Density estimation neural networks (DENNs) represent a form of artificial neural network designed to provide an efficient approach to the Bayesian estimation of a probability density on a model parameter space, conditioned on an empirical observation of the underlying system. Despite their efficiency and potential, DENNs remain underutilized for parameter estimation in mathematical modeling. In this work, we aim to boost the accessibility of the DENN approach by providing a user-friendly introduction and code that makes it easy for users to harness existing, cutting-edge DENN software. Furthermore, we insert an easily-implemented preliminary data simulation step that reduces the computational demands of the approach and empirically demonstrates that it maintains the accuracy of parameter estimation for a stochastic oscillator model.

We investigate the spatiotemporal dynamics of a tri-trophic food chain model incorporating a strong Allee effect on the prey and a fear effect on the middle predator. The model's well-posedness is established through the positivity and boundedness of solutions. We derive all equilibria and examine their local stability, revealing saddle-node and transcritical bifurcations under varying parameter conditions. The analysis demonstrates how shifts in the Allee threshold and fear intensity induce bistability, coexistence, or extinction. Numerical simulations highlight diffusion-driven instabilities and complex Turing patterns, including labyrinthine formations and unexpected "leaser slime" structures-resembling those observed in fungi and algae in aquatic systems. These findings reveal the crucial role of behavioral and ecological feedbacks in shaping pattern formation and species persistence.

, also known as , is a prevalent cause of infectious diarrhea in United States healthcare facilities. Spread through the fecal-oral route and often through contact with spores on contaminated surfaces, can cause severe diarrhea, stomach pain, and colitis. Most individuals can mount an effective immune response, but older populations, immunocompromised individuals, and those taking antibiotics have a higher risk of being colonized by . While extensive research has been conducted in hospital-based settings to improve understanding of the transmission of this bacteria, few studies apply mathematical models in the context of long-term care facilities. This work introduced a mathematical model using a system of ordinary differential equations to represent transmission dynamics in assisted living facilities, with their interactive nature and high risk factors. The equations included four resident classes (susceptible, colonized, diseased, and isolated) and three pathogen-carrying classes (high-traffic areas, low-traffic areas, and healthcare workers' hands) to simultaneously capture the movement between classes and track spore density on environmental reservoirs and healthcare workers' hands, including their contributions to disease spread. Parameter estimation using data from the Emerging Infections Program at the Centers for Disease Control and Prevention was completed and was followed by sensitivity analyses to quantify the impact of varying these parameters and their impact on incidence. Mitigation strategies, including frequent disinfection, increased healthcare worker hand hygiene compliance, a lower ratio between residents and healthcare workers, and increased resident screening had the greatest impact on reducing the incidence of .

Accurately predicting the discharge coefficient (C) is fundamental to the hydraulic design and performance of side weirs. In this study, we introduced a novel artificial intelligence (AI) framework to enhance the prediction accuracy of C for two-cycle trapezoidal labyrinth side weirs. Using a comprehensive laboratory dataset, three distinct machine learning models (MLMs), Support Vector Machine (SVM), Artificial Neural Network (ANN), and Gene Expression Programming (GEP), were developed and rigorously compared with application of the Γ-test technique for sensitivity analysis, systematically identifying the five most influential geometric and hydraulic parameters (Fr, $ \frac{\text{L}}{\text{B}} $, $ \frac{{\text{L}}_{\text{e}}}{\text{L}} $, $ \frac{{\text{Y}}_{\text{1}}\text{-P}}{\text{P}} $, α) to serve as model inputs. The model's efficacy was evaluated across training, testing, and validation phases using multiple statistical metrics: Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Coefficient of Determination (R), and the Maximum Developed Discrepancy Ratio (C). The results demonstrated that the three MLMs are effective predictive tools. However, the ANN model, specifically an MLP5-7-1 architecture utilizing Atan and Identity activation functions optimized with the BFGS 385 algorithm, significantly outperformed the others. It achieved superior results (e.g., validation phase: RMSE = 0.0061, MAE = 0.0003, R = 0.9301, C = 5.22), confirming its highest predictive accuracy and robustness. This research conclusively shows that MLMs, particularly ANN, offer a highly precise and efficient method for predicting Cd in complex hydraulic structures.

The effectiveness of isolation strategies against emerging infectious diseases (EIDs) is critically undermined by two interacting factors: Limited resource capacity and imperfect public compliance, yet their combined impact remains poorly quantified. We develop an ordinary differential equation (ODE) model incorporating a saturation function for resource limits and a compliance parameter ($ \epsilon $) to quantify their nonlinear interaction. Theoretical analysis reveals a resource-driven backward bifurcation, indicating that reducing a basic reproduction number $ R_0 $ below 1 is necessary but may be insufficient for disease elimination when isolation capacity is critically low. Numerically, we identify a counterintuitive paradox: High compliance amplifies the infection risk when isolation resources are severely constrained. The simulation results classify the dynamic regimes under various parameter settings and reveal the qualitative impact of different isolation strategies. The study finds that increasing isolation resources, combined with a certain level of compliance, significantly reduces the infection risk and aids in disease control. Notably, specific transmission patterns emerge when isolation resources are inadequate, resulting in elevated infection risks even when compliance is high. Our results underscore the imperative of synchronizing resource allocation with behavioral interventions, particularly during early outbreak stages, providing a framework for precision public health strategies.

Despite significant advances in image processing, achieving human-like semantic understanding and explainability remains a formidable challenge. Current deep learning models excel at feature extraction but lack the ability to reason about relationships, interpret context, or provide transparent decision-making. To address these limitations, we propose the adaptive neuro-symbolic framework with dynamic contextual reasoning (ANS-DCR), a novel architecture that seamlessly integrates neural networks with symbolic reasoning. ANS-DCR introduces four key innovations: 1) A contextual embedding layer (CEL) that dynamically converts neural features into structured symbolic embeddings tailored to the scene's context; 2) hierarchical knowledge graphs (HKGs) that encode multi-level object relationships and update in real-time on the basis of neural feedback; 3) an adaptive reasoning engine (ARE) that performs scalable, context-aware logical reasoning; and 4) an explainable decision-making module (EDM) that generates human-readable explanations, including counterfactuals, enhancing interpretability. This framework bridges the gap between pattern recognition and logical reasoning, enabling deeper semantic understanding and dynamic adaptability. We demonstrate ANS-DCR's efficacy in complex scenarios such as autonomous driving, where it accurately interprets traffic scenes, predicts behaviors, and provides clear explanations for decisions. Experimental results show superior performance in semantic segmentation, contextual reasoning, and explainability compared with state-of-the-art methods. By combining the strengths of neural and symbolic paradigms, ANS-DCR sets a new benchmark for intelligent, transparent, and scalable image processing systems, offering transformative potential for applications in robotics, healthcare, and beyond. The source code of the proposed ANS-DCR is at github.com/livingjesus/ANS-DCR.

This review examines recent developments in modeling the interaction between tumor cells and the immune system, with a specific focus on the application of delay differential equations (DDEs). The models serve as crucial tools to simulate and predict the immune response to tumor proliferation, thus facilitating a more effective evaluation of clinical and therapeutic strategies before their implementation. This approach enables the hypothetical testing of various interventions, thus resulting in significant time and resource savings. The central theme is the integration of DDEs to represent biologically realistic time delays. These delays-inherent in biological processes such as the activation and migration of immune cells to the tumor site-are essential for a more accurate and dynamic representation of the system. Furthermore, this document acknowledges the inherent limitations of these mathematical models, which are simplified representations of complex biological phenomena by nature. The precision and practical utility of these models depend on the use of biologically plausible delay formulations, the validation of parameters with empirical data, and the alignment of model predictions with clinical outcomes. Ultimately, this work underscores the considerable potential and significant challenges of employing mathematical models as a bridge between theoretical understanding and applied oncology.

This paper develops a mathematical framework for life and health insurance premium calculation under epidemic conditions, incorporating age-structured population dynamics and disease compartments. We proposed a compartmental epidemic model with three age groups and four states (susceptible, infectious, recovered, deceased) to reflect heterogeneity in disease progression and risk exposure. The model captures differential mortality and morbidity risks across age groups and infection states, enabling dynamic adjustment of insurance premiums. By integrating actuarial principles with epidemic-driven transition probabilities, we derived explicit premium formulas and validated them through numerical simulations. Our results demonstrate that age stratification and detailed infection stages significantly impact premium pricing, particularly for older populations with higher mortality risks. Sensitivity analysis reveals that recovery and mortality rates are key drivers of premium variability. The framework provides insurers with a robust tool for pandemic risk assessment, ensuring solvency while maintaining affordability.

Climatic factors exert a substantial influence on both biotic and abiotic components of marine ecosystems, significantly affecting the abundance and spatial distribution of fish species. In this study, we introduced a stochastic modeling framework, grounded in stochastic differential equations (SDEs), to analyze the temporal dynamics of sea surface temperature and its relationship with the abundance of Mahi Mahi () in a region of the Colombian Pacific coast. Model parameters such as sea surface temperature, fish stock, and catch per unit effort for the period 2000 to 2012 were estimated using the maximum likelihood method, implemented via the Euler-Maruyama numerical scheme. The model's performance was assessed using empirical data through numerical simulation, cross-validation, and sensitivity analysis, demonstrating its applicability and robustness in capturing key ecological dynamics.

The classification of rare skin diseases faces significant data scarcity challenges due to the difficulty in acquiring clinical samples and the high cost of annotation, which severely hinders the training of deep neural network-based models. Few-shot learning has emerged as a cutting-edge solution, with its core capability being the identification of novel disease classes using limited annotated samples to mitigate data insufficiency. However, most existing methods fail to fully leverage the statistical information from base classes to calibrate the distribution of few-shot classes, thereby optimizing classifier inputs. Two critical research challenges remain: (1) accurately estimating the true distribution of few-shot classes with minimal samples, and (2) selecting appropriate base class information for effective distribution calibration. To address these challenges, we propose SADC (skin disease classification via adaptive distribution calibration), a new few-shot learning framework incorporating multi-scale feature extraction and adaptive sample calibration. First, our multi-scale feature extraction strategy employs feature descriptor matrices and composite metrics to optimize multi-dimensional, multi-directional feature representations, enabling precise similarity computation between base-class and few-shot samples. Second, the adaptive sample calibration strategy constructs weight matrices based on sample similarity to automatically select optimal base-class samples with adaptive weights for distribution calibration, ensuring alignment between calibrated distributions and true unbiased distributions. Experimental results demonstrated that SADC achieves state-of-the-art performance across three public dermatology datasets (ISIC2018, Derm7pt, and SD198), showing significant improvements over existing methods. The framework's innovation lies in its dual-strategy approach to distribution-aware few-shot learning, advancing the frontier of data-efficient medical image analysis.

Adaptive immunity, performed by T and B lymphocytes, seeks total virus elimination through specific recognition of viral antigens. It has been shown that innate or adaptive immune response regulation variations are associated with an excessive immune response, leading to tissue damage with an increased risk of complications and death. This article is a novel contribution focused on models that represent pathogenic interactions with humans. In our case, the objective was to build and analyze a mathematical model for SARS-CoV-2 infection in the human host, including elements of respiratory cell dynamics, viral particles, and immune-responding cells. The methodology developed considered modeling by means of ordinary differential equations, validation by comparing referenced studies, and sensitivity analysis with respect to the variables considered. Finally, a comparison of simulation models was performed, verifying that an increase in viral particles increases the response of some adaptive immune system cells in the human host.

Boars, being one of the most widely spread ungulates worldwide, have a widely recognized important role in the balance of natural environment and forests. Since large boar populations severely damage crops and cause serious traffic accidents, they are widely hunted, thereby also representing a relevant economic resource. In the model presented here, the species is at times considered ravaging, enabling it to be kept in check, while on the other hand, it must be preserved from extinction as a protected species. We considered an idealized, relatively simple situation in which rangers of the park where the boars are hosted manage this animal population size when they extrude into the surrounding areas through the woods perimeter. Modeling this situation involves considering not the whole boar population, but only those that are involved in the spillover, i.e., those living in proximity of the woods edge. The theoretical investigation and the simulations revealed the existence of a transcritical bifurcation relating the two viable equilibria, coexistence, and the ranger-free point. Also, the possible onset of persistent oscillations via a Hopf bifurcation is shown, leading to periodic recalling of rangers to contain the spillovers. On the other hand, a better regime was obtained by reducing the environment's resources for the wild boars, which stabilized the the boar population at constant level, with a reduced presence of the rangers, reducing the costs of their periodic recalling.

Prior research has explored co-infections that involve two respiratory viruses, yet triple infections remain poorly elucidated. With the COVID-19 pandemic and seasonal epidemics of respiratory syncytial virus (RSV) and influenza, understanding the dynamics of triple infections is critical for public health preparedness. The simultaneous circulation of influenza A virus (IAV), RSV, and SARS-CoV-2 presents a significant public health burden, particularly among vulnerable populations such as children, the elderly, and immunocompromised individuals. Comprehending the interactions among these viruses is crucial to improve our capacity to forecast and curb disease outbreaks. This study addresses the escalating concern over the interaction of multiple respiratory viruses by introducing a simple mathematical model to analyze triple infection dynamics involving IAV, RSV, and SARS-CoV-2. The central question addressed in this study is how variations in infection rates influence each virus's duration and peak viral load in a triple-infection scenario. We find distinct regimes where each virus can dominate and suppress the viral load and duration of the remaining two viruses. We derive an analytical expression for the dependence of the critical infection rate of one virus on the infection rates of the other two viruses. While the model will need to be extended to realistically capture in vivo viral dynamics, this analysis helps provide insight into the complex dynamics of multiple virus infections.

In this article, we proposed a simplified mathematical model of primary tumor growth that involves four cell populations: Two types of cancer cells with different levels of immunogenicity, and the immune response in its two components, innate and adaptive. By varying the proliferation rate of non-immunogenic cancer cells and the innate immune stimulation parameter, and applying biparametric numerical continuation techniques, we identified distinct stability regions that revealed scenarios of tumor escape and latency. A closed curve of supercritical Hopf bifurcation points was also detected, delineating the parameter region in which limit cycles emerged. By examining the population maxima of each cell type at steady state, we identified parameter values at which both immunogenic and non-immunogenic tumor cell populations remain in stable equilibrium at modest levels, sustained by an immune response that does not escalate to intensities associated with immunological damage.

This study present a comparative modeling framework for COVID-19 dynamics using stationary and non-stationary transition probabilities within a Markov decision process (MDP). Stationary transitions assume constant rates, while non-stationary transitions capture time-dependent behaviors driven by policy interventions or behavioral changes. We develop a seven-compartmental epidemiological model, derive transition probabilities from binomial and multinomial processes, and implement time-dependent parameterizations to reflect real-world dynamics. Mathematical models for both stationary and non-stationary transition frameworks are developed and simulated over a 365-day period to emphasize dynamic variations in epidemic outcomes. Our findings highlight the significance of non-stationary modeling in accurately representing the dynamic characteristics of pandemic situations and provide recommendations for optimizing public health interventions under uncertainty. This comparative analysis offers useful information for epidemiological modeling and decision making in dynamic risk environments.

The combined effects of ecological and disease characteristics are examined in eco-epidemiological models, which incorporate infectious illnesses into interaction models. We assumed in this article that the prey population is somewhat infected, and the predator benefits more from eating susceptible prey than from feeding on infected prey. Infected and susceptible prey are equally competitive for resources, and the predator consumes both at the same rate. We employed polar blow-up and time-scale desingularization techniques to tackle the singularity caused by frequency-dependent disease transmission at the origin in our model. For simplicity, we considered the linear functional response for interactions between prey and predators. We aimed to determine the influence of fear of predation on the eco-epidemiological system. According to our findings, there are two ways in which predation fear might support the coexistence of three populations: stable coexistence and oscillatory coexistence. Furthermore, our finding remained unchanged if we eliminated two presumptions: that susceptible and infected prey compete equally for resources and that predators consume both prey at identical rates. We also compared the outcomes by taking into account the growth with positive density dependency (Allee effect) and arrived at the same conclusion.

This study presents an ordinary differential equation (ODE) based hybrid kinetic-metabolic model to predict the time evolution of biomass, glucose, hyaluronic acid (HA), and lactic acid during fermentation by subsp. . The model incorporates simplified metabolic pathways and estimates the qualitative dynamics of internal, unmeasured metabolites involved in glycolysis, biomass synthesis, and HA production. Special emphasis is placed on the energetic molecules ATP/ADP, as well as the coenzymes NADH/NAD, which are involved in redox reactions. These molecules have been shown to play regulatory roles in metabolism. The model predictions closely match the experimental data and provide insights into how varying glucose levels affect intracellular metabolic fluxes.

The vascular tumor growth model proposed by Pinho et al. has gained attention in studies of the effect of anti-angiogenic therapy. In the present work, we extend Pinho's model to a reaction-diffusion model with different cell growth behaviors to evaluate the individual and combined effects of chemotherapy, anti-angiogenic therapy, and immunotherapy across different stages of vascular cancer. Analysis of the model includes the existence and stability of up to six different equilibria with bifurcations that define the transitions between them. By establishing conditions for the stability of the cancer-free equilibrium, we numerically explore different dynamics of cancer relapse. This includes examining the timing and frequency of relapse and identifying thresholds for critical treatment parameters. Furthermore, the numerical simulations of the extended model show that in the advanced stages of cancer, the integration of chemotherapy, immunotherapy, and anti-angiogenic therapy is essential for effective control of vascular cancer and reduces the overall duration of treatment.

Early warning signals are vital in predicting critical transitions in complex dynamical systems. For behavioral epidemiology systems in particular, this includes shifts in vaccine sentiments that may precede disease outbreaks. Conventional statistical indicators, such as variance and lag-1 autocorrelation, often struggle in noisy environments and may fail in real-world scenarios. In this study, we leveraged universal signals of critical slowing down to train deep learning classifiers, specifically using long short-term memory (LSTM) and residual neural network (ResNet) architectures, for detecting early warning signals in disease-related social media time series. These classifiers were trained on simulated data from a stochastic coupled behavior-disease model with additive Lévy noise, a non-Gaussian noise that better reflects the heavy-tailed nature of real-world fluctuations. Our results show that these classifiers consistently outperform conventional indicators in both sensitivity and specificity on theoretical data while delivering quantitatively clear results that are easier to interpret on empirical data. Integrating deep learning with real-time social media monitoring offers a powerful tool for preventing disease outbreaks through proactive public health interventions.

The necessity of modeling the dynamics of infectious disease spread is driven by the imperative to accurately predict epidemics and assess the efficacy of control measures, such as isolation and quarantine. Conventional compartmental SIR and SEIR models have been widely used for predicting the course of epidemics, but they have limitations due to their inability to account for dynamic isolation. Research frequently recognizes the assumptions underlying these models but rarely provides justification for their validity within the specific contexts where they are applied. In this paper, we propose a novel approach based on the concept of a working set, which we utilize as a subset of agents actively involved in social contact and potential transmission. Our adapted working set model incorporates isolation states for susceptible and infected agents, enabling dynamic adjustment of the transmission rate according to the current size of the Working Set. The incorporation of a time window parameter enables the identification of current contacts and the identification of superspreaders, an important component for the optimization of epidemiological measures. Experimental results and comparative analysis showed that, compared to the SIR and SEIR models, the adapted working set model provides a more detailed and realistic tool for analyzing the spread of infection under dynamic control measures. Our model accounts for contact heterogeneity and allows a better assessment of the impact of isolation. The presented approach integrates resource management principles from computer systems with epidemiological models, providing a flexible and realistic tool for evaluating and optimizing infectious disease control measures. In addition, a practical analysis of established models reveals fundamental modeling principles that can be adapted to different scenarios.

In this study, we present a unified mathematical model for tuberculosis (TB) that integrates key interventions: Mask use and media campaigns to raise community awareness and promote vaccine booster uptake. The model also incorporates slow-fast disease progression and limited treatment capacity. A mathematical analysis was conducted to determine the existence and stability of equilibrium points. From the mathematical analysis on the stability criteria of the TB-free equilibrium point, we show that TB can be eradicated if the basic reproduction number is below one. However, due to insufficient treatment capacity, a backward bifurcation may occur when the reproduction number equals one, enabling the coexistence of endemic and disease-free equilibria even when the reproduction number is below one. The parameter estimation is based on TB incidence data per 100,000 individuals in Indonesia. Sensitivity analysis reveald that although both interventions are effective, media campaigns combined with vaccine boosters are more impactful in reducing TB transmission than the use of masks. Numerical simulations further suggest the possibility of periodic outbreaks, indicating potential seasonal TB patterns. To explore adaptive intervention strategies, we extended the model using an optimal control framework. Our findings suggested that combined implementation of face masks and media campaigns is more effective than using either alone, particularly when the likelihood of rapid disease progression increases.