The present investigation delves into the intricate dynamics of diabetic population, accounting for genetic, hereditary, social, environmental, and lifestyle determinants in the progression from prediabetes to diabetes. The model encompasses comorbidities, articulated through a suite of six nonlinear differential equations. Employing numerical methodologies alongside comprehensive stability and sensitivity analyses, it unveils nuanced insights into both biological and social interactions. Theoretical discoveries are vividly illustrated, and the model’s credibility is attested through empirical validation. Conclusions drawn from the findings underscore pivotal parameters, endowing invaluable perspectives on the dynamical system in concert with stability elucidations. © 2025 L&H Scientific Publishing, LLC. All rights reserved

Hyperlipidemia is recognized as a significant health concern in the human body. In this study, a novel mathematical framework is developed to investigate targeted therapeutic strategies for reducing hyperlipidemia through a fifth-compartment mathematical model. The model consists of five compartments: the liver, blood, gallbladder, intestine, and tissue. To address hyperlipidemia, direct drug administration into the bloodstream is incorporated. Potential treatments for lowering cholesterol levels in the blood and tissue are explored, contributing to advancements in medical research. Sensitivity analysis is performed to determine the impact of various parameters on equilibrium stability. Stability tests evaluate the model’s long-term stability, ensuring greater accuracy in predictive modeling. The variation in cholesterol levels and drug concentration over time is analyzed using MATLAB software, with graphical results demonstrating a gradual decline in cholesterol levels following drug administration. Both analytical and numerical assessments confirm the model’s effectiveness in characterizing cholesterol transport and optimizing therapeutic strategies for hyperlipidemia management.

Limited progress in the mathematical modeling of cholesterol transport systems is hampering novel therapeutic interventions. This issue is addressed by the present study through precise design, employing four compartmental models to elucidate cholesterol dynamics in the comprehensive bloodstream. Disparities in medical advancements, particularly in cholesterol-related pathophysiology, are aimed to be bridged, advancing medical science and patient care outcomes. Therapeutic strategies for reducing blood cholesterol are explored by the model, with parameter influences on equilibrium stability revealed through sensitivity analysis. System parameters are effectively manipulated by imposing sensitivity analysis, and pinpointing areas for model refinement. Stability analysis contributes to diverse realistic models, confirming local asymptotic stability. Model efficacy in studying cholesterol transport dynamics is supported by analytical and numerical assessments. The study concludes with the present model validation to substantiate it by comparing the present outcomes with the existing ones.

The regulatory influence of the parathyroid hormone (PTH) is exerted on bone, which serves as a vital reservoir of calcium within the body. While various aspects of bone growth, turnover, and mechanisms operate independently of gonadal hormones, a crucial role is assumed by sex steroids, particularly estrogen, in maintaining bone equilibrium in adults. In order to unravel the underlying mechanisms of bone remodeling mediated by PTH, a distinguished mathematical model of this intricate process is presented. Subsequently, the temporal effects of Plasma PTH and PTH external dosages are investigated using the proposed mathematical model. Among the limited repertoire of approved and accessible anabolic treatments for severe osteoporosis, daily injections of PTH stand out. This pharmaceutical marvel possesses a unique dual action, capable of acting either anabolically or catabolically, contingent upon the mode of administration. The study aims to accurately predict osteogenic responses to introduced and endogenous PTH, incorporating factors such as TGF-?, RANKL, and bisphosphonates in osteoblast–osteoclast signaling, as well as considering PTH’s influence on the gland and the regulatory roles of Runx2, pCREB, and Bcl2 in osteoblast apoptosis and bone volume effects. Through diverse methods including illustrative depictions, numerical simulations, sensitivity analysis, and stability analysis, this work seeks to comprehend how PTH therapy impacts bone volume, enhancing its therapeutic relevance. © World Scientific Publishing Company.

The focus of the current investigation lies in the formulation and analysis of a dynamic model depicting cancer growth, incorporating the joint influences of chemotherapy and immune system augmentation. The primary emphasis of this study revolves around the analysis of the dynamic behaviour within a living-cell closed carcinoma system, specifically one devoid of external vitamin support, with a particular exploration of chaos dynamics. Subsequently, the authors aim to scrutinize the pivotal impact of infused vitamins in attaining stable system dynamics through the application of chaos control techniques. The formulated model exhibits fundamental mathematical properties, revealing a spectrum of co-dimension one and co-dimension two bifurcations. The identification of specific bifurcation types relies on algebraic criteria techniques, where conditions necessary and sufficient for bifurcation types are developed. Notably, these criteria are distinct from traditional approaches based on the characteristics of the eigenvalues of the Jacobian matrix, instead relying on coefficients derived from characteristic equations. The accuracy of the analytical conclusions is validated through numerical findings, elucidating diverse bifurcation structures. The article enriches its contribution by delving into the control of chaos through the reinforcement of the internal immune system and the maintenance of the biological system’s stability. This work culminates in proposing future directions aimed at advancing a more realistic approach to eradicating cancer. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

A mathematical model examines the temporal effects of plasma parathyroid hormone (PTH) and external dosages on bone remodeling. Sex steroids, especially estrogen, crucially maintain bone equilibrium. Daily PTH injections, with dual anabolic or catabolic action, are prominent for severe osteoporosis. The study predicts osteogenic responses to PTH, considering factors like TGF-? (Transforming Growth Factor-?), RANKL (RANK Ligand), bisphosphonates, PTH’s influence on the gland, and regulatory roles of Runx2 (Runt-related transcription factor 2), pCREB (Phosphorylation of cAMP response element-binding protein), and Bcl2 (B-cell lymphoma 2). Utilizing methods such as numerical simulations and sensitivity analysis, it comprehends how PTH therapy impacts bone volume, enhancing its therapeutic relevance

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Despite stupendous advancement of medical science, yet mankind gets perplexed when it comes to cancer. As yet cancer poses substantial threat to life as it is lethal in some cases due to its complexity and heterogeneity. With the objective of increasing the potency of cancer treatment, scientists are now focussing on combination therapy such as chemovirotherapy. In the current study, an updated and realistic mathematical model embracing different facets like uninfected tumour cells, infected tumour cells, free virus particles, chemotherapeutic agent, tumour specific immune cells and virus specific immune cells is advocated. In addition to mutual interactions between cells, diffusion phenomenon plays a vibrant role on account of their mobility. All these biological and physical processes are embodied in the novel mathematical model. Stability analysis corresponding to the temporal system along with its sub-models undergoing comparative study is performed. Suitable numerical methods are adopted for the model outcome followed by their exhaustive delineations. Spatial distributions are visualized using graphical manifestations. Sensitized parametric variation is illustrated pictorially. The study concludes that with proper management of model parameters so that cancerous tumours can be eradicated from the body using chemovirotherapy effectively.

Through the utilization of a proactive approach, interaction with human immunodeficiency virus type I (HIV-1) is facilitated, enabling the sequential stages of its fusion mechanism to be navigated successfully. As a result, the efficient infiltration of a target CD4 T helper cell within the host organism by the virus is achieved. The onset of the virus’s replication cycle is initiated through this infiltration. As a retrovirus, the orchestration of the conversion of its single-stranded viral RNA genome into a more stable double-stranded DNA structure by HIV-1 is observed. Integration of this newly formed DNA with the host cell’s genetic material occurs. This pivotal transformation of the integrated pro-viral DNA into fully functional messenger RNA (mRNA) is facilitated by the host enzyme RNA polymerase II (Pol II). The central focus of the present ongoing research involves the construction of a meticulous mathematical framework consisting of a system of nonlinear differential equations. The investigation of the impact of a Tat inhibitor on the suppression of the transcriptional activity of HIV-1 is the aim of this inquiry. The perspective of an optimal control problem is assumed for this investigation. Furthermore, the assessment of the efficacy of the Tat inhibitor as a potential therapeutic intervention for HIV-1 infection is undertaken. Integration of a one-dimensional impulsive differential equation model, which determines a mathematically derived maximum concentration of the elongating complex (P), is employed for this assessment. The crucial aspect of this investigation is the consideration of the optimal timing between successive dosages. A comparative analysis is conducted to evaluate the distinct effects of continuous dosing versus impulse dosing of the Tat inhibitor. Numerical analysis is employed to contrast the outcomes of these dosing strategies. The present findings highlight that impulsive dosing demonstrates superior effectiveness compared to continuous dosing in the inhibition of HIV-1 transcription. Ultimately, the model’s parameter sensitivities are visualized through graphical representations. These visualizations serve to enhance the understanding of the underlying physiological and biochemical processes within this intricate system.

The global emergence of COVID-19 and its widespread transmission posed a formidable challenge for the global medical community. While vaccinations succeeded in mitigating the severity and fatality of the infection, a new challenge emerged: addressing transmission in the presence of comorbidities. A comprehensive mathematical model has been developed to address this issue, incorporating elements such as nonpharmaceutical interventions, vaccination strategies, comorbidity factors, limited healthcare resources, and the impact of nosocomial transmission. This updated model is formulated as a set of nonlinear partial differential equations under the category of reaction-diffusion models, aiming to provide a more accurate representation of the dynamics and interactions involved in spreading infectious diseases in a given population. The methodology employed involves a comprehensive analysis of the master model system’s qualitative characteristics, focussing on the stability of its constituent subsystems. The model’s dynamical system is subjected to numerical solutions, enabling a detailed exploration of its behaviour under various conditions. A rigorous parametric variation is carried out to understand the model’s response to different parameter values. The novelty of this research is rooted in its pioneering approach to bridging the gap between theory and real-world observations. By rigorously validating theoretical results against empirical experimental data, the research aims to provide valuable insights into the dynamics of the ongoing pandemic. The outcomes generated by the present model system are expected to offer a deeper and more comprehensive understanding of the pandemic’s behaviour and transmission patterns, playing a pivotal role in advancing the field of theoretical modelling.