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.

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.

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-β (Transforming Growth Factor-β), 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.

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-β, 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.

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.

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 (P2), 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.

In order to overcome the obstruction of cell membranes in the path of drug delivery to diseased cells, the applications of electric pulses of adequate potency are designated as electroporation. In the present study, a mathematical model of drug delivery into the electroporated tissue is advocated, which deals with both reversibly and irreversibly electroporated cells. This mathematical formulation is manifested through a set of differential equations, which are solved analytically, and numerically, according to the complexity, with a pertinent set of initial and boundary conditions. The time-dependent mass transfer coefficient in terms of pores is used to find the drug concentrations through reversibly and irreversibly electroporated cells as well as extracellular space. The effects of permeability of drug, electric field and pulse period on drug concentrations in extracellular and intracellular regions are discussed. The threshold value of an electric field (E > 100 V cm−1) to initiate drug uptake is identified in this study. Special emphasis is also put on two cases of electroporation, drug dynamics during ongoing electroporation and drug dynamics after the electric pulse period is over. Furthermore, all the simulated results and graphical portrayals are discussed in detail to have a transparent vision in comprehending the underlying physical and physiological phenomena. This model could be useful to various clinical experiments for drug delivery in the targeted tissue by controlling the model parameters depending on the tissue condition.

The objective of the present study is to mathematically model the integrated kinetics of drug release in a polymeric matrix and its ensuing drug transport to the encompassing biological tissue. The model embodies drug diffusion, dissolution, solubilization, polymer degradation and dissociation/recrystallization phenomena in the polymeric matrix accompanied by diffusion, advection, reaction, internalization and specific/nonspecific binding in the biological tissue. The model is formulated through a system of nonlinear partial differential equations which are solved numerically in association with pertinent set of initial, interface and boundary conditions using suitable finite difference scheme. After spatial discretization, the system of nonlinear partial differential equations is reduced to a system of nonlinear ordinary differential equations which is subsequently solved by the fourth-order Runge–Kutta method. The model simulations deal with the comparison between a drug delivery from a biodegradable polymeric matrix and that from a biodurable polymeric matrix. Furthermore, simulated results are compared with corresponding existing experimental data to manifest the efficaciousness of the advocated model. A quantitative analysis is performed through numerical computation relied on model parameter values. The numerical results obtained reveal an estimate of the effects of biodegradable and biodurable polymeric matrices on drug release rates. Furthermore, through graphical representations, the sensitized impact of the model parameters on the drug kinetics is illustrated so as to assess the model parameters of significance.

In this paper, a mathematical model of drug release from polymeric matrix and consequent intracellular drug transport is proposed and analyzed. Modeling of drug release is done through solubilization dynamics of drug particles, diffusion of the solubilized drug through the polymeric matrix in addition to reversible dissociation/recrystallization process. The interaction between drug-receptor, drug-plasma proteins along with other intracellular endosomal events is modeled. This leads to a mixed system of partial and ordinary differential equations with associated pertinent set of initial and boundary conditions. Furthermore, besides the stability of the proposed model, several sub-models are also studied for their stability criteria. Prominence is provided to the reduced model system having requisite relevance to the original system where quasi steady state approximation (QSSA) theory is utilized. For the model to be potent enough to generate appropriate predictive results for drug delivery, the stability properties of equilibrium in the mathematical model are analyzed both analytically and numerically. Numerical simulation in the embodiment of graphical representations speaks about various vital characteristics of the underlying physical phenomena along with the importance and sensitized impact of the model parameters controlling significant biological functions. Probed new therapies and clinical procedures could be assessed considering the present mathematical model and its analysis as the basis framework in order to effectively enhance therapeutic efficacy and improved patient compliance. The present study confirms the necessity of stability analysis study so that advocated mathematical model can effectively complement the real physiological behavior of pharmacokinetics.

Local drug delivery system is a thrust area of current research and it possesses clinical implications so long as therapy is concerned. The purpose of the present study is to frame primarily an appropriate mathematical model for drug release from a porous polymeric matrix to biological tissues through endocytosis. Drug release phenomenon is described by taking into account both solubilisation dynamics of solid drug and diffusion of solubilised drug through porous polymeric matrix. In the tissue medium, reversible dissociation/association together with internalization processes of drug are involved. The model under consideration is duly formulated by a system of partial differential equations. These equations are solved analytically with the appropriate choice of initial, boundary and interface conditions as well. In order to establish the potency of the proposed model, the simulated results are compared with corresponding experimental data and found remarkable agreement so as to validate the applicability of the model considered. A quantitative analysis is carried out at the end through numerical simulation based on the values of all the model parameters in order to illustrate the behaviour of drug concentrations with time under various situations. The sensitivity of most of the model parameters on drug concentrations and drug masses is also put on record for the purpose of applicability of the drug release model under consideration.

Nowadays, electroporation is widely used to deliver drug in different parts of the human body. In electroporation, long time low voltage electric pulses are used to transport drug molecules into the cell through the cell membrane. A mathematical model of drug delivery into the electroporated cells is advocated in the current study which deals with both reversible and irre-versible cells. Furthermore, the model depicts temperature change in the tissue as an outcome of electroporation due to the effect of an electric pulse. This leads to a set of ordinary differential equations and a system of partial differential equations which are solved numerically with appropriate initial and boundary conditions. Drug distribution in both reversible and irreversible cells are illustrated through graphical representations which help in the apprehension of the underlying physical phenomena. Temporal change in the electroporated tissue temperature is also noted analytically.

The primary aim of liposomal drug delivery is to wisely modulate the drug delivery system in order to target diseased tissues. Temperature-sensitive liposomes function as a prospective weapon to combat toxic side effects corresponding to direct infusion of anticancer drugs. The main objective of the present study is to model liposomal drug release, subsequent drug transport in solid tumour along with integrated actions of tumour cell surface and endosomal events. Generalized mathematical model for liposomal drug delivery is proposed in which vital physical phenomena, such as kinetics of liposome-encapsulated drug, free drug release from liposomes, transport of both liposomal drug and free drug into the tumour compartment, plasma clearance, protein-drug interactions, drug-tumour cell receptor interactions, internalization of drug through endocytosis along with corresponding endosomal events. The model is expressed through a system of coupled partial differential equations along with appropriate set of initial, interface and boundary conditions which is solved numerically. Simulated results are compared with respective existing experimental data to demonstrate the potency and reliability of the proposed model. Graphical representations of time variant concentration profiles are illustrated to understand the underlying phenomena in details. Moreover, the model speaks for the sensitivity of important drug kinetic parameters, such as advection coefficients, drug release coefficient, plasma clearance rate and internalization parameters through graphical portrayals. The proposed model and the simulated results act as a tool in designing a more effective drug delivery system for cancerous tumours.

In modern days, biodegradable polymeric matrix used as the kingpin of local drug delivery system is in the center of attention. This work is concentrated on the formulation of mathematical model elucidating degradation of drug-loaded polymeric matrix followed by drug release to the adjacent biological tissues. Polymeric degradation is penciled with mass conservation equations. Drug release phenomenon is modeled by considering solubilization dynamics of drug particles, diffusion of the solubilized drug through polymeric matrix along with reversible dissociation/recrystallization process. In the tissue phase, reversible dissociation/association along with internalization processes of drug are taken into account. For this, a two-phase spatio-temporal model is postulated, which has ensued to a system of partial differential equations. They are solved analytically with appropriate choice of initial, interface and boundary conditions. In order to reflect the potency of the advocated model, the simulated results are analogized with corresponding experimental data and found laudable agreement so as to validate the applicability of the model considered. This model seems to foster the delicacy of the mantle enacted by important drug kinetic parameters such as diffusion coefficients, mass transfer coefficients, particle binding and internalization parameters, which is illustrated through local sensitivity analysis.

The present study aims to provide a comprehensive mathematical model for drug release from microparticles to the adjacent tissues. In the elucidation of drug release mechanisms, the role of mathematical modelling has been depicted thereby facilitating the development of new therapeutic drug by a systematic approach, rather than expensive experimental trial-and-error methods. In order to study the whole process, a two-phase mathematical model describing the dynamics of drug transport in two coupled media is presented. Drug release is described taking into consideration both solubilisation dynamics of drug crystallites and diffusion of the solubilised drug through the microparticle. In the coupled media, reversible dissociation/recystallisation processes are taking place. The model has led to a system of partial differential equations that are solved analytically. The model points out the important roles played by the diffusion, mass-transfer and reaction parameters, which are the main architects behind drug kinetics across two layers. The dependence of drug masses on the main parameters is also analysed.

Local drug delivery has received much recognition in recent years, yet it is still unpredictable how drug efficacy depends on physicochemical properties and delivery kinetics. The purpose of the current study is to provide a useful mathematical model for drug release from a drug delivery device and consecutive drug transport in biological tissue, thereby aiding the development of new therapeutic drug by a systemic approach. In order to study the complete process, a two-layer spatio-temporal model depicting drug transport between the coupled media is presented. Drug release is described by considering solubilisation dynamics of drug particle, diffusion of the solubilised drug through porous matrix and also some other processes like reversible dissociation / recrystallization, drug particle-receptor binding and internalization phenomena. The model has led to a system of partial differential equations describing the important properties of drug kinetics. This model contributes towards the perception of the roles played by diffusion, mass-transfer, particle binding and internalization parameters.