In existing models of pathogen evolution under voluntary vaccination, vaccination behaviour is typically assumed to be static, overlooking the conflicts between individual and collective interests. To address this, we develop a multi-scale model within a game-theoretic framework that couples the dynamic processes of disease transmission, vaccination decision-making, and pathogen evolution. By reformulating vaccine uptake as a time-dependent strategic choice and incorporating a heterogeneous update mechanism, we establish the evolutionary model of vaccination strategies. Numerical results show that heterogeneous updating rules will capture autonomous population decision-making. The co-evolutionary dynamics are highly sensitive to the evolutionary adaptation rates and exhibit an asymmetric relationship between population and evolutionary dynamics. At the early stage of a vaccination scenario, we observe that imitation behaviour dominates due to limited information. Enhancing vaccine uptake is effective in limiting the emergence of detrimental variants, and the long-term evolutionary outcome may favour the emergence of specialist variants. This framework provides quantitative methods and key methodological advancements for multi-scale mathematical modelling in the context of pathogen evolution.

During the COVID-19 pandemic in China, non-pharmaceutical interventions (NPIs) have been changed from the containment to the dynamic zero-case policy (DZCP) to achieve the goal of zero COVID in approximately 40 days, and then to reopening. Epidemic characteristic metrics are similar for multiple outbreaks but with markedly different peak values, timings and final sizes. In order to quantify the sensitivity of epidemic dynamics to the effectiveness of NPIs, and to guide intervention strategies for future outbreaks the data-driven modeling approach is adapted. In particular, to examine to which process or which parameter is sensitive, we used analytic techniques to identify and analyze major changes in epidemiological indices caused by small changes in characteristic metrics. By comparing basic statistics for 80 outbreaks induced by three different strains, we show that the controlled epidemic trajectories of COVID-19 epidemics depend entirely on the efficacy of NPIs: the infectivity of a strain has little relevance, even for the most infectious strains. Thus, minor changes in the strength of NPIs will lead to huge differences in epidemiological indicators such as outbreak peak value and outbreak clearing time.

Prevention and control strategies play an essential role in reducing new infections of infectious diseases, while uniform and instantaneous responses to disease outbreaks are assumed in traditional epidemic models. To model the adaptive public health policies that account for hysteresis effects and heterogeneous control measures, we proposed an SIQR epidemic model with multiple two-threshold structures to describe the prevalence-dependent real-time control strategies. The formulated model incorporated the Preisach operator which effectively characterizes time-varying control strategies that depend on the historical evolution of a disease. We established the global stability of this steady-state set through an innovative analytical approach by constructing a series of Lyapunov functions corresponding to different branches of the hysteresis operator. Specifically, our results indicate that: (i) heterogeneity facilitates the convergence of system trajectories to a point within the continuum of steady-state, and (ii) greater heterogeneity leads to higher peaks of infected individuals during the evolution of an epidemic and more severe disease prevalence in the long term. Our findings reveal that the hysteresis phenomenon in control events may lead to existence of a continuum of endemic equilibria, significantly altering the effectiveness of disease suppression strategies. These results provide a theoretical foundation for designing adaptive control policies, emphasizing the importance of incorporating past intervention effects in epidemic modeling.

Existing single-scale HIV models often overlook conflicts between within-host and population-level treatment strategies. To resolve this, we develop a multi-scale model with infection-age structure to optimize antiretroviral therapy (ART) initiation timing and efficacy through multi-objective optimization. By reformulating initiation time as a control variable in the system, we establish the existence of Pareto-optimal solutions through variational analysis and provide a theoretical characterization of the optimal treatment schedule. Numerical analyses reveal that optimal ART strategies must dynamically adapt to evolving priorities between individual outcomes and population-level benefits: early high-efficacy ART maximizes individual benefits, while delayed initiation optimizes population outcomes under persistent post-treatment risks. Post-treatment behavioural shifts influence this balance: reduced sexual activity supports earlier initiation, while high-risk populations require precisely timed delayed administration to balance transmission control and treatment costs. This framework provides quantitative principles for reconciling scale-specific treatment priorities.

While cold weather’s impact on influenza-like illness (ILI) is well-studied, heat effects remain underexplored. This study examines the relationship between heat and influenza risk across China, examining regional variations and quantifying heat-attributable disease burden. We analyzed weekly ILI surveillance data and temperature records from 333 Chinese cities (2010–2018), focusing on warm seasons (May–October). Using distributed lag non-linear models combined with the generalized additive quasi-Poisson regression model, we assessed non-linear heat-ILI associations in each city. China was divided into seven geographical regions to evaluate how economic development, healthcare resources, and population composition modify the adaptation of ILI to heat effects. The merged results of heat effects on ILI vary in seven regions. Nationally, influenza risk increases when temperatures exceed the region-specific minimum risk temperature. In the northeast, northern, northwest, and southern regions, risk rises significantly at temperatures of 17 °C, 16.4 °C, 14.7 °C, and 21.9 °C respectively. Conversely, in the central and southwest regions, risk declined with rising temperatures before showing minimal subsequent growth. In the eastern region, risk increased steadily above 11.8℃, with only a slight decline at extreme high temperatures. These patterns collectively highlight the pronounced effect of high temperatures on ILI epidemics. Elderly population proportion significantly modified heat effects in most regions. Population size modified ILI’s heat adaptability in Northwest/Southwest, while the proportion of low-income populations across cities influenced Northern/Southern/Northwest/central regions. Quantitative analysis revealed that in 139 cities, temperature spikes contributed to an estimated 3,128,629 excess ILI cases in total. This nationwide study demonstrates that heat effects were associated with influenza-like illness risk across China, with variations across regions. These regional differences in heat-ILI associations varied with local healthcare conditions, economic status, social environment, and population age structure. The findings support the need for region-specific public health interventions that consider these modifying factors.

During disease outbreaks, individuals adopt preventive measures based on perceived infection risk and the associated costs. To simultaneously capture the effects of social influence and personal evaluation on behavioural adaptation, we propose a co-evolutionary model that integrates both imitation and aspiration mechanisms into a classical disease transmission framework. We analyze the dynamic behaviour of this full model, as well as two limiting cases governed purely by imitation or aspiration dynamics. Our analysis shows that Hopf bifurcations and periodic oscillations arise only from including both decision-making mechanisms; a single mechanism alone cannot produce them, highlighting the importance of incorporating multiple behavioural change processes. Moreover, the inclusion of aspiration dynamics alters the threshold for disease elimination, which can be reduced by tuning behavioural costs, effectiveness, or aspiration levels, thus promoting disease eradication. Notably, we identify scenarios in which increasing the basic reproduction number (in the absence of behaviour change) paradoxically leads to a reduction in the endemic infection level. To evaluate the empirical relevance of our framework, we calibrate it using real-world data on COVID-19 cases/tests and subway usage from New York City (later-stage pandemic) and Tokyo (early-stage pandemic) for illustrative fitting. The mixed-mechanism model, or its limiting forms (pure aspiration for NYC, pure imitation for Tokyo), better replicate observed behavioural and epidemiological patterns. These differences reflect varying levels of public awareness and reliance on social cues across pandemic stages. Overall, our study highlights the importance and flexibility of incorporating diverse behavioural pathways to enhance the realism of infectious disease models.

Understanding how collective behaviors evolve during outbreaks is essential for accurate prediction and effective intervention design, yet it remains insufficiently explored across diverse socioeconomic contexts. We propose a Behavior-Coupled Physics-Informed Neural Network (BC-PINN), establishing a flexible, data-driven framework to learn heterogeneous behavioral responses across varying socioeconomic environments. Following BC-PINN training, we further employ symbolic regression to recover mechanistic functions governing behavioral evolution, enabling interpretable learning and pattern identification. Applying our framework to COVID-19 data across 58 countries, we uncover two distinct behavioral evolution patterns strongly associated with levels of socioeconomic development. Specifically, we observe a significant transition around a Human Development Index (HDI) of 0.8, where countries above this value exhibit Benefit-Driven Behavior (BDB), while those below display Barrier-Inhibition Behavior (BIB). Simulation results suggest that if countries exhibiting BIB transitioned to BDB, both epidemic peaks and cumulative infections at peak are consistently decreased in several countries. Moreover, within the BDB group, behavioral response intensity positively correlates with GDP per capita. These findings underscore the profound impact of economic inequality on collective behavioral responses during pandemics, offering valuable insights and practical guidance for crafting tailored public health strategies across diverse socioeconomic contexts.

The spread of epidemics is profoundly affected by adaptive behavior change in response to perceived infection risks. In this study, we propose an SIS epidemic model that integrates behavioral dynamics driven by imitation process based on the game theory. The model also incorporates a density-dependent factor to represent the efficacy of behavior change and saturated recovery to describe the limitation of medical resources. We theoretically analyze the existence and local stability for all equilibria and examine the global stability of disease-free equilibrium and endemic equilibria under certain thresholds. We further calculate conditions for the occurrence of backward bifurcation and Hopf bifurcation, showing that bistability and periodic oscillations are possible. It is worth noting that in consideration of saturated recovery, the basic reproduction number R_0=1 may not be threshold for disease eradication, which enriches the transmission dynamics. By comparing the model without behavior change or saturation recovery, we obtain that saturation recovery, though insufficient to bring about periodicity, can serve as the trigger for bistability, while combination of behavior change and saturated recovery brings about periodic oscillations. Our findings suggest that enhancing public sensitivity to perceived infection risks and expanding medical resources are critical for mitigating outbreaks.

Adaptive therapy is a novel cancer treatment strategy that proposes to tackle cancer drug resistance by leveraging resource competition between drug-sensitive and resistant cells. Because the underlying mathematical mechanisms of adaptive therapy remain unclear, determining the most effective rates of intervention is a significant challenge. In this paper, we propose a competition model incorporating fixed-time periodic tumor measurements with impulsive interventions performed if the number of tumor cells exceeds a threshold value. For the proposed model, we find a novel type of periodic solutions. Specifically, we demonstrate the existence of various (ℓ, m)T boundary periodic solutions and rigorously analyze their stability. Using bifurcation theory, we further prove the existence and stability of positive periodic solutions. Further, we perform numerical simulations to study these bifurcations with respect to key parameters such as the threshold value (TV) and the monitoring period (T). Numerical studies find that time to treatment failure exhibits a nonlinear dependence on the killing rate of drug-sensitive cells, i.e., it initially increases, reaches a plateau, and subsequently declines as the killing rate increases, revealing that maximizing the killing rate does not yield optimal therapeutic outcomes. The finding indicates that incorporating a threshold can extend patient's survival time, implying the therapeutic benefit of threshold-based adaptive therapy for tumor control.

Vaccination is widely used to control the spread of infectious diseases. However, there is substantial evidence suggesting that the perceived immunity from vaccine drives the decrease of risk-awareness among vaccinated individuals, which results in the paradoxical increase of transmission risk, consequently leads to the counter-intuitive outcomes in disease control in the case of low vaccine effectiveness. In this study, we propose a novel method to quantify the counter-intuitive effects of vaccination in a complex environment with integrated dynamic factors. To this end, we begin by linking shifts in risk awareness to human behavior changes, and subsequently, we couple the evolution dynamic of human behaviors with the disease transmission dynamic system by applying game theory within a multi-scale model, distinguishing between vaccinated and unvaccinated populations. We calibrate the proposed model using vaccination data and epidemic data for the Omicron variant of COVID-19, spanning from 23/12/2021 to 20/09/2022 with two epidemic waves in Tokyo Metropolis, Japan. The calibration results provide the quantitative evidence that the vaccinated group exhibits a slower rate of risk perception to adjust their behaviors and a higher mental threshold for behavioral changes needed to protect themselves against infection. The model estimates that if vaccinated individuals had maintained the same level of NPI adherence and behavioral responses to infection prevalence as unvaccinated individuals, the accumulative confirmed cases during the study period would have been reduced by approximately 40% (although the exact proportion depends on the chosen time window). This finding highlights the importance of post-vaccination behavioral responses. Additionally, our analysis suggests an optimal vaccination coverage that minimizes accumulative infections by balancing the protective effects and the counter-intuitive effects of mass vaccination. Therefore, it’s of significant meaning to collect and disclose the epidemic data with vaccination status and timely evaluate the effective transmission among the population involving human behavior dynamics for designing the proper vaccination strategy, and improve the vaccine effectiveness and maintain high risk-awareness for vaccinated individuals would still be the key to flatten the epidemics.

In classical Filippov systems, the switching between vector fields is typically assumed to occur instantaneously upon crossing the switching manifold. In practice, however, there is inevitably a time delay between observing the system's state and implementing control actions, such as delays in reporting infection data and enacting interventions in epidemiological models. Until now, a general classification of planar Filippov systems with a single switch has not been developed. By employing a Poincar & eacute; map and asymptotic matching techniques, we investigate the properties of periodic solutions induced by time delays, including their uniqueness, stability, amplitude and location in the state space. We then apply this framework to an epidemiological model and analyse its global dynamics. Our results show that, under certain sufficient conditions, the epidemic system stabilizes either at an equilibrium of subsystems or at a periodic orbit induced by the delayed threshold policy, depending on the chosen threshold levels. If the threshold level is set too high or too low, the delay has little effect on the long-term dynamics; in contrast, intermediate thresholds may lead to oscillations in the disease cycle. These findings highlight that timely reporting of infection data, prompt implementation of control measures and careful selection of threshold levels are critical for mitigating the spread of infectious diseases.

The dynamic model that couples micro- and macro-level factors holds significant importance for studying diseases with long incubation periods, which also presents a major challenge in multiscale modeling. Based on the nested models, we propose a multiscale model with bidirectional coupling between population-level transmission and within-host viral progression, by including the feedback from the population scale to the individual scale mediated by perceived disease-induced mortality and its effect on treatment adherence. Suppose a saturating incidence and a linear mortality law, we rigorously analyze the dynamics of the proposed system and reveal multiple endemic equilibria and local bifurcations (backward, fold, and Hopf). Numerical continuation further identifies codimension-two organizing centers (Bogdanov-Takens points) and global bifurcations, including homoclinic bifurcations and fold bifurcations of limit cycles. These give rise to parameter regions exhibiting two distinct forms of bistability: the coexistence of two equilibria and the coexistence of an equilibrium with a stable periodic orbit. By contrast, a corresponding unidirectional model exhibits comparatively simple dynamics and predicts that sufficiently high drug efficacy alone will achieve elimination. The bidirectional model, however, yields richer asymptotical and transient behavior and indicates that population-level eradication generally requires simultaneous, substantial improvements in both drug efficacy and medication adherence. These findings show that the multiscale model with bidirectional coupling can change epidemic outcomes and produce more realistic predictions for control of chronic infections.

This study aims at exploring a general and adaptive control strategy to confront the rapid evolution of an emerging infectious disease ('Disease X'), drawing lessons from the management of COVID-19 in China. We employ a dynamic model incorporating age structures and vaccination statuses, which is calibrated using epidemic data. We therefore estimate the cumulative infection rate (CIR) during the first epidemic wave of Omicron variant after China relaxed its zero-COVID policy to be 82.9% (95% CI: 82.3%, 83.5%), with a case fatality rate (CFR) of 0.25% (95% CI: 0.248%, 0.253%). We further show that if the zero-COVID policy had been eased in January 2022, the CIR and CFR would have decreased to 81.64% and 0.205%, respectively, due to a higher level of immunity from vaccination. However, if we ease the zero-COVID policy during the circulation of Delta variant from June 2021, the CIR would decrease to 74.06% while the CFR would significantly increase to 1.065%. Therefore, in the face of a 'Disease X', the adaptive strategies should be guided by multiple factors, the 'zero-COVID-like' policy could be a feasible and effective way for the control of a variant with relative low transmissibility. However, we should ease the strategy as the virus matures into a new variant with much higher transmissibility, particularly when the population is at a high level of immunity.

In this paper, we formulate a delayed water borne pathogen model incorporating overexposure, investigate threshold dynamics, and analyze the impact of overexposure and delay on disease transmission. Threshold dynamics are characterized by the basic reproduction number R0. The model exhibits backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium for R0 values below 1, indicating that the disease may persist even when R0<1. Furthermore, we theoretically and numerically examined the existence of Hopf bifurcation in absence of time delay and the results reveal that the overexposure induces rich dynamics, including stability switches, endemic bubble and multiple limits cycles. For the delayed model, by regarding time delay as the bifurcation parameter, the local and global Hopf bifurcation have been carried out to show influence of time delay on model dynamics. Numerical simulations confirm that delay induces stability switches and coexistence of multiple periodic solutions. Our findings indicate that overexposure and time delay are responsible for the model's complex dynamics, complicating disease control efforts.

Overcompensation, defined as recovery beyond a baseline state, arises from both hormetic and hydra effects, illustrating adaptive responses to stress. The overcompensation framework of a re-evaluated fishery resource management model was examined through nonlinear growth patterns based on logistic or Ricker models, emphasizing population size over carrying capacity. This complete overcompensation model’s threshold conditions reveal an interplay between hydra and hormetic effects. Also, when dividing a population into distinct subgroups, such as susceptible and infected classes in disease transmission, the population size can be modelled as a function of the basic reproduction number ( R 0 ). A threshold condition of R 0 allows examination of how disease infectivity triggers hydra or hormetic effects and, also, development of a partial overcompensation model that elucidates the internal mechanisms of overcompensation. Analysis of data from 24 groups of U-shaped or inverted U-shaped dose–response curves validated the dose–response curves. The simplified modelling approach developed revealed the mechanisms underlying hydra and hormetic effects, highlighting the importance of strong growth or regenerative capabilities, overcompensatory responses (strong nonlinearity), mild external stimuli (weak stressors) and the baseline population size. Our new analytical techniques for overcompensation modelling can be adapted to many fields, including tumour treatment and toxicology.

A general type of piecewise-smooth dynamic system with two thresholds is analyzed in this work. We define some fundamental notions for the proposed system such as oscillating space and real/virtual equilibrium, which generalizes the counterparts for the planar Filippov system. Moreover, we show that under certain conditions the planar switched system with two thresholds generates a novel limit cycle, and analyze the properties of this periodic solution such as existence, stability, amplitude and period. Interestingly, the existence and stability of this periodic solution in the oscillating space are consistent with the pseudo-equilibrium of the corresponding planar Filippov system. Hence, we establish the connection between the planar switched system with two thresholds and the planar Filippov system. Finally, we apply the modeling and analytical approaches to a piecewise-smooth epidemic model with density-dependent interventions, describing the control measure that is triggered when the number of infected individuals increases and reaches a critical level while being suspended when it decreases down to another level. We prove that the epidemic model stabilizes at either the endemic equilibrium of the free system (the one not under control) or the new periodic solution induced by the two thresholds, depending on the threshold levels. The two-threshold measure is able to suppress the number of infected individuals during the evolution of an infectious disease.

A novel multiscale model is formulated to examine the co-evolution among behavioral dynamics, disease transmission dynamics and viral dynamics, in which perceived risk act as a bridge for realizing the bidirectional coupling of between-host dynamics and within-host dynamics. The model is validated by real data and exhibits rich dynamic behaviors including the periodic oscillations of the solutions, the discordance of transmission dynamics and viral dynamics. It is observed that new infections may increase with improving treatment efficacy, which may reveal the hidden mechanisms why it is hard to eliminate HIV/AIDS infection only with the strategy of treatment. If increasing treatment efficacy but without improving diagnosis rate, “nearly elimination” phenomenon may happen when the risk threshold for behavior changes is low, in which the number of new infections may drop to a relatively low level but increase again to a relatively high level after a period of time as people may hardly keep their awareness and increase their high risk behaviors. The findings indicate that the intervention measures should be implemented both at individual level and population level to realize “ending the AIDS”.

The spread of epidemics is inevitably influenced by a wide range of random factors, resulting in complex and diverse dynamics. This study develops a stochastic epidemic model driven by the Black-Karasinski process to investigate the impact of environmental noise on disease extinction and uniform persistence. Two key quantities are formulated to characterize the conditions for disease persistence and extinction, with rigorous analysis of their relationship. Sufficient conditions for the existence of a stationary distribution and disease extinction are established. Additionally, an optimal control problem is proposed for the stochastic model to achieve a dynamic balance between disease control and socio-economic requirements, with the optimal control derived. Notably, our findings reveal that the persistence-determining quantity for the stochastic system is higher than the basic reproduction number of the deterministic system. Most intriguingly, we find that noise favors to the persistence of the disease. Such a stochasticity-induced persistence effect challenges the conventional conclusion that large noise suppresses disease prevalence. Numerical studies further examine the effect of random factors on the quantities and the dynamic behavior especially when unity is between the two quantities. Additionally, the effect of noise intensity on extinction probability and the noise distributions associated with the transition from extinction to persistence is explored through numerical simulations. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Host defense and pathogen virulence interact and mutually shape each other's evolution. Host-pathogen co-evolutionary outcomes have potentially significant impacts on population dynamics and vice versa. To investigate host-pathogen interactions and explore the impact of micro-level co-evolutionary outcomes on macro-level epidemics, we develop a co-evolutionary model with a combined host-defense strategy. Our results illustrate that host-pathogen co-evolution may induce infection cycling and lead to the vanishing of the disease-induced hydra effect, whereas pathogen mono-evolution strengthens the hydra effect in both range and magnitude. As the recovery rate increases, we find a counter-intuitive effect of increased disease prevalence due to host-pathogen co-evolution: the disease is first highly infectious and lethal, then highly infectious but with low lethality. Such diverse outcomes suggest that this combined co-evolutionary and epidemiological framework holds great promise for a better understanding of infection.