Introduction

Chinese healthcare institutions were mandated to establish fever clinics following the SARS epidemic 2003 to enhance epidemic prevention and control systems. The Coronavirus Disease 2019 (COVID-19) pandemic underscored significant operational vulnerabilities in these clinics. Alleviation of the epidemic prevention policies in late 2022 by introducing the “Twenty Measures to Optimise Prevention and Control” 1 and the “New Ten Measures”,2 coupled with increasing viral transmissibility, resulted in a surge in infections, overwhelming primary healthcare institutions. The long-standing underinvestment in fever clinics exacerbated these challenges, resulting in clinic closures and refusals to accept patients.3 Consequently, local governments advised the public to avoid visiting primary healthcare facilities,4 causing these clinics to fail in their screening and diversion functions. This impediment to the hierarchical diagnosis and treatment system forced many febrile patients to visit fever clinics in non-primary healthcare institutions. However, these non-primary clinics encounter challenges such as layout inefficiencies and inadequate equipment, among other issues,5 hindering the establishing of an effective treatment system during large-scale infectious disease outbreaks. Additionally, the patient influx created medical surges and secondary complications, such as healthcare rationing6 and nosocomial infections.7 In response, the State Council’s Joint Prevention and Control Mechanism issued policies mandating the establishment of fever clinics at all eligible healthcare institutions above the secondary level and prohibiting arbitrary closures.8,9 Meanwhile, local governments increased their investments in fever clinics and consultation rooms to address public health limitations.10

In fact, during the COVID-19 pandemic, fever clinics in other countries also faced numerous challenges. For example, Singapore’s fever clinics — Public Health Preparedness Clinics (PHPCs) struggled with shortages of medical resources.11 In Japan, infectious disease clinics turned away febrile patients due to capacity limitations.12,13 Similarly, fever clinics in Italy were overwhelmed by an influx of patients, disrupting the hierarchical diagnosis and treatment system and leaving critically ill febrile patients without adequate care.14 These challenges highlight that the development and optimization of fever clinics have become a pressing issue in the field of public health.

Currently, with the gradual waning of the COVID-19 pandemic in China, discussions on the future direction of fever clinics in the post-pandemic era have resurfaced. Multiple healthcare managers and clinicians have expressed concerns regarding the operational cost-benefit balance owing to the decreased number of febrile patients and the general reluctance of medical staff to rotate in fever clinics,15 causing stagnation in various medical alliances. The underperformance of fever clinics is detrimental to the monitoring, prevention, and orderly functioning of the hierarchical diagnosis and treatment system for infectious diseases, as evidenced by subsequent peaks in Influenza A16 and Mycoplasma pneumoniae infections.17 Considering the potential threat of a future “Disease X”,18 ensuring the continuous operation and efficient response of fever clinics is a crucial and urgent issue.

Literature Review

Previous studies of fever clinics have typically focused on topics such as architectural design,19 nursing management,20,21 and resource allocation.22 Few scholars have investigated the operations of fever clinics and their critical issues. For example, Ding Hui23 comprehensively reviewed the complex processes and cross-infection issues in fever clinics, detailing the processes of pre-examination, triage, diagnosis, treatment, and prevention, and proposed optimization strategies and recommendations. Few researchers have examined fever clinic operations from the perspective of multi-stakeholder collaborations involving public participation. Scholars typically focused on single entities, such as the government,24 healthcare institutions,25 or patients.26 However, the efficient and high-quality operation of fever clinics within a tiered medical system requires coordination between various levels of healthcare institutions and government, in addition to the public’s role in supervising these operations.27 Especially, previous studies have confirmed that enhancing public health literacy and awareness of supervisory participation can promote the exercise of supervisory power,28 thereby encouraging the active involvement of all stakeholders in coordinated responses to epidemic prevention and control.

Certain scholars have acknowledged the importance of government supervision and incentives as factors influencing decision-making.29,30 Nonetheless, government supervision and incentives require dynamic adjustment to adapt to changes in different scenarios. For example, Fan31 introduced a dynamic incentive mechanism that considered changes in the social environment and epidemic prevention and control situations in a simulation experiment. A comparison of decision changes under dynamic and static incentive mechanisms demonstrated that the dynamic mechanism more effectively addressed varying social environments and epidemic prevention situations. Current research predominantly explores potential influencing factors from the interaction perspectives between government incentives, public supervision, and healthcare institutions;31 however, it lacks sufficient consideration of specific factors such as nosocomial infections7 and medical surges causing healthcare congestion.32

Researchers often focus on single scenarios, such as pre-epidemic normalcy or emergency states.6 However, the risks and impact of fever vary across scenarios due to epidemics’ cyclical nature. Only a few scholars have recognized the necessity of considering multiple temporal contexts. Xiao Hongju33 analyzed the differences in work characteristics and management models of fever clinics during routine prevention periods and epidemic outbreaks and proposed operational mechanisms and strategies tailored to different scenarios. Moreover, previous research has overlooked issues such as rigidity in transitioning between “normal” and “emergency” scenarios, a lack of flexibility and mobility, and a disconnection between peacetime and wartime modes. A significant gap exists in research on the “transitional scenario” of health emergencies.

Regarding research methods, scholars commonly adopt evolutionary game theory for studies involving multiple agents, particularly when data is difficult to obtain or quantify directly.31,34 The collaborative response to epidemic control in China’s fever clinics involves various stakeholders, including the government, primary and non-primary healthcare institutions, and the public. These stakeholders engage in strategic decision-making characterized by bounded rationality and complex interactive game dynamics, aligning with the assumptions of dynamic evolutionary game theory. Nevertheless, evolutionary game studies specifically targeting fever clinics are scarce.

From a theoretical perspective, previous scholars have primarily relied on evolutionary game theory and collaborative governance theory for multi-stakeholder interactions. The evolutionary game theory can simulate the strategy choices of parties under bounded rationality and accurately predict their final stable evolutionary strategies using replicator dynamic equations.35,36 The collaborative governance theory advocates the participation of multiple stakeholders in the governance of public affairs through cooperation to maximize governance effectiveness.37 Combining these two theories within a tiered medical system allows further analyses of the games and collaborative interactions among fever clinics at different levels.

In summary, previous research reveals several gaps:

Lack of focus on fever clinics at different levels of healthcare institutions and insufficient consideration of multi-stakeholder collaboration, particularly concerning government and public supervision. The non-dynamic, static consideration of key factors, such as government supervision, subsidies, and public participation across different scenarios, ignores the need for dynamic adjustments. Limited application of multi-stakeholder evolutionary game simulation methods in fever clinic studies. Insufficient consideration of typical risk factors in fever clinics during infectious disease outbreaks.

Therefore, we aimed to address these gaps by focusing on multi-stakeholder collaborative governance to ensure efficient epidemic prevention and control. We examined strategy choices in the collaborative governance of fever clinics involving the government, primary and non-primary healthcare institutions, and the public along with integrating a hierarchical diagnosis and treatment system. Moreover, we examined risk factors, such as nosocomial infections and medical surges. After that, we constructed a tripartite evolutionary game model across different scenarios to simulate the impact of dynamic adjustments in government subsidies, punishments, and public supervision on stakeholders’ decisions. This study examines the evolution of key stakeholders’ roles in fever clinics during the COVID-19 pandemic, exploring multi-party collaborative governance strategies for epidemic prevention and control. These strategies are designed to ensure the provision of equitable and orderly healthcare services to the public, offering scientific foundations for managing future potential infectious disease outbreaks, such as “Disease X”.

Materials and Methods Problem Description

This study addressed the challenges encountered by fever clinics in participating in collaborative epidemic prevention and control, particularly regarding chain reactions and secondary issues arising from inefficiencies in the tiered healthcare system. We constructed an evolutionary game model involving different stakeholders under various scenarios and analyzed the evolutionarily stable strategies of each entity during the collaborative operation of fever clinics, as shown in Figure 1. The following primary questions were addressed:

What emergency strategies should fever clinics adopt at different levels of healthcare institutions for pandemic prevention and control in different scenarios? How do government incentives and punishments influence the strategic choices of entities in an emergency response chain during sudden public health events? How does public supervision, considered a variable in the tripartite evolutionary game model of “government-primary healthcare institutions-non-primary healthcare institutions”, affect the strategic choices of the three entities?

Figure 1 Theoretical framework of tripartite evolutionary game model.

Model Assumptions

Assumption 1: The evolutionary game model involves three stakeholders: primary healthcare institutions, non-primary healthcare institutions, and governments. Each stakeholder operates with bounded rationality and makes independent decisions to maximize their benefits. In addition, they can learn and adapt to environmental changes by adjusting their strategies accordingly.

Assumption 2: Primary healthcare institutions can adopt two strategies for fever clinics in epidemic prevention: actively participate in the coordination of epidemic prevention and control (probability x) or negatively participate (probability 1 - x, where 0 ≤ x ≤ 1). All patients with fever were assumed first to visit a primary healthcare institution (all were considered infectious). Primary healthcare institutions can promptly diagnose and treat all patients and refer severe cases to non-primary healthcare institutions. “Active participation in collaboration” involves operating fever clinics and engaging in epidemic control, with operating costs and benefits denoted as Cz1 and R1. During participation, they incur costs (Cr) but gain benefits from treating patients with fever (E1), receiving subsidies from non-primary healthcare institutions (Mt), and receiving government subsidies (αSg, where α is the distribution coefficient). “Negatively participating in collaboration” involves negatively running fever clinics, refusing patients with fever, and referring all patients to non-primary healthcare institutions, avoiding the risk of in-hospital infections but facing a public discovery probability (L1, 0 ≤ L1 ≤ 1), resulting in average losses (P1L1) and government punishments (W1). They also gained additional benefits (V1) from other services.

Assumption 3: Non-primary healthcare institutions have the same strategic options: actively participating in epidemic response coordination (probability y) or negatively participating (probability 1 - y, where 0 ≤ y ≤ 1). “Actively participating in coordination” involves operating fever clinics and promptly treating patients with severe fever referred from primary institutions. The costs and benefits are Cz2 and R2. Actively participating in the coordination incurs costs (C2) but yields benefits from treating patients with fever (E2) and receiving government subsidies (1 - α) Sg, in addition to subsidies to primary institutions (Mt). “Negatively participating in coordination” involves negatively running fever clinics, avoiding costs but gaining additional benefits (V2) from other services. However, they face in-hospital infection risks (K) with average losses (Kb, where b is infection probability) and risks of patient complaints and government supervision (P2, W2), with probabilities (L2, 0 ≤ L2 ≤ 1), resulting in average losses (P2L2, W2L2). If both institution types in a tiered healthcare system participate, they gain synergy benefits (T), such as efficient resource use and improved treatment, distributed by a coefficient (r, 0 ≤ r ≤ 1). If primary institutions do not participate, non-primary institutions risk resource strain, thereby incurring basic losses (C3). If neither participates, additional losses (C4) occur, with losses varying with resource–strain probability (d).

Assumption 4: Assume the total number of patients (h) includes a proportion of mild cases (q, 0 ≤ q ≤ 1). The maximum capacity of a non-primary healthcare institution is defined as n. Non-primary institutions face a resource strain when h exceeds n. If primary institutions participate in the coordination, non-primary institutions effectively avoid strain, expressed mathematically as h (1 - q) ≤ n. This assumes that primary institutions share the patient load, thus reducing the burden on non-primary institutions. Conversely, if primary institutions do not participate, the probability (d) of exceeding n reflects the risk of resource strain without assistance, which is influenced by factors such as patient number fluctuations and actual capacity limits.

Assumption 5: The government can choose between strict and lenient supervision strategies, with probabilities z and 1 - z (0 ≤ z ≤ 1), respectively. Strict supervision involves government monitoring of the participation of primary and non-primary healthcare institutions in coordinated pandemic response efforts to ensure effective emergency health management. The government’s basic operational costs and benefits are Cg and Rc, respectively. When choosing strict supervision, the government incurs monitoring costs Cs, imposes punishments W1 and W2 on institutions that do not participate, and provides incentive subsidies Sg to those that do, with a distribution coefficient α (α ∈ [0,1]). Lenient supervision means that the government does not monitor the participation of healthcare institutions in the pandemic response unless there are public reports of misconduct. The public and government will respond if the institutions’ lack of participation causes a social impact (probability L3, L3 ∈ [0,1]), leading to corresponding losses P3L3, such as reputational damage, punishments from higher authorities, and the maintenance of logistics and supplies. Additionally, under lenient supervision, the government will provide subsidies for Sg due to concerns regarding its reputation, regardless of healthcare institutions’ participation in the pandemic response. The government will gain social benefits Ec from the active collaboration of healthcare institutions only if both primary and non-primary institutions choose to participate in a pandemic response.The relevant parameters above are shown in Table S1. In addition, based on these assumptions, the expected payoffs of primary healthcare institutions, non-primary healthcare institutions, and the government under different strategy choices are summarized in Table 1.

Table 1 The Payoff Matrix of the Evolutionary Game Model Between Government, Primary, and Non-Primary Healthcare Institutions

Model Analysis Stability Analysis of Primary Healthcare Institutions’ Strategies

Based on the aforementioned research hypotheses, we constructed a game payoff matrix with mixed strategies for primary healthcare institutions, non-primary institutions, and the government, as shown in Table 1.

Evolutionary Equilibrium Strategy Trend of Primary Healthcare Institutions

The payoff matrix of the mixed strategy game in Table 1 is as follows:

The expected benefit for the primary healthcare institution in choosing the “actively participate in collaboration” strategy is , and the expression is as follows:

(1)

The expected return for choosing the “negatively participate in collaboration” strategy is , and the expression is as follows:

(2)

The average expected return of the primary healthcare institution is . The expression is as follows:

(3)

Therefore, the replication dynamic equation of primary healthcare institutions is as follows:

(4)

The first-order derivatives of x and the set G(z) are as follows:

(5)

(6)

Using the stability theorem of differential equations, the probability of a primary healthcare institution choosing to actively participate in collaboration in a steady state must satisfy F(x) = 0 and d(F(x))/dx < 0. Since ∂G(z)/∂z= W1-W1L1+αSg > 0, G(z) is an increasing function for z. Thus, when z = z = (V1+Cr-W1L1-L1P1-E1hq+qhMt-rTy)/(W1-W1L1+αSg), G(z)=0, at which point d(F(x))/dx ≡ 0 and the primary healthcare institution cannot determine a stabilization strategy. According to Eq. (4) ~ (6), When z < z*, G(z)<0, at which point dF(x)/dx|x=0<0 and x=0 is the Evolutionary Stable Strategy (ESS), which is defined as a strategy that is uninvadable under evolutionary dynamics.38 Conversely, when z > z*, G(z) > 0, at which point dF(x)/dx|x=1<0 and x = 1 is the ESS. The evolutionary phase diagram of the primary healthcare institution’s strategy is shown in Figure 2.

Figure 2 Phase diagram of the evolution of primary healthcare institution.

Volume A1 (VA1) represents the probability that primary healthcare institutions choose to participate negatively in the collaboration, as shown in Figure 2. Volume A2 represents the probability that they will actively participate in collaboration. These are calculated as follows:

(7)

(8)

(9)

Corollary 1: The likelihood of primary healthcare institutions actively participating in coordinated pandemic response efforts is negatively correlated with the costs of operating fever clinics, the additional benefits of non-cooperation, and the subsidies obtained through patient referrals. It is positively correlated with the benefits of coordinating with non-primary healthcare institutions, government subsidies, the average loss due to patient complaints regarding non-cooperation, and revenue from treating patients with fever.

Proof: The probability x of the primary healthcare institution to choose the “actively participate in collaboration” strategy, ie, , is obtained by taking the partial derivatives of , , , , , and . The following was calculated:

(10)

(11)

(12)

(13)

(14)

(15)

Therefore, is an increasing function of , , and . That is, as , , and increase, gradually increases, and the probability of the primary healthcare institution choosing the “actively participate in collaboration” strategy increases. Alternatively, as , , and increase, gradually decreases.

Evolutionary Equilibrium Strategy Trend of Non-Primary Healthcare Institutions

The expected benefits for non-primary healthcare institutions in choosing to “actively participate in collaboration” and “negatively participate in collaboration” are and , respectively.

(16)

(17)

Therefore, the replication dynamic equation of a non-primary healthcare institution is as follows:

(18)

The replication dynamic equation of a non-primary healthcare institution is as follows:

(19)

The first-order derivatives of x and the set J(z) are as follows:

(20)

(21)

Using the stability theorem of differential equations, the probability of a non-primary healthcare institution choosing to participate actively in collaboration in a steady state must satisfy F(y) = 0 and d(F(y))/dy < 0. Since ∂ J(z)/∂z =W2+Sg-W2L2-αSg > 0, J(z) is an increasing function for z. Thus, when z = z = (C4d-V2-C2+Kb+W2L2+L2P2+Tx-C4dx-rTx)/(W2L2+αSg-W2-Sg), J(z) = 0, at which point d(F(y))/dy = 0 and the non-primary healthcare institution cannot determine a stabilisation strategy. According to Eq. (19) ~ (21), When z < z*, J(z) < 0, at which point dF(y)/dy|y = 0 < 0 and y = 0 is the ESS. Conversely, when z > z*, J(z) > 0, at which point dF(y)/dy|y =1 < 0 and y = 1 is the ESS. An evolutionary phase diagram of the strategy of non-primary healthcare institutions is shown in Figure 3.

Figure 3 Phase diagram of the evolution of non-primary healthcare institution.

Volume B1 (VB1) represents the probability that non-primary healthcare institutions will choose to actively participate in the collaboration, as shown in Figure 3. Volume B2 (VB2) represents the probability of negative participation in the collaboration. These are calculated as follows:

(22)

(23)

Corollary 2: The probability of non-primary healthcare institutions participating in a coordinated pandemic response is negatively correlated with the costs of maintaining fever clinics and the additional benefits of non-cooperation. It is positively correlated with government subsidies, coordination benefits with primary healthcare institutions, losses from hospital-acquired infections, the probability of such infections, and the average loss from patient complaints during non-cooperation.

Proof: The probability y that the non-primary healthcare institution will choose the “actively participate in collaboration” strategy, ie, , is obtained by taking the partial derivatives of , , , and . The following was calculated: , , , , , and . Therefore, as , , and increase, gradually increases, and the probability of the non-primary healthcare institution choosing the “actively participate in collaboration” strategy increases. Alternatively, as and increase, gradually decreases.

Government Strategy Stability Analysis

The government’s expected benefits V31 as cooperators, V32 as defectors, and average expected benefits V33 are as follows:

(24)

(25)

(26)

Therefore, the replicator dynamic equation for the government is as follows:

(27)

Taking the first-order derivative of F(z) yields the following:

(28)

The set K(y) is as follows:

(29)

Using the stability theorem of differential equations, the probability of the government choosing strict supervision in the steady state must satisfy F(z) = 0 and d(F(z))/dz < 0. Since ∂ K(y)/∂y = W2+Sg-W2L2-αSg+L3P3x>0, K(y) is an increasing function of y. Thus, when y=y=[(Cs-W1-W2-Sg+W1L1+W2L2-L3P3) +x(W1-W1L1+αSg)]/(W2L2+αSg-L3P3x-W2-Sg), K(y) = 0, at which point d(F(z))/dz ≡ 0 and the government cannot determine a stabilization strategy. According to Eq. (27) ~ (29), when y > y*, K(y) > 0, at which point d(F(z))/dz|z=0<0 and z=0 is the ESS. Conversely, when y < y*, K(y) < 0, at which point d(F(z))/dz|z =1 < 0 and z = 1 is the ESS. An evolutionary phase diagram of the government is shown in Figure 4.

Figure 4 Phase diagram of the evolution of government.

As shown in Figure 4, volume C1 (VC1) represents the probability that the government will choose to supervise strictly. Volume C2 (VC2) indicates the probability of strict supervision. These are calculated as follows:

(30)

(31)

Corollary 3: The probability of strict government supervision positively correlates with the average loss from patient complaints under lax supervision and negatively correlates with supervision costs.

Proof: The probability y that the government will choose the “strict supervision” strategy, ie, , is obtained by taking the partial derivatives of and . According to Eq. (30) ~ (31), the following is calculated: , . Therefore, as increases, gradually increases, and the probability of the government choosing the “strict supervision” strategy increases. Alternatively, as increases, gradually decreases.

Evolutionary Stability Analysis of the System

According to Friedman38 (1991), ESSs occur only among pure strategies, excluding mixed strategies. Therefore, eight pure strategy points can be obtained: B1 (0, 0, 0), B2 (1, 0, 0), B3 (0, 1, 0), B4 (0, 0, 1), B5 (1, 1, 0), B6 (1, 0, 1), B7 (0, 1, 1), and B8 (1, 1, 1). The Jacobian matrix of the three-party evolutionary game system is as follows:

(32)

Among this:

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

Based on Lyapunov’s theorem,39 the stability of a replicated dynamic system can be determined by the eigenvalues of a Jacobian matrix composed of a set of replicated dynamic equations, as Eq. (13) ~ (41). The equilibrium point of the ESS occurs if all the eigenvalues of the Jacobian matrix are negative. Table 2 presents the stability analysis of the eight pure-strategy equilibria.

Table 2 Equilibrium Point Stability Analysis

Table 2 shows that the equilibrium point B8 (1,1,1) has both positive and negative eigenvalues, rendering it evolutionarily unstable The equilibrium points B1 (0,0,0), B2 (1,0,0), and B3 (0,1,0) hold if government supervision costs are very high, which is unrealistic. Thus, this study focuses on four evolutionarily stable points that satisfy the premise conditions and reflect real-world scenarios: B4 (0,0,1), B5 (1,1,0), B6 (1,0,1), and B7 (0,1,1).

Equilibrium Point B4 (0,0,1)

In normal scenarios with low pandemic risk, the public is unable to play an effective role in supervision and reporting. Therefore, the government becomes the leading force in coordinating emergency health responses. In the absence of public supervision and the threat of emergencies, primary healthcare institutions benefit more from not participating in coordinated responses, ie, E1hq-Cr+Sgα<V1-W1+Mthq-L1P1. Similarly, non-primary healthcare institutions also benefit more from not participating, as follows: (1-α) Sg-C2<V2-Kb-L2P2-W2-C4d, they are more likely to choose non-participation. Consequently, W1-Cr-V1+αSg+L1P1+E1hq-Mthq<0, W2-C2+Sg-V2+C4d-αSg+Kb+L2P2<0. Additionally, under normal scenarios, the cost of government supervision is relatively low. Assuming it is much lower than the punishments for healthcare institutions, that is, -Sg-L3P3< (1- L1) W1+(1- L2) W2-Cs, all eigenvalues at the equilibrium point B4 (0,0,1) are less than 0, making it a stable point.

Equilibrium Point B6 (1,0,1)

As the number of patients with an infectious disease increases, the epidemic shifts to a transitional scenario. The public has become increasingly aware of the importance of coordinated emergency health responses and supervision, and the probability of the government choosing to supervise also rises. As the first point of contact, primary healthcare institutions encountering more patients during the pandemic must balance the need to avoid public scrutiny losses with the relatively low cost of participating in coordinated responses at this stage. Thus, the net benefit of their participation outweighs non-participation, ie, V1-W1-L1P1+Mthq<E1hq-Cr+αSg. Due to this stage, the epidemic risk is still manageable or not fully recognized, and normal healthcare orders are mostly unaffected. Non-primary healthcare institutions find the costs of non-participation lower and continue their strategy under normal scenarios: (1-α) Sg+W2+(1-r) T-C2<V2-Kb-L2P2. Meanwhile, although public awareness of supervision is gradually increasing, the overall level is still relatively low. Assuming that the government’s punishment for non-primary healthcare institutions is much greater than their supervision costs and subsidies, and (1-L2)*W2-Cs-(1-α) Sg>0. B6 (1,0,1) emerges as a stable point in the evolutionary game.

Equilibrium Point B7 (0,1,1)

As the epidemic progresses and the number of patients rises, primary healthcare institutions, the first point of contact, often negatively participate owing to limited capacity and risk management, ie, E1hq-Cr+αSg+rT<V1-W1-L1P1+Mthq. However, this negative response leads to an influx of patients into non-primary healthcare institutions. As the final line of defense, if these institutions avoid the responsibility of treating patients it could lead to immeasurable serious consequences and punishments. Therefore, they are compelled to actively participate in epidemic control, ie, V2-Kb-L2P2-C4d<(1-α) Sg+W2-C2. Furthermore, it is assumed that when -αSg < (1-L1) W1-Cs, the equilibrium point B7 (0,1,1) will become a stable point.

Equilibrium Point B5 (1,1,0)

In the “emergency scenario”, public awareness of the importance of multiparty collaboration for health and safety increases. Public supervision and reporting by healthcare institutions increases. Thus, even with minimal government supervision, the benefits of active epidemic control participation for these institutions are higher: V1-(W1+P1)L1+Mthq<E1hq-Cr+rT and V2-Kb-(W2+P2)L2<(1-r)T-C2. Consequently, the probability of active participation increases. The government tends to adopt a more lenient supervisory role since both types of institutions choose collaboration.

Parameter Setting

We used MATLAB R2023a to conduct a simulation based on the actual values to verify the effectiveness of the evolutionary stability strategy. We analyzed the evolution process of the tripartite agent behavior and its sensitivity to changes in key parameters under different scenarios.

Regarding parameter settings, we set the monthly operating cost, income, and other related parameters of fever clinics under normal scenarios according to Tian40 (2017), ensuring that the ESS points meet stable and balanced conditions under various scenarios. We combined the current consumer price index and expert consultation and assigned different values to the relevant parameters in other scenarios by approximate conversion, as shown in Table 3. And, the initial willingness of the primary healthcare institutions, non-primary healthcare institutions, and government to participate x = y = z = 0.5.

Table 3 Parameter Settings in Different Scenarios

Results Dynamic Evolution Results

Based on the aforementioned values of the relevant parameters satisfying the stability conditions (Table 3). We validated the robustness of the results by testing different iteration numbers (10, 50, and 100). Specifically, when the iterations were set to 10, 50, and 100, the model’s output remained consistent, indicating that the evolutionary process had already converged, and further increases in iterations did not significantly affect the outcome. Based on these results and consistent with prior studies, we ultimately selected 50 iterations as a sufficient and reliable representation of the evolutionary process.41 The strategy combinations of the tripartite evolutionary game finally converge to B4 (0,0,1), B6 (1,0,1), B7 (0,1,1), and B5 (1,1,0), indicating that the simulation is consistent with the analysis of strategic stability and has guiding significance for a coordinated response to epidemic prevention and control. Figure 5 shows the results of this evolution.

Figure 5 Parameter simulation of equilibrium points (evolved 50 times). (a) Converge to point B4 (0,0,1); (b) Converge to point B6 (1,0,1); (c) Converge to point B7 (0,1,1); and (d) Converge to point B5 (1,1,0).

Sensitivity Analysis

We conducted a parameter sensitivity analysis to explore the effects of government subsidy intensity (Sg), punishment intensity (W1, W2), and public supervision probability (L1, L2) on the evolution of coordinated response strategies for primary and non-primary healthcare institutions participating in epidemic prevention and control.

The Impact of Government Subsidy Intensity

Under normal scenarios (B4 (0,0,1)), the level of government subsidy intensity, Sg, was set to 10, 30, 50, and 70 for the simulation. The results are shown in Figure 6. As government subsidies increased, the tripartite strategy combination gradually changed from (0,0,1) to (1,0,1) and (1,1,1) and finally ended with (1,1,0) with an increase in government subsidies. Increasing government subsidies during the evolution process increases the probability of primary and non-primary healthcare institutions actively participating in the coordinated response to epidemic prevention and control. However, in the later stages of the evolution, government subsidies exceeding a certain threshold result in negative supervision.

Figure 6 The effect of government subsidy intensity (Sg=10, 30, 50, 70) on the behavior evolution of three stakeholders under normal scenarios (B4 (0,0,1)). (a) The trend of Sg; (b) The evolutionary process of Sg.

The Impact of Government Punishment Intensity

In the simulation of different scenarios, the severity of punishments W1 and W2 by the government was set to (5, 10, 20, 30) and (20, 30, 40, 50), respectively. Under normal scenarios (B4 (0,0,1)), an increase in government punishment will incline primary and non-primary healthcare institutions to participate in a coordinated response to epidemic prevention and control, as shown in Figure 7a and b. In the pre-transition scenario (B6 (1,0,1)), as the government’s punishment continues to increase, the strategic choices of non-primary healthcare institutions and the government fall into a cycle (when the government chooses the “1” strategy, non-primary healthcare institutions choose the “0” strategy), and both tend toward the ESS (1,1,0) (Figure 7c). In the post-transition scenario, when the government’s punishment exceeds a certain threshold, the strategy of active participation in epidemic prevention and control and coordinated response tends to stabilize. The government constantly hesitates between the “0” strategy and the “1” strategy but finally chooses negative supervision (as shown in Figure 7d).

Figure 7 The effect of government punishment intensity on the behavior evolution of three stakeholders under different scenarios. (a) Impact of W1 under normal scenarios (B4 (0,0,1)); (b) Impact of W2 under normal scenarios (B4 (0,0,1)); (c) Impact of W2 under pre-transition scenarios (B6 (1,0,1)); (d) Impact of W1 under post-transition scenarios (B7(0,1,1)).

The Impact of Public Supervision Probability

The public supervision probabilities L1 and L2 under normal scenarios (B4 (0,0,1)) were set as (0.1, 0.3, 0.5, 0.7) and (0.05, 0.35, 0.65, 0.95), respectively. During the evolution process, the increase in L1 under normal (B4 (0,0,1)) will gradually cause primary healt