-periodic. In particular, b=ψt=∏0≤τk<t1/1+ckexp−∫0txβdβ is also a T-periodic solution of (27), where ψT+=ψ0+=1. The necessary and sufficient condition for (29) to have only one T-periodic solution is ∫0Tψtfdt=0, that is,
(34)∫0T∏0≤τk<t11+ckexp−∫0txβdβfdt=0.
Assumey¯1>1/Tln∏k=1q1/1+hk and regard y2 as the bifurcation referential value. This condition suggests that ξ∗=1/Tln∏k=1q1/1+gk+x21θy1¯ is the bifurcation value of the positive periodic solution bifurcated from the semitrivial periodic solution θy1,0. Assume that y2t=ξ+p2t, where p¯2=0. Then,
(35)a21t=∏0≤τk<t1+gkexp∫0tp2+ξ∗−x21θy1ds∈PCT.
Assumea11 to be the only T-periodic solution of the following linear oscillation differential equation:
(36)ȧ11t=y1−2x11θy1a11t−x12tθy1a21t,t≠τk,k∈N,a11τk+=1+hka11τk,t=τk,k∈N.
Then(37)a11t=−∫0tGt,sx12θy1a21sds,where
(38)Gt,s=At1−At−1A−1s,0<s<t≤T,At+T1−AT−1A−1s,0<t≤s≤T,Gt−kT,s−jT,kT<t≤kT+T,jT<s≤jT+T,k∈N,j∈N,and At=∏0≤τk<tt+hkexp∫0ty2−2x11θy1ds.
Note thata21t>0 exists if t>0. Since
(39)At=∏k=1q1+hkexp∫0Ty1−2x11θy1ds=∏k=1q1+hkexp∫0Ty1−x11θy1dsexp∫0T−x11θy1ds=exp−∫0Tx11θy1ds<1,Gt,s>0.
Then, for anyt, a11t<0.
Defineξ1=x21a11t+x22ta21t¯. The main conclusions are stated in the following hypothesis.Hypothesis 5.
Assumeyit. xij∈PCTi,j=1,2 is valid in terms of y¯1>1/Tln∏k=1q1/1+hk and x21θy1>1/Tln∏k=1q1+gk. For any t>0, there is xijt>0. Then,
(i)
Ifξ1>0, then there exists a constant c0>0, which is sufficiently small to ensure that, in terms of every y2ty2t∈PCT existing under the restriction of ξ∗<y¯2<ξ∗+c0, Model (2) has a periodic solution β,λ∈PCT×PCT, which can guarantee βt<θy1 and λt>0 as long as t>0.
(ii)
Ifξ1<0, there exists a constant d0>0, which is sufficiently small to ensure that, in terms of every y2ty2t∈PCT existing under the restriction of ξ∗−d0<y¯2<ξ∗, Model (2) has a periodic solution β,λ∈PCT×PCT, which can guarantee βt<θy1 and λt>0 as long as t>0.Proof 1.
In Model (2), after conversion, we have a1=βt−θy1a2=λt. Then,
(40)ȧ1t=y1−2x11θy1a1−x12θy1a2+g1a1,a2,t≠τk,k∈N,ȧ2t=y2−x21θy1a2+g2a1,a2,t≠τk,k∈N,a1τk+=1+hka1τk,t=τk,k∈N,a2τk+=1+gka2τk,t=τk,k∈N,where g1a1,a2=−x11a12−x12a1a2 and g2a1,a2=−x21a1a2−x22a22.Definep2t=y2t−ξ, where ξ=y¯2. Then, y2=p2+ξ and p¯2=0. Then, we write (40) as follows:
(41)ȧ1t=y1−2x11θy1a1−x12θy1a2+g1a1,a2,t≠τk,k∈N,ȧ2t=p2−x21θy1a2+ξa2+g2a1,a2,t≠τk,k∈N,a1τk+=1+hka1τk,t=τk,k∈N,a2τk+=1+gka2τk,t=τk,k∈N.Its linear homogeneous function can be written as follows:(42)δ̇1t=y1−2x11θy1δ1−x12θy1δ2,t≠τk,k∈N,δ̇2t=p2−x21θy1δ2,t≠τk,k∈N,δ1τk+=1+hkδ1τk,t=τk,k∈N,δ2τk+=1+gkδ2τk,t=τk,k∈N.Noting thatx21θy1¯>1/Tln∏k=1q1+gk, we have
(43)c22¯=p2−x21θy1¯=−x21θy1¯<1Tln∏k=1q11+gk,and
(44)c11¯=y1−2x11θy1¯=y1−x11θy1¯−x11θy1¯=1Tln∏k=1q11+hk−x11θy1¯≠1Tln∏k=1q11+hk.We apply (41) to obtain the relatively compact linear operator L:PCT×PCT→PCT×PCT. Based on L, we write (41) as the following equivalent operator function
(45)a1,a2=ξL∗a1,a2+Ga1,a2,where
(46)L∗a1,a2=L1−x12θy1L2a2,L2a2and
(47)Ga1,a2=L1−x12θy1L2g2a1,a2+g1a1,a2,L2g2a1,a2.Here,L∗:PCT×PCT→PCT×PCT is linear and relatively compact and G:PCT×PCT→PCT×PCT is quasi-equicontinuous and relatively compact and meets the condition G=∘a1,a2PCT when a1,a2PCT→o.Regardξ in (45) as a real value parameter. As long as ξ∈R, any nontrivial periodic solution a1a2≠0,0 of (45) on PCT×PCT can generate a periodic solution β,λ=a1+θy1,a2 in the collaboration and coexistence model, where y¯2=ξ. 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
(48)δ1,δ2=ξL∗δ1,δ2,δ1,δ2≠0,0,ξ∈Rhas a nontrivial solution.Assume that ifξ∈R, δ1,δ2∈PCT×PCT is a T-periodic solution of (48). By the definition of L∗, δ1,δ2 is the solution to the following equation
(49)δ̇1t=y1−2x11θy1δ1−x12θy1δ2,t≠τk,k∈N,δ̇2t=p2+ξ−x21θy1δ2,t≠τk,k∈N,δ1τk+=1+hkδ1τk,t=τk,k∈N,δ2τk+=1+gkδ2τk,t=τk,k∈N,and vice versa. Based on Hypothesis 3(i), if ξ=ξ∗, where
(50)ξ∗=1Tln∏k=1q11+gk+x21θy1¯,(49) indicates that (45) has the only nontrivial T-periodic solution on PCT×PCT. In view of Hypothesis 3(ii), (41) has the only T-periodic solution on PCT×PCT and bifurcation occurs only under the simple eigenvalue ξ∗. Then, bifurcation of the sectionally continuous periodic solution of (45) can be obtained. Apart from ξ∗;0,0, all the other solutions are nontrivial.The expanded form ofξ;a1,a2 in the neighbourhood of ξ∗;0,0 suggests that the periodic solution of (45) is equivalent to the positive periodic solution of Model (2). In fact, when ε is sufficiently small and aij∈PTT, then
(51)ξ=ξ∗+ξ1ε+⋯,ai=ai1ε+ai2ε2+⋯,i=1,2.Upon substituting the above equation into (41), and if the coefficients ε and ε2 on both ends are the same, then
(52)ȧ11=y1−2x11θy1a11−x12θy1a21,t≠τk,k∈N,ȧ21=p2+ξ∗−x21θy1a21,t≠τk,k∈N,a11τk+=1+hka11τk,t=τk,k∈N,a21τk+=1+gka22τk,t=τk,k∈N,(53)ȧ12=y1−2x11θy1a12−x12θy1a22−x11a112−x12a11a21,t≠τk,k∈N,ȧ22=p2+ξ∗−x21θy1a22+ξ1a21−x21a11a21−x22a212,t≠τk,k∈N,a12τk+=1+hka12τk,t=τk,k∈N,a22τk+=1+gka22τk,t=τk,k∈N.Hence,a11,a21∈PCT×PCT is the solution of (48). A special solution meeting the condition of a210=1 is
(54)a21t=∏0≤τk<t1+gkexp∫0tp2+ξ∗−x21θy1ds>0,a11t=−∫0tGt,sx12θy1a21sds.Hence, in terms oft, a11t<0 is valid. Applying Hypothesis 2 to the second equation in (53), we have ξ=x21a11+x22a21¯.Ifξ1>0, then there exists a bifurcation γ+=ξ;a1,a2⊆R×PCT×PCT, where ξ>ξ∗, and as long as t>0, then a1t<0 and a2t>0. When ξ becomes sufficiently close to ξ∗, it is equivalent to the positive periodic solution of Model (2). Therefore, there exists a c0>0 that is sufficiently small to ensure that the bifurcation C+=ξ;β,λ∈R×PCT×PCT can be solved and meets the condition, such that when ξ∗<ξ<ξ∗+c0, if t>0, then βt<θy1 and λt>0. Since ξ∗<ξ<ξ∗+c0 is equivalent to ξ∗<y¯2<ξ∗+c0, Hypothesis 5(i) is valid. If ξ1<0, it is valid similarly that there exists a constant d0>0 that is sufficiently small to ensure that the solution β,λ∈PCT×PCT of Model (2) meets the condition such that when ξ∗−d0<y¯2<ξ∗, if t>0, then βt<θy1 and λt>0. The proof is complete.From Hypothesis5, we note that ξ1 is a measure of the probability that the two personnel groups coexist. ξ∗ is a value of the positive periodic solution bifurcated from the semitrivial periodic solution θy1,0. If ξ1>0, then y¯2>ξ∗. From Hypothesis 1, we see that the unsteady manifold of the semitrivial positive periodic solution θy1,0 increases by one dimension. According to the steadiness exchange principle, when ξ1>0, the periodic solution y¯2;β,λ exhibits linear asymptotic steadiness. If ξ1<0, then the bifurcated periodic solution is unsteady.


## 4. Application Examples

During the implementation process of complex mega infrastructure projects, many factors, both internal and external, can exert an influence. The engineering personnel groups are not isolated from one another but coexist in a complex engineering system. Once the risk of infection occurs, a chain reaction will be triggered. Therefore, it is necessary to balance the structure of the personnel groups and formulate targeted strategic precautions. The periodic collaboration and coexistence model with the oscillation effect introduced in this study obtains three types of components, namely, a nonnegative periodic solution, a trivial periodic solution0,0, and semitrivial periodic solutions θy1,0 and 0,θy2. Based on the nontrivial T-periodic solution bifurcated from the semitrivial periodic solution, the conditions necessary to maintain stability of the trivial periodic solution and the nontrivial periodic solution are obtained. In addition, the conditions obtained suggest that all the trivial periodic solutions and nontrivial periodic solutions are linearly unsteady. This indirectly indicates that personnel groups in the complex major engineering systems discussed in Model (2) can coexist sustainably. The conditions of Hypothesis 2 imply that oscillating periodic collaborative management can prevent one engineering personnel group from being fully assimilated or disappearing after the risk of infection occurs, guaranteeing sustainable production of the major engineering project.To prove that the internal positive periodic solution bifurcated from the semitrivial periodic solution is stable and steady, we present the following examples and numerical simulation results.We can use Model (2) to describe periodic collaborative management of two coexisting personnel groups within a complex major engineering system. Since the risk of infection occurs on a discontinuous basis, Model (2) is reasonable. For the convenience of discussion, we assume that an engineering personnel group in any period (such as 1 year) is managed for a finite time period independently. In the other case, two coexisting engineering personnel groups are managed collaboratively during the same period. We can express the oscillation effect as follows:
(55)β̇t=βtη+sin2πt−x11βt−x12λt,t≠τk,k∈Z+,λt=λtξ+cos2πt−x21βt−x22λt,t≠τk,k∈Z+,βτk+=1−p1βτk,t=τk,τk=122k−1,k∈Z+,λτk+=1−p2λτk,t=τk,τk=122k−1,k∈Z+,where βt and λt stand for the density of the two coexisting engineering personnel groups at a certain time point η, ξ and xiji,j=1,2 are positive constants, T=1, and 0≤p1<1,0≤p2<1 represent the degrees of collaboration between the two engineering personnel groups. The two components of interest in this research are both nonnegative solutions. The two engineering personnel groups can coexist.Assume thatη=1, ξ=1, x11=0.4, x12=0.2, x21=0.5, x22=0.3, and β0+,λ0+=0.4,0.5. Set p1 and p2 as control variables. System (55) has a semitrivial periodic solution with asymptotic stability when there is no management collaboration p1=p2=0.Figure2 presents the time-sequence comparison chart of the two groups in Model (55) under the condition of nonoscillating collaborative management. Figure 3 shows the time-sequence comparison chart of the two groups in Model (55) under the condition of oscillating collaborative management. As observed from Figures 2 and 3, β is the dominant group. If periodic collaborative management is pursued, that is, p1=0.3 and p2=0.15, then the conditions of Hypothesis 2 η>ln1/1−p1+x12/x221−ln1/1−p2,ξ>ln1/1−p2+x21/x11η−ln1/1−p2 and the conditions of Hypothesis 5 are valid. The two engineering personnel groups coexist in a steady periodic solution. Under the condition and measure of the probability that the two personnel groups coexist, oscillating collaborative management can stop group λ from being assimilated or disappearing. In fact, when there is no oscillating collaboration and β is the dominant group, and if p2 is relatively small, the influence of β will decline with increasing p1. When the latter continues to increase to a certain extent, the two engineering personnel groups can coexist in a steady periodic solution, fulfilling the management goal.Figure 2
The time-sequence comparison chart of the two groups without oscillating collaborative management.Figure 3
The time-sequence comparison chart of the two groups under oscillating collaborative management.Ifp1 keeps increasing and when η<ln1/1−p1+x12/x221−ln1/1−p2, group β, as stated in Hypothesis 1(iii), tends to be assimilated or disappear, while group λ will periodically oscillate. If the oscillating collaboration value is large, excessive management might lead to the assimilation or disappearance of both groups. Therefore, to guarantee the coexistence of the two kinds of engineering personnel groups, determining an appropriate collaboration and coexistence management strategy is vital.


## 5. Conclusions

Based on the structural characteristics of major engineering personnel groups under the risk of infection, we consider how to inhibit the risk of infection. Based on the theories of dynamic, life cycle, and oscillation, we develop a periodic collaboration and coexistence management model with the oscillation effect. We perform mathematical analysis to show how periodic oscillating collaborative management prevents the full assimilation or disappearance of one engineering personnel group when the risk of infection is triggered, and how the personnel groups in a complex major engineering system can remain balanced to guarantee the sustainability of project construction. We derive the following findings: (1) When the degree of oscillating collaboration is high, excessive management might result in severe imbalance of the personnel structure and impede the complex major engineering project from progressing normally. (2) When infection breaks out, the dominant group will be weakened as collaborative management of the group increases. In addition, when the degree of collaboration reaches a certain degree, the two engineering personnel groups can coexist over a steady period. (3) When two engineering personnel groups exist in a steady period, collaborative management under the oscillating model can prevent the engineering personnel groups from being assimilated or disappearing even under the risk of infection.In summary, we do not only reveal the mechanism of group collaboration in a complex major engineering system under the risk of infection and enriches the major engineering collaborative management theory but also build a periodic collaboration and coexistence management model with the oscillation effect. The model can provide references for the collaborative management of major engineering systems under the risk of infection. More importantly, it can prevent and curb the occurrence of large-scale infection risk, thus safeguarding the structural stability of the complex major engineering system. Taken together, our research findings complement and supplement the pertinent theories and provide useful references for improving collaborative management in complex mega infrastructure projects. However, this study is in the theoretical research stage of mechanism revelation and model establishment. We will continue to enrich this theory in future research and try to do some research on simulation and experiment.



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*Source: 9846074-2018-10-02.xml*