The persistence of covalently closed circular DNA (cccDNA) in the nuclei of HBV-infected hepatocytes plays a critical role in the pathogenesis of chronic hepatitis B (CHB), and hepatitis B surface antigen (HBsAg) levels can be considered a surrogate marker for cccDNA quantification. In this paper, a mathematical model is proposed to mimic cccDNA kinetics in infected hepatocytes, and the basic reproduction rate of cccDNA is obtained. It is found that the backward bifurcation occurs. Importantly, the model predictions match well the clinical data from 96 newly treated CHB patients, collected at the authors' hospitals. Specifically, 18 patients (23.96 %) achieved HBsAg serologic negative conversion (SNC), excluding 5 patients with serologic relapse. Ten patients were predicted to be uncertain, including two clinically confirmed patients with SNC. Excluding the uncertain patients, our model gives a concordance rate of 93.75 % (15/16). Our results suggest that the baseline HBsAg, HBV DNA and hepatitis e surface antigen statuses are the major factors related to the accuracy of the model for the prediction of negative conversion. Therefore, our model is helpful to design effective clinical withdrawal indicators.

Neuroinflammation immediately follows the onset of ischemic stroke. During this process, microglial cells are activated in and recruited to the tissue surrounding the irreversibly injured infarct core, referred to as the penumbra. Microglial cells can be activated into two distinct phenotypes; however, the dynamics between the detrimental M1 phenotype and beneficial M2 phenotype are not fully understood. Using phenotype-specific cell count data obtained from experimental studies on middle cerebral artery occlusion-induced stroke in mice, we employ sparsity-promoting system identification techniques combined with Bayesian statistical methods for uncertainty quantification to generate continuous and discrete-time predictive models of the M1 and M2 microglial cell dynamics. The resulting sparse, data-driven models explain the data using constant and linear terms. Results emphasize an initial M2 dominance followed by a takeover of M1 cells, capture potential long-term dynamics of microglial cells, and suggest a persistent inflammatory response.

In this paper we explore the impact of latency delay and infection by Mycobacterium tuberculosis in the environment on the spread of tuberculosis in a population. We first derive a delay differential equation model with environmental indirect transmission. We address the well-posenedness and identify the basic reproduction number R of the model. We then discuss the equilibria and their stability in terms of the composite threshold parameter R which determine whether or not the tuberculosis will go extinct of persist in the popolaiton: the disease free equilibrium is globally stable if R<1, and it becomes unstable if R>1. In the latter case, there exists a unique endemic equilibrium, which is locally asymptotically stable when τ is sufficiently small; furthermore, we obtain the conditions for the existence of Hopf bifurcation around the endemic equilibrium. The condition implies that the interplay of the latency delay and infection of Mycobacterium tuberculosis in the environment may contribute not only to the TB's persistence but also the way it persists: either as an constant pattern (endemic equilibrium) or as a periodic pattern (oscillation around the endemic equilibrium). We also discuss the epidemiological implication of the mathematical results.

We propose a compartmental model for epidemiology wherein the population is split into groups with either comply or refuse to comply with protocols designed to slow the spread of a disease. Parallel to the disease spread, we assume that noncompliance with protocols spreads as a social contagion. We begin by deriving the reproductive ratio for a deterministic version of the model, and use this to fully characterize the local stability of disease free equilibrium points. We then append the deterministic model with stochastic effects, specifically assuming that the transmission rate of the disease and the transmission rate of the social contagion are uncertain. We prove global existence and nonnegativity for our stochastic model. Then using suitably constructed stochastic Lyapunov functions, we analyze the behavior of the stochastic system with respect to certain disease free states. We demonstrate all of our results with numerical simulations.

Spike frequency adaptation is a key characteristic of spiking neurons. To examine this form of adaptation, we introduce a higher-order fractional leaky integrate and fire model. In this model, the exponent of the fractional derivative can range from one (representing an ordinary first order derivative) to two. In this regime, the impact of the past membrane potential on the present potential is inhibitory leading to spike frequency adaptation. We also analyze spike frequency adaptation in response to noisy input current and show that spike frequency adaptation is reinforced as the intensity of noisy input increases.

Interspecies interactions within ecosystems generate intricate ecological networks and spatial structures. To mitigate predation risks during ecological engagement, species frequently adopt adaptive survival strategies such as refuge concealment. This study develops a bi-trophic food-chain turbidostat model incorporating multiple time delays and refuge protection mechanisms to systematically investigate how critical parameters influence population dynamics and evolutionary patterns. Through rigorous stability analysis of system equilibria, we establish sufficient conditions for equilibrium stability and characterize parameter perturbation effects on system dynamics. Our bifurcation analysis reveals that both transcritical and Hopf bifurcations emerge when refuge parameters approach critical thresholds, demonstrating how parameter variations can transition population growth patterns from stable equilibrium to sustained oscillations. Notably, our refuge parameter analysis demonstrates the dual-edged nature of protective strategies: both excessive and insufficient refuge utilization destabilize population equilibrium. By employing center manifold and normal form theory, we quantitatively assess the nonlinear dynamics near bifurcation points and derive stability criteria for emergent periodic solutions. The temporal analysis further uncovers that time delays induce Hopf bifurcations when surpassing critical values, generating persistent population oscillations that endanger ecological stability. Numerical simulations across multiple parameter regimes consistently validate our theoretical predictions.

Due to the presence of residual effects of pesticides, repeated spraying of pesticides has a cumulative lethal effect on pests which has not been clearly expounded in the existing literature. In this paper, we start by depicting the cumulative lethal rate of pests caused by repeated pesticide spraying. Although the cumulative lethal rate function is complex, our analysis gives an integral invariant of the cumulative killing-rate function, which plays a crucial role in the dynamical analysis of the Logistic single-population growth model that we preferred as a direct application, and helps us obtain a complete dynamical conclusion including the existence, uniqueness and stability of periodic solutions. We derive a threshold of pesticide spraying period for the eventual extinction of the pest population. By combining our theoretical findings and numerical simulations, in accordance with the frequency and cumulative killing-rate function of pesticide spraying, pesticide spraying strategies can be determined to achieve effective pest control within a predetermined time.

The present study is devoted to the precise characterization of dynamic transitions within a continuous two-dimensional ecological framework, induced by a bifurcation module. This investigation aims to enhance our understanding of ecological evolution by disentangling the individual and interactive influences of the double Allee effect and hunting cooperation. Dynamics of all non-negative equilibria are investigated to disclose the degenerate nature of them. Bifurcation points are systematically identified, as their determination is critical for devising strategies aimed at stabilizing ecological systems or averting species extinction, particularly within networks influenced by dual Allee effects and cooperative predation. Both local and global bifurcation analyses, distinguished by the number of relevant parameters and the qualitative behaviour have been explored to capture the system's intricate dynamics. A comprehensive overview of current numerical bifurcation analysis techniques and their ecological applications is provided. Emphasis is placed on the computational challenges encountered in detecting extinction-driven bifurcations and in identifying diverse attractor landscapes and phase transitions. Particular attention has been given to the numerical difficulties posed by both weak and strong forms of the Allee effect, and potential resolutions to these methodological bottlenecks are proposed. Through this structured analysis, an in-depth understanding of species interaction dynamics within bifurcation-driven ecological models is sought, aiming to elucidate complex behaviours that are often obscured in conventional studies of large bifurcating ecological networks.

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 R. The model exhibits backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium for R values below 1, indicating that the disease may persist even when R<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.

Chemical communication is a fundamental component of predator-prey interactions, significantly influencing prey vigilance and predator hunting efficiency. Despite its ecological importance, prevailing mathematical models commonly treat odour as a static or externally imposed parameter, thereby overlooking its inherently dynamic nature and the reciprocal feedback between chemical cues and population dynamics. In this study, we present a novel three-dimensional model incorporating odour concentration as a dynamic variable, directly influenced by predator density and species interactions. Our model captures odour production, its natural decay through environmental processes such as wind-mediated dissipation, and the resultant non-consumptive effects on prey, including stress-induced reductions and alterations in vigilance behaviour. We rigorously analyse the qualitative dynamics of the model. Specifically, we perform stability analyses of all biologically feasible equilibria and detect various bifurcation phenomena, such as transcritical and Hopf bifurcations. The dynamics of the system are also examined across various two-parametric spaces. Extensive numerical simulations corroborate our theoretical findings and facilitate a detailed investigation of parameter influences. Furthermore, we compute the normalized forward sensitivity index to quantify the relative impact of model parameters.

In this work, we extend the nested modeling framework to establish an immuno-epidemiological model for Hand Foot and Mouth Disease (HFMD). This model intricately links host-to-host transmission, virus release, and recovery rates to within-host immune system dynamics. We introduce an ordinary differential equation (ODE)-based within-host model and an partial differential equation (PDE)-based between-host model. Utilizing sensitivity-based locally structural identifiability methods, including principal component analysis (PCA) and eigenvalue analysis, we systematically prioritize parameters from least to most identifiable. Furthermore, we perform globally structural identifiability analysis using differential algebra methods, demonstrating global identifiability of within-host parameters under specific conditions. Leveraging these results, we accurately estimate model parameters using experimental data. Monte Carlo simulations (MCS) reveal practical unidentifiable parameters, indicating that the multi-scale immuno-epidemiological HFMD model is both locally and practically unidentifiable. Finally, sensitivity analysis reveals that within host kinetics considerably influences the temporal dynamics of the HFMD model at the population level.

Tuberculosis (TB) remains a critical global public health challenge, particularly in high-burden regions like Guangdong Province, China. This study develops an integrated framework combining generalized additive models (GAM) and non-autonomous dynamical modeling to elucidate the synergistic effects of environmental and socioeconomic factors on TB transmission dynamics. Utilizing weekly TB case data, air quality index (AQI), absolute humidity (AH), and holiday indicators from Guangdong (2014-2019), GAM quantified nonlinear lagged effects of environmental exposures (AQI, AH) and aperiodic drivers (holidays) on incidence. Results revealed that a 10-unit increase in AQI elevated TB risk by 3.8% (95% CI: 1.2-6.5%), while AH exhibited a negative regulatory effect on transmission. Holiday-related population aggregation amplified case fluctuations by 37% (p < 0.01), with post-holiday rebounds up to 68%. These time-varying parameters were incorporated into a non-autonomous SEIR model with recurrence mechanisms. The basic reproduction number R was estimated at 1.9 (95% CI: 1.2-2.6). Bifurcation analysis confirmed global stability of the disease-free equilibrium when R < 1 and endemic persistence when R > 1. Sensitivity analysis identified infection rate and relapse probability as dominant drivers of transmission intensity. The model predicted a declining long-term trend (-2.6% annually) but persistent winter-spring seasonality. This hybrid approach providing a quantitative tool for optimizing intervention strategies. Key recommendations include reducing airborne pollutants, enhancing surveillance, and targeting relapse prevention to mitigate endemic persistence.

This paper deals with a stochastic nutrient-phytoplankton (NP) model with the impacts of pH and global warming, where the stochastic environmental disturbance is characterized by the logarithmic Ornstein-Uhlenbeck (LOU) process. In the deterministic NP model, we investigate the existence of possible equilibria and analyze their local and global stability. Additionally, by utilizing sensitivity analysis technique, it is shown that phytoplankton density and nutrient concentration are highly sensitive to global warming and pH. In the stochastic NP model, we derive the sufficient conditions of exponential extinction and persistence in the mean of phytoplankton, prove the existence of a stationary distribution, and give the specific expression of the probability density under some appropriate conditions. Ecologically, via numerical simulations, we find that the variation in global warming and pH can generate new influence mechanisms for the interactions between nutrient and phytoplankton within the deterministic and stochastic environments. One of the most interesting results is that an appropriate increase or decrease in pH value is beneficial for inhibiting the occurrence of phytoplankton blooms. This study may provide some new ideas for understanding the dynamic mechanisms of phytoplankton growth in natural aquatic environments.

In silico clinical trials offer a powerful tool for overcoming several limitations of traditional clinical trials. Conventional trials are time- and resource-intensive, typically designed to assess average effects across a population while being restricted to studying the impact of a fixed treatment protocol. In contrast, in silico trials are cost-effective, flexible in their design, and able to explore heterogeneity in treatment response. These trials generally rely on expert-developed and data-calibrated mechanistic mathematical models and the identification of model parameterizations that satisfy biological or clinical constraints. With the growing availability of multi-scale and high-resolution clinical data, it is the opportune time to thoughtfully consider how machine learning (ML) methods can enhance the feasibility, interpretability, and reliability of these in silico trials. In this perspective piece, we explore both the opportunities and the challenges of introducing ML tools at various stages of this process, from biomarker identification to interpreting the results of the trial. We argue that in the hands of an expert modeler, the thoughtful application of ML tools can result in more accurate and informative in silico clinical trials that may potentially accelerate drug development and find the right drug/protocol for the right patient.

In this paper, we study the dynamics of biological invasion through complementary modeling frameworks in the context of nonlocal resource-driven dispersal. During the very early stage of invasion, when only a few individuals are present, demographic variability is crucial: extinction may occur even under favorable average conditions. To capture this, we use a branching-process approximation that provides explicit formulas for extinction probabilities, survival conditions, and mean extinction times. At larger scales and higher densities, invasion is described by a deterministic system of nonlinear integro-differential equations. For this system, we establish well-posedness and derive lower and upper bounds on the asymptotic spreading speed. A unifying threshold parameter T, defined as the spectral radius of a next-generation operator, characterizes invasion outcomes: if T≤1, extinction occurs; if T>1, the invader persists and spreads. Importantly, the threshold derived from the early-stage approximation coincides with that of the deterministic model, thus providing a consistent criterion for invasion success. Finally, numerical simulations illustrate the transition between extinction and persistence and highlight how resource-driven dispersal shapes invasion speed.

Interventions such as vaccinations, treatment et cetera are usually the gold standard of disease control, as measured by reducing the reproduction number below unity. However, in practice, few diseases are reduced below this eradication threshold and instead persist despite active intervention campaigns. We propose an epidemic model of rabies with a saturated incidence rate that represents "soft" interventions such as public-awareness campaigns, animal curfews, fences etc. We prove local and global stability results based on the reproduction number. However, numerical simulations suggest that eradication is unlikely to occur using current practices. We thus investigate the effect of altering the saturated incidence term using "soft" interventions and show that near-eradication can be achieved even when the reproduction number exceeds unity. Soft interventions such as public-awareness campaigns, reducing contacts, animal curfews and fences can have a greater effect on eradicating rabies than current vaccination programs.

The effectiveness of oncolytic virotherapy is significantly affected by several elements of the tumour microenvironment, which reduce the ability of the virus to infect cancer cells. In this work, we focus on the influence of hypoxia on this therapy and develop a novel continuous mathematical model that considers both the spatial and epigenetic heterogeneity of the tumour. We investigate how oxygen gradients within tumours affect the spatial distribution and replication of both the tumour and oncolytic viruses, focusing on regions of severe hypoxia versus normoxic areas. Additionally, we analyse the evolutionary dynamics of tumour cells under hypoxic conditions and their influence on susceptibility to viral infection. Our findings show that the reduced metabolic activity of hypoxic cells may significantly impact the virotherapy effectiveness; the knowledge of the tumour's oxygenation could, therefore, suggest the most suitable type of virus to optimise the outcome. The combination of numerical simulations and theoretical results for the model equilibrium values allows us to elucidate the complex interplay between viruses, tumour evolution and oxygen dynamics, ultimately contributing to developing more effective and personalised cancer treatments.

Acute myeloid leukemia (AML) is characterized by the uncontrolled proliferation of abnormal myeloid cells in the bone marrow and peripheral blood. In this study, we develop a stochastic differential equation model to capture the dynamic interactions among hematopoietic, osteoblastic, and leukemic cell populations within the bone marrow microenvironment. We calibrate model parameters using clinical data via an optimal control framework. Our study provides a mathematical framework to investigate how leukemic cells may remodel the heterogeneous bone marrow niche through dynamic interactions with osteoblastic lineages. This remodeling process disrupts both the quantity and functional capacity of hematopoietic populations, thereby offering insights into how leukemic-niche interactions may contribute to AML treatment failure. Furthermore, we evaluate the efficacy of combination therapies (traditional chemotherapy with targeted therapy) and compare their therapeutic outcomes. These findings offer a theoretical foundation for optimizing clinical strategies and advancing personalized treatment approaches for AML.

Particle-based simulations are an essential tool for the study of biochemical systems for scales between molecular/Brownian dynamics and the reaction-diffusion master equation. These simulations utilise proximity-based reaction conditions and are typically limited to elementary (mass-action) kinetics. We present a novel framework for directly simulating non-elementary bimolecular kinetics in a particle-based framework. By mimicking the behaviour of a third implicit reactant, we adapt non-elementary reaction conditions, previously restricted to trimolecular chemical interactions, to biomolecular reactions for the first time. We implement our approach in an event-driven simulation, which we validate by reproducing Michaelis-Menten kinetics. We then demonstrate its utility by simulating the classical Goldbeter model of circadian oscillations completely at the level of individual molecules. This model features multiple non-elementary reactions and requires the incorporation of several existing simulation techniques. Our method accurately reproduces the target non-elementary kinetics, without simulating the implied underlying fast elementary reactions, thereby significantly reducing the computational cost. This work expands the class of reaction networks accessible to particle-based simulations and provides a practical alternative to explicitly simulating all elementary steps in systems where quasi-steady-state approximations are applicable.

We study a Keller-Segel system with nonlinear chemical gradient. The system contains two parameters c > 0 and ε > 0 and has a family of equilibria (u,v) with u 0 and s > 0 and sufficiently small ε > 0; if u 0 such that for c ≥ c*, such a solution exists for all s > 0 and sufficiently small ε, while for 0 < c < c*, such a solution exists only for s in a disconnected set of (0, ∞) which includes two connected components (0,s) and (s^,∞), where 0

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