SEIR Evolutionary Game Model Applied to the Evolution and Control of the Medical Waste Disposal Crisis in China during the COVID-19 Outbreak

Abstract

The behavioral choices and speculative psychology of the participants in medical waste disposal can lead to the evolution of the medical waste disposal crisis, which has a great impact on and represents a potential threat to environmental safety as well as public health. This study constructs the SEIR evolutionary game model based on the theory of propagation dynamics and evolutionary game and explores the game relationship between local governments and medical waste disposal enterprises. Then it analyzes the propagation threshold and evolutionary path of medical institutions’ speculative psychology under different behavioral decisions of both subjects and explores the process and law of system evolution to a benign stable state and conducts a multi-situated simulation analysis. The results showed that the number of infected states in medical institutions varies in a positive direction with the propagation threshold of their speculative psychology, and when the propagation threshold is greater than 1, the speculative psychology of medical institutions will spread widely in the system. The “strict regulation and high-quality disposal” behavior of local governments and disposal enterprises could effectively eliminate the speculative psychology of medical institutions, the number of infected medical institutions could gradually evolve to 0, then the further evolution of the medical waste disposal crisis could be prevented. The formation of an integrated, coordinated, and mutually constraining crisis governance mechanism should improve the government’s regulatory capacity and efficiency, develop attractive and deterrent reward and punishment policies to guide disposal enterprises to high-quality disposal, and contribute to the compliant disposal of medical waste in medical institutions.

1. Introduction

In early 2020, COVID-19 began to explode on a large scale around the world, with medical institutions around the world facing great pressure (Kulkarni et al., 2020; Wu, et al., 2020; Yu, et al., 2020) [1,2,3]. China was the first country to have a large-scale outbreak. According to the official statistical data of the National Health Commission of the People’s Republic of China (NHC) and the Ministry of Ecology and Environment of the People’s Republic of China (MEE), the amount of medical waste generated in 2019 was about 1,180,000 tons, while with the outbreak of COVID-19, the production of medical waste in 2020 and 2021 was 1,260,000 tons and 1,400,000 tons, with an average growth rate of about 14.85%. These huge amounts of medical waste bring new challenges to medical waste management systems (El-Ramady et al., 2021; Chen et al., 2021; Yoon et al., 2022) [4,5,6], and pose a serious threat to social, economic, and environmental sustainability (Ilyas et al., 2020; Etim et al., 2022; Lv et al., 2022) [7,8,9].

Under the established regional centralized disposal system for medical waste in China (Chen et al., 2021) [5], local governments, medical waste disposal enterprises, and medical institutions are important participants in the medical waste disposal chain. Among them, local governments play a dominant role through conducting qualification audits, bidding, and pricing for disposal enterprises, and medical institutions passively accept medical waste disposal services (Lv et al., 2022) [9]. However, the state has promulgated regulations such as “Medical Waste Management Regulations” and local governments have also introduced medical waste management policies and epidemic response plans to provide policy protection for the standardized disposal of medical waste (Yang et al., 2020; Ma et al., 2020) [10,11]. However, due to information asymmetry and cost constraints, it is difficult for local governments to implement effective regulation on disposal enterprises. As a result, disposal enterprises take advantage of their information superiority to operate illegally, reducing the quality of medical waste disposal, and resulting in frequent crises in the disposal chain (Aung et al., 2019; Mallick et al., 2021) [12,13].

It is worth noting that local governments and disposal enterprises are the key links in the disposal chain, while medical institutions are the source of medical waste generation and are responsible for the timely and proper disposal of medical waste. Specifically, this includes the timely registration of medical waste sources, types, weights, transfer times, disposal methods, final destinations, etc. (Maalouf et al., 2021) [14]. However, driven by market interests, medical institutions do not follow the standardized disposal process of medical waste, and the dumping of medical waste speculation occurs (Zhang et al., 2020; Al-Khatib et al., 2020) [15,16]. If the local governments and disposal enterprises choose to implement negative behavior, this will seriously intensify the speculative psychology of medical institutions, so that the speculative psychology in the group of medical institutions spreads unchecked, further aggravating the formation of the medical waste disposal crisis.

In fact, scholars have explored a mechanism for coordinating the interests of multiple subjects in the system by an evolutionary game, and the application in the field of waste disposal research has been relatively mature, including the recycling of waste electrical appliances (Li et al., 2022) [17], disposing of medical waste (Lv et al., 2022) [9], and the recycling of construction waste (Liu et al., 2021; Sun et al., 2022) [18,19]. However, in order to make the construction of the evolutionary game model more realistic, numerous scholars have tried to combine the evolutionary game model with an infectious disease model. Infectious disease dynamics represent an important method for conducting theoretical quantitative studies, the core idea of which is to analyze the process of disease development, reveal epidemic patterns, and predict trends (Unkel et al., 2012) [20]. This method has been widely applied to the study of individual mental state transmission patterns because of the similarities between individual mental state changes and spread mechanisms of infectious disease models in populations (Tian, 2015; Zhao et al., 2017) [21,22]. Yang et al. (2021) [23] and Qi et al. (2022) [24] agreed that combining the evolutionary game model with the infectious disease model could better portray the emotional infection and evolutionary state of the subject in the actual situation, and combined the SEIR (Susceptible-Exposed-Infectious-Removed) model with the evolutionary game model to conduct an in-depth study on the psycho-emotional state of the public subject. Therefore, combining the infectious disease model with the evolutionary game model is preferable to describe the interaction between local governments and disposal enterprises in the process of medical waste disposal in China and the mechanism of their influence on the evolution of speculative psychological behavior of medical institutions. Based on the above analysis, this study constructs an SEIR evolutionary game model with the local governments, disposal enterprises, and medical institutions, and analyzes the internal evolutionary process of the game behavior between the local governments and disposal enterprises and the conditions for the system to evolve to a stable state. On this basis, we explore the influence mechanism of different behavioral strategies of local governments and disposal enterprises on the speculative psychology of medical institutions, then discuss the process and law of the evolution of the whole system to a benign stable state. Finally, we present suggestions to manage the medical waste disposal crisis in China. The main problems to be solved in this study are as follows:

(1)

What are the internal evolutionary processes and characteristics of the game behavior between local governments and disposal enterprises?

(2)

How do different behavioral strategies between local governments and disposal firms affect the evolutionary process and path of speculative psychology of medical institutions?

(3)

Which measures should be taken to enable the medical waste disposal crisis in China to be efficiently controlled?

Compared with the existing research results, the main contributions of this study are as follows: (1) As the COVID-19 pandemic continues to spread and rebound, the study of the medical waste disposal crisis is timely and provides theoretical support for local governments to design effective crisis management measures. (2) This study describes the influence mechanism of interaction behavior between local governments and disposal enterprises on the spread process and evolution path of medical institutions’ speculative psychology through a computational experimental simulation, which is significant for studying how to inhibit the further evolution of the medical waste disposal crisis. (3) This study innovatively combines the SEIR infectious disease model with the evolutionary game model in the field of medical waste disposal crisis management which extends the application research of the SEIR evolutionary game model.

The remainder of this study is structured as follows. Section 2 summarizes the related literature. Section 3 describes the problem and gives the related assumptions. Section 4 establishes and analyzes the SEIR model and the evolutionary game model. Section 5 simulates the evolution of the medical waste disposal crisis through a computational experimental method. Section 6 presents the main conclusions and limitations.

2. Literature Review

According to the research purposes of this study, the related literature can be divided into two categories: medical waste and medical waste disposal crisis.

2.1. Medical Waste Research

Medical waste is a special pollutant produced by medical institutions in the course of medical, preventive, and related medical activities (Lee. 1991) [25], and its main sources are hospitals, clinics, health centers, diagnostic and research laboratories, autopsy centers, transfusion and hemodialysis centers, nursing homes, and mortuaries (Marinkovic et al., 2008) [26]. Or et al. (1994), Mato et al. (1999), and Askarian et al. (2004) [27,28,29] classified medical waste into five categories, as shown in

Table 1. Basic classification of medical waste.

Waste Category	Components
Infectious waste	Medical waste carrying pathogenic microorganisms with the risk of spreading infectious diseases
Damaging waste	Abandoned medical sharps that can stab or cut the human body
Pathological waste	Human waste and medical laboratory animal carcasses generated in the process of treatment
Pharmaceutical waste	Obsolete, deteriorated, or contaminated discarded drugs
Chemical waste	Corrosive, waste chemicals

. However, in the reality of recurring epidemics, scholars have focused on the generation and disposal of medical waste. Most scholars have used SVR, ARIMA, and multiple linear regression (MLR) models to predict the generation of medical waste (Ceylan et al., 2020; Cetinkaya et al., 2020; Wang et al., 2021) [30,31,32]. Some scholars have also studied the drivers of medical waste generation and found that regional economic growth (Pu et al., 2021; Ma et al., 2021) [33,34], quality of waste management (Tsai et al., 2022) [35], etc., are important factors influencing medical waste generation. Moreover, traditional medical waste disposal includes a multitude of disposal methods, such as incineration, landfilling, and chemical treatments (Dharmaraj et al., 2018) [36]. In particular, incineration and landfilling as the main methods of medical waste disposal have become the focus of academic discussion. For example, Zhao et al. (2021) [37] evaluated and compared five medical waste incineration technologies using energy recovery analysis (ERA), life cycle assessment (LCA), and life cycle coasting methods (LCC). Tirkolaee et al. (2022) [38] attempted to address the landfill location selection problem for healthcare waste using a novel decision support system. However, these traditional methods may cause serious damage to the environment. (Kenny et al., 2021) [39]. Therefore, scholars have conducted innovative research on alternative disposal technologies for medical waste based on the technological perspective, such as pyrolysis (Su et al., 2021; Dharmaraj et al., 2021) [36,40], ion gasification (Erdogan et al., 2021) [41], etc. These innovations in disposal technology have led to a qualitative improvement in the quality and efficiency of medical waste disposal.

2.2. Medical Waste Disposal Crisis Research

The frequent occurrence of medical waste disposal violations worldwide has made the medical waste disposal crisis a hot issue for academic research. In summary, the research mainly covers the reasons of crisis, manifestations, and management disposal. In terms of the reasons for the emergence of the crisis, Ragazzi et al. (2020) [42] identified the inadequate management in the process of medical waste disposal as an intrinsic reason, while Acharya et al. (2021) [43] and Mohamed et al. (2022) [44] attributed the disruption of the disposal chain to the proliferation of medical waste. Meanwhile, scholars have studied the external reasons of the medical waste disposal crisis. Defects in the regional management system (Mihai et al., 2020) [45], the absence of the public health function of the government (Zhang, 2021) [46], and the lack of substantive design of the legal system (Wang et al., 2021) [47] have had a significant impact on the formation of the medical waste disposal crisis. In terms of the manifestations of the medical waste crisis, the proliferation and undesirable disposal of medical waste poses a considerable threat to society, the economy, and the environment. The main impact was the severe deterioration of air quality (Adams, 2020; Berman et al., 2020) [48,49], public hygiene threats (Mihai, 2020) [45], public health risks (Goswami et al., 2021) [50], destruction of aquatic biosystems (Mohamed et al., 2022) [44], and a severe shock to trade and investment worldwide (Ghadir et al., 2022) [51]. In terms of medical waste disposal crisis management, scholars’ research has mainly focused on the government-led crisis governance model of multiple participation and joint response. For example, Zaher et al. (2021) [52] used the SWOT analysis to find that key components of crisis governance included efficient government regulation, integrated utilization of public–private partnerships, and a global workforce of excellence. Lv et al. (2022) [9] constructed a two-sided game model between the local governments and medical waste disposal enterprises and found that the government’s static reward and dynamic punishment strategy could better improve the disposal quality to prevent the occurrence of a disposal crisis. Kuhlmann et al. (2022) [53] suggested the importance of coordination mechanisms and collaboration models between government departments and task units for crisis management. Weber et al. (2019) [54] proposed the importance of raising public awareness of environmental issues and facilitating change towards more sustainable practices within local communities.

In summary, the medical waste disposal crisis has become a common focus of scholars, who have conducted rich research on the generation, disposal, disposal crisis manifestations of medical waste, and the management of the crisis. However, it is not difficult to find the following limitations: (1) Although there is literature emphasizing the regulatory role of government in medical waste disposal crises, further research is needed on how government departments can efficiently regulate and guide the evolution of disposal crises to a benign state. (2) Although the related literature has used evolutionary game to study the pathways in which the government and multiple parties collaborate in crisis management, few scholars have combined the infectious disease model with evolutionary game to study the impact of the interaction behavior of local governments and disposal enterprises on the speculative psychology of medical institutions.

3. Problem Description and Underlying Assumptions

In the process of medical waste disposal in China, the main participants are local governments, medical waste disposal enterprises, and medical institutions. Government regulatory departments, such as environmental protection, health, and price departments, mainly issue regulatory policies and use feasible technological means to regulate and manage the fees of related links. Medical institutions and other production units are placed in a centralized classification of medical waste, while disposal enterprises are responsible for more links, coupled with the higher disposal costs of each link, and thus more hidden trouble spots. Therefore, the regulation of local governments and the policy implementation of disposal enterprises, as well as the game of both subjects become the key to influence the crisis of medical waste disposal. The speculative psychology of medical institutions has also shifted with the gaming behavior of both subjects, which in turn affects the further evolution of the medical waste disposal crisis. Based on this, the following assumptions are proposed in this study.

Assumption 1: According to the basic nature of the evolutionary game (Smith, 1974) [55], local governments and disposal enterprises will choose the maximum benefit strategy based on Pareto-optimal, and different combinations of behavioral decisions will cause medical institutions to shift the degree of standardization of medical waste disposal, and the continuous game of both subjects’ decisions will make the system evolve to the desired stable state. Based on the study of Yang et al. (2021) [23], medical institutions are classified into four state types: medical institutions $S(t)$ with a potentially speculative mentality, medical institutions $E(t)$ that have been infected by speculative psychology and have not yet developed speculative behavior, medical institutions $I(t)$ with speculative psychology that generate speculative behavior not in accordance with the standardized processes for the disposal of medical waste and the dumping medical waste, and medical institutions $R(t)$ that eliminate speculation due to high-quality treatment by disposal enterprises or strict regulatory strategies implemented by the local governments.

Assumption 2: Referring to the study of Lv et al. (2022) [9], it is argued that the governments ‘regulatory strategy may be strict or relaxed due to the local government’s manpower, costs, and the degree of cooperation of the disposal enterprises. Where $x_1$ is the probability of strict regulation and $x_2$ is the probability of relaxed regulation ($0\le x_1,x_2\le 1$ and $x_1+x_2=1$). Considering cost and profit claims, disposal enterprises’ behavior can be high-quality disposal, but may also be low-quality disposal that includes the implementation of lower standards and a violation of standards, and even transit, discard, etc. Where $y_1$ is the probability of high-quality disposal and $y_2$ is the probability of low-quality disposal ($0\le y_1,y_2\le 1$ and $y_1+y_2=1$).

Assumption 3: According to Deng et al. (2021) [56], it is assumed that when local governments choose strict regulation and disposal enterprises choose low-quality disposal, local governments must be able to detect the low-quality disposal behavior of disposal enterprises, at which time the cost of strict regulation is $C_{g1}$ and the fine received from the disposal enterprises is $F$. $R_3$ is the combined benefits of strict regulation by local governments including reputation and commendations from higher levels of government and $R_1$ is the environmental benefits brought to the local governments by the high-quality disposal of the disposal enterprises. Assume that the cost is $C_{g2}$ when the local governments choose the relaxed regulatory strategy, and the probability of being able to find the low-quality disposal behavior is $a$ when a disposal enterprise with low-quality disposal needs to pay an environmental management cost $C_{g3}$ regardless of whether the local governments strictly regulate it or not.

Assumption 4: Referring to the study of Rocha et al. (2019) [57], it is assumed that the basic benefit of the disposal enterprises providing the service is $R_2$, the low-quality disposal cost is $C_{b1}$, and the high-quality disposal cost is $C_{b2}$. Disposal enterprises with low-quality disposal found by the local governments will be punished $F$ mainly include liquidated damages and a certain compensation for environmental pollution. In addition, disposal enterprises providing high-quality services will receive incentives from local governments for $W$.

Assumption 5: Based on infection rates in the infectious disease model studied by Qi et al. (2022) [24], the speculative psychology state shift of medical institutions is shown in Figure 1. Among them, set with the rebound of the epidemic and the increase in medical waste, the number of medical institutions grows at a proportional N growth rate when local governments choose relaxed regulations, and the medical institutions develop a speculative state of psychology and transitions from state $S$ to $E$ with probability $x_2+a$. Conversely, the local governments adopt strict regulations to make medical institutions eliminate speculation and move from state $S$ to $R$ with probability $x_1+\beta$. Moreover, the disposal enterprises with low-quality disposal will turn the speculative psychology of medical institutions to produce speculative behavior and switch from state $E$ to $I$ with the probability of $y_2+\gamma$. Prompted by a sense of social responsibility, the speculative psychology of the medical institutions will be eliminated and switch from state $I$ to $R$ with probability $\theta$. On the contrary, the disposal enterprises with high-quality disposal will regulate the medical waste handed over by medical institutions, which will make medical institutions eliminate speculation to some extent and move from state $E$ to state $R$ with the probability of $y_1+\mu$.

By combining the strategies of local governments and high and low-quality disposal as described in the above scenario, we can obtain the strategy benefit combination matrix of the two participating subjects, as shown in

Table 2. The payment matrix for the evolutionary game model.

Players		Disposal Enterprises
		High-Quality Disposal	Low-Quality Disposal
Local Governments	Strict regulation	$R_1+R_3-C_{g1}-W$, $R_2-C_{\mathrm{b}2}+W$	$R_3-C_{g1}+F-C_{g3}$, $R_2-C_{\mathrm{b}1}-F$
	Relaxed regulation	$R_1-C_{g2}-W$, $R_2-C_{b2}+W$	$-C_{g2}-C_{g3}+aF$, $R_2-C_{b1}-aF$

.

4. Model Construction and Analysis

4.1. SEIR Evolutionary Game Model Construction

Based on the evolutionary game analysis method of replication dynamics of Friedman (1998) [58], this study describes the evolutionary process of behavioral strategies of local governments and disposal enterprises by constructing replication dynamics equations. Assume that the expected utilities of local governments with strict and relaxed regulatory strategies at moment $t$ are $E_{u1}$ and $E_{u2}$, and their average expected utility is $\overline{E_u}$. The calculation is as follows.

$$E_{u1}=y_1(R_1+R_3-C_{g1}-W)+y_2(R_3-C_{g1}+F-C_{g3})$$

(1)

$$E_{u2}=y_1(R_1-C_{g2}-W)+y_2(-C_{g2}-C_{g3}+aF)$$

(2)

$$\overline{E_u}=x_1{E}_{u1}+x_2{E}_{u2}$$

(3)

Based on the evolutionary stabilization strategy of the replication dynamic equation, we can obtain that the replication dynamic equation of the local governments is:

$$F\left(x\right)=dx/dt=x_1{x}_2\left[-y_1\left(1-a\right)F+R_3+\left(1-a\right)F-C_{g1}+C_{g2}\right]$$

(4)

Similarly, the replication dynamic equation of the disposal enterprises is obtained as:

$$F\left(y\right)=dy/dt=y_1{y}_2[x_1\left(1-a\right)F-C_{b2}+W+C_{b1}+aF]$$

(5)

A two-dimensional dynamical system Q can be formed from the differential Equations (4) and (5), as shown in Equation (6).

$$\{\begin{array}{l}F\left(x\right)=dx/dt=x_1{x}_2\left[-y_1\left(1-a\right)F+R_3+\left(1-a\right)F-C_{g1}+C_{g2}\right]\\ F\left(y\right)=dy/dt=y_1{y}_2[x1\left(1-a\right)F-C_{b2}+W+C_{b1}+aF]\end{array}$$

(6)

Based on the above assumptions, the following dynamic equation system representing the group change of medical institutions in each state is constructed to describe the process of the speculative psychological evolution of different behavioral strategies of participating subjects. The system of ordinary differential equations is obtained as follows.

$$\{\begin{array}{l}\frac{dS}{dt}=N-(\alpha +x_2)SI-(x_1+\beta )S\\ \frac{dE}{dt}=(\alpha +x_2)SI-(y_1+\mu )E-(y_2+\gamma )E\\ \frac{dI}{dt}=(y_2+\gamma )E-\theta I\\ \frac{dR}{dt}=\theta I+(x_1+\beta )S+(y_1+\mu )E\end{array}$$

(7)

4.2. Stability Analysis of Game Strategies of Local Governments and Disposal Enterprises

In order to determine the stable equilibrium state; the stability of each equilibrium point can be derived based on the local stability of the matrix of the differential dynamical system Friedman (1998) [58]. Therefore, the construction of the matrix of the system yields Equation (8):

$$\begin{array}{l}J=\left(\begin{array}{cc}\frac{\partial F\left(x\right)}{\partial x_1}& \frac{\partial F\left(x\right)}{\partial y_1}\\ \frac{\partial F\left(y\right)}{\partial x_1}& \frac{\partial F\left(y\right)}{\partial y_1}\end{array}\right)\\ =\left(\begin{array}{cc}(1-2x_1)\left[-y_1\left(1-a\right)F+R_3+\left(1-a\right)F-C_{g1}+C_{g2}\right]& -x_1\left(1-x_1\right)\left(1-a\right)F\\ y_1\left(1-y_1\right)\left(1-a\right)F& \left(1-2y_1\right)\left[x_1\left(1-a\right)F-C_{b2}+W+C_{b1}+aF)\right]\end{array}\right)\end{array}$$

(8)

This study will further explore the evolutionary path of the game model of participating subjects in the process of medical waste disposal. From the Liapunov discriminant (Pai, 1981) [59], we can obtain that the evolutionary stability point $(ESS)$ of the constraint of all the eigenvalues of the $Jacobian$ matrix is negative $\left(\lambda <0\right)$.

The above local equilibrium points are substituted into Equation (8) to obtain the eigenvalues of the system $Jacobian$ matrix, as shown in

Table 3. Eigenvalues of the $Jacobian$ matrix.

Equilibrium Point	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$
$\left(0,0\right)$	$R_3+\left(1-a\right)F-C_{g1}+C_{g2}$	$-C_{b2}+W+C_{b1}+aF$
$\left(1,0\right)$	$-\left[R_3+\left(1-a\right)F-C_{g1}+C_{g2}\right]$	$\left(1-a\right)F-C_{b2}+W+C_{b1}+aF$
$\left(0,1\right)$	$R_3-C_{g1}+C_{g3}$	$C_{b2}-W-C_{b1}-aF$
$\left(1,1\right)$	$-R_3+C_{g1}-C_{g2}$	$-\left[\left(1-a\right)F-C_{b2}+W+C_{b1}+aF\right]$

.

In summary, in the dynamic system of the game between local governments and disposal enterprises, the evolutionary game equilibrium of the behavioral strategies of both subjects is influenced by various factors. The following scenario analysis was conducted on the stability of the equilibrium point in the system:

Scenario 1 When $C_{g1}-C_{g2}>R_3+\left(1-a\right)F$ and $C_{b2}-C_{b1}>W+aF$, the additional expenditures of strict regulation by local governments cannot be compensated by benefits such as commendations from superior governments, which will prompt them to choose relaxed regulation. Meanwhile, when the additional expenditures of high-quality disposal by disposal enterprises cannot be compensated by the incentives of local governments, which will prompt disposal enterprises to choose low-quality disposal strategy. Concurrently, $\left(0,0\right)$ becomes the only local asymptotic stabilization point and evolutionary stabilization strategy within the two-dimensional dynamical system.

Scenario 2 When $R_3+\left(1-a\right)F>C_{g1}-C_{g2}$ and $C_{b2}-C_{b1}>W+F$, the local governments are motivated to choose the strict regulation strategy when the benefits of strict regulation exceed their additional expenditures. The incentive amount of the local governments has little effect on compensating the additional expenditures of high-quality disposal for the disposal enterprises. For the objectives of maximizing their own interests, the disposal enterprises will ignore the governments’ strategy and choose a low-quality disposal strategy. Eventually, $\left(1,0\right)$ becomes the only local asymptotic stabilization point and evolutionary stabilization strategy within the two-dimensional dynamical system.

Scenario 3 When $R_3<C_{g1}-C_{g2}$ and $C_{b2}-C_{b1}<W+aF$, the local governments choose strict regulation which is not enough to support larger regulatory expenditures. In the long-term, local governments are overwhelmed with financial burdens and have to turn to relaxed regulation. Local governments’ incentives obtained by disposal enterprises for high-quality disposal can largely compensate for the additional expenditures, and they are not worried about paying penalty expenses when low-quality disposal is detected by local governments regulation, so disposal enterprises gradually evolve to high-quality disposal strategy. Eventually, $\left(0,1\right)$ becomes the only local asymptotic stabilization point and evolutionary stabilization strategy within the two-dimensional dynamical system.

Scenario 4 When $R_3>C_{g1}-C_{g2}$ and $C_{b1}-C_{b2}+W+F>0$, the local governments receive more praise from higher levels of government than its additional expenditures of strict regulation and the local governments gradually evolve toward the strict regulation strategy. Eventually, $\left(1,1\right)$ becomes the only local asymptotic stabilization point and evolutionary stabilization strategy within the two-dimensional dynamical system.

4.3. Threshold Solution and Stability Analysis of Equilibrium Point

4.3.1. Zero-Infection Balance Point for Speculative Psychology in Medical Institutions

In the process of the speculative psychological transformation of medical institutions, when both local governments and disposal enterprises can choose reasonable strategies, medical institutions are all non-speculative psychologists in the system, reaching the zero-infection equilibrium point. Concurrently, the zero-infection equilibrium point of the differential equation is $Q_0$$(\frac{N}{x_1+\beta },0,0)$.

The $Jacobian$ matrix of Equation (7) is presented as:

$$J=\left(\begin{array}{ccc}-(\alpha +x_2)I-(x_1+\beta )& 0& -(\alpha +x_2)S\\ (\alpha +x_2)I& -(y_1+y_2+\mu +\gamma )& (\alpha +x_2)S\\ 0& y_2+\gamma & -\theta \end{array}\right)$$

(9)

Substituting $(\frac{N}{x_1+\beta },0,0)$ into the $Jacobian$ matrix, we can obtain

$$J(Q_0)=\left(\begin{array}{ccc}-(x_1+\beta )& 0& -(\alpha +x_2)\frac{N}{x_1+\beta }\\ 0& -(y_1+y_2+\mu +\gamma )& (\alpha +x_2)\frac{N}{x_1+\beta }\\ 0& y_2+\gamma & -\theta \end{array}\right)$$

(10)

The characteristic equation of $J(Q_0)$ is obtained as:

$$f(\lambda )=(\lambda +x_1+\beta )[{\lambda }^2+(\theta +y_1+y_2+\mu +\gamma )\lambda +(y_1+y_2+\mu +\gamma )\theta -(\alpha +x_2)\frac{N(y_2+\gamma )}{x_1+\beta }]$$

(11)

An eigenroot of $J(Q_0)$ is solved for ${\lambda }_1=-(x_1+\beta )$. The other two eigenvalues, ${\lambda }_2$ and ${\lambda }_3$, are determined by the equation ${\lambda }^2+(\theta +y_1+y_2+\mu +\gamma )\lambda +(y_1+y_2+\mu +\gamma )\theta -(\alpha +x_2)\frac{N(y_2+\gamma )}{x_1+\beta }=0$. Thus, it is obtained that when $(y_1+y_2+\mu +\gamma )\theta >(\alpha +x_2)\frac{N(y_2+\gamma )}{x_1+\beta }$, it means that the eigenvalue ${\lambda }_2$, ${\lambda }_3$ is negative or has a negative real part. When $(y_1+y_2+\mu +\gamma )\theta <(\alpha +x_2)\frac{N(y_2+\gamma )}{x_1+\beta }$, it means that the eigenvalues ${\lambda }_2$, ${\lambda }_3$ are one positive and one negative, and when $(y_1+y_2+\mu +\gamma )\theta =(\alpha +x_2)\frac{N(y_2+\gamma )}{x_1+\beta }$, it means that there is an eigenvalue of 0.

Therefore, according to Liapunov’s stability discriminant theorem (Pai, 1981) [59], it can be shown that there are three negative real parts of Equation (11) only if $(y_1+y_2+\mu +\gamma )\theta >(\alpha +x_2)\frac{N(y_2+\gamma )}{x_1+\beta }$, indicating that the zero-infection equilibrium point $Q_0$$(\frac{N}{x_1+\beta },0,0)$ is globally asymptotically stable.

4.3.2. The Propagation Balance Point of Speculative Psychology in Medical Institutions

When local governments and disposal enterprises fail to choose reasonable strategies, there exists an infection equilibrium point $Q_1(S^{\ast },E^{\ast },I^{\ast })$ in the system. It may be noticed in Equation (7) that the former three differential equations all do not contain the variable R. Thus, it is sufficient to consider only the former three equations. The propagation equilibrium points of the medical institutions’ speculative psychology must all satisfy: $dS/dt=0$, $dE/dt=0$, $dI/dt=0$, there is:

$$\{\begin{array}{l}{S}^{\ast }=\frac{(y_1+y_2+\mu +\gamma )\theta }{(\alpha +x_2)(y_2+\gamma )}\\ E^{\ast }=\frac{N}{y_1+y_2+\mu +\gamma }-\frac{\theta (x_1+\beta )}{(\alpha +x_2)(y_2+\gamma )}\\ I^{\ast }=\frac{N(y_2+\gamma )}{(y_1+y_2+\mu +\gamma )\theta }-\frac{x_1+\beta }{\alpha +x_2}\end{array}$$

(12)

The propagation equilibrium point is $Q_1(S^{\ast },E^{\ast },I^{\ast })$. Substituting $Q_1$ into the matrix, the calculation is obtained:

$$J(Q_1)=\left(\begin{array}{ccc}-(\alpha +x_2)I^{\ast }-(x_1+\beta )& 0& -(\alpha +x_2)S^{\ast }\\ (\alpha +x_2)I^{\ast }& -(y_1+y_2+\mu +\gamma )& (\alpha +x_2)S^{\ast }\\ 0& y_2+\gamma & -\theta \end{array}\right)$$

(13)

Similarly, it can be solved that all three eigenvalues of $J(Q_1)$ have negative real parts. Therefore, it can be shown that the equilibrium point $Q_1(S^{\ast },E^{\ast },I^{\ast })$ of the speculative psychological propagation in medical institutions is globally asymptotically stable. The propagation threshold $R_0=\frac{(\alpha +x_2)(y_2+\gamma )N}{(x_1+\beta )\theta (y_1+y_2+\mu +\gamma )}$ is found to be influenced by the strategy choices of local governments and medical waste disposal enterprises, and different propagation thresholds could predict the evolution of speculative psychology in medical institutions.

5. Situational Simulation and Numerical Simulation Analysis

By simulating the actual development process of medical waste disposal crisis, we analyzed the behavioral decision-making choices of local governments and disposal enterprises as well as the evolution of speculative psychology of medical institutions under different situational states. The initial values of the parameters refer to the actual situation of a medical waste disposal enterprises in Zhenjiang City, Jiangsu Province combined with the development of the main subjects in the process of medical waste disposal. The unified unit was thousand yuan, and the relevant parameters were assigned as follows.

5.1. The Numerical Simulation Results under Scenario 1

The equilibrium condition for the game behavior of both subjects at this point are $C_{g1}-C_{g2}>R_3+\left(1-a\right)F$ and $C_{b2}-C_{b1}>W+aF$, suppose $C_{g1}=50$, $C_{g2}=30$, $R_3=10$, $a=0.6$, $F=20$, $C_{b2}=300$, $C_{b1}=250$, $W=20$. The initial value states of local governments and disposal enterprises are 0.4, 0.6, and 0.8, respectively, and the evolution time $T=\left[0,1\right]$. The numerical simulation results are shown in Figure 2.

Concurrently, suppose $N\approx 0.4$, $\theta =0.2$, $\alpha =0.5$, $\beta =0.1$, $\gamma =0.6$, $\mu =0.1$. The initial states of various types of medical institutions are 0.4, 0.2, 0.2, 0.2, and the evolution time $T=\left[0,30\right]$. The numerical simulation results are shown in Figure 3.

The simulation results in Figure 2a,b show that whatever the initial states of local governments and disposal enterprises are, they eventually evolve to 0, namely, the stable strategies are relaxed regulation and low-quality disposal. Meanwhile, the evolutionary stability point for the combination of local governments and disposal enterprises strategies is (0,0). Figure 3a,b reveals that the choice of low-quality disposal strategy by disposal enterprises leads to a speculative psychology among medical institutions, and the number of medical institutions with $E$ status gradually increases. However, the local governments’ relaxed regulatory policy will further prompt medical institutions to put their speculative mentality into action and generate speculative behavior. Moreover, the number of medical institutions in the system with status $E$ gradually decreases, and the number with status $I$ shows an increasing trend. There is a serious phenomenon of opportunism in the system, leading to more and more disorder in the operation of medical institutions and disposal enterprises, ultimately making the medical waste disposal crisis experience further vicious evolution.

5.2. The Numerical Simulation Results under Scenario 2

The equilibrium condition for the game behavior of both subjects are $R_3+\left(1-a\right)F>C_{g1}-C_{g2}$ and $C_{b2}-C_{b1}>W+F$, suppose $R_3=10$, $a=0.6$, $F=30$, $C_{g1}=50$, $C_{g2}=40$, $C_{b2}=300$, $C_{b1}=250$, $W=20$, $F=20$. The initial value states of local governments and disposal enterprises are 0.4, 0.6, and 0.8, respectively, with evolution time $T=\left[0,1\right]$. The numerical simulation results are shown in Figure 4.

Concurrently, suppose $N\approx 0.4$, $\theta =0.2$, $\alpha =0.2$, $\beta =0.6$, $\gamma =0.5$, $\mu =0.1$. The initial states of various types of medical institutions are 0.4, 0.2, 0.2, 0.2, and the evolution time $T=\left[0,30\right]$. The numerical simulation results are shown in Figure 5.

The simulation results in Figure 4a,b show that whatever the initial states of local governments and disposal enterprises are, they eventually evolve to 1. However, the incentive of local governments has little effect on compensating the extra cost of high-quality disposal, and the disposal enterprises will choose low-quality disposal based on interest maximization goal, which eventually evolves to 0. The evolutionary stability point of the strategy combination of local governments and disposal enterprises is (1,0), namely, the stable strategies are strict regulation and low-quality disposal. Figure 5a,b reveals that the choice of disposal enterprises for low-quality disposal means that there are still some medical institutions who put speculative psychology into action at the initial stage, and the number of medical institutions in $I$ status gradually increases. While in the later period, the local governments’ strict regulatory policy will prompt medical institutions to eliminate speculative behavior and dispose medical waste according to the standardized process. Concurrently, the number of medical institutions with status $E$ in the system gradually decreases, while the number of medical institutions with status $I$ shows a decreasing trend. Finally, there is a small amount of opportunistic behavior phenomenon in the system.

5.3. The Numerical Simulation Results under Scenario 3

The equilibrium condition for the game behavior of both subjects are $R_3<C_{g1}-C_{g2}$ and $C_{b2}-C_{b1}<W+aF$, suppose $R_3=20$, $a=0.6$, $F=30$, $C_{g1}=50$, $C_{g2}=20$, $C_{b2}=300$, $C_{b1}=260$, $W=20$, $F=50$. The initial value states of local governments and disposal enterprises are 0.4, 0.6, and 0.8, respectively, evolution time $T=\left[0,1\right]$. The numerical simulation results are shown in Figure 6.

Concurrently, suppose $N\approx 0.4$, $\theta =0.2$, $\alpha =0.2$, $\beta =0.2$, $\gamma =0.2$, $\mu =0.4$. The initial states of various types of medical institutions are 0.4, 0.2, 0.2, 0.2, and the evolution time $T=\left[0,30\right]$. The numerical simulation results are shown in Figure 7.

The simulation results through Figure 6a,b show that whatever the initial state of the local governments is, its evolutionary trend does not change and eventually evolves to 0. Moreover, disposal enterprises gradually evolve to high-quality disposal and eventually evolve to 1. The evolutionary stability point of the combination of local governments and disposal enterprises strategy is (0,1), namely, the stable strategies are relaxed regulation and high-quality disposal. Figure 7a,b reveals that when the disposal enterprises choose high-quality disposal of medical waste, the disposal enterprises supervise the medical waste handed over by medical institutions, and thus the medical institutions with $E$ status in the system show a decreasing trend. Moreover, when local governments choose relaxed regulation, it can prompt some medical institutions to take risks in pursuit of high profits and put speculative psychology into action; the $I$ state medical institutions gradually stabilize at a particular value. At this moment, the number of medical institutions with this state gradually decreases, and the number of medical institutions with state $I$ shows a decreasing trend but stabilizes at a specific value, and some opportunistic behavior phenomena still exist in the system.

5.4. The Numerical Simulation Results under Scenario 4

The equilibrium condition for the game behavior of both subjects are $R_3+\left(1-a\right)F>C_{g1}-C_{g2}$ and $C_{b2}-C_{b1}>W+F$, suppose $R_3=40$, $a=0.6$, $F=50$, $C_{g1}=50$, $C_{g2}=40$, $C_{b2}=300$, $C_{b1}=250$, $W=20$, $F=50$. The initial value states of local governments and disposal enterprises are 0.4, 0.6, and 0.8, respectively, evolution time $T=\left[0,1\right]$. The numerical simulation results are shown in Figure 8.

Concurrently, suppose $N\approx 0.4$, $\theta =0.2$, $\alpha =0.2$, $\beta =0.6$, $\gamma =0.2$, $\mu =0.6$. The initial states of various types of medical institutions are 0.4, 0.2, 0.2, 0.2, and the evolution time $T=\left[0,30\right]$. The numerical simulation results are shown in Figure 9.

The simulation results in Figure 8a,b show that whatever the initial state of local governments is, its evolutionary trend does not change and eventually evolves to 1. Moreover, under the strict control of the government, the behavior of disposal enterprises gradually evolves to become high-quality. The evolutionary stability point of the strategy combination of local governments and disposal enterprises is (1,1), namely, the stable strategies are strict regulation and high-quality disposal. Figure 9a,b reveals that disposal enterprises choose high-quality disposal; thus, the medical institutions in the system status show a decreasing trend. Moreover, when local governments choose to strictly regulate, most disposal enterprises will not choose to take risks and gradually turn to standardized treatment of medical waste under the dual constraints of local government’s strict regulation as well as the sense of social responsibility. At this point, the number of medical institutions in the system with status gradually decreases, and the number of medical institutions with status $I$ infinitely tends to 0. The irregularities of disposal of the participating subjects in the disposal chain are rectified, and the phenomenon of speculative behavior in the system almost eliminates.

5.5. Results Discussion

In the game system between local governments and disposal enterprises, the behavioral interaction mechanisms between the two subjects are complex and diverse. (0,0), (0,1), (1,0), (1,1) are possible to be ESS when specific conditions are satisfied. It is consistent with the findings of Yuan et al. (2022) [60] in other area. Among them, when $R_3+(1-a)F>C_{g1}-C_{g2}$ and $C_{b2}-C_{b1}>W+F$, the local governments gradually evolve towards a strict regulatory strategy, while the disposal enterprises gradually evolve to a high-quality disposal strategy under the high-pressure control of the government. (1,1) is the equilibrium stabilization point, indicating that the local governments choose the strategy of “strict regulation” and the disposal enterprises choose the strategy of “high quality disposal”, which is the most ideal state.

In addition, the above simulation results show that different interaction behaviors between local governments and disposal enterprises have different effects on the speculative psychology of medical institutions. This is a further extension of the study by Lv et al. (2022) [9]. The interaction of local governments’ “relax regulation” and disposal enterprises’ “low-quality disposal” will trigger medical institutions to generate opportunistic psychology in the sorting and recycling process, prompting them to put their speculative psychology into action and then generate speculative behavior. Consequently, this will lead to more chaos in the operation of medical institutions and disposal enterprises, making the medical waste disposal crisis further evolve. This is consistent with the study by Yang et al. (2021) [10] in another area. Compared with the “relaxed regulation and high-quality disposal” behavior of local governments and disposal enterprises, the “strict regulation and low-quality disposal” behavior has more significant effect on the short-term mitigating effect of medical waste disposal crisis. However, under such interactive behavior, although local governments have chosen the “strict regulation” strategy, disposal enterprises still pursue high returns through low-quality disposal strategy, which can lead to cyclical fluctuations in the speculative psychology of medical institutions as time evolves. Inevitably, the medical waste disposal crisis will remain a vicious evolution. The interaction between local governments’ “strict regulation” and disposal enterprises’ “high quality disposal” will have a greater deterrent effect on medical institutions, prompting them to dispose of medical waste timely and efficiently. At the same time, if local governments implement proactive regulatory strategies and disposal enterprises comply with the treatment and continuously improve the quality of waste disposal, the medical waste disposal crisis could be effectively controlled. The research in [52,53] showed similar results. In summary, the local governments’ “strict regulation” and disposal enterprises’ “high-quality disposal” is the most preferable solution to eliminate the speculative mentality of medical institutions, improve the quality of medical waste disposal, and achieve the effective containment of the medical waste disposal crisis.

6. Conclusions

Taking the control of the medical waste disposal crisis as the research perspective, this study constructed a medical waste disposal crisis model by combining evolutionary game and an SEIR infectious disease model. Then, we explored the game relationship of strategic choices between local governments and disposal enterprises and discussed the evolutionary path of the speculative psychological propagation state of medical institutions under different behavioral decisions of both subjects. Based on the simulation of different scenarios with parameter assignment, we analyzed the conditions and processes of system evolution to a benign steady state. The research results showed that: (1) The interaction behavior of local governments’ “relaxed regulation” and disposal enterprises’ “low-quality disposal” will prompt the widespread spread of speculative psychology in medical institutions, and ultimately, the medical waste disposal crisis will continue to evolve viciously. (2) Compared with the “relaxed regulation and high-quality disposal” behavior of local governments and disposal enterprises, the “strict regulation and low-quality disposal” behavior has a more significant effect on the short-term mitigating effect of medical waste disposal crisis. However, as time evolves, the speculative mentality of medical institutions will fluctuate cyclically, and the medical waste disposal crisis will still evolve viciously. (3) The interaction behavior of “strict regulation” by local governments and “high-quality disposal” by disposal enterprises maximizes the effectiveness of long-term containment of the medical waste disposal crisis. The system gradually evolves to a benign and stable state. To summarize, the medical waste disposal crisis will be effectively curbed.

Given the above, this study proposes the following management opinions.

First, local governments should improve the efficiency of supervision, and consider the use of big data, blockchain, and other technical means as well as guide the public, the media, and other multiple subjects to supervise the disposal enterprises’ behavior. Second, local governments should design reasonable reward and punishment mechanisms to guide disposal enterprises to high-quality disposal behavior by increasing the amount of fines and incentives (such as giving tax incentives, etc.) to eliminate the speculative mentality of medical institutions. Third, under the premise of an efficient cooperation mechanism formed by local governments and disposal enterprises, local governments should also propaganda the importance of high-quality disposal of medical waste for environmental safety as well as public health using various social media platforms to stimulate the compliance willingness of medical institutions.

However, it should be noted that this study also has some limitations that need to be further improved and strengthened. On the one hand, the nature of evolutionary games determines the information asymmetry among game subjects, and the design of a shared mechanism of governance with the participation of other stakeholders (e.g., the public and the media) deserves further discussion. On the other hand, this study focused on four states of the emergence of speculative psychology in the medical institutions without subdividing their states. Future study will further subdivide the state of medical institutions and investigate the multi-dimensional dynamic relationship between the different states of medical institutions to optimize the SEIR epidemic model.