Search Results (91)

In this study, we propose a compartmental mathematical model that considers two interacting populations (citrus plants and insect vectors) and investigate the transmission dynamics of Huanglongbing in citrus crops. This disease is caused by the bacterium Candidatus Liberibacter asiaticus and is vectored by the psyllid Diaphorina citri. The disease is modeled under the following three main assumptions: there is vital dynamics with constant recruitment rates of citrus plants, the force of infection in both populations is a spatially dependent function varying with geographic location, and there is a spatial displacement of the vectors. In the main results of the paper, we formulate a coupled ordinary and partial differential equation system with initial and zero flux boundary conditions, establish the existence and uniqueness of solutions to the proposed model by applying semigroup theory, and introduce a numerical approximation of the system. Moreover, we develop a stability and persistence analysis. From the analytical point of view, we calculate the basic reproduction number $R_0$ and prove three facts: the disease-free equilibrium is globally asymptotically stable when $R_0<1$; the disease-free equilibrium is globally asymptotically stable when $R_0>1$; and the hybrid system exhibits uniform persistence of infection when $R_0>1$. In addition, we present some numerical examples. Full article

Background: Tuberculosis remains a major global public health challenge. Mathematical models are essential for strategic planning and evaluation of tuberculosis control programs, while addressing socioeconomic risk factors has proven key to accelerating incidence declines. Therefore, this study quantitatively assesses the impact of socioeconomic interventions on tuberculosis incidence in Kazakhstan. Methods: A modified SIR compartmental model was developed in Python 3.12 to simulate tuberculosis transmission dynamics. Parameters were calibrated using the Nelder–Mead simplex algorithm, and predictive performance was evaluated via hold-out validation. Scenario-based projections were generated to explore the impact of socioeconomic improvements on future tuberculosis incidence. Results: The calibrated SIR model demonstrated strong predictive accuracy, achieving a mean absolute percentage error of 2.3%. The sensitivity analysis revealed that the model is robust to moderate socioeconomic perturbations, with healthcare funding and unemployment rate as the primary uncertainty drivers. Scenario simulations showed that enhanced financial assistance for tuberculosis patients produced the largest effect beyond baseline. Optimization results indicate that 7.4% rise in GDP per capita, 10.2% increase in healthcare funding, 23.1% and 19.1% reductions in poverty and unemployment rates, and 40.2% growth in tuberculosis patient financial support relative to 2024 are sufficient to achieve the WHO’s End TB Strategy 2030 target. Conclusions: The model offers a valuable tool for tuberculosis forecasting and intervention evaluation, highlighting the synergistic role of socioeconomic measures in achieving global elimination goals. Full article

This article introduces and thoroughly examines a novel deterministic compartmental model of COVID-19 dynamics. The model uniquely incorporates compartments for symptomatic and asymptomatic individuals alongside an environmental reservoir for the pathogen. It also accounts for a steady inflow of infected visitors and a steady outflow from the removed class. The mathematical soundness of the model is established by identifying the invariant region and ensuring positivity of solutions. Notably, during surges of infected visitors, certain classes maintain positive minimum values. We analytically determine endemic equilibrium points and prove the global stability of the disease-free equilibrium. Sensitivity analysis highlights the significant roles of transmission rates and asymptomatic individuals in disease spread. Simulation results corroborate the theoretical findings and provide additional insights into the model’s predictive capabilities. Full article

This paper presents a comprehensive mathematical analysis of a novel compartmental model describing the dynamics of dispersed water pollutants and their interaction with two distinct host populations. The model is formulated as a system of integro-differential equations that incorporates multiple distributed delays to realistically account for time lags in the infection process and pollutant transport. We rigorously establish the biological well-posedness of the model by proving the non-negativity and ultimate boundedness of solutions, confirming the existence of a positively invariant feasible region. The analysis characterizes the long-term behavior of the system through the derivation of the basic reproduction number ${\mathcal{R}}_0^d$, which serves as a sharp threshold determining the system’s fate. For the model without delays, we prove the global asymptotic stability of the infection-free equilibrium (IFE) when ${\mathcal{R}}_0\le 1$ and of the endemic equilibrium (EE) when ${\mathcal{R}}_0>1$. These stability results are extended to the distributed-delay model by using sophisticated Lyapunov functionals, demonstrating that ${\mathcal{R}}_0^d$ universally governs the global dynamics: the IFE (${\mathcal{E}}_0^d$) is globally asymptotically stable (GAS) if ${\mathcal{R}}_0^d\le 1$, while the EE (${\mathcal{E}}_d^{\ast }$) is GAS if ${\mathcal{R}}_0^d>1$. Numerical simulations validate the theoretical findings and provide further insights. Sensitivity analysis identifies the most influential parameters on ${\mathcal{R}}_0^d$, highlighting the recruitment rate of susceptible individuals, exposure rate, and pollutant shedding rate as key intervention targets. Furthermore, we investigate the impact of control measures, showing that treatment efficacy exceeding a critical value is sufficient for disease eradication. The analysis also reveals the inherent mitigating effect of the maturation delay, demonstrating that a delay longer than a critical duration can naturally suppress the outbreak. This work provides a robust mathematical framework for understanding and managing dispersed water pollution, emphasizing the critical roles of multi-source contributions, time delays, and targeted interventions for environmental sustainability. Full article

Antibody-drug conjugates (ADCs) have risen in prominence over the past 15 years, with numerous regulatory approvals in oncology. A complicating factor in the development of ADCs is the presence of numerous analytes with unique pharmacologic properties. Following administration, ADCs are present in the body as the intact ADC, unconjugated antibody, and liberated payload. Due to heterogeneity in conjugation and in vivo deconjugation rates, the drug-to-antibody ratio (DAR) changes with time. Each of these molecular species has unique pharmacokinetic (PK) and pharmacodynamic (PD) properties that should be understood and characterized. One approach that is frequently applied is the development of in silico mathematical models to characterize and predict the PK/PD of ADCs. In this review, we summarize key mechanisms controlling the PK/PD of ADCs. This provides context for a detailed discussion of the array of PK/PD models that have been applied for ADCs, ranging from empirical compartmental models all the way through system-level models, such as physiologically based pharmacokinetics (PBPK) and cell-level PK/PD models. We provide a critical discussion of the strengths, weaknesses, and utility of each of these model structures. Full article

In this article, mathematical modeling and stability analysis of Lyme disease and its transmission dynamics using Caputo fractional-order derivatives is presented. The compartmental model has been formulated to analyze the spread of Borrelia burgdorferi virus through tick vectors and mammalian hosts. The feasible region is established, and the boundedness of the model is verified. Analytically, the disease-free equilibrium and the basic reproduction number $\left({\Re }_0\right)$ has been determined to assess outbreak potential. By virtue of the fixed-point theory, the existence and uniqueness of solutions has been established. The numerical simulations are obtained via the Runge–Kutta 4 method, demonstrating the model’s ability to capture realistic disease progression. Finally, sensitivity analysis and control strategies (tick population reduction, host vaccination, public awareness, and early treatment) are evaluated, revealing that integrated control measures significantly reduce infection rates and enhance recovery. Full article

Background: Monkeypox (Mpox) is a re-emerging zoonotic disease caused by the monkeypox virus (MPXV). Transmission occurs primarily through direct contact with lesions or contaminated materials, with sexual transmission playing a significant role in recent outbreaks. In 2022, Mpox triggered a major global outbreak and was declared a Public Health Emergency of International Concern (PHEIC) by the World Health Organization (WHO), prompting renewed interest in effective control strategies. Methods: This study developed a compartmental SEIR-based model to assess the epidemiological impact of key interventions, including vaccination and isolation, while incorporating critical epidemiological parameters. Sensitivity analyses were conducted to examine (1) disease dynamics in relation to the basic reproduction number, and (2) how different parameters influence the curve of symptomatic infections. Real-world continental-scale data were used to validate the model and identify the parameters that most significantly affect epidemic progression and potential control of Mpox. Results: Results showed that the basic reproduction number was most influenced by the recovery rate, vaccination rate, vaccine effectiveness, and transmission rates of symptomatic and asymptomatic individuals. In contrast, the progression of symptomatic cases was highly sensitive to the case fatality rate and incubation rate. Conclusions: These findings highlight the importance of integrated public health strategies combining vaccination, isolation, and early transmission control to mitigate future Mpox outbreaks. Full article

According to the World Health Organization (WHO), globally, cervical cancer ranks as the fourth most common cancer in women, with around 660,000 new cases in 2022. In the same year, about 94 percent of the 350,000 deaths caused by cervical cancer occurred in low- and middle-income countries. This paper focuses on the dynamics of HPV by modeling the interactions between four compartments, as follows: S(t), the number of susceptible females; I(t), females infected with HPV; X(t), females infected with HPV but not yet affected by cervical cancer (CCE); and V(t), females infected with HPV and affected by CCE. A compartmental model is formulated to analyze the progression of HPV, ensuring all key mathematical properties, such as existence, uniqueness, positivity, and boundedness of the solution. The equilibria of the model, such as the HPV-free equilibrium and HPV-present equilibrium, are analyzed, and the basic reproduction number, $R_0$, is computed using the next-generation matrix method. Local and global stability of these equilibria are rigorously established to understand the conditions for disease eradication or persistence. Sensitivity analysis around the reproduction number is carried out using partial derivatives to identify critical parameters influencing $R_0$, which gives insights into effective intervention strategies. With appropriate positivity, boundedness, and numerical stability, a new stochastic non-standard finite difference (NSFD) scheme is developed for the proposed model. A comparison analysis of solutions shows that the NSFD scheme is the most consistent and reliable method for a stochastic fractional delay model. Graphical simulations are presented to provide visual insights into the development of the disease and lend the results to a more mature discourse. This research is crucial in highlighting the mathematical rigor and practical applicability of the proposed model, contributing to the understanding and control of HPV progression. Full article

Agri-food supply chains have experienced notable changes in recent decades, with tomatoes (Solanum lycopersicum) maintaining their status as a key global crop in terms of both production and consumption. These supply chains comprise a complex network of stakeholders—including producers, processors, distributors, and retailers—who collectively ensure the delivery of tomatoes from farms to consumers. This study develops mathematical models of agri-food tomato supply chains (AFTSCs) and examines their behavior through stability analysis and dynamic simulations based on a compartmental approach. Furthermore, the environmental impact is evaluated using a sustainability index, to which the waste diversion rate is introduced. This metric is defined as the proportion of diverted waste (i.e., materials recycled, reused, or composted) relative to the total waste generated, thereby enabling the quantification of sustainability performance within the system. Finally, a sensitivity analysis is conducted on the proposed dynamical models to validate and reinforce the findings. Full article

Graphic design and image processes have a vital role in information technologies and safe, memorable learning activities, which can meet the need for modern and visual aids in the field of education. In this article, the concepts of comparison and competition have been presented using grades or numbers obtained for two different intelligence quotient (IQ) classes of students. The two classes are categorized as learners having textual (un-visualized) and visualized aids. We use the results and outcomes of the random sampling data of the two classes in the parameters of four different, competitive, two-compartmental mathematical models. One of the compartments is for students who only learn through textual learning, and the other one is for students who have access to visualized text resources. Four of the mathematical models were solved numerically, and their grades were obtained by different iterations using the data of the mean of different random sampling tests taken for thirty months; each sampling involved thirty students. The said data are also drawn by using a neural network approach, showing the fitting curves for all the data, the training data, the validation data, and the testing data with histogram, aggression, mean square error, and absolute error. The obtained dynamics are also compared with neural network dynamics. The results of the scenario pointed out that the best results (determined through high grades) were obtained among the students of visual aid learners, as compared to textual and conventional learners. The visualized resources, constructed within the mathematics syllabus domain, may help to upgrade multidimensional mathematical education and the learning activities of intermediate-level students. For this, the findings of the present study are helpful for education policymakers: there is a directive to focus on visual-based learning, utilizing data from various surveys, profile checks, and questionnaires. Furthermore, the techniques presented in this article will be beneficial for those seeking to build a better understanding of the various methods and ideas related to mathematics education. Full article

This study presents an extended approach to compartmental modeling of infectious disease spread, focusing on regional heterogeneity within affected areas. Using classical SIS, SIR, and SEIR frameworks, we simulate the dynamics of COVID-19 across two major regions of Ukraine—Dnipropetrovsk and Kharkiv—during the period 2020–2024. The proposed mathematical model incorporates regionally distributed subpopulations and applies a system of differential equations solved using the classical fourth-order Runge–Kutta method. The simulations are validated against real-world epidemiological data from national and international sources. The SEIR model demonstrated superior performance, achieving maximum relative errors of 4.81% and 5.60% in the Kharkiv and Dnipropetrovsk regions, respectively, outperforming the SIS and SIR models. Despite limited mobility and social contact data, the regionally adapted models achieved acceptable accuracy for medium-term forecasting. This validates the practical applicability of extended compartmental models in public health planning, particularly in settings with constrained data availability. The results further support the use of these models for estimating critical epidemiological indicators such as infection peaks and hospital resource demands. The proposed framework offers a scalable and computationally efficient tool for regional epidemic forecasting, with potential applications to future outbreaks in geographically heterogeneous environments. Full article

Poverty is a multifaceted phenomenon impacting millions globally, defined by a deficiency in both material and immaterial resources, which consequently restricts access to satisfactory living conditions. Comprehensive poverty analysis can be accomplished through the application of mathematical and modeling techniques, which are useful in understanding and predicting poverty trends. These models, which often incorporate principles from economics, stochastic processes, and dynamic systems, enable the assessment of the factors influencing poverty and the effectiveness of public policies in alleviating it. This paper introduces a mathematical compartmental model to investigate poverty within a population ($\psi \left(t\right)$), considering the effects of immigration, crime, and incarceration. The model aims to elucidate the interconnections between these factors and their combined impact on poverty levels. We begin the study by ensuring the mathematical validity of the model by demonstrating the uniqueness of a positive solution. Next, it is shown that under specific conditions, the probability of poverty persistence approaches certainty. Conversely, conditions leading to an exponential reduction in poverty are identified. Additionally, the semigroup associated with our model is proven to possess the Feller property, and its distribution has a unique invariant measure. To confirm and validate these theoretical results, interesting numerical simulations are performed. Full article

Huanglongbing (HLB), a globally devastating citrus disease, demands sophisticated mathematical modeling to decipher its complex transmission dynamics and inform optimized disease management protocols. This investigation develops an innovative compartmental framework that simultaneously incorporates two critical factors in HLB epidemiology: saturated removal rates of infected citrus trees and behavioral bias in vector movement patterns. Our study delves into the dynamics of non-spatial systems by analyzing the basic reproduction numbers, equilibria, bifurcation phenomena, and the stability of these equilibria. Additionally, we explore the impact of spatial factors on system stability. Results indicate that when the basic reproduction number $R_0<1$, the system may exhibit bistable behavior, while $R_0>1$ leads to a unique stable equilibrium. Notably, vector bias significantly enhances the likelihood of forward bifurcation, and the delay in the removal of diseased trees increases the risk of backward bifurcation. However, reaction–diffusion processes do not alter the stability of the system’s equilibria, and the spatial system lacks complex dynamic properties. This research offers valuable insights into the mechanisms driving HLB transmission and provides a foundation for developing effective control strategies. Full article

Encephalitis, a severe neurological condition caused by human-to-human (H2H) transmitted viruses, such as herpes simplex virus (HSV), requires a rigorous mathematical framework to understand its transmission dynamics. This study develops a nonlinear compartmental model, SEITR (Susceptible–Exposed–Infected–Treated–Recovered), to characterize the progression of viral encephalitis. The basic reproduction number (R₀) is derived using the next-generation matrix method, serving as a threshold parameter determining disease persistence. The local and global stability of the disease-free and endemic equilibria are established through a rigorous mathematical analysis. Additionally, a sensitivity analysis quantifies the impact of key parameters on R₀, offering more profound insights into their mathematical significance. Numerical simulations validate the theoretical results, demonstrating the system’s dynamical behavior under varying epidemiological conditions. This study provides a mathematically rigorous approach to modeling viral encephalitis transmission, filling a gap in the literature and offering a foundation for future research in infectious disease dynamics. Full article

This study presents a comprehensive analysis of zoonotic disease transmission dynamics between baboon and human populations using both deterministic and stochastic modeling approaches. The model is constructed with a symmetric compartmental structure for each species—susceptible, infected, and recovered—which reflects a biological and mathematical symmetry between the two interacting populations. Public health control strategies such as sterilization, restricted food access, and reduced human–baboon interaction are incorporated symmetrically, allowing for a balanced evaluation of their effectiveness across species. The basic reproduction number (${\mathcal{R}}_0$) is derived analytically and examined through sensitivity indices to identify critical epidemiological parameters. Numerical simulations, implemented via the Euler–Maruyama method, explore the influence of stochastic perturbations on disease trajectories. Statistical tools including Maximum Likelihood Estimation (MLE), Mean Squared Error (MSE), and Power Spectral Density (PSD) analysis validate model predictions and assess variability across noise levels. The results provide probabilistic confidence intervals and highlight the robustness of the proposed control strategies. This symmetry-aware, dual-framework modeling approach offers novel insights into zoonotic disease management, particularly in ecologically dynamic regions with frequent human–wildlife interactions. Full article