In this paper, we investigate the dynamics of a new chronic hepatitis B model through environmental-driven infection in heterogeneous and homogenous space. The model includes spatial diffusion, five time delays, and a general incidence function. The well-posedness of solutions, including the existence of global solutions, uniqueness, non-negativity, ultimate boundedness, is established. The reproduction numbers for virus infection, CTL immune response, antibody immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By employing the Lyapunov functionals and linearization methods, the threshold conditions on the global stability of the infection-free equilibrium are obtained; and the threshold conditions on the uniform persistence and the global stability of the equilibria for immune-free, antibody response, CTL response and antibody and CTL responses, respectively, are established. The occurrence of Hopf bifurcation of the spatially homogeneous equilibria is also established. An application is provided, and numerical simulations are carried out to illustrate the analytical results and verify the conjectures. Our findings reveal the rich and complex patterns of HBV infection dynamics in vivo, including regular and irregular periodic oscillations. Irregular oscillations may signify immune disorders and allergies. Also, from numerical simulations, it follows that the majority of hepatitis B infection could escape chronic infection.
A method is presented for registering MRA and 4D Flow images of large blood vessels (e.g., aorta, brachiocephalic, left common carotid, and left subclavian) with the aim of combining information from both modalities to calculate blood flow properties within these vessels. In particular, the vessel surface segmented from an MRA image is aligned with the blood velocity field extracted from a corresponding 4D Flow image. The registration algorithm is driven by aligning vessel centerlines extracted from the two images. The proposed method is robust to small deformations, partial omissions of vessel surfaces, and noise in the blood velocity field. It is tested on image pairs from 8 patients with single-ventricle physiology. The quality of the resulting alignment is assessed using (i) histograms of distances between corresponding vessel centerlines, (ii) estimates of blood flow using the aligned and unaligned segmentations (as extracted from the MRA image), and (iii) visual comparisons of the aligned and unaligned segmentations together with the (possibly deficient) segmentation extracted from the corresponding 4D Flow image. Following registration, medians of the distance histograms between corresponding centerlines decrease by an average of 80.6%, blood flow estimates at systole increase at all three measured locations along the aorta for every patient, and visualizations show a significant improvement in the alignment between corresponding segmentations.
Cancer Antigen 15-3 (CA 15-3) is a glycoprotein used to monitor breast cancer progression and treatment response. This study analyses five years of CA 15-3 measurements from a male patient with metastatic breast cancer using an integrated mathematical and data-driven approach. A phenomenological model combining an oscillatory component, capturing sinusoidal fluctuations, with an exponential term representing clearance, treatment response or tumor growth, was fitted to the longitudinal data. Key parameters-including baseline level, rates of increase and decay, oscillation frequency and amplitude-were estimated, providing a structured characterization of the biomarker's temporal dynamics. In parallel, data-driven analysis using variational mode decomposition (VMD) was employed to isolate intrinsic frequency components and detect latent transitions represented in the Hilbert Spectrum (HS) that were not captured by the parametric model. The results revealed a stable sinusoidal pattern with a 170-day period persisting for three cycles-an atypical finding, as CA 15-3 variations typically reflect disease progression or treatment effects rather than a predictable oscillatory rhythm. Oscillations emerged with the start of metastatic phase therapy, with the sinusoidal pattern linked to palbociclib, though the underlying biological mechanism remains unclear and warrants further investigation. The model accurately reproduced these periodic fluctuations and overall trends, showing strong agreement with measured values across five years. Complementary VMD/HS analysis identified instantaneous frequency components linked to metastatic onset and therapy intervals, enhancing the temporal resolution of structural changes. Together, the combined modelling approach improves the detection and interpretation of dynamic biomarker behaviour and supports more personalized strategies for cancer monitoring.
The Human Immunodeficiency Virus (HIV) has been a major focus of research for several decades due to its multiple transmission pathways, its high virulence and the absence, to date, of both curative and preventive treatments. Although sexual transmission remains by far the most dominant route, mother-to-child (vertical) transmission-which may occur during pregnancy, childbirth or breastfeeding-is also significant and accounts for approximately 15%-30% of cases in the absence of antiretroviral therapy (ART). In this work, we develop a heterosexual HIV transmission model that incorporates vertical transmission, arising from sexual contact. The model is analysed both qualitatively and quantitatively. We establish the global asymptotic stability of the disease-free equilibrium when the basic reproduction number, $\mathscr{R}_{0}$, is less than one. When $\mathscr{R}_{0}$ is greater than one, we show the existence, uniqueness and local asymptotic stability of a unique endemic equilibrium. To perform numerical simulations, we construct a nonstandard finite difference scheme, which preserves the positivity, boundedness and stability properties of the continuous model. Numerical results are presented to support the theoretical findings and to assess the impact of abstinence, safe sexual practices, ART intervention and some joint public health strategies in reducing HIV infections. In particular, we found that if the men's and women's yearly contact rates are less than 120 and 100, respectively, HIV will be eliminated from the community.
An impulsive differential equation model of cancer treatment by radiation therapy (RT) is studied. Analytical results for the model's persistence and eradication of cancer cell volumes are obtained to illuminate the dynamics between tumor growth and RT. It is also shown that, although periodic solutions may exist, they are necessarily unstable. A modified model is then proposed, assuming that RT is more effective than the first model assumes. In addition to similar results as for the original model, conditions are obtained under which periodic solution exists and is globally stable, showing the possibility that regression can occur in periodicity. Numerical simulations are provided to confirm the results.
Biofilm infections on medical implants are difficult to eradicate because insufficient nutrient availability promotes antibiotic-tolerant persister cells that survive treatment and reseed growth. Existing mathematical models usually omit nutrient-dependent phenotypic switching between proliferative and persister states. Without this mechanism, models cannot capture how environmental conditions control the balance between active growth and dormancy, which is central to biofilm persistence. We present a continuum model that couples nutrient transport with the dynamics of proliferative bacteria, persisters, dead cells and extracellular polymeric substances. The switching rates between proliferative and persister phenotypes depend on local nutrient concentration through two thresholds, enabling adaptation across nutrient-poor, intermediate and nutrient-rich regimes. Simulations show that nutrient limitation produces a high and sustained proportion of persister cells even when biomass is reduced, whereas nutrient-rich conditions support reversion to proliferative growth and lead to greater biomass. The model also predicts that persister populations peak at times that vary with nutrient availability, and these peaks coincide with turning points in biofilm growth, identifying critical intervention windows. By directly linking nutrient availability to phenotypic switching, our model reveals mechanisms of biofilm persistence that earlier models could not capture, and it points toward strategies that target nutrient-driven adaptation as a means to improve the control of implant-associated infections.
From 1999 to 2022, nearly 300,000 people died from overdoses involving prescription opioids. Commonly used for the management of acute post-surgical pain, prescription opioids remain a leading cause of opioid misuse and abuse. Among surgical populations, patients who undergo hand surgery have especially high rates of prolonged post-operative opioid use and opioid use disorder (OUD). We built a compartmental model describing the progression from opioid exposure to OUD in hand surgery patients and parameterized it using empirical patient data. Using this model, we investigated the functional relationship between the number of opioid prescriptions received and opioid abuse. We found that, depending on the response curve used to describe opioid addiction risk, different numbers of prescriptions pose higher risks than others, advancing our understanding of how prescription patterns influence OUD development.
Tear film (TF) breakup is a key driver of understanding dry eye disease, and estimating TF thickness and osmolarity from fluorescence (FL) imaging typically requires solving computationally expensive inverse problems. We propose an operator learning framework that replaces traditional inverse solvers with neural operators trained on simulated TF dynamics. This approach offers a scalable path toward rapid, data-driven analysis of tear film dynamics.
In Single Photon Emission Computed Tomography (SPECT), the image reconstruction process involves many tunable parameters that have a significant impact on the quality of the resulting clinical images. Traditional image quality evaluation often relies on expert judgment and full-reference metrics such as Mean Squared Error and Structural Similarity Index. However, these approaches are limited by their subjectivity or the need for a ground-truth image. In this paper, we investigate the usage of a No-Reference Image Quality Assessment method in SPECT imaging, employing the Perception-based Image QUality Evaluator (PIQUE) score. Precisely, we propose a novel application of PIQUE in evaluating SPECT images reconstructed via filtered backprojection using a parameter-dependent Butterworth filter. For the optimization of filter's parameters, we adopt a kernel-based Bayesian optimization framework grounded in reproducing kernel Hilbert space theory, highlighting the connections to recent greedy approximation techniques such as $P$- and $f$-greedy. Experimental results in a concrete clinical setting for SPECT imaging show the potential of this optimization approach for an objective and quantitative assessment of image quality, without requiring a reference image.
This study proposes using accumulated neurotoxicity, defined as the time integral of Aβ oligomer concentration, as a biomarker for neuronal aging. A relationship between biological age and accumulated neurotoxicity is proposed. Numerical analysis guided the development of a new analytical solution linking the biological and calendar ages of neurons. The effects of Aβ monomer and oligomer half-lives—key indicators of proteolytic efficiency—on biological age are examined. Both constant and age-dependent (exponentially increasing) half-life scenarios are considered. The findings indicate that increasing the half-life of Aβ monomers and oligomers with age accelerates biological aging. Reducing Aβ monomer production is shown to slow biological aging, with a linear relationship established between these two quantities. Additionally, biological age is found to depend linearly on the half-deposition time of Aβ oligomers into senile plaques. The model demonstrates that biological age is irreversible, providing a theoretical explanation for why plaque-clearing therapies cannot reverse established cognitive impairment. The model also demonstrates that biological age is path-dependent rather than state-dependent.
We investigate the competition of two cell types in an epithelial layer driven by differences in their mechanical properties. A simple one-dimensional model of a cell layer is represented as a chain of overdamped elastic springs with turnover of cells described as stochastic birth and death events. First, we investigate the effects of size-dependent cell division probability in establishing equilibrium density of a single cell type. Then we focus on the competition of mechanically different cells as a simple model of invasive cancer. We show that a sharp size threshold for cell division leads to equilibrium density dependent on the cell stiffness and is different from the biological equilibrium density resulting from the balance of birth and death rates. In a system composed of two distinct cell types, mechanical differences lead to invasion waves. We derive an analytical approximation for the travelling wave speed and show that the competitive advantage of cells with different stiffness is determined by the relationship between the mechanical and biological equilibrium density in the cell layer.
We propose a system of ordinary differential equations to model the mouse immune response of two key immune cell types (CD4+ and CD8+ cells) to an established triple-negative breast cancer tumor while being treated with immunotherapy drugs of anti-PD-1 and anti-CTLA-4 immune checkpoint inhibitors. The model incorporates longitudinal positron emission tomography image data from a series of experiments where immunotherapy treatment was given in combination or separately. Control data optimization estimates the immune-tumor response of a general mouse burdened with breast cancer. Collaborative input designated the location of treatment effects that were further parameterized. The results indicate quantifiable differences in parameter values that differentiate immunotherapy responder and nonresponder groups. Treatment parameters are first determined from single and then from combination immunotherapy data. Structural identifiability is used to classify the identifiability of the parameters, while Sobol sensitivity analysis is employed to narrow the key treatment interactions of the model. From the constrained treatment model, we can accurately predict tumor volume changes for most treatment data, which strengthens our methodology while highlighting key interactions.
Scratch assays are affordable methods developed for sampling wound healing in a laboratory setting. Thanks to these assays, it is possible to investigate the dynamical structure of cell migration and proliferation, which play a central role in the healing process of the wound. Johnston et al. (2015, BMC Syst. Biol., 9, 38) use scratch assay data to estimate migration and proliferation parameters in a Fisher-type model. The present study is a new attempt to interpret the same data using a non-local continuum approach that incorporates cell-cell adhesion. The non-local part of our model includes two different force functions representing different types of cell adhesion. Using these functions, we estimate the parameters involved in the diffusive and adhesive motion. The original and our model give similarly good agreement with the experimental data for their respective (optimal) parameter sets but the estimated diffusion coefficients differ significantly between both sets. Consequently, Johnston et al.'s data, and thus their experimental methodology, are incapable of providing guidance on the effect of cell-cell adhesion in wound healing.
An infectious disease such as COVID-19 posed a threat to public health worldwide due to its high infection rate and its further mutation into novel variants. Vaccination serves as a vital tool to interrupt its transmission cycle and far-reaching effects. However, the effectiveness of vaccination depends upon a well-planned strategy. This study explores the comparison between full and partial vaccination strategies using a novel fractional SVIR mathematical model with Caputo fractional derivative. The model categorizes vaccinated individuals into two groups: partially and fully vaccinated class. To account for limited medical resources and virus reemergence, we adopt the Holling type III saturated treatment function for treatment rate. In the analysis, we first show well posedness of model solutions. Further, we discuss the stability of the two equilibria exhibited by the system: DFE (Disease Free Equilibrium) and EE (Endemic Equilibrium). It is shown that the DFE is locally asymptotically stable when R0 < 1, and EE is locally asymptotic stable by Routh-Hurwitz criterion. Moreover, both the equilibrium points are proved to be globally asymptotically stable under certain conditions with the help of appropriate Lyapunov function. Numerical simulations are also performed to validate the analytical findings using MATLAB. The quantification of effects of partial and full vaccination reveals that full vaccination results in higher percentage of recovered population, making it evident that policymakers and professionals should focus on the implications of effective full vaccination among susceptible individuals.
We consider a reaction network of the Wnt pathway endowed with mass-action kinetics. Using concepts in the theory of robustness and stability within Chemical Reaction Network Theory and advances in decomposing reaction networks, we perform a systematic analysis of the structural, structo-kinetic and kinetic properties of this pathway. We show that the network can be systematically decomposed into a set of subnetworks and we use elements matrix theory to study their stability properties. Considering the positive stoichiometric classes, we obtain the analytical expressions of the positive steady states and we identify three species with absolute concentration robustness within the core of the destruction complex of the pathway. We identify nonnegative stoichiometric classes that admit boundary steady states. We construct the non-associative commutative algebra associated with the system and combine algebraic and algorithmic approaches to characterize its structural properties, construct its subalgebras and show how they relate to the existence of boundary initial conditions that admit boundary steady state solutions. We also show the existence of a category of subalgebras that generate unbounded solutions for most of the nonnegative initial conditions.
Delineating the dynamics of highly lethal anthrax disease in a biosecured livestock farm and impact of anthrax vaccination is presented through a modified deterministic $SIRBV$ model incorporating nonlinear ratio-dependent disease transmission rate. The basic reproduction number $(R_{0})$ of the system is computed and employed to explore the existence and asymptotic stability around the steady states of the system. The system experiences transcritical bifurcation at the disease-free steady state for $R_{0} = 1$. Waning of recovery-derived immunity and vaccination-derived immunity trigger backward bifurcation causing reemergence of anthrax in livestock. The dynamical behaviors of the fractional order system express that increased immunological memory will benefit to cut down the eradication time of anthrax transmission from the system. Numerical simulations suggest that appropriate vaccination and comprehensive biosecurity protocols would help to prevent the anthrax transmission and control the disease-induced deaths of cattle.
The study of the dynamics of an infectious disease is fundamental to understanding its community spread. These include obtaining estimates for transmission rates, recovery rates and the average number of secondary cases per infectious case (reproduction number). Social behaviors, control measures, environmental conditions and long recovery times result in time varying parameters. Further, imperfect data and many uncertainties lead to inaccurate estimations. This is particularly true in third-world countries, where a greater proportion of people with mild infections may not seek medical treatment. Data on the prevalence of COVID-19 provides an excellent source for case studies to analyse time-dependent parameters. Using Sri Lankan COVID-19 data, we demonstrate how one could utilize Itô stochastic differential equations with a gamma distribution correction to estimate disease transmission parameters as a function of time. As we illustrated here, the model is well-suited for forecasting the dates of peak prevalence and the number of new cases using the estimated parameters.
The normal function of the brain depends on adequate oxygen supply. Oxygen deprivation (hypoxia) can result in irreversible damage to neurons within minutes. Cooling (hypothermia) of brain tissue can reduce the rate of damage, and is used in surgeries where blood flow to the brain is interrupted, such as aortic arch reconstruction. Hypothermia affects several factors that influence tissue oxygen levels, including oxygen consumption rate, diffusivity and solubility. The goal of the present work is to predict the effects of hypothermia on the partial pressure of oxygen in brain tissue. The dependence on temperature of parameters governing oxygen transport is estimated from literature data. A theoretical model based on the Krogh cylinder configuration is used to predict the effects of hypothermia on the distribution of oxygen partial pressure in the cylindrical tissue region surrounding a capillary. For a given blood flow rate and inflowing oxygen level, tissue oxygen levels are shown to increase with decreasing temperature. Although oxygen diffusivity in tissue declines with hypothermia, the reduction in oxygen consumption leads to a net increase in predicted oxygen levels. Tissue hypoxia resulting from reductions in blood flow rate can be ameliorated by reductions in temperature. For example, if blood flow is reduced to 36% of normal, temperature reduction by $2.3^\circ{C}$ can increase tissue oxygen levels above the hypoxic range. The results support the use of hypothermia to reduce brain damage under conditions of reduced blood flow.
Controlling the elevated levels of methane ($\mathrm{CH}_{4}$) in the atmosphere is crucial to tackling the problem of climate change. Both rice paddies and livestock farming are substantial contributors to this elevated methane. The integrated rice-livestock farming system is an agricultural practice designed to optimize the use of agricultural waste, while concurrently boosting rice and livestock productivity. Achieving the dual objectives of food security and mitigating climate change demands formulation and implementation of strategies that are aimed at managing the methane emissions from the rice-livestock farming system. This study introduces a nonlinear mathematical model of the emission and mitigation of methane in the integrated rice-livestock farming system. Through qualitative analysis, the model's dynamic behavior is thoroughly explored, identifying conditions for reduction and stabilization of atmospheric methane concentrations. Model parameters are estimated using secondary data on atmospheric methane concentration, rice yield and livestock population. A sensitivity analysis is presented to evaluate the influence of variations in crucial parameters on the system's behavior. Numerical simulations are conducted to confirm the validity of the theoretical results.
This study presents a mathematical model describing cloned hematopoiesis in chronic myeloid leukemia $\text{(CML)}$ through a nonlinear system of differential equations. The primary objective is to understand the progression from healthy hematopoiesis to the chronic and accelerated-acute phases in myeloid leukemia. The model incorporates intrinsic cellular division events in hematopoiesis and delineates the evolution of chronic myeloid leukemia into five compartments: cycling stem cells, quiescent stem cells, progenitor cells, differentiated cells and terminally differentiated cells. Our analysis reveals the existence of three distinct non-zero steady states within the dynamical system, representing healthy hematopoiesis, the chronic phase and the accelerated-acute stage of the disease. We investigate the local and global stability of these steady states and provide a characterization of the hematopoietic states based on this analysis. Additionally, numerical simulations are included to illustrate the theoretical results.
