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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Apr 16;16:12516. doi: 10.1038/s41598-026-42052-7 Search in PMC Search in PubMed View in NLM Catalog Add to search Evolutionary game model for public health emergency management in universities Jing Wei Jing Wei 1 School Infermary, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353 China Find articles by Jing Wei 1 , Jiawei Zhou Jiawei Zhou 2 Shandong Institute of Standardization, Jinan, 250014 China Find articles by Jiawei Zhou 2 , Li Zheng Li Zheng 3 School of Mathematics, China University of Mining and Technology, Xuzhou, 221116 Jiangsu China Find articles by Li Zheng 3 , Dong minyi Dong minyi 4 Qilu Institute of Technology, School of Computer and Information Engineering, Jinan, 250200 China Find articles by Dong minyi 4 , Yitong Xiao Yitong Xiao 5 Shandong University of Traditional Chinese Medicine , College of Traditional Chinese Medicine, Jinan, 250355 China Find articles by Yitong Xiao 5 , Ma Boyuan Ma Boyuan 6 Paris Curie BUCT Engineer School, Beijing University of Chemical Technology, Beijing, China Find articles by Ma Boyuan 6 , Qiang He Qiang He 7 Tianjin University of Traditional Chinese Medicine, Tianjin, 301617 China Find articles by Qiang He 7, ✉ , Lili Zhang Lili Zhang 8 Tongfang Knowledge Network Digital Technology Co., Ltd., Beijing, 100083 China Find articles by Lili Zhang 8, ✉ , Lin Song Lin Song 8 Tongfang Knowledge Network Digital Technology Co., Ltd., Beijing, 100083 China Find articles by Lin Song 8, ✉ Author information Article notes Copyright and License information 1 School Infermary, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353 China 2 Shandong Institute of Standardization, Jinan, 250014 China 3 School of Mathematics, China University of Mining and Technology, Xuzhou, 221116 Jiangsu China 4 Qilu Institute of Technology, School of Computer and Information Engineering, Jinan, 250200 China 5 Shandong University of Traditional Chinese Medicine , College of Traditional Chinese Medicine, Jinan, 250355 China 6 Paris Curie BUCT Engineer School, Beijing University of Chemical Technology, Beijing, China 7 Tianjin University of Traditional Chinese Medicine, Tianjin, 301617 China 8 Tongfang Knowledge Network Digital Technology Co., Ltd., Beijing, 100083 China ✉ Corresponding author. Received 2024 Oct 13; Accepted 2026 Feb 24; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13086918 PMID: 41991557 Abstract Public health crises, such as the COVID-19 pandemic, demand effective emergency management within universities, involving collaboration among university departments, students, and Public opinion channel. This study aims to construct and simulate a tripartite evolutionary game model capturing the dynamic strategic interactions of these stakeholders during crises. Utilizing an evolutionary game theory model, the study examines their decision-making behaviors under bounded rationality and incomplete information. Vensim software simulations reveal key findings: (1) moderate government penalties effectively encourage timely information disclosure by university departments; (2) reducing the costs of student complaints influences media verification behaviors; and (3) addressing students’ optimistic biases significantly impacts strategic decision-making across all parties. These insights provide actionable recommendations for universities, including fostering collaborative governance, improving emergency response mechanisms, and enhancing stakeholder communication. This research contributes to the theoretical understanding of stakeholder dynamics and offers practical strategies to enhance the emergency management efficiency of higher education institutions during public health crises. Supplementary Information The online version contains supplementary material available at 10.1038/s41598-026-42052-7. Subject terms: Public health, Quality of life Introduction The outbreak of COVID-19: motivation and context The outbreak of COVID-19 and other major public health crises has profoundly affected public health, social stability, and economic development worldwide. Universities, as densely populated and highly interconnected institutions, hold a pivotal role in managing such emergencies. However, higher education institutions face distinct challenges in this regard, such as the unpredictable scale of crises, the dissemination of misinformation, and the dynamic interplay of stakeholder strategies, including those of university departments, students, and Public opinion channel,referring to decentralized digital platforms such as Weibo, campus forums, anonymous message boards, etc., that amplify sentiment and information within and beyond university communities, and differ significantly from traditional centralized news media. (To clarify terminology, we use "public opinion channel" instead of “news media” to reflect the unique dynamics in Chinese university crisis communication. This term encompasses both campus-based and broader social platforms, distinguishing it from institutionally managed traditional media. The decentralized, participatory, and often anonymous nature of these channels enables rapid sentiment amplification and poses unique verification challenges). While previous studies have extensively explored emergency management mechanisms, most have focused on external stakeholders, such as government authorities and healthcare providers, overlooking the significant role of internal stakeholders in higher education. Internal stakeholders, such as university-related departments and students, play critical roles in shaping institutional responses to public health emergencies. This study addresses this gap by examining how strategic interactions among these internal stakeholders evolve during crises and how they collectively influence emergency outcomes. Research background and existing gaps Public health emergencies, officially defined in China in 2003 following the SARS outbreak, have driven significant advancements in crisis management systems 1 . Over the years, research has focused predominantly on external dimensions of crisis management, including optimizing emergency medical facilities 2 , designing cost-effective warning systems 3 , and improving crisis investments 4 . However, the role of internal stakeholders, particularly in higher education, remains underexplored 5 . Recent studies have highlighted the potential influence of online sentiment and individual behavior on emergency outcomes. For instance, Li et al. 6 demonstrated the impact of social media sentiments on crisis management, while Zhang et al. 7 found that online sentiment could enhance public understanding of emergencies. Within universities, the strategic decisions and sentiments of students have been shown to shape institutional responses, yet little research has been conducted to explore this dynamic in detail. Furthermore, Wang et al. 8 analyzed factors influencing the dissemination of public sentiment during crises, emphasizing the importance of stakeholder strategies, but these findings fail to capture the complex interactions among internal stakeholders in universities. Game theory, particularly evolutionary models, offers a powerful framework for analyzing stakeholder strategies in public health emergencies. For example, Fan 9 applied evolutionary game theory to evaluate decision-making processes in emergency management systems, and Alam et al. 10 demonstrated the effectiveness of isolation strategies using game models. However, few studies have utilized evolutionary game theory to examine internal stakeholder interactions in higher education institutions, representing a critical gap in understanding crisis management within this context. The role and capacity of universities in crisis management In the context of public health emergencies, especially during the pandemic, Chinese universities are not only education providers, but also key units of “localized management” (Shudi Guanli) in the national emergency response system. Unlike open campuses in some areas, universities typically operate as densely populated, semi-gated communities with primary legal and administrative responsibility for the health and safety of students and staff. Why universities handle emergencies: Universities act as the "first line of defense". High student density requires rapid, localized response mechanisms that external government agencies cannot provide immediately. During the pandemic, universities were required to implement closed-loop management, regulate campus access and oversee daily health monitoring. This administrative autonomy makes the university the primary decision-making body (player 1) in our game model, rather than the local government as a supervisor (parameter P). Preparation and organizational skills: To effectively carry out this management, universities rely on specialized organizational structures: Top-down command system: Establish a "leading group for epidemic prevention and control", usually led by the university president or party secretary, responsible for coordinating the overall strategy and resource allocation. Functional departments: The school infirmary (or university hospital) is a professional medical unit responsible for temperature screening, isolation and transfer, and vaccination. At the same time, the Student Affairs Department (made up of counselors) manages the information network to ensure that health data and student sentiment are reported daily. Emergency response mechanisms: Universities are equipped with emergency plans that integrate logistics, security, and medical support, allowing them to immediately transition from regular teaching to emergency management models in the event of a crisis. This hierarchy and resource allocation capacity form the underlying assumptions of our model, demonstrating the role of universities as capable participants in the evolutionary game. Research objectives and contributions This study aims to address these gaps by constructing a tripartite evolutionary game model involving university-related departments, students, and Public opinion channel as key participants. The model captures the dynamic decision-making processes of these stakeholders under conditions of bounded rationality and incomplete information. By incorporating internal dynamics and leveraging simulation tools, this research provides both theoretical and practical insights into improving emergency management in higher education. This study makes three core contributions: (1) Theoretical — construction of a tripartite evolutionary game model among internal stakeholders in universities; (2) Analytical — simulation-based exploration of behavior evolution under crisis scenarios; and (3) Practical — offering strategic recommendations for optimizing emergency management. Modeling framework and structure To address the outlined objectives, this research employs a tripartite evolutionary game model to simulate real-world scenarios in higher education. By manipulating parameters such as government penalties and complaint costs, the study visualizes stakeholder behavior under various crisis conditions. This approach not only enhances the theoretical foundation of emergency management research but also ensures practical applicability in guiding universities toward more effective crisis management strategies. The analytical framework and structure are illustrated in Fig. 1 . Fig. 1. Open in a new tab Modeling roadmap of stakeholder evolution under public health emergencies. Methodology and simulation design This section focuses on model development procedures, parameter assignments, and simulation execution, separate from theoretical underpinnings now covered in Section “ Incomplete information ”. Theoretical background and conceptual model This section outlines conceptual foundations and participant strategy logic prior to formal modeling. Evolutionary game theory is particularly suited for analyzing dynamic strategic interactions among agents operating under bounded rationality and incomplete information. Traditional game theory assumes that agents are fully rational and make decisions based on maximizing utility. However, in real-world scenarios—especially during public health emergencies—agents often have limited cognitive capabilities and access to incomplete information, necessitating the use of evolutionary game theory 11 , 12 . This framework enables the study of how strategies evolve over time based on relative payoffs, rather than assuming optimal decision-making from the outset. In this study, the decision-making processes of university-related departments, students, and Public opinion channel during public health emergencies are characterized by interdependent strategies and evolving behaviors. Previous research has established that the interactions between institutional actors and media platforms are critical to shaping public perceptions during crises 3 . Furthermore, the inclusion of students as active participants reflects their role as information providers and evaluators, as supported by studies on online sentiment analysis and youth risk perception during emergencies 2 . The tripartite model constructed in this study builds upon the principles of evolutionary game theory to capture the complex dynamics among these stakeholders. The following theoretical underpinnings justify the assumptions made in the model: Bounded rationality and behavioral adaptation Agents in the model—university-related departments, students, and social media actors—operate under bounded rationality, a concept introduced by Simon (1957), which suggests that decision-makers, constrained by limited cognitive capacity and incomplete information, tend to adopt “satisficing” strategies rather than seeking globally optimal solutions. In our model, this is reflected in agents’ reliance on past experiences, heuristic learning, and imitation-based adaptation instead of strict utility maximization 4 . In parallel, the model incorporates incomplete information, which refers to stakeholders’ inability to fully observe or verify other players’ payoffs, intentions, or the actual state of public health events. For instance, students may be unaware of whether universities are deliberately concealing information, and social media actors may lack access to reliable sources for verification. Within this framework, bounded rationality provides a theoretical foundation for employing evolutionary dynamics instead of classical optimization models. Stakeholders iteratively adjust their strategies based on observed outcomes, often imitating others or modifying behavior incrementally, rather than calculating the most advantageous course of action in a single step. For example, a student’s decision to report a university incident is not based on perfectly predicting all potential consequences, but rather on perceived outcomes from similar prior situations. Moreover, the assumption of incomplete information is embedded in the subjective and probabilistic nature of stakeholder behavior. Social media actors, for example, may not be able to verify the integrity of university disclosures in real time. This uncertainty justifies the use of evolutionary modeling to capture dynamic and path-dependent shifts in strategies, rather than relying on static, one-shot decision-making frameworks. 2. Incomplete information Stakeholders often lack access to complete and accurate information during public health crises, leading to suboptimal and adaptive decision-making processes 11 . This assumption is particularly relevant for Public opinion channel platforms, which must decide whether to verify information before reporting, balancing verification costs against potential traffic revenue. 3. Dynamic strategic interactions The model assumes that the strategies of university departments, students, and Public opinion channel co-evolve over time, influenced by mutual interactions and feedback mechanisms. Evolutionary game theory provides a robust framework for studying such dynamic systems, allowing for the identification of stable strategy sets (ESS) under various parameter configurations 6 Model assumptions and justification While the model is constructed as a repeated interaction framework, it does not explicitly incorporate the fact that students may be silenced or penalized after initial complaints, especially when dealing with sensitive topics under authoritative pressure. Future improvements may involve adding participation decay functions, dropout probabilities, or finite-horizon dynamics to reflect these realistic institutional constraints. University-Related Departments. The decision to disclose information or partially conceal it is influenced by factors such as government penalties, costs of investigation, and potential accountability measures. Studies have shown that institutional transparency is often contingent on external pressures, including media scrutiny and regulatory oversight 13 , 14 . Students. Students serve as both information gatherers and providers, incurring costs for reporting issues but also gaining rewards (e.g., recognition or incentives). Their behavior is shaped by cognitive biases, such as over-optimism, which can lead to delayed or inconsistent actions during crises 2 . These assumptions align with findings on youth risk perception and behavioral inconsistencies in public health emergencies 3 . Public opinion channel. The “public opinion channel,” represented by digital media and social platforms (e.g., Weibo, campus forums). Given the limited role of traditional news outlets in Chinese university crisis reporting, this category better reflects the decentralized, peer-amplified nature of information dissemination in China’s contemporary media ecosystem. These platforms shape stakeholder behavior through indirect pressure and information amplification, though their verification mechanisms vary.Public opinion channel platforms face a trade-off between verification costs and the potential gains from rapid reporting. This assumption reflects the economic and reputational considerations that drive media behavior, as highlighted in studies on the role of digital platforms during emergencies 5 . The factors included in the model—government penalties (P), student complaint costs (C3), and student optimism bias (V)—are derived from behavioral public administration and crisis management literature. Government penalties represent an institutional constraint mechanism common in top-down Chinese governance structures, while student complaint costs capture the real and perceived effort or risk in engaging with formal channels. Optimism bias is supported by risk perception studies 15 , which indicate youth are likely to underestimate personal risk, especially in distant or abstract crisis scenarios. It is important to clarify that the current model employs a stylized form of infinite or continuous interaction among the three agents, which is a common simplification in evolutionary game theory. This does not imply that each individual student, university official, or media actor repeatedly engages in a single event. Rather, it reflects an ongoing process of collective behavioral adjustment across similar incidents and over time. However, we acknowledge that in reality —in contrast, other possible factors, such as party membership or academic performance, are excluded due to difficulty in quantifying, low public observability, or lack of consistent empirical associations with emergent acts in higher education settings. Despite the oversight role played by education authorities (such as branches of the Ministry of Education), the model is constructed as a repetitive framework of interactions, which does not explicitly incorporate the fact that students may be silenced or punished after the initial complaint, especially when dealing with sensitive topics under authority pressure, while direct interaction with students in early crises is limited, so they are not explicitly modeled. Future improvements may involve adding participation decay functions, dropout probabilities, or finite-horizon dynamics to reflect these realistic institutional constraints. Justification for tripartite evolutionary game model The decision to use a tripartite evolutionary game model is based on the recognition that public health emergency management requires the simultaneous consideration of multiple interdependent stakeholders. While prior research has primarily focused on bilateral interactions (e.g., government-public or government-media), the inclusion of students introduces an additional layer of complexity, reflecting their dual role as both contributors to and beneficiaries of emergency management strategies. The tripartite model allows for the examination of: The impact of government penalties on university disclosure strategies and Public opinion channel verification behaviors. The influence of student reporting costs and optimism bias on stakeholder dynamics. The co-evolution of strategies across all three groups, enabling the identification of stable equilibrium points and the conditions under which they occur. This methodological approach extends previous work by integrating: Cognitive and Behavioral Dynamics: Incorporating parameters such as optimism bias and verification costs to reflect real-world decision-making complexities. Multi-Stakeholder Interactions: Simultaneously modeling the strategic choices of universities, students, and Public opinion channel to provide a holistic understanding of public health emergency management. Dynamic Simulations: Using Vensim software to simulate strategy evolution and visualize the effects of parameter adjustments on stakeholder behaviors. Model assumptions and model construction The evolutionary game framework was chosen to reflect gradual adaptation and learning under uncertainty—features more realistic than classical equilibrium game models in the context of emergency response. Unlike rational-choice models, agents in this study revise their behavior based on perceived outcomes over time, allowing the simulation to reproduce realistic non-linear shifts and delayed reactions often observed in campus emergencies.This evolutionary game model in this study involves three main agents: university-related departments, students, and Public opinion channel. The behavior sets for each group are as follows: the behavior set for university-related departments includes (timely disclosure, partial concealment), the behavior set for students includes (report and complain, not report and complain), and the behavior set for Public opinion channel includes (verify before reporting, not verify and report). Power asymmetry is reflected indirectly in the model by assigning differential payoff impacts. For example, universities bear larger penalties (P, T) for concealment compared to students’ cost of complaint (C3), and news platforms face reputational trade-offs. This asymmetry shapes strategic incentives without adding new formal variables. Additionally, it is assumed that during the evolutionary game process, university-related departments, students, and Public opinion channel all make behavioral strategy choices under conditions of bounded rationality and incomplete information, and the following assumptions are made: Assumption 1 The proportion of university-related departments choosing the strategy of timely disclosure is denoted as x (0 ≤ x ≤ 1), and the proportion choosing partial concealment is denoted as 1 − x . The proportion of students choosing the strategy of reporting and complaining is denoted as y (0 ≤ y ≤ 1), and the proportion choosing not to report and complain is denoted as 1 − y . The proportion of Public opinion channel choosing the strategy of verifying before reporting is denoted as z (0 ≤ z ≤ 1), and the proportion choosing not to verify and report is denoted as 1 − z . Assumption 2 In the event of a sudden public health emergency, university-related departments incur a cost, denoted as C 1 , for conducting a thorough investigation and timely disclosure of the event’s truth. If the local government harbors overconfidence or makes a misjudgment regarding the situation and chooses partial concealment, it incurs a cost, denoted as C 2 ( C 1 > C 2 ).It implies that even if the university chooses timely disclosure, verification and complaints may still occur due to information asymmetry and time lag. Public opinion channels need to expend costs to verify the information before they can confirm its authenticity; they cannot know the university’s strategy beforehand. Similarly, due to bounded rationality and panic processing, students may not immediately perceive the university’s disclosure as truthful, leading to protective complaints even under honest disclosure scenarios." In this case, if students complain to the government regulatory department or if Public opinion channel’s verification exposes the partial concealment by university-related departments, the latter will face penalties from the government regulatory department, such as administrative dismissal, accountability, or official warnings, denoted as P . Assumption 3 Students, as agents choosing to complain about the actions of university-related departments during public health emergencies, also serve as information gatherers and providers. When students report and complain about the partial concealment by university-related departments (e.g., through channels like the university president’s mailbox, the mayor’s mailbox, or the State Council’s "Internet + Inspection" platform), they incur a cost, denoted as C 3 , and receive a reward, denoted as R (e.g., extra credits, cash rewards, or honorary titles). The logic behind their risk response behavior is based on risk perception and assessment. The overall risk perception of students lags behind that of the general youth population. Students tend to perceive greater psychological distance from public health emergencies that occur in "distant places," often exhibiting a greater optimistic bias, denoted as V . When students choose not to complain, the partial concealment by university-related departments poses a potential risk, and the students’ perception of this potential risk is denoted as S 1 . In a situation where students exhibit cognitive and behavioral inconsistencies during public health emergencies within a framework of multiple information competitions, and they lean towards certain approaches for dealing with public health emergencies, it can lead to a game of consequences arising from the adverse outcomes of choosing partial concealment, resulting in a potentially greater contribution to the university. In this case, students’ perception of the potential risk is denoted as S 2 , as they perceive a greater potential contribution. These assumptions form the foundation for the construction of the evolutionary game model in this study, which aims to capture the dynamics of behavior strategies among university-related departments, students, and Public opinion channel in the context of sudden public health emergencies. Assumption 4 Public opinion channel possesses rapid dissemination capabilities and extensive influence. By reporting on public health emergencies, Public opinion channel can gain profits denoted as I . When universities disclose information, if Public opinion channel chooses to verify the authenticity of the information before reporting, it incurs a cost, denoted as C 4 . If Public opinion channel chooses not to verify and directly publishes the information disclosed by the university on its media platform, it incurs a cost denoted as C 5 ( C 4 > C 5 ). Public opinion channel’s verification and reporting on public health emergencies can provide valuable information M 1 to students and can also bring more traffic value L to Public opinion channel. Assumption 5 Timely disclosure of the truth of events by universities can provide valuable information denoted as M 2 to students, reminding them to take self-preventive measures and avoid causing panic on campus. If universities conceal the truth of the event, students’ life, health, and property security may be threatened, and the harm caused to students when universities conceal the truth is denoted as D 1 . When universities engage in partial concealment, and Public opinion channel verifies and reports afterward, Public opinion channel will disseminate the factual truth it has uncovered to the public. As the truth of the event circulates through Public opinion channel, its impact on public opinion grows. In this scenario, universities may face additional accountability actions from relevant government departments, denoted as T . Based on the above assumptions, a tripartite evolutionary game payoff matrix was constructed for university-related departments, students, and Public opinion channel, as shown in Table 1 . Table 1. Tripartite evolutionary game payment income matrix of relevant functional departments, students and Public opinion channel of universities. Public opinion channel Student University-related departments Stakeholder portfolio (university-related departments, students, public opinion channels) Verifying before reporting z Reporting and complaining y Timely disclosure x ( M 2 + M 1 − C 3 , I − C 4 + L , − C 1 ) Partial concealment 1 − x ( M 1 + R − C 3 − D 1 , I − C 4 + L , − C 2 − P − T ) Not to verifying before reporting 1 − z Timely disclosure x ( M 2 − C 3 , I − C 5 , − C 1 ) Partial concealment 1 − x ( R − C 3 − D 1 , I − C 5 , − C 2 − P ) Verifying before reporting z Not to reporting and complaining 1 − y Timely disclosure x ( M 2 + M 1 − V, I − C 4 + L , − C 1 ) Partial concealment 1 − x ( M 1 − D 1 − S 1 − S 2 − V , I − C 4 + L , − C 2 − P − T ) Not to verifying before reporting 1 − z Timely disclosure x ( M 2 − V, I − C 5 , − C 1 ) Partial concealment 1 − x ( − D 1 − S 1 − S 2 − V , I − C 5 , − C 2 ) Open in a new tab Equilibrium points in the evolutionary process Let the probability of university-related departments choosing timely disclosure be denoted as x (0 ≤ x ≤ 1), and the probability of choosing partial concealment be denoted as (1 − x ). The probability of students choosing to complain and report is denoted as y (0 ≤ y ≤ 1), and the probability of choosing not to complain and report is denoted as (1 − y ). The probability of Public opinion channel choosing to verify before reporting is denoted as z (0 ≤ z ≤ 1), and the probability of choosing not to verify and report is denoted as (1 − z ). The expected payoffs for university-related departments’ strategies of timely disclosure and partial concealment are denoted as EA 1 and EA 2 , respectively. The expected payoffs for Public opinion channel’s strategies of verifying before reporting and not verifying and reporting are denoted as EB 1 and EB 2 , respectively. The expected payoffs for students’ strategies of complaining and reporting and not complaining and reporting are denoted as EC 1 and EC 2 , respectively. The average expected payoff for university-related department strategy choices ( ), the average expected payoff for Public opinion channel strategy choices ( ), and the average expected payoff for student strategy choices ( ) can be expressed as follows: 1 2 3 4 5 6 7 8 9 According to the Malthusian dynamic equation 16 , the growth rate of the number of timely disclosure strategies by university-related departments is equal to EA 1 minus the average payoff ( ), where t represents time. This yields the replicator dynamic equation for university-related departments. 10 Following the same logic, the replicator dynamic equation for students can be expressed as: 11 the replicator dynamic equation for Public opinion channel can be expressed as: 12 It should be noted that the evolution of public opinion channels is mainly driven by exogenous market parameters. This reflects the role of the media as an independent external watchdog, whose motivation to verify depends on the value of the news rather than the specific behavior of universities. However, system interactions remain strong: the media’s strategies, as a key environmental variable, directly determine the universities’ payoff structure and evolutionary path, which in turn affects the evolution of students. Based on the three equations mentioned above, we can derive a three-dimensional dynamical system, namely: 13 Proposition 1 The equilibrium points of this system are (0,0,0), (0,1,0), (0,0,1), (1,0,0), (1,0,1), (1,1,0), (0,1,1), (1,1,1), and (A,B,C). Proof For the three-dimensional dynamical system L, let dx / dt = 0, dy / dt = 0, dz / dt = 0. It is evident that (0,0,0), (0,1,0), (0,0,1), (1,0,0), (1,0,1), (1,1,0), (0,1,1), and (1,1,1) are equilibrium points of the system. This establishes the existence of 8 local equilibrium points for system L. Equilibrium point and stability analysis The stability of equilibrium points in evolutionary dynamics can be derived from the local stability analysis of the Jacobian matrix of the dynamical system 17 . Therefore, the stability of equilibrium points is determined by calculating the eigenvalues of the Jacobian matrix. 14 In equation, the coefficients a 11 , a 12 , a 13 , a 21 , a 22 , a 23 , a 31 , a 32 , and a 33 are defined as follows: 15 16 17 18 19 20 21 22 23 According to the Lyapunov indirect method 18 , it is known that when all the eigenvalues of the Jacobian matrix are negative, the local equilibrium points represent Evolutionarily Stable Strategies (ESS). Therefore, the eigenvalues of the Jacobian matrix corresponding to the 8 local equilibrium points are as shown in Table 2 below. Table 2. Equilibrium point stability analysis. Equilibrium point Jacobian matrix eigenvalues Real symbol λ 1 λ 2 λ 3 E 1 (0,0,0) − C 1 + C 2 R − C 3 + S 1 + S 2 + V C 5 − C 4 + L (−,*,*,) E 2 (0,0,1) − C 1 + P + T + C 2 R − C 3 + S 1 + S 2 + V C 4 − L − C 5 (*,*,*) E 3 (0,1,0) P − C 1 + C 2 − R + C 3 − S 1 − S 2 − V C 5 − C 4 + L (*,*,*) E 4 (1,0,0) C 1 − C 2 2 R − C 3 + V C 5 − C 4 + L (+ ,*,*) E 5 (1,1,0) C 1 − P + C 2 − R + C 3 − S 1 − S 2 − V C 5 − C 4 + L (*,*,*) E 6 (1,0,1) C 1 − P – T + C 2 2 R − C 3 + V C 4 − L − C 5 (*,*,*) E 7 (0,1,1) − C 1 + P + T + C 2 − R + C 3 − S 1 − S 2 − V C 4 − L − C 5 (*,*,*) E 8 (1,1,1) C 1 − P − T + C 2 − 2 R + C 3 − V C 4 − L − C 5 (*,*,*) Open in a new tab "+" indicates a positive eigenvalue; "−" indicates a negative eigenvalue; "*" indicates that the sign of the eigenvalue depends on specific parameter values. Based on the determinant and trace of the Jacobian matrix, we can determine the local stability at each equilibrium point. From Table 2 , it can be observed that for the 8 potential equilibrium points, the real part of the eigenvalue λ1 corresponding to E 4 (1,0,0) is always positive. Therefore, E 4 (1,0,0) cannot be a local equilibrium point. For the remaining 7 cases, analyzing the conditions for all eigenvalues to be negative, we can derive the corresponding conditions for the stability of the 7 stable strategy combinations. When C 2 > C 1 , R < C 3 − S 1 − S 2 − V , and L < C 4 − C 5 , the replicator dynamic system has a stable point at E1(0,0,0). When P + T + C 2 < C 1 , R < C 3 − S 1 − S 2 − V , and L > C 4 − C 5 , the replicator dynamic system has a stable point at E2(0,0,1). When C 1 > P + C 2 , R > C − S 1 − S 2 − V , and L < C 4 − C 5 , the replicator dynamic system has a stable point at E3(0,1,0). When C 1 < P − C 2 , R > C 3 − S 1 − S 2 − V , and L < C 4 − C 5 , the replicator dynamic system has a stable point at E5(1,1,0). When C 1 < P + T − C 2 , R < , and L > C 4 − C 5 the replicator dynamic system has a stable point at E6(1,0,1). When P + T + C 2 < C 1 , R > C 3 − S 1 − S 2 − V , and L > C 4 − C 5 , the replicator dynamic system has a stable point at E7(0,1,1). When C 1 < P + T − C 2 , R > , and L > C 4 − C 5 , the replicator dynamic system has a stable point at E8(1,1,1). Evolutionary simulation study For parameter assignment, most studies directly assign values, and this study also chooses direct assignment based on previous studies and actual situations. However, it is worth noting that the main focus of this study is not on precise parameter allocation, but on understanding the trend changes in the probability of strategy selection among university departments, students, and Public opinion channel in sudden public events.Based on the above analysis, this study, building on this foundation, adopts Zhu’s approach of setting the government regulatory penalties at 0.4, 0.6, and 0.8, and while taking actual conditions into account, uses an iterative method to continuously derive new values from the old values of the variables, ultimately determining the initial values of each parameter as: C 1 = 0.5, C 2 = 0.7, P = 0.8, C 3 = 0.3, R = 0.6, V = 0.6, S 1 = 0.2, S 2 = 0.3, I = 0.5, C 4 = 0.6, C 5 = 0.4, M 1 = 0.5, L = 0.7, M 2 = 0.6, D 1 = 0.4, T = 0.3. On this basis, by adjusting the government’s punitive measures P to be Current 1: P = 0.4,current 2: P = 0.6 19 , current 3: P = 0.8, we can observe from Figs. 2 , 3 , 4 and Table 3 that as the government’s punitive measures increase, the probability of Public opinion channel verifying and reporting after decreases, the probability of timely disclosure strategy chosen by relevant university departments increases, but the probability of students reporting and complaining initially increases and then decreases. Fig. 2. Open in a new tab P affects the strategic choices of relevant functional departments of universities (Current 1: P = 0.4, current 2: P = 0.6, current 3: P = 0.8). Fig. 3. Open in a new tab The impact of P on the choice of Public opinion channel strategy (Current 1: P = 0.4, current 2: P = 0.6, current 3: P = 0.8). Fig. 4. Open in a new tab The influence of P on students’ strategy choice (Current 1: P = 0.4, current 2: P = 0.6, current 3: P = 0.8). Table 3. Impact of government punitive measures. Parameter P Probability of Timely disclosure (x) Probability of media verification (z) Probability of student complaints (y) 0.4 0.56 0.78 0.62 0.6 0.68 0.65 0.74 0.8 0.72 0.53 0.49 Open in a new tab The simulation findings demonstrate a counterintuitive dynamic: as the severity of the government’s punitive measures (parameter (P)) increases, the probability of Public opinion channel verifying and reporting decreases. This trend can be attributed to the increased risks and costs associated with such verification, which discourages Public opinion channel platforms from taking proactive steps to ensure the accuracy of information. From the perspective of Public opinion channel, verifying the accuracy of information before publication incurs a cost ( C 4 ), which increases with the potential penalties for publishing incorrect or misreported information. When punitive measures are heightened, the expected payoff for avoiding verification ( I − C 5 ) may surpass that of thorough verification ( I − C 4 + L ). This shifts Public opinion channel’s strategic preference toward non-verification, as reflected in the replicator dynamic equation: F ( z ) = z (1 − z )( C 5 − C 4 + L) . Here, the interplay of C 4 , C 5 , and L becomes crucial. If the traffic revenue L generated by unverified reporting is sufficient to offset the penalties or reputational damage, Public opinion channel may find it rational to forego verification altogether.This behavior underscores the delicate balance required in designing punitive policies. Excessive penalties can inadvertently suppress the role of Public opinion channel as a watchdog, reducing their willingness to engage in the verification process. Fine-tuning penalties P to an optimal range that incentivizes accurate reporting without creating disincentives is crucial. To explore this further, we incorporated a sensitivity analysis by adjusting the penalty value ( P = 0.4, 0.6, 0.8 ). The results, visualized in Figs. 3 and 4 , illustrate that moderate punitive measures ( P = 0.6 ) lead to the most balanced outcomes. This level encourages timely information disclosure by universities while maintaining a sufficient probability of Public opinion channel verification. Beyond this threshold ( P > 0.6 ), the diminishing verification probability indicates that penalties begin to outweigh the benefits of accurate reporting. Simulation results reveal an interesting strategic interaction: as the severity of government penalties increases, the probability of verification by public opinion channels first rises to a certain point and then declines (Fig. 3 ). This phenomenon should be interpreted from the perspective of strategic substitution and resource efficiency, rather than direct risk transfer. Decrease in the necessity of verification: severe government penalties act as a strong deterrent, forcing universities to disclose information promptly with high probability. In a high-transparency environment driven by strict penalties, the likelihood of media uncovering concealed actions (“exclusive news”) is significantly reduced; Cost–benefit analysis: verification involves specific fixed costs, including human resources (investigative and verification personnel), technical resources, and the time cost that may cause news reports to lose their "first-mover advantage"; Rational abandonment: when universities are compelled by the government to maintain integrity, the marginal utility of media verification decreases. Rational media will avoid incurring fixed costs for redundant verification. Therefore, strict government regulation effectively “substitutes” for the media’s verification role, leading to a decline in the probability of verification.While the probability of timely disclosure by universities increases (Fig. 2 ). However, student complaints increase initially and then decrease (Fig. 4 ).This finding indicates that moderate penalties (e.g., ( P = 0.6)) incentivize universities to disclose information without overly discouraging media verification. Excessive penalties ( P = 0.8) deter Public opinion channel from verification due to increased risks, thus undermining its role in validating information.Consistency with Song L, Yu Z, He Q’s findings on government social management functions, which emphasized moderate penalties to incentivize disclosure 13 . By adjusting the cost of student complaints, denoted as C 3 , with the following settings: Current 1: C 3 = 0.3, Current 2: C 3 = 0.2, Current 3: C 3 = 0.1. As shown in Figs. 5 , 6 , 7 and Table 4 , it can be observed that as the cost of student complaints decreases, the probability of timely disclosure by relevant departments of universities remains unchanged. However, the probability of reporting by Public opinion channel after verification decreases, while the probability of students making complaints and reports increases. This indicates that reducing the cost of student complaints not only has no effect on the probability of relevant university departments choosing the strategy of timely disclosure but also reduces the probability of Public opinion channel reporting without verification.This result highlights the trade-off between lowering barriers for student participation and maintaining the integrity of media reporting. Policies should aim to balance C 3 to maximize constructive student engagement without compromising media standards. Fig. 5. Open in a new tab C 3 affects the strategic choices of relevant functional departments in universities (Current 1: C 3 = 0.3, current 2: C 3 = 0.2, current 3: C 3 = 0.1). Fig. 6. Open in a new tab The influence of C 3 on the choice of Public opinion channel strategy (Current 1: C 3 = 0.3, current 2: C 3 = 0.2, current 3: C 3 = 0.1). Fig. 7. Open in a new tab The influence of C 3 on students’ strategy choice (Current 1: C 3 = 0.3, current 2: C 3 = 0.2, current 3: C 3 = 0.1). Table 4. Influence of student complaint Costs C 3 . Parameter C 3 Probability of timely disclosure (x) Probability of media verification (z) Probability of student complaints (y) 0.3 0.69 0.66 0.54 0.2 0.69 0.6 0.67 0.1 0.69 0.54 0.79 Open in a new tab By adjusting the student optimism bias, denoted as V , with the following settings: Current 1: V = 0.2, Current 2: V = 0.4, Current 3: V = 0.6. As shown in Figs. 8 , 9 , 10 and Table 5 , it can be observed that with an increase in student optimism bias, the probability of relevant university departments choosing the strategy of timely disclosure decreases, as does the probability of Public opinion channel reporting after verification and the probability of students making complaints and reports. It can be inferred that student optimism bias significantly impacts relevant university departments, Public opinion channel, and students themselves. Therefore, high optimism bias ( V = 0.6) creates complacency among students, reducing pressure on universities and media to act responsibly. Lowering V can significantly improve the accountability and responsiveness of all stakeholde.Liu et al.'s study, which also showed the significant role of public sentiment and optimism in public health emergencies 2 . Fig. 8. Open in a new tab V ’s impact on the strategic choices of relevant functional departments in universities (Current 1: V = 0.2, current 2: V = 0.4, current 3: V = 0.6). Fig. 9. Open in a new tab V Impact on Public opinion channel Strategy Choice (Current 1: V = 0.2, current 2: V = 0.4, current 3: V = 0.6). Fig. 10. Open in a new tab V Influence on students’ strategy choices (Current 1: V = 0.2, current 2: V = 0.4, current 3: V = 0.6). Table 5. Effect of student optimism bias V . Parameter V Probability of timely disclosure (x) Probability of media verification (z) Probability of student complaints (y) 0.2 0.75 0.72 0.68 0.4 0.66 0.61 0.55 0.6 0.57 0.49 0.43 Open in a new tab The results confirm the effectiveness of balancing government penalties, student complaint mechanisms, and optimism bias in shaping stakeholder strategies, directly addressing the study’s objectives. By extending the application of evolutionary game theory to multi-stakeholder scenarios within higher education during public health emergencies, this study provides a novel framework for understanding behavioral evolution. Specifically, the incorporation of optimism bias and media dynamics offers fresh insights into how strategic behaviors evolve under crisis conditions. Policymakers are advised to implement moderate penalties ( P = 0.6) to strike a balance between incentivizing timely disclosure and discouraging unverified media reporting. Meanwhile, universities should prioritize reducing student optimism bias through targeted education and awareness campaigns, thereby fostering greater accountability and responsiveness across all stakeholders during crises. Evaluation and validation of the model The credibility and robustness of the tripartite evolutionary game model were evaluated and validated through multiple approaches: Internal consistency validation via simulation The model assumptions and equations were validated through iterative simulation experiments using Vensim software. Parameters such as government penalties P , student complaint costs C 3 , and optimism bias V were varied systematically to observe their influence on the strategic behaviors of the three stakeholders: university departments, students, and Public opinion channel. The simulation results consistently reflected logical trends aligned with theoretical expectations, such as the increase in timely information disclosure by universities under moderate punitive measures. 2. Parameter sensitivity analysis A comprehensive sensitivity analysis was performed to assess the impact of key parameters on the model’s stability and outcomes. For example, when P was varied from 0.4 to 0.8, a significant yet consistent shift in stakeholder strategies was observed (Figs. 2 – 4 ). The sensitivity results demonstrated that the model effectively captures the dynamic interplay among stakeholders across varying contextual conditions, enhancing its reliability. 3. Comparative validation against prior studies Wang Gaoling and Bie Ruo’e's study highlighted the effectiveness of moderate government penalties in improving institutional transparency 20 . Similar trends were observed in our model, reinforcing its alignment with established theoretical frameworks. 4. Scenario testing The model was subjected to scenario-based validation, simulating real-world public health crises such as COVID-19 outbreaks. Hypothetical yet plausible scenarios were constructed, incorporating variations in stakeholder behavior and external factors like information dissemination speed. The outcomes demonstrated the model’s applicability in predicting stakeholder strategies under diverse emergency scenarios. 5. Validation limitations While the model demonstrated high internal consistency and alignment with theoretical principles, certain limitations must be acknowledged: The model relies on parameter values assigned based on prior studies or logical estimations. Real-world data integration could further enhance its empirical validity.Certain factors, such as external media influence and inter-university collaboration, were not explicitly included in the current iteration of the model and could be explored in future studies. Moreover, while the model adopts a continuous-time evolutionary framework, many real-world campus crises are finite in duration and stakeholder participation is subject to exit, fatigue, or suppression. In particular, students may only have a single opportunity to voice dissent before being silenced or penalized. To address this, future iterations of the model may incorporate a probabilistic dropout function, expected round limits, or risk-adjusted participation mechanisms. Conclusion The research results indicate that in different contexts, the decision-making behaviors of the three parties interact with each other and continuously evolve towards different stable strategy sets. Using Vensim software for simulation analysis, it is found that moderate government penalties can effectively increase the probability of relevant university departments choosing to disclose information promptly. Reducing the cost of student complaints will increase the probability of Public opinion channel choosing not to verify and report. Lowering students’ optimistic bias will effectively influence the choice of behavior strategies among university management departments, students, and Public opinion channel in the context of public health emergencies. In theoretical terms, this article focuses on emergency strategies for public emergencies in higher education. It covers different strategies of higher education, academia, and Public opinion channel under public emergencies, covering the entire process of decision-making optimization for different strategy subjects. The research results are an important supplement and improvement to the existing knowledge system in the field of decision-making optimization for public emergencies in higher education, promoting the interdisciplinary integration of applied statistics, management, and game theory. In practical terms, this study is conducted through simulation and parameter assignment, and is deeply integrated with management practice. The research results will help higher education institutions accurately evaluate the handling strategies of different parties in public emergencies and make scientifically reasonable decisions, which has important application value for promoting the high-quality development of higher education under sudden public events.Therefore, based on the research results, the following recommendations are made for the management of public health emergencies: Establish a comprehensive management system: Government departments should improve emergency management mechanisms, establish mechanisms for collaborative work, implement strict accountability and liability systems, enhance information sharing, moderately increase penalties for neglect of duty and other misconduct by relevant university departments, and establish reward and punishment mechanisms, including moderate rewards for Public opinion channel verification and reporting. Expand student complaint channels and reduce the cost of student complaints. For example, provide online reporting platforms or offer more educational resources to cultivate students’ awareness of emergency management and supervision. Relevant university departments should emphasize the dissemination of knowledge related to public health emergencies, enhance students’ abilities to collect and discern relevant information, increase students’ perception of danger, and further promote their active participation. Meanwhile, Public opinion channel plays a role in guiding public opinion and should be encouraged to take on the social responsibility of being "watchdogs," thereby encouraging the active participation of students and Public opinion channel and promoting multi-party collaborative governance. Enhance the sense of responsibility of government departments in public health emergencies in universities and implement accountability and liability systems. Disseminate knowledge about public health emergencies to students, strengthen their awareness of emergency management, and enhance their ability to participate in emergency management, thereby reducing students’ optimistic bias. For example, organize knowledge competitions related to public health emergencies. Students, as individuals who may report emergencies related to universities, are also collectors and providers of information about events. They have the right to know and the right to supervise. Relevant university departments should appropriately reward students who provide correct information, further increasing the enthusiasm for internal supervision. Enhance Public opinion channel’s Role as a Responsible Stakeholder.Public opinion channel, as an essential participant in emergency management, should be incentivized to act as a "social watchdog." Policies should emphasize the importance of verification before reporting to reduce the spread of misinformation during public health crises. These could include financial incentives for platforms that prioritize verification and public recognition for media outlets demonstrating excellence in responsible journalism. The integration of automated verification technologies or partnerships with fact-checking organizations could further support Public opinion channel in reducing costs while maintaining reporting quality. Public health emergencies require the collective participation of relevant university departments, Public opinion channel, and students. This paper only considers the partial impact factors of management departments, Public opinion channel, and students on public health emergencies and does not fully represent the influence of various factors in real life, such as the influence of school clinics on the selection of public emergency strategies in management departments. It also does not consider the influence of online evaluations on various gaming entities. The impact of online evaluation on emergency response strategies for higher education emergencies is significant, and its real-time nature is more helpful in helping to solve public emergencies in their early stages and ensuring timely problem-solving. Therefore, constructing an evolutionary game model of relevant university departments, Public opinion channel, and students under the influence of online evaluations is a future research direction 18 , 21 , 22 . Despite the oversight role played by education authorities (such as branches of the Ministry of Education), the model is constructed as a repetitive framework of interactions, which does not explicitly incorporate the fact that students may be silenced or punished after the initial complaint, especially when dealing with sensitive topics under authority pressure, while direct interaction with students in early crises is limited, so they are not explicitly modeled. Future improvements may involve adding participation decay functions, dropout probabilities, or finite-horizon dynamics to reflect these realistic institutional constraints. While the model is simplified, it captures recurring behavioral patterns observed during campus emergencies in China — particularly the interplay between institutional opacity, student mistrust, and media risk aversion. These dynamics, though stylized, help university administrators anticipate tipping points in information management and complaint handling. In particular, the model suggests that excessive penalties may inadvertently discourage third-party verification, undermining transparency goals. Supplementary Information Below is the link to the electronic supplementary material. Supplementary Material 1 (18KB, zip) Supplementary Material 2 (75.2MB, mov) Acknowledgements We thank the participants of the study. Author contributions Conceptualization, Data curation, Investigation: Jing Wei; Methodology, Writing– original draft: Ma Boyuan, Jiawei Zhou and Li Zheng, Dong minyi, Yitong Xiao; Writing – review & editing: Ma Boyuan, Jing Wei, Lin Song, Lili Zhang and Jiawei Zhou. Supervision, Funding acquisition and Project administration: Qiang He, Lili Zhang. Funding This research received no external funding. Data availability All data generated or analysed during this study are included in this published article (and its supplementary information files). Vensim PLE Version 6.4’s URL is https://vensim.com/software/#Vensim . Citespace V5.5 R2’s URL is http://cluster.cis.drexel.edu/~cchen/citespace/download/ . The data related to my research has not been stored in publicly available repositories.The data related to my research is included in the data in the article/supply. Declarations Competing interests The authors declare no competing interests. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Contributor Information Qiang He, Email: [email protected]. Lili Zhang, Email: [email protected]. Lin Song, Email: [email protected]. References 1. State Council of the People’s Republic of China. Emergency Regulations on Public Health Emergencies. 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Supplementary Materials Supplementary Material 1 (18KB, zip) Supplementary Material 2 (75.2MB, mov) Data Availability Statement All data generated or analysed during this study are included in this published article (and its supplementary information files). Vensim PLE Version 6.4’s URL is https://vensim.com/software/#Vensim . Citespace V5.5 R2’s URL is http://cluster.cis.drexel.edu/~cchen/citespace/download/ . The data related to my research has not been stored in publicly available repositories.The data related to my research is included in the data in the article/supply. 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