HIV-1 remains a formidable global health challenge, as complete viral eradication is still unattainable despite considerable advances in combination antiretroviral therapy (cART). To address this, we develop a stochastic differential equation (SDE) model that incorporates environmental noise into a classical HIV-1 infection dynamics framework, establishing two key advances. Mathematically, we derive the stochastic basic reproduction number R s and establish the corresponding threshold dynamics: when R s < 1 (under mild conditions), the infection is almost surely cleared, whereas for R s > 1 , the virus persists stochastically, following an ergodic stationary distribution. Epidemiologically, we demonstrate that environmental noise profoundly influences HIV-1 dynamics and reaffirm the central role of cART. Using optimal control theory, we evaluate three intervention strategies: Strategy 1 (cART enhancement), Strategy 2 (immune modulation), and Strategy 3 (a combined cART-immune approach). Both statistical indicators and dynamical outcomes confirm that Strategy 3 provides a clear advantage in promoting rapid viral suppression by integrating cART enhancement with immune modulation. Moreover, we observe that this combined intervention remains highly effective even under stringent cost constraints, and further reductions in intervention cost could improve its cost-efficiency. These results provide not only a novel theoretical framework for understanding HIV-1 infection dynamics, but also actionable clinical insights for optimizing treatment protocols, underscoring the critical importance of cost considerations in HIV-1 management.

This paper develops an integrated stochastic-deterministic framework to model competition between two cultural traditions. The deterministic component incorporates logistic growth, frequency-dependent transmission, attrition, and bidirectional conversion, revealing two fundamental dynamical regimes: monostable coexistence and bistable dominance, with explicit thresholds governed by conversion asymmetry. The stochastic extension introduces multiplicative environmental noise, enabling analysis of cultural resilience under uncertainty. Analytically, we derive explicit conditions for equilibrium existence, invasion growth rates quantifying cultural competitiveness, and stochastic extinction thresholds. We establish rigorous criteria for persistence and prove the existence of a unique stationary distribution characterizing long-term cultural diversity. Numerical simulations validate theoretical predictions and demonstrate how environmental noise induces regime shifts that overcome historical lock-in in bistable systems. Key findings reveal that: (1) conversion asymmetries often outweigh transmission advantages in shaping long-term cultural configurations; (2) environmental variability critically modulates the probability of coexistence; and (3) pluralistic configurations exhibit higher resilience to stochastic shocks than monocultural dominance. This integrated framework provides a rigorous, quantifiable foundation for understanding the dynamics of cultural competition, with implications for cultural policy and intercultural dialogue in an increasingly volatile global environment.

This study develops a fractional-order SEIR model incorporating memory effects through two distinct indices: infectivity memory (1-alpha(1)) for exposed individuals and recovery memory (1-alpha(2)) for infected individuals. We establish a comprehensive theoretical framework in which the sic reproduction number R0 serves as a sharp threshold for global dynamics: when R-0 < 1, disease-free equilibrium (DFE) is globally asymptotically stable, whereas when R-0> 1, a unique endemic equilibrium (EE) becomes globally asymptotically stable. Through sensitivity analysis, we quantify how R-0 and equilibrium states respond to variations in memory indices. Numerical simulations across four epidemiological scenarios reveal dual roles of memory effects: dynamic modulators that reshape transmission trajectories, and as strategic regulators that fine scenario-specific control patterns. These include synergistic suppression through coordinated memory reduction in favorable conditions, and competitive modulation requiring balanced memory adjustment in high-transmission settings. Our findings demonstrate that memory indices provide a mathematically-guided framework for precision public health, enabling tailored interventions that adapt to specific transmission-recovery patterns.

This paper provides a rigorous mathematical resolution of the open global stability problem for a “shock-and-kill” model of HIV-1/SIV infection in brain reservoirs recently formulated by Roda et al. (2021). The model explicitly incorporates the effects of latency-reversing agents and enhanced immune clearance of reactivated cells. We derive an explicit formula for the basic reproduction number ℛ_0 , which serves as the sole threshold parameter governing viral eradication versus persistence and integrates infection pathways from both productive and latent compartments. By combining the next-generation matrix approach with an extended graph-theoretic Lyapunov method for multigraphs with parallel arcs, we rigorously establish that the disease-free equilibrium is globally asymptotically stable when ℛ_0 ≤ 1 , whereas a unique productive equilibrium exists and is globally asymptotically stable when ℛ_0 > 1 . To resolve the sign-indefinite quadratic perturbations induced by structurally distinct parallel transmission arcs—a fundamental bottleneck of classical graph-theoretic Lyapunov schemes—we develop a refined composite Lyapunov framework equipped with hierarchically calibrated parameters. Systematic asymptotic scaling and multi-parameter tuning eliminate indefinite cyclic quadratic interactions, securing strict negative definiteness of the Lyapunov derivative and overcoming key limitations of conventional graph-based methods. These global stability results provide a definitive mathematical answer to whether therapeutic interventions guarantee viral eradication or lead to persistent brain-reservoir infection. Furthermore, they furnish a rigorous theoretical foundation for the “shock-and-kill” strategy and establish mathematically precise conditions to guide the design of safe, effective interventions for eliminating HIV-1/SIV from CNS reservoirs.

Biological processes inherently operate with memory: past infections modulate present transmissibility, and cumulative immune experience shapes future recovery. Conventional epidemic models, however, largely neglect such history-dependent mechanisms. To address this gap, we develop a fractional SIR model with dual memory kernels-for infectivity and recovery-extending the framework introduced by Angstmann et al. This formulation captures the non-Markovian dynamics of transmission and immunity. We prove that the basic reproduction number R0 serves as a sharp threshold: the disease-free equilibrium is globally stable when R0<1, and a unique endemic equilibrium emerges and remains globally stable when R0>1, extending classical threshold theory to coupled memory effects. Moreover, we identify context-dependent intervention strategies by modulating memory indices: reducing recovery memory or enhancing infectivity memory suppresses spread in low-mortality settings, while the reverse approach is effective under high mortality. This work transforms memory effects from mathematical concepts into actionable targets for epidemic control, bridging mechanistic modeling with public health strategy.

This paper presents a novel dual-stochastic network epidemic model of SIW type that systematically incorporates two complementary stochastic mechanisms. The first is a logarithmic Ornstein-Uhlenbeck (log-OU) process, which captures long-term, correlation-structured fluctuations in transmission rates while preserving positivity-a key advantage over standard OU processes that can yield biologically infeasible negative rates. The second is multiplicative white noise, which represents short-term, uncorrelated environmental variability. Mathematically, we first establish the existence and uniqueness of a global positive solution. We then derive a sharp extinction threshold ℛ_0^E using spectral radius analysis of the next-generation matrix under stochastic perturbations, and prove the existence of a stationary distribution via Lyapunov function construction and Itô calculus, thereby characterizing the long-term persistence regime of the disease. Unlike deterministic thresholds that only depend on mean parameters, our stochastic thresholds ℛ_0^E and ℛ_0^S explicitly incorporate noise intensities σ _i , ρ _i and reversion rates κ _i , revealing how environmental variability can either destabilize or stabilize endemic states. Epidemiologically, the model introduces a multi-scale stochastic framework that bridges short-term random shocks with longer-term transmission trends, a feature often omitted in conventional models. Numerical simulations on scale-free networks reveal how network heterogeneity interacts with stochasticity to alter outbreak thresholds and persistence patterns. As a direct application, the model is calibrated to COVID-19 surveillance data from Ethiopia, where it outperforms deterministic equivalents and quantifies the differential influence of environmental clearance, pathogen release, and contact heterogeneity on epidemic outcomes. These results provide not only theoretical insights into stochastic epidemic dynamics but also quantitative support for targeted intervention strategies in real-world public health practice.

In this paper, we propose a novel fractional-order SIRS (frSIRS) model incorporating infection forces under intervention strategies, developed through the framework of generalized continuous-time random walks. The model is first transformed into a system of Volterra integral equations to identify the disease-free equilibrium (DFE) state and the endemic equilibrium (EE) state. Additionally, we introduce a new F V - 1 method for calculating the basic reproduction number R 0 . Through several examples, we demonstrate the broad applicability of this F V - 1 method in determining R 0 for fractional-order epidemic models. Next, we establish that R 0 serves as a critical threshold governing the model's dynamics: if R 0 < 1 , the unique DFE is globally asymptotically stable; while if R 0 > 1 , the unique EE is globally asymptotically stable. Furthermore, we apply our findings to two fractional-order SIRS (frSIRS) models incorporating infection forces under various intervention strategies, thereby substantiating our results. From an epidemiological perspective, our analysis reveals several key insights for controlling disease spread: (i) when the death rate is high, it is essential to increase the memory index; (ii) when the recovery rate is high, decreasing the memory index is advisable; and (iii) enhancing psychological or inhibitory effects-factors independent of the death rate, recovery rate, or memory index-can also play a critical role in mitigating disease transmission. These findings offer valuable insights into how the memory index influences disease outbreaks and the overall severity of epidemics.

In this paper, we investigate the spatiotemporal dynamics of a fractional-order predator-prey reaction-diffusion model (frPDE) with Holling-type III functional response. We prove that this is not the integer-order reaction-diffusion model, but the frPDE model exhibits fraction-diffusion-induced instability (i.e., Turing instability), which is induced by the fractional-order and diffusion together. Furthermore, via numerical simulations, the frPDE model dynamics exhibits both fractional-order and diffusion controlled Turing pattern formation, which shows that the dynamics of the frPDE model is not simple, but rich and complex.

We develop a stochastic human immunodeficiency virus type 1 (HIV-1) infection model to analyze combination antiretroviral therapy (cART) dynamics in the brain microenvironment, explicitly accounting for two infected cell states: (1) productively infected and (2) latently infected populations. The model introduces two key epidemiological thresholds $ - \mathcal{R}_{c1} $ (productive infection) and $ \mathcal{R}_{c2} $ (latent infection) -- and defines the stochastic control reproduction number as $ \mathcal{R}_c = \max\{\mathcal{R}_{c1}, \mathcal{R}_{c2}\} $. Our analysis reveals three distinct dynamical regimes: (1) viral extinction ($ \mathcal{R}_c < 1 $): the infection clears exponentially with probability one; (2) latent reservoir dominance ($ \mathcal{R}_c = \mathcal{R}_{c2} > 1 $): the system almost surely converges to a purely latent state, characterizing stable viral reservoir formation; (3) persistent productive infection ($ \mathcal{R}_c = \mathcal{R}_{c1} > 1 $): the infection persists indefinitely with a unique stationary distribution, for which we derive the exact probability density function. And numerical simulations validate these theoretical predictions, demonstrating how environmental noise critically modulates HIV-1 dynamics in neural reservoirs. Our results quantify the stochastic balance between productive infection, latency establishment, and cART efficacy, offering mechanistic insights into viral persistence in the brain.

Seismic data acquisition is often affected by various types of noise, which degrade data quality and hinder subsequent interpretation. Recovery of seismic data becomes particularly challenging in the presence of strong noise, which significantly impacts both data accuracy and geological analysis. This study proposes a novel single-encoder, multiple-decoder network based on Nash equalization (SEMD-Nash) for effective strong noise attenuation in seismic data. The main contributions of this method are as follows: First, we design a shared encoder-multi-decoder architecture, where an improved encoder extracts key features from the noisy data, and three parallel decoders reconstruct the denoised seismic signal from different perspectives. Second, we develop a multi-objective optimization system that integrates three loss functions-Mean Squared Error (MSE), Perceived Loss, and Structural Similarity Index (SSIM)-to ensure effective signal reconstruction, high-order feature preservation, and structural integrity. Third, we introduce the Nash Equalization Weight Optimizer, which dynamically adjusts the weights of the loss functions, balancing the optimization objectives to improve the models robustness and generalization. Experimental results demonstrate that the proposed method effectively suppresses strong noise while preserving the geological characteristics of the seismic data.

Infectious disease superspreading is a phenomenon where few primary cases generate unexpectedly large numbers of secondary cases. Superspreading, is frequently documented in epidemiology literature, and is considered a consequence of heterogeneity in transmission. Since understanding the risks of superspreading became a rising concern from both statistical modelling and public health aspects, the R package modelSSE provides comprehensive analytical tools to characterize transmission heterogeneity. The package modelSSE integrates recent advances in statistical methods, such as decomposition of reproduction number, for modelling infectious disease superspreading using various types and sources of contact tracing data that allow models to be grounded in real-world observations. This study provided an overview of the theoretical background and implementation of modelSSE, designed to facilitate learning infectious disease transmission, and explore novel research questions for transmission risks and superspreading potentials. Detailed examples of classic, historical infectious disease datasets are given for demonstration and model extensions.

It is well known that there is nonexistence of Turing instability in the integer-order Holling type-IV reaction-diffusion predator-prey model. In this paper, we investigate the spatiotemporal dynamics of the effect of the fractional-order derivative and diffusion on the reaction-diffusion predator-prey model with Holling type-IV functional response. Additionally, we give the conditions that Turing instability occurs in the model, which is induced by the fractional-order and diffusion together. Via numerical simulations, we present the evolutionary processes that involve organism distribution and the interaction of spatially distributed prey with local diffusion, and find that the model dynamics exhibits rich and complex Turing patterns.

In this paper, we investigate the dynamics of a class of discrete predator–prey model with alternative prey. We prove the boundedness of the solution, the existence and local/global stability of equilibrium points of the model, and verify the existence of flip bifurcation and Neimark-Sacker bifurcation. In addition, we use the maximum Lyapunov exponent and isoperimetric diagrams to verify the existence of periodic structures namely Arnold tongue and the shrimp-shaped structures in bi-parameter spaces of a class of predator–prey model.

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International Journal of BiomathematicsAccepted Papers No AccessDynamics of an Echinococcosis transmission modelChun Peng, Kai Wang, and Weiming WangChun Peng, Kai Wang Search for more papers by this author , and Weiming Wang Search for more papers by this author https://doi.org/10.1142/S1793524524500499Cited by:0 (Source: Crossref) PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Remember to check out the Most Cited Articles in IJB! Check out new Biomathematics books in our Mathematics 2018 catalogue! Featuring author Frederic Y M Wan and more! FiguresReferencesRelatedDetails Recommended Accepted Papers Metrics History Received 25 February 2024 Accepted 6 April 2024 PDF download

Plant epidemics are often associated with weather-related variables. It is difficult to identify weather-related predictors for models predicting plant epidemics. In the article by Shah et al., to predict Fusarium head blight (FHB) epidemics of wheat, they explored a functional approach using scalar-on-function regression to model a binary outcome (FHB epidemic or non-epidemic) with respect to weather time series spanning 140 days relative to anthesis. The scalar-on-function models fit the data better than previously described logistic regression models. In this work, given the same dataset and models, we attempt to reproduce the article by Shah et al. using a different approach, boosted regression trees. After fitting, the classification accuracy and model statistics are surprisingly good.

Industrial development has made air pollution increasingly severe, and many respiratory diseases are closely related to air quality in terms of infection and transmission. In this work, we used the classic stochastic susceptible–infectious–recovered (SIR) model to reflect the spread of respiratory disease, coupled with the diffusion process of air pollutants to the infectious disease model, and we investigated the impact of various environmental noises on the process of disease transmission and air pollutant diffusion. The value of this study lies in two aspects. Mathematically, we define threshold ℛ_1^s for extinction and threshold ℛ_2^s for persistence of the disease in the stochastic model ( ℛ_2^s<ℛ_1^s ) when the parameters are constant, and we show that (i) when ℛ_1^s is less than 1, the disease will go to stochastic extinction; (ii) when ℛ_2^s is larger than 1, the disease will persist almost surely and the model has a unique ergodic stationary distribution; (iii) when ℛ_1^s is larger than 1 and ℛ_2^s is less than 1, the extinction of the disease has randomness, which is demonstrated through numerical experiments. In addition, we derive the exact expression of the probability density function of the stationary distribution by solving the corresponding Fokker–Planck equation under the condition of disease persistence and analyze the effects of random noises on stationary distribution characteristics and the disease extinction. Epidemiologically, the change of the concentration of air pollutants affects the conditions for disease extinction and persistence. The increase in the inflow of pollutants and the increase in the clearance rate have negative and positive impacts on the spread of diseases, respectively. We found that an increase in random noise intensity will increase the variance, reduce the kurtosis of distribution, which is not conducive to predicting and controlling the development status of the disease; however, large random noise intensity can also increase the probability of disease extinction and accelerates disease extinction. We further investigate the dynamic of the stochastic model, assuming that the inflow rate switches between two levels by numerical experiments. The results show that the random noise has a significant impact on disease extinction. The data fitting of the switching model shows that the model can effectively depict the relationship and changes in trends between air pollution and diseases.

Objectives: Vaccination and the emergence of the highly transmissible Omicron variant changed the fate of the COVID-19 pandemic. It is very challenging to estimate the number of lives saved by vaccination given the multiple doses of vaccination, the time-varying nature of transmissibility, the waning of immunity, and the presence of immune evasion. Methods: We established a S-SV-E-I-T-D-R V-E-I-T-D-R model to simulate the number of lives saved by vaccination in six states in the United States (U.S.) from March 5, 2020, to March 23, 2023. The cumulative number of deaths were estimated under three vaccination scenarios based on two assumptions. Additionally, immune evasion by the Omicron and loss of protection afforded by vaccination or infection were considered. Results: The number of deaths averted by COVID-19 vaccinations (including three doses) ranged from 0.1540.295% of the total population across six states. The number of deaths averted by the third dose ranged from 0.008-0.017% of the total population. Conclusions: Our estimate of death averted by COVID-19 vaccination in the U.S. was largely in line with a previous estimate (at a level of 0.15-0.20% of the total population). We found that the additional contribution of the third dose was small but significant.

In this paper, we establish a new fractional order SIS (frSIS) model by way of continuous time random walk. The value of this study lies in two aspects. Mathematically, we provide a framework for the global stability of the frSIS model, and prove that the basic reproduction number R0 can be used to govern the dynamics of the frSIS model. If R0<1, the disease-free equilibrium of the model is globally asymptotically stable; if R0>1, the endemic equilibrium of the model is globally asymptotically stable. And epidemiologically, we find that, in order to control the spread of the disease, we must decrease the death rate and the average infectious period to make the disease go to extinction, which can provide us with some useful control strategies to regulate disease dynamics.