Symbiotic Mechanism of Multiple Subjects for the Resource-Based Disposal of Medical Waste in China in the Post-Pandemic Context

Abstract

In the post-pandemic era, the continuous growth in the rate of medical waste generation and the limited capacity of traditional disposal methods have posed a double challenge to society and the environment. Resource-based disposal is considered an efficient approach for solving these problems. Previous studies focused on the methods of medical waste disposal and the behavior of single stakeholders, lacking consideration of cooperation among different stakeholders. This study establishes an evolutionary game model of the resource-based disposal of medical waste to analyze the behavioral decision evolution of governments, medical institutions, and disposal enterprises. This study also explores the influencing factors in the achievement of the symbiotic state and investigates the conditions that participants need to meet. The results show that joint tripartite cooperation can be achieved when the subsidies and penalties from governments are sufficient, as well as the efficiency of resource-based disposal, which can effectively promote the evolution of the three subjects from the state of “partial symbiosis” to the state of “symbiosis”. However, the resource-based classification level cannot directly change the symbiotic state of the system due to the goal of minimizing cost and risk. When evolutionary subjects have reached the state of “symbiosis”, the improvement in the classification level can enhance the willingness of disposal enterprises to choose the resource-based classification strategy. Under such circumstances, governments reduce their corresponding level of intervention. At this time, the whole system is in a more idealized symbiotic state.

1. Introduction

Medical waste is mainly produced from medical activities, such as preventive health care [1]. In China, the rapid development of the health care sector led to a sharp increase in the volume of medical waste. The unexpected outbreak of the COVID-19 pandemic exacerbated the situation [2], and with the ongoing small-scale virus transmission in the post-pandemic era, this has not yet been alleviated [3]. According to the annual report published by the Ministry of Ecology and Environment of China, the annual generation of medical waste has surged from 250,000 tons to 1.79 million tons in the last decade, and it is predicted to rise to 2.496 million tons in 2023 [4], which may lead to a “waste siege” dilemma and pose a threat to society and the environment [5]. Due to its complicated components, infectiousness, and toxicity, medical waste requires closed-loop handling under government supervision, which is usually sorted by medical institutions and transported to disposal enterprises for terminal treatment [6]. There are normally two kinds of approaches to medical waste disposal: thermal treatment (incineration and pyrolysis) and landfill [7]. However, certain cities are facing the issue of limited landfill capacity, giving rise to a crisis in disposal [8]. Because of the conflict between the continuous growth of waste generation and insufficient capacity of traditional disposal methods, this paper argues that resource-based disposal can solve the dilemma to a certain extent, since it can convert the valuable part of medical waste into renewable resources, with residues being treated in a hazard-free manner [9,10]. At the same time, the energy generated in the disposal process can also be used [11].

Pilot implementations of resource-based disposal projects have begun in China. For example, three waste-to-energy plants have been built in Xiamen, which can dispose of around 250,000 discarded masks per day and generated 48 million kWh of electricity in 2020. However, resource-based disposal projects remain controversial [12]. The goals of the stakeholders are not aligned. While governments are concerned about the environmental and social impact of the projects, medical institutions and disposal enterprises focus more on the profit obtained, that is, the return on investment. Their decisions influence each other, and agreement is reached only when their interests are all satisfactorily met. Therefore, it is crucial to study, in depth, how to realize the joint participation of all parties (e.g., governments, medical institutions, and disposal enterprises) in resource-based disposal projects to maximize both economic and environmental benefits and achieve sustainable development [13]. Based on the symbiotic definition of “synergistic coexistence” proposed in biology [14,15] and the concept of “cooperation for mutual benefit” extended to the social sciences [16], we define the joint participation status as “symbiosis”. Similar to the symbiotic mechanism for rural waste management that lies in the synergistic possibilities and collaborative benefits [17], the crucial point of the symbiotic mechanism for resource-based disposal is to resolve the benefits contradiction among three parties in order to reach mutually profitable cooperation, which is also a multi-object game process.

Many scholars hitherto have paid much attention to the problems of generation control and the efficient disposal of medical waste, including the prediction of quantity and generation rate [18,19], the technical optimization of classification, transportation, and disposal [20,21,22,23,24], the selection of disposal sites [25,26], the management mechanisms [27], the economic benefits [28], and the environmental impact [29]. However, much of the existing research emphasizes centralized and harmless disposal in public health emergencies, aiming to reduce hazards such as disease transmission and environmental pollution [30]. By contrast, studies on the resource-based disposal of medical waste are rarely found. Currently, resource-based disposal has been investigated in studies of solid waste, including municipal solid waste [31], food waste [32], construction waste [33,34], and electronic waste [35], with a focus on how waste can be transformed into new resources through special technological treatment, then re-produced and sold [31]. Additionally, scholars have explored the feasibility and optimization of policies [36,37], technical pathways [38], and operational mechanisms [39]. In the healthcare field, the application of resource-based disposal is more stringent and difficult due to the potentially hazardous nature of medical waste. The value of its application has been demonstrated by Chauhan et al. [40] through exploring the economic benefits gained from the resource-based treatment of medical waste. Several scholars have investigated the technological pathways for the implementation of resource-based disposal, including the reuse of packaging, plastic, and metal particles in other production areas [41,42,43,44], and the conversion of medical waste into energy such as electricity [45]. Moreover, some studies have examined the efficiency of existing resource-based disposal programs for medical waste, for example, evaluating the contribution of electricity and district heating generated by waste-to-energy plants in some countries [46,47,48].

To sum up, the operational mechanism of resource-based disposal projects has been studied for other types of waste; however, it has rarely been studied with regard to medical waste. Although existing studies demonstrate the feasibility and prospects of developing the resource-based treatment of medical waste, they mostly concentrated on the disposal methods and behaviors of disposal enterprises. The cooperation among different stakeholders, including disposal enterprises, governments, and medical institutions, has been overlooked, as have been the difficulties and influencing factors in resource-based projects.

To achieve the maximum joint effort of all stakeholders and solve the practical problems of management, in this paper, we introduce evolutionary game theory. Evolutionary game theory considers the limited rationality and information asymmetry among participants, describing the dynamic evolutionary process behind the behavioral decisions of the decision-making groups and their interaction mechanisms [49]. At present, evolutionary game theory has been widely applied in the research of pandemic control [50,51], waste disposal management [52], waste recycling management [53,54], and other areas. For example, Lv et al. [55] provided insights into different reward and penalty models by establishing a game system between governments and disposal enterprises, proposing the ideal static reward and dynamic penalty model of medical waste management. Ma et al. [56] developed a dynamic evolutionary game model of construction enterprises and recycling enterprises and found that subsidies to construction enterprises were essential to promote construction waste recycling in China. To better describe the decisions of stakeholders, many scholars have combined evolutionary game theory and symbiosis theory in the field of industrial symbiosis to study the interaction mechanisms of game subjects. For example, Shan et al. [57] and Zhao et al. [58] established evolutionary game models of stakeholders in industrial symbiotic systems. They explored the influence of different factors on the reduction of low-carbon emissions and analyzed the interaction mechanism of the subjects’ strategies. Zhang et al. [59] constructed a multi-group symbiotic evolutionary model of an innovation ecosystem and analyzed the dynamic mechanism to achieve equilibrium. To summarize, an ideal symbiotic status means that multiple subjects can achieve benign interaction and cooperation in dynamic evolution and mutually benefit from each other. From the literature listed above, this paper finds that few studies apply evolutionary game theory to the resource-based disposal of medical waste, not to mention analyze the symbiotic mechanism of the situation where multiple subjects are involved.

According to the existing literature, some limitations should be specified. Firstly, most studies on resource-based disposal focus on the behaviors of disposal enterprises and governments, overlooking their cooperation with medical institutions. The three parties’ behavioral strategies and interaction mechanisms are also rarely discussed. Secondly, few studies combine symbiotic theory and evolutionary game theory to study the symbiotic mechanisms and influencing factors of resource-based disposal.

To address the research gaps mentioned above, this study incorporates the three subjects and considers the resource-based factors of medical waste disposal to examine the multiparty symbiosis. This study constructs a tripartite evolutionary game model, and then explores the internal evolutionary process of the three subjects. Finally, the symbiotic state of the three parties and the essential conditions in different scenarios are analyzed. The main problems to be solved in this study are as follows:

(1)

What is the symbiotic state among governments, medical institutions, and disposal enterprises, and what are its characteristics?

(2)

What factors influence symbiosis among governments, medical institutions, and disposal enterprises?

(3)

How best can we achieve and maintain the symbiosis among governments, medical institutions, and disposal enterprises?

Based on the existing research results, the main contributions of this study are as follows:

(1)

Different from the existing studies on medical waste disposal during the pandemic [60], this study investigates the resource-based disposal of medical waste in the post-pandemic era in order to broaden the theoretical research perspective.

(2)

This study establishes a tripartite evolutionary game model, which consists of governments, medical institutions, and disposal enterprises, and combines it with symbiosis theory. The study also divides the subjects’ states into three major categories and investigates the symbiotic mechanism of multiple subjects. Additionally, the results obtained widen the application of symbiosis theory and evolutionary game theory.

(3)

This study sets several parameters affecting the symbiotic state. From the results, corresponding countermeasures to three subjects can be proposed, respectively. Hence, it might be said the paper provides insights for the future development of the resource-based disposal of medical waste.

The paper is structured as follows. Section 2 establishes a tripartite evolutionary game model. Section 3 explores the symbiotic state of multiple subjects and equilibrium points in different scenarios. Section 4 conducts a numerical simulation, analyzes the effects of different factors and the paths toward achieving symbiosis, and discusses these themes. In Section 5, conclusions, limitations, and suggestions for future research are presented.

2. Problem Description and Underlying Assumptions

2.1. Problem Formulation

The core participants in the resource-based disposal of medical waste can be listed as governments, medical institutions, and disposal enterprises. Among them, the specific division of work is as follows: On the generation side of medical waste, medical institutions pre-sort the waste generated in medical activities. The disposal enterprises are on the final side of medical waste, undertaking the main work of resource-based disposal. Both of their behaviors are directed by governments as a matter of policy. Based on the basic process flow and evolutionary game theory, this study sets the strategic choices for the three main subjects as follows.

(1)

The strategies of governments are “insist” and “not insist”, abbreviated as $A_1,A_2$, respectively. The “insist” strategy refers to the requirement for governments to intervene in the behavior of medical institutions and disposal enterprises. Governments provide medical institutions and disposal enterprises with incentive subsidies when they actively participate in resource-based classification or disposal processes. In contrast, governments will penalize them accordingly. The “not insist” strategy means that governments do not take any action.

(2)

The strategies of medical institutions are “classify strictly” and “classify simply”, abbreviated as $B_1,B_2$, respectively. The “classify strictly” strategy refers to medical institutions sorting the generated waste according to resource-based criteria, with the valuable part of medical waste being sorted and categorized in advance. Under this strategy, medical institutions have to pay for staff training, sorting equipment, and other costs. The amount of sorted waste meeting resource-based disposal standards correlates with their sorting efforts. The “classify simply” strategy means that medical institutions only sort according to the conventional criteria.

(3)

The strategies of disposal enterprises are “agree” and “disagree”, abbreviated as $C_1,C_2$, respectively. The “agree” strategy refers to disposal enterprises being willing to pay additional costs for equipment updates and personnel training based on conventional disposal. Medical waste is converted into other resources or energy through technical methods. The profits generated from resource-based disposal will be returned to governments and medical institutions in the form of tax and price reductions, respectively. The “disagree” strategy means that disposal enterprises only carry out conventional medical waste disposal.

2.2. Basic Assumptions of the Model

Based on the above description of the subjects’ behavior strategies, the following assumptions are made in this study.

Assumption 1: In the “natural” environment, governments, medical institutions, and disposal enterprises all have limited rationality, considering their decision-making behaviors according to their interests. In a repeated game process with three subjects, when governments select the “insist” strategy, they are able to identify the behavioral strategies of both sides accurately and will reward and punish the behaviors of medical institutions and disposal enterprises related to resource-based disposal.

Assumption 2: For governments, the probability of choosing strategy $A_1$ is $x \left(0\le x\le 1\right)$, and the probability of choosing strategy $A_2$ is $1-x$. The primary revenue of governments in medical waste disposal is $R$. Referring to the regulatory cost, it is considered as $T$ when governments do not insist on resource-based disposal. Conversely, the regulatory cost of insisting on resource-based disposal is $KT$, which is influenced by the degree of adherence to resource-based disposal $K$. Meanwhile, when governments choose the “insist” strategy, they will moderately subsidize the behaviors of medical institutions and disposal enterprises related to resource-based disposal, with the subsidy of $K{P}_1$ and $K{P}_2$, respectively. Accordingly, governments will also punish medical institutions and disposal enterprises that demonstrate a reluctance to participate in resource-based disposal, the degree of which is $K{F}_1$ and $K{F}_2$, respectively.

Assumption 3: For medical institutions, the probability of choosing strategy $B_1$ is $y \left(0\le y\le 1\right)$, and the probability of choosing strategy $B_2$ is $1-y$. Referring to Chen et al. [61], the additional classification cost of strict classification is $\frac{1}{2}L{H}^2$. $L$ and $H$ refer to the cost coefficient of medical waste classification and the level of effort in classification according to resource-based criteria, respectively. To simplify the model calculation, it is assumed that the amount of medical waste with resource-based disposal potential is $D=1+HJ$, where “$1$” refers to the amount of medical waste under the “classify simply” strategy and $J$ refers to the sensitivity coefficient of medical waste volume with respect to the level of sorting effort. In addition, the basic disposal fee of medical waste paid by medical institutions to disposal enterprises is $G$. The potential loss of an administrative penalty for simple classification is $K{F}_1$, and the potential gain of incentive subsidy received for strict classification is $K{P}_1$.

Assumption 4: For disposal enterprises, the probability of choosing strategy $C_1$ is $z \left(0\le z\le 1\right)$, and the probability of choosing strategy $C_2$ is $1-z$. Referring to the model design practice of Chen et al. [62], the conventional disposal cost of medical waste is $W$, and the additional input cost for resource-based disposal is $Q{D}^2$, where $Q$ means the cost coefficient affected by the waste volume of resource-based disposal. Through the resource-based disposal of medical waste, additional economic benefits, $MD$, can be created, and $M$ indicates the benefit coefficient affected by resource-based disposal efficiency. When governments accurately identify the resource-based disposal behaviors for medical waste, the benefits transferred in the form of tax and price reduction to the governments and medical institutions by disposal enterprises are $\alpha MD$ and $\beta MD$, respectively. Conversely, the benefits transferred to governments and medical institutions with a probability of $\theta$ are $\theta \alpha MD$ and $\theta \beta MD$, respectively. In addition, the incentive subsidy for enterprises participating in resource-based disposal is $K{P}_2$, and the penalty for enterprises that refuse to carry out resource-based disposal is $K{F}_2$.

The definitions of the assumed parameters are listed in

Table 1. Definition of assumed parameters.

Subject	Parameter	Definition
Governments	$R$	Primary revenue of disposal
	$T$	Regulatory costs under a “not insist” strategy
	$K$	Degree of adherence to resource-based disposal
	$P_1$	Subsidies for medical institutions’ resource-based classification
	$P_2$	Subsidies for disposal enterprises’ resource-based disposal
	$F_1$	Penalties for medical institutions not carrying out resource-based classification
	$F_2$	Penalties for waste disposal enterprises not carrying out resource-based disposal
Medical institutions	$H$	Level of effort in resource-based classification
	$J$	Sensitivity coefficient of the waste volume
	$D$	The amount of medical waste with resource-based disposal potential
	$G$	Basic disposal fee
	$L$	Cost coefficient of medical waste classification
Disposal enterprises	$W$	Conventional disposal cost
	$Q$	Cost coefficient affected by the waste volume
	$M$	Benefit coefficient affected by disposal efficiency
	$\theta$	Probability of benefits being transferred
	$\alpha$	Benefits transferred to governments
	$\beta$	Benefits transferred to medical institutions

as follows.

3. Construction and Analysis of the Model

3.1. Model Parameters and Payment Matrix

Based on the above research assumptions and parameter settings, the payment matrix of the three subjects under different strategies is shown in

Table 2. Tripartite game payment matrix.

Strategy Portfolio	Governments	Medical Institutions	Disposal Enterprises
$(A_1,B_1,C_1)$	$R-KT-K{P}_1-K{P}_2+\alpha MD$	$-G-\frac{1}{2}L{H}^2+K{P}_1+\beta MD$	$G-W-Q{D}^2+K{P}_2+\left(1-\alpha -\beta \right)MD$
$(A_1,B_1,C_2)$	$R-KT+K{F}_2$	$-G-\frac{1}{2}L{H}^2$	$G-W-K{F}_2$
$(A_1,B_2,C_1)$	$R-KT+K{F}_1-K{P}_2+\alpha M$	$-G-K{F}_1+\beta M$	$G-W-Q+K{P}_2+\left(1-\alpha -\beta \right)M$
$(A_1,B_2,C_2)$	$R-KT+K{F}_1+K{F}_2$	$-G-K{F}_1$	$G-W-K{F}_2$
$(A_2,B_1,C_1)$	$R-T+\mathsf{\alpha }\mathsf{\theta }\mathrm{MD}$	$-G-\frac{1}{2}L{H}^2+\beta \theta MD$	$G-W-Q{D}^2+\left(1-\alpha -\beta \right)\mathsf{\theta }MD$
$(A_2,B_1,C_2)$	$R-T$	$-G-\frac{1}{2}L{H}^2$	$G-W$
$(A_2,B_2,C_1)$	$R-T+\alpha \theta M$	$-G+\beta \theta M$	$G-W-Q+\left(1-\alpha -\beta \right)\mathsf{\theta }M$
$(A_2,B_2,C_2)$	$R-T$	$-G$	$G-W$

.

3.2. Construction of Tripartite Evolutionary Game Model

According to the Malthusian dynamic equation theorem [63], the expected return and the replicating dynamic equations of the three subjects can be calculated.

Assuming that the expected and average expected returns of government “insist” and “not insist” strategies are $E_{A1}$, $E_{A2}$, and $\overline{E_A}$, respectively, the equations are calculated as follows.

$$\begin{array}{ll}{E}_{A1}=& yz\left(R-KT-K{P}_1-K{P}_2+\alpha MD\right)\\ & +y\left(1-z\right)\left(R-KT+K{F}_2\right)+\left(1-y\right)z\left(R-KT+K{F}_1-K{P}_2+\alpha M\right)\\ & +\left(1-y\right)\left(1-z\right)\left(R-KT+K{F}_1+K{F}_2\right)\end{array}$$

(1)

$$\begin{array}{ll}{E}_{A2}=& yz\left(R-T+\alpha \theta MD\right)+y\left(1-z\right)\left(R-T\right)\\ & +\left(1-y\right)z\left(R-T+\alpha \theta M\right)+\left(1-y\right)\left(1-z\right)\left(R-T\right)\end{array}$$

(2)

$$\overline{E_A}=x{E}_{A1}+\left(1-x\right)E_{A2}$$

(3)

Based on the evolutionary stabilization strategy of the replication dynamic equation, the replicating dynamic equation of governments $U\left(x\right)$ can be calculated as:

$$\begin{array}{ll}U\left(x\right)=& x\left(E_{A1}-\overline{E_A}\right)=x\left(1-x\right)[\left(1-K\right)T+z\alpha M\left(yD+1-y\right)\left(1-\mathsf{\theta }\right)\\ & +\left(1-y\right)K{F}_1+\left(1-z\right)K{F}_2-yzK{P}_1-zK{P}_2]\end{array}$$

(4)

Similarly, the replicating dynamic equation of medical institutions $U\left(y\right)$ and the replicating dynamic equation of disposal enterprises $U\left(z\right)$ can be derived as follows.

$$U\left(y\right)=y\left(1-y\right)-\frac{1}{2}L{H}^2+xK{F}_1+xzK{P}_1+\left[x+\left(1-x\right)\theta \right]z\left(D-1\right)\beta M$$

(5)

$$U\left(z\right)=z\left(1-z\right)-yQ{D}^2-\left(1-y\right)Q+xK{F}_2+xK{P}_2+\left[x+\left(1-x\right)\theta \right]\left[y\left(D-1\right)+1\right]\left(1-\alpha -\beta \right)M$$

(6)

3.3. Analysis of Evolutionary Stabilization Strategies of Different Subjects

According to Equations (4)–(6), the first-order derivatives of $\frac{dU\left(x\right)}{dx}$, $\frac{dU\left(y\right)}{dy}$, and $\frac{dU\left(z\right)}{dz}$ are shown below.

$$\begin{array}{ll}\frac{dU\left(x\right)}{dx}=& \left(1-2x\right)[\left(1-K\right)T+z\alpha M\left(yD+1-y\right)\left(1-\theta \right)\\ & +\left(1-y\right)K{F}_1+\left(1-z\right)K{F}_2-yzK{P}_1-zK{P}_2]\end{array}$$

(7)

$$\frac{dU\left(y\right)}{dy}=\left(1-2y\right)-\frac{1}{2}L{H}^2+xK{F}_1+xzK{P}_1+\left[x+\left(1-x\right)\theta \right]z\left(D-1\right)\beta M$$

(8)

$$\begin{array}{ll}\frac{dU\left(z\right)}{dz}=& \left(1-2z\right)-yQ{D}^2-\left(1-y\right)Q+xK{F}_2+xK{P}_2\\ & +\left[x+\left(1-x\right)\theta \right]\left[y\left(D-1\right)+1\right]\left(1-\alpha -\beta \right)M\end{array}$$

(9)

To analyze government strategy, according to the stability theorem of the replicating dynamic equation, the process of strategy adjustment tends to a steady state only when $U\left(x\right)=0$ and $\frac{dU\left(x\right)}{dx}<0$ simultaneously. Based on Equations (4) and (7), we can calculate the result below:

$$z^{*}=\frac{\left(1-K\right)T+\left(1-y\right)K{F}_1+K{F}_2}{K{F}_2+yK{P}_1+K{P}_2-\left(yD-y+1\right)\left(1-\theta \right)\alpha M}$$

(10)

The evolutionary stabilization strategy of governments can be analyzed as follows. When $z=z^{*}$ is met, both $A_1$ and $A_2$ have the potential to evolve into the governments’ stabilization strategy. If $z\ne z^{*}$, there are two cases in which $A_1$ evolves to the governments’ stabilization strategy when $z>z^{*}$, while $A_2$ evolves to the governments’ stabilization strategy when $z<z^{*}$.

Similarly, let $U\left(y\right)=0$ and $\frac{dU\left(y\right)}{dy}<0$, then $x^{*}=\frac{\frac{1}{2}L{H}^2-z\theta \left(D-1\right)\beta M}{K{F}_1+zK{P}_1+z\left(1-\theta \right)\left(D-1\right)\beta M}$. To discuss different cases, we can find that $B_1$ evolves to the medical institutions’ stabilization strategy when $x>x^{*}$, while $B_2$ evolves to the medical institutions’ stabilization strategy when $x<x^{*}$. It is possible for both $B_1$ and $B_2$ to evolve into the medical institutions’ stabilization strategy when $x=x^{*}$.

Repeating the same steps for disposal enterprises gives the following result. For the strategy of disposal enterprises, $y^{*}=\frac{-Q+xK{F}_2+xK{P}_2+\left[x+\left(1-x\right)\theta \right]\left(1-\alpha -\beta \right)M}{Q{D}^2-Q-\left(D-1\right)\left[x+\left(1-x\right)\theta \right]\left(1-\alpha -\beta \right)M}$. $C_1$ evolves to a stabilization strategy when $y>y^{*}$. Conversely, $C_2$ evolves to a stabilization strategy when $y<y^{*}$. Both $C_1$ and $C_2$ tend to evolve to a stabilization strategy when $y=y^{*}$.

3.4. Equilibrium Point Stability Analysis

According to the stability principle of differential equations, let $U\left(x\right)=0,U\left(y\right)=0,U\left(z\right)=0$. There are eight equilibrium points: $E_1\left(0,0,0\right)$, $E_2\left(1,0,0\right)$, $E_3\left(0,1,0\right)$, $E_4\left(0,0,1\right)$, $E_5\left(0,1,1\right)$, $E_6\left(1,0,1\right)$, $E_7\left(1,1,0\right)$, and $E_8\left(1,1,1\right)$. Friedman [63] proposed that the evolutionary stability strategy can be derived from the stability of the Jacobian matrix, which is shown below.

$$\begin{array}{l}J=\left[\begin{array}{ccc}\frac{\partial F\left(x\right)}{\partial x}& \frac{\partial F\left(x\right)}{\partial y}& \frac{\partial F\left(x\right)}{\partial z}\\ \frac{\partial P\left(y\right)}{\partial x}& \frac{\partial P\left(y\right)}{\partial y}& \frac{\partial P\left(y\right)}{\partial z}\\ \frac{\partial G\left(z\right)}{\partial x}& \frac{\partial G\left(z\right)}{\partial y}& \frac{\partial G\left(z\right)}{\partial z}\end{array}\right]\\ =\left[\begin{array}{ccc}\left(1-2x\right)\left[\begin{array}{c}\left(1-K\right)T-yzK{P}_1-zK{P}_2\\ +\left(1-y\right)K{F}_1+\left(1-z\right)K{F}_2\\ +z\left(yD+1-y\right)\left(1-\theta \right)\alpha M\end{array}\right]& x\left(1-x\right)\left[\begin{array}{c}z\left(1-\theta \right)\alpha MD\\ -z\left(1-\theta \right)\alpha M\\ -zK{P}_1-K{F}_1\end{array}\right]& x\left(1-x\right)\left[\begin{array}{c}y\left(D-1\right)\left(1-\theta \right)\alpha M\\ +\left(1-\theta \right)\alpha M\\ -K{F}_2-yK{P}_1-K{P}_2\end{array}\right]\\ y\left(1-y\right)\left[\begin{array}{c}K{F}_1+zK{P}_1\\ +\left(1-\theta \right)z\beta MD\\ -\left(1-\theta \right)z\beta M\end{array}\right]& \left(1-2y\right)\left[\begin{array}{c}-\frac{1}{2}L{H}^2+xK{F}_1+xzK{P}_1\\ +\left(x-\theta x+\theta \right)z\beta MD\\ -\left(x-\theta x+\theta \right)z\beta M\end{array}\right]& y\left(1-y\right)\left[\begin{array}{c}\left(x-\theta x+\theta \right)\beta MD\\ -\left(x-\theta x+\theta \right)\beta M\\ +xK{P}_1\end{array}\right]\\ z\left(1-z\right)\left[\begin{array}{c}K{F}_2+K{P}_2+\\ \left[y\left(D-1\right)+1\right]\left(1-\alpha -\beta \right)M-\\ \theta \left[y\left(D-1\right)+1\right]\left(1-\alpha -\beta \right)M\end{array}\right]& z\left(1-z\right)\left[\begin{array}{c}-Q{D}^2+Q+\\ \left(x-\theta x+\theta \right)\left(1-\alpha -\beta \right)MD-\\ -\left(x-\theta x+\theta \right)\left(1-\alpha -\beta \right)M\end{array}\right]& \left(1-2z\right)\left[\begin{array}{c}-yQ{D}^2-\left(1-y\right)Q+xK{F}_2+xK{P}_2\\ +\left(x-\theta x+\theta \right)\left(1-\alpha -\beta \right)yMD\\ +\left(1-y\right)\left(x-\theta x+\theta \right)\left(1-\alpha -\beta \right)M\end{array}\right]\end{array}\right]\end{array}$$

(11)

According to the Lyapunov criterion, it is known that the equilibrium point is evolutionarily stable (ESS) when all of the eigenvalues of the Jacobian matrix are negative, which means $\lambda <0$. The equilibrium point is non-stable when positive eigenvalues exist. The Jacobian matrix eigenvalues are obtained by substituting each of the eight equilibrium points into the Jacobian matrix, which are shown in

Table 3. Eigenvalues and stability of the Jacobian matrix.

Equilibrium Point	Eigenvalues			Stability
	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$
$E_1\left(0,0,0\right)$	$-\left(K-1\right)T+K{F}_1+K{F}_2$	$-\frac{1}{2}L{H}^2$	$-Q+\left(1-\alpha -\beta \right)\theta M$	Uncertain
$E_2\left(1,0,0\right)$	$\left(K-1\right)T-K{F}_1-K{F}_2$	$-\frac{1}{2}L{H}^2+K{F}_1$	$-Q+K{F}_2+K{P}_2+\left(1-\alpha -\beta \right)M$	Uncertain
$E_3\left(0,1,0\right)$	$-\left(K-1\right)T+K{F}_2$	$\frac{1}{2}L{H}^2$	$-Q{D}^2+\left(1-\alpha -\beta \right)\theta MD$	Unstable
$E_4\left(0,0,1\right)$	$-\left(K-1\right)T+K{F}_1-K{P}_2+\left(1-\theta \right)\alpha M$	$-\frac{1}{2}L{H}^2+\left(D-1\right)\theta \beta M$	$Q-\left(1-\alpha -\beta \right)\theta M$	Uncertain
$E_5\left(1,1,0\right)$	$\left(K-1\right)T-K{F}_2$	$\frac{1}{2}L{H}^2-K{F}_1$	$-Q{D}^2+K{F}_2+K{P}_2+\left(1-\alpha -\beta \right)MD$	Uncertain
$E_6\left(1,0,1\right)$	$\left(K-1\right)T-\left(1-\theta \right)\alpha M-K{F}_1+K{P}_2$	$-\frac{1}{2}L{H}^2+K{F}_1+K{P}_1+\left(D-1\right)\beta M$	$Q-K{F}_2-K{P}_2-\left(1-\alpha -\beta \right)M$	Uncertain
$E_7\left(0,1,1\right)$	$-\left(K-1\right)T+\left(1-\theta \right)\alpha MD-K{P}_1-K{P}_2$	$\frac{1}{2}L{H}^2-\left(D-1\right)\theta \beta M$	$Q{D}^2-\left(1-\alpha -\beta \right)\theta MD$	Uncertain
$E_8\left(1,1,1\right)$	$\left(K-1\right)T-\left(1-\theta \right)\alpha MD+K{P}_1+K{P}_2$	$\frac{1}{2}L{H}^2-K{F}_1-K{P}_1-\left(D-1\right)\beta M$	$Q{D}^2-K{F}_2-K{P}_2-\left(1-\alpha -\beta \right)MD$	Uncertain

.

According to the eigenvalues of the Jacobian matrix in Table 3, it can be found that the eigenvalue ${\lambda }_2$ of equilibrium point $E_3$ is always greater than 0. Therefore, $E_3\left(0,1,0\right)$ is an unstable point. Since the mathematical relationship between the parameters is complicated, the eigenvalues of the remaining seven equilibrium points cannot be determined. As a result, each point may evolve into a stable equilibrium point. Combined with the research of Du et al. [64] on symbiotic logic, this study divides the evolution of equilibrium points into the following three scenarios and analyzes them separately.

Scenario 1: When governments, medical institutions, and disposal enterprises reach an agreement regarding the goal of resource-based disposal, the state of “symbiosis” is formed, and the corresponding equilibrium point is $E_8\left(1,1,1\right)$. $E_8$ evolves into a stable equilibrium when ${\lambda }_1,{\lambda }_2,{\lambda }_3$ are all negative, which means $\frac{1}{2}L{H}^2-K{F}_1-K{P}_1-\left(D-1\right)\beta M<0$, $\left(K-1\right)T-\left(1-\mathsf{\theta }\right)\alpha MD+K{P}_1+K{P}_2<0$, $Q{D}^2-K{F}_2-K{P}_2-\left(1-\alpha -\beta \right)MD<0$ in detail. Under such circumstances, as long as medical institutions improve their efforts in terms of classification, disposal enterprises enhance the efficiency of resource-based disposal, and governments increase their regulatory powers, then governments will tend to choose the “insist” strategy, medical institutions will tend to choose the “classify strictly” strategy, and disposal enterprises will tend to choose the “agree” strategy. Consequently, three subjects are capable of achieving the symbiotic state.

Scenario 2: When all subjects are unsatisfied with their interests and refuse to carry out resource-based behaviors, this state can be described as “non-symbiosis”, and the corresponding equilibrium point is $E_1\left(0,0,0\right)$. If $E_1$ evolves into a stable equilibrium point, it has to satisfy $-\left(K-1\right)T+K{F}_1+K{F}_2<0$ and $-Q+\left(1-\alpha -\beta \right)\theta M<0$. The key factors are governments’ low adherence to resource-based behaviors and disposal enterprises’ low efficiency of resource-based disposal. At this point, all subjects are bound to fail to achieve symbiosis even with the joint effort of medical institutions and disposal enterprises in resource-based disposal.

Scenario 3: When only one or two subject(s) among governments, medical institutions, and disposal enterprises are involved in resource-based behaviors, while the remaining subject(s) do not due to unsatisfied interests, this state is regarded as “partial symbiosis”. A variety of factors affect the choice between these three subjects, including the degree of governments’ insistence on resource-based disposal, the reward and punishment for the behaviors of medical institutions and disposal enterprises, the amount of medical waste sorted by medical institutions according to resource-based criteria, and the efficiency of resource-based disposal by disposal enterprises. Only by adjusting the different factors, it is possible to promote the evolution of the tripartite subjects from “partial symbiosis” to “symbiosis”.

4. Situational Simulation and Numerical Simulation Analysis

According to the above analysis of the evolution path of the equilibrium point in different scenarios, we reveal that the tripartite subjects of resource-based medical waste disposal may evolve into “symbiosis”, “ non-symbiosis”, or “partial symbiosis”. In order to visualize the evolutionary path of the symbiotic state of the three subjects and analyze the influence of different parameters on the symbiotic state of the three subjects, in this section we shall further numerically investigate and discuss the choice of evolutionary strategy with the help of the Matlab R2021b simulation tool. The analysis is based on the evolutionary game model designed above, combined with the actual data of medical waste disposal in Zhenjiang City in China in 2021 to assign values to the relevant parameters.

4.1. Analysis of the Evolutionary Path of Partial Symbiosis

In this section, initial assignment to the system is based on the less desirable “partial symbiosis” state. When $\left(K-1\right)T-K{F}_1-K{F}_2<0$, $-\frac{1}{2}L{H}^2+K{F}_1<0$, $-Q+K{F}_2+K{P}_2$$+\left(1-\alpha -\beta \right)M<0$ are met, the values of the initial parameters are as shown in

Table 4. Initial value of parameters.

Variable	Initial Value	Variable	Initial Value
$T$	10	$J$	0.8
$K$	1.2	$L$	15
$P_1$	0.5	$Q$	15
$P_2$	0.5	$M$	18
$F_1$	1	$\alpha$	0.2
$F_2$	1	$\beta$	0.1
$H$	0.8	$\theta$	0.3

. The uniform unit is 10,000 CNY.

The values are substituted into the model and the initial willingness of governments, medical institutions, and disposal companies are set as 0.3, 0.5, and 0.7, respectively. Evolutionary time $T$ = [0, 2] is used to verify the state of the three-party symbiosis. The simulation results show that the evolutionarily stable state of the system is $E_2\left(1,0,0\right)$, i.e., the three subjects are in a partially symbiotic state of “insist”, “simple classification”, and “disagree”. At this time, the probability of both medical institutions and disposal enterprises choosing the resource-based strategy always develops toward 0, even if the probability of the government insisting on resource-based disposal gradually approaches 1. Figure 1b indicates that the process of the governments’ strategy evolves to 1 slowly, and there is an obvious inflection point in the process of medical institutions and disposal enterprises’ strategy evolving to 0. Therefore, there is room for the evolution of the three subjects to the “symbiosis” state.

4.2. Analysis of Factors Influencing the Symbiotic State of the Three Subjects

Combined with the previous analysis, it demonstrates that the evolution of the tripartite subjects to a “symbiosis” state has room for development and is influenced by several factors. Therefore, in this section, numerical simulations and analyzes will be conducted for the degree of governments’ adherence to resource-based disposal $K$, governments’ rewards and punishments $P_1$, $P_2$, $F_1$, $F_2$ for medical institutions and disposal enterprises, the degree of resource-based disposal effort in resource-based classification $H$, and the efficiency $M$ for disposal enterprises’ resource-based disposal, respectively.

4.2.1. Effect of $P_1$ and $P_2$ on the Results of Evolutionary Game

Gradually adjust the amount of governments’ subsidy $P_1$ for strict classification by medical institutions and $P_2$ for resource-based disposal by disposal enterprises, taking the value of “$P_1=0.5$, $P_2=0.5$”, “$P_1=10$, $P_2=10$”, ” $P_1=20$, $P_2=20$”, respectively, evolutionary time $T=[0,10]$. The simulation results are shown in Figure 2a. From Figure 2a,b, it can be seen that the three subjects are initially in the state of “partial symbiosis”, as $P_1$ and $P_2$ gradually increase to 10 simultaneously, $x$ always converges to 1, and the speed of convergence accelerates with the increase in $P_1$ and $P_2$. In such a scenario, the willingness of medical institutions and disposal enterprises to resource-based behavior is gradually enhanced. However, if disposal enterprises cannot gain satisfactory revenue and give up resource-based disposal, medical institutions will also lose the willingness to resource-based disposal classification because they cannot get enough revenue return. Consequently, the choice of two subjects enters a vicious circle. As is shown from Figure 2c,d, the behavior of medical institutions and disposal enterprises appears to be cyclically oscillating, and both of them cannot form a stable state. When $P_1$ and $P_2$ continue to increase to 20, the behavioral strategies of the three subjects in this system will converge to 1, that is, the system equilibrium point finally converges to $E_8\left(1,1,1\right)$, and the three subjects are in a “symbiosis” state. From the simulation results, it can be seen that the amount of government subsidy $P_1$ for strict classification by medical institutions and $P_2$ for resource-based disposal by disposal enterprises will influence the evolution path of the symbiotic state of the three subjects. When the amount of subsidy reaches a sufficiently high level, it can effectively motivate medical institutions and disposal enterprises to implement resource-based behavior and evolve into the “symbiosis” state of the three subjects.

4.2.2. Effect of $F_1$ and $F_2$ on the Results of the Evolutionary Game

The amount of government penalty for the simple classification of medical institutions $F_1$ and the amount of penalty for disposal enterprises that refuse resource-based disposal can be adjusted by taking the value of “$F_1=1$, $F_2=1$”, “$F_1=5$, $F_2=5$”, and “$F_1=20$, $F_2=20$”, respectively. The simulation results are shown in Figure 3. With the gradual increase in $F_1$ and $F_2$ from 1 to 5, respectively, the stable point of the system evolves from $E_2\left(1,0,0\right)$ to $E_5\left(1,1,0\right)$. When $F_1$ and $F_2$ are sufficiently large and reach 20 simultaneously, the system will evolve to $E_8\left(1,1,1\right)$. Throughout the process, $x$ always converges to 1, and the convergence rate speeds up. It is found that the amount of government penalty for the simple classification of medical institutions $F_1$ and the amount of penalty for disposal enterprises that refuse resource-based disposal have an impact on the evolution path of the symbiotic state of the tripartite subjects. When the penalty increases, it can promote the participation of the tripartite subjects, increasing the number from one party to two parties, and finally allowing the three parties to evolve to the “symbiosis” state.

4.2.3. Effect of $M$ and $\theta$ on the Results of the Evolutionary Game

With other parameters unchanged, the efficiency of resource-based medical waste disposal, $M$, and the probability of resource-based disposal benefits being identified by governments, $\theta$, are taken as “$M=18$, $\theta =0.3$”, “$M=25$, $\theta =0.4$”, and “$M=50$, $\theta =0.5$”, the simulation results are shown in Figure 4. When the efficiency of resource-based disposal, $M$, gradually increases, the evolutionary path of the system changes even if the probability of benefits being identified by governments, $\theta$, increases simultaneously. When “$M=18$, $\theta =0.3$” increases to “$M=25$, $\theta =0.4$”, $z$ converges to 1 and the system evolves to $E_5\left(1,1,0\right)$, at which time disposal enterprises agree to carry out resource-based disposal and generate certain benefits. If the values of $M$ and $\theta$ reach 50 and 0.5, respectively, the system evolves to $E_8\left(1,1,1\right)$. Throughout the process, $x$ always converges to 1, and the convergence speed accelerates with the increase in $M$ and $\theta$. To conclude, with regard to the simulation results, we can find that $M$ and $\theta$ have an impact on the evolution path of the symbiotic state of the three subjects. When the efficiency of resource-based disposal and the probability of resource-based disposal benefits being identified by governments increase, this can promote the participation of the three subjects, increasing the number from one party to two parties in the “partial symbiosis” state, and finally evolving to the “partial symbiosis” state.

4.2.4. Effect of $K$, $H$, and $J$ on the Results of the Evolutionary Game

Combined with the above analysis, the degree of adherence of governments to resource-based classification, $K$, the level of effort in resource-based classification, $H$, and the sensitivity coefficient of waste volume with regard to the level of classification effort, $J$, were adjusted for simulation, with values of “$K=1.2$, $H=0.8$, $J=0.8$ ”, “$K=1.5$, $H=0.85$, $J=0.85$”, and “$K=2$, $H=0.9$, $J=0.9$”, respectively. The simulation result of the system evolution is shown in Figure 5. Even if the values of $K$, $H$, and $J$ increase, $x$, $y$, $z$ always evolve to 0, and the stability point of the system evolves to $E_1\left(0,0,0\right)$. The behavior strategies of the three subjects are “insist”, “strictly classify”, and “disagree”, and the system is in the state of “non-symbiosis” in this situation.

In order to investigate the symbiotic path of the system, the values of the parameters in the symbiotic state above are combined, taking $F_1=20$, $F_2=20$, $P_1=20$, $P_2=20$, $M=5$ and adjusting the values of $K$, $H$ and $J$. The evolution results are shown in Figure 6. $x$, $y$, $z$ always converge to 1, and the stable point of the system evolves to $E_8\left(1,1,1\right)$, which means the three subjects are in a “symbiotic” state. In the process of evolution, the convergence speed of $y$, $z$ accelerates with increasing values, while the convergence speed of $x$ slows down. From the simulation results, it can be seen that under the synergistic effect of other factors, increasing governments’ adherence to resource-based disposal, $K$, the level of medical institutions’ effort in resource classification, $H$, and the sensitivity coefficient of waste volume with regard to the level of classification, $J$, can greatly enhance the willingness of medical institutions and disposal enterprises to carry out resource-based behavior. As a consequence, governments will not intervene too much, forming an ideal “symbiosis” state among the tripartite subjects.

4.3. Results and Discussion

This study demonstrates the different symbiotic states of the three parties in the resource-based disposal of medical waste. Figure 1 shows that when the system evolves to the stable equilibrium point $E_2\left(1,0,0\right)$, only the government insists on the resource-based strategy, and the other subjects do not participate. The “partial symbiosis” state is consistent with the state in the industrial symbiosis system proposed by He et al. [65] and the initial state of the food waste resource-based disposal project proposed by Filimonaua et al. [66]. The reason for the inability of the three parties to reach the “symbiosis” state is that there are some differences of interest between the government and enterprises [64]. To be specific, firstly, in the context of the increased volume of post-pandemic medical waste and contemporary environmental pressure, the government, as a seeker of social well-being, is willing to promote such programs in order to achieve a win–win situation for both the economy and the environment [9]. Secondly, under the current medical waste disposal scheme in China, the selection of enterprises for medical waste disposal and the fee rates are decided by the government through public bidding, which is of a certain public interest, and thus the fees have not reached a high level [6]. In this case, the high cost of resource-based disposal may add to the budgetary burden of disposal enterprises, leading them to disagree with expanding their resource-based operations in order to maintain maximum revenue. Finally, medical institutions are only responsible for the sorting and temporary storage of medical waste, with cost and risk minimization being their main objectives [42]; therefore, the resource-based disposal of medical waste is not the primary choice for medical institutions.

Just as Glew et al. [67] and Costa et al. [68] demonstrated the role of government subsidies and supervision on the efficiency of waste reuse in symbiotic systems in other fields, the simulation results of this study suggest a positive impact of government subsidies and supervision on the promotion of symbiosis among multiple subjects in resource-based medical waste disposal projects. Similarly, this study confirms the effect of resource-based disposal efficiency on system stability in terms of the disposal side, which is in line with the study by Huang et al. [69] in other fields. However, these findings not only ignore the influence of the waste generation side of the process on the evolution of the symbiotic state of the system, but also do not consider the synergistic effects of the various factors. According to the results given in the previous section, the “partial symbiosis” state is influenced by the incentives and penalties of governments, the disposal efficiency of the disposal enterprises, and the probability that governments identify the resource-based benefits. When these factors reach a high level, they can have a positive effect on the willingness of each subject to participate in resource-based behavior and promote the evolution of the tripartite subjects to the “symbiosis” state. In addition, considering the generation of medical waste, we found that only the level of resource-based classification and the cost-sensitivity coefficient of medical institutions had little effect on the symbiotic evolution of the system. When other factors are taken into account, the results show that the classification level at the generation side can effectively increase the willingness of medical institutions and disposal enterprises to choose the “strictly classify” and “agree” strategies, and can promote the system toward the ideal “symbiosis” state, which does not require excessive government intervention.

Globally, many countries have been pushing for recovering energy or resources from waste using reward–penalty mechanisms [70], which is an effective approach, in line with the direction of Chinese policy development. However, this study has certain limitations. Firstly, only three stakeholders in resource-based disposal are selected. Considering the fact that household medical waste is also a major source [54], other stakeholders such as citizens also deserve further exploration to improve the operation mechanism of the resource-based disposal chain. Secondly, this paper does not consider the possible speculative behavior of the participants in the implementation process, such as illegal recycling [71]. Thirdly, resource-based medical waste disposal has yet to become widespread in China, and the subsequent development warrants further empirical research.

5. Conclusions

In order to explore the symbiosis mechanism of multiple subjects in the post-pandemic context, this study establishes a tripartite evolutionary game model among governments, medical institutions, and disposal enterprises, compares the different symbiosis states of multiple subjects, analyzes the factors that affect the evolution of the system from the “partial symbiosis” state to the “symbiosis” state, and explores the choices of behavioral strategy and dynamic evolution paths of multiple subjects. To summarize, conclusions and recommendations are drawn as follows.

(1)

The resource-based medical waste disposal system can be divided into three states: “no symbiosis”, “partial symbiosis”, and “symbiosis”. Considering the current reality in China, most of the resource-based medical waste disposal projects are still in the state of “partial symbiosis” due to the differences in interests between governments and enterprises. In this situation, the government chooses the “insist” strategy, while medical institutions choose the “simply classify” strategy and disposal enterprises choose the “disagree” strategy. There is a clear inflection point in the evolution of the strategy choices of medical institutions and disposal enterprises, which shows that there is room for the development of tripartite symbiosis. Therefore, the resource-based disposal project of medical waste needs some guidance from the government to promote the evolution of the strategic choice of the three parties (insist, strictly classify, and agree), in order to realize the “symbiosis” of the three parties.

(2)

The evolution of the tripartite subjects involved in resource-based medical waste disposal from the “partial symbiosis” to “symbiosis” state is directly influenced by two factors. On the one hand, government subsidies and penalties can promote the symbiosis of the three parties. On the other hand, the resource-based disposal efficiency of the disposal enterprises and the probability of the government identifying the benefits of resource-based disposal also have an influence. When these two factors reach a high level, the tripartite subjects are locked in the symbiosis state. Therefore, in order to achieve and maintain symbiosis, this paper puts forward the following suggestions. The government should build a platform for the resource-based disposal of medical waste and improve the corresponding operation mechanism. Moreover, the government needs to encourage each subject to participate in resource-based disposal projects through tax incentives and other policies, and appropriately supervise and punish those who do not participate. By strengthening industry guidance, the implementation of resource-based disposal projects can be promoted and the development of the symbiotic relationship between medical institutions and disposal enterprises is guided. Disposal enterprises should take measures such as equipment renewal and technological innovation to expand production capacity and improve the efficiency and revenue of resource-based disposal, in order to actively promote the development of the symbiotic relationship between the three parties.

(3)

The level of resource-based classification of medical institutions and their cost-sensitive coefficients do not directly promote the tripartite system from “partial symbiosis” to “symbiosis”. However, when the system is in a “symbiotic” state, it can effectively improve the willingness of medical institutions and disposal companies to choose the resource-based strategy in cooperation with other factors, which can also promote the system to gradually evolve to the ideal “symbiosis” state, requiring less government intervention. In order to achieve this state, medical institutions need to improve the training budget for the personnel involved in resource classification and enhance the level of the relevant resource classification standards. Moreover, they ought to specify and improve the classification system of medical waste, so as to promote the stable development of the resource-based classification network.

In the future, there are more research directions to be explored. First, in the process of resource-based medical waste disposal, other participating subjects, such as citizens, can be considered in the construction of an evolutionary game model, in order to explore the synergistic symbiosis mechanism of multiple subjects. Secondly, it is important to consider the hazards of realistic speculative behavior with regard to the resource-based disposal chain. In addition, as the main research context of this paper is China, it is worth noting that the generalizability of this study in other countries needs to be further examined.