Stability Analysis Of Equilibria Research Articles

In Vivo Dynamics of HIV-1 Infection With Impaired Antibody Immunity and Three General Infection Mechanisms

In this paper we investigate two generalized human immunodeficiency virus type-1 (HIV-1) dynamics models with impaired antibody immunity. The models include both latently and actively infected cells. Three infection mechanisms are incorporated into the models, viral infection mechanism (VIM), latent cellular infection mechanism (CIM) and active CIM. The three infection rates are provided by generic nonlinear functions. The second model includes three types of distributed time delays. We find that our models are biologically feasible. The global stability analysis of equilibria are performed and found the basic reproduction ratio (R0) as a threshold parameter. Using Lyapunov method we show that, the virus-free equilibrium is globally asymptotically stable when R0≤1 and the virus-persistence equilibrium is globally asymptotically stable when R0>1. Sensitivity analysis on R0 is studied. To support our theoretical results we provide some numerical simulations. We have demonstrated that R0 is influenced by all three of the infection types, and that if one of them were ignored, R0 would be underestimated. This might lead to inadequate medication effectiveness that aims to remove HIV-1 from the body. The effects of time delay and impaired antibody immunity on HIV-1 progression are examined. According to our research, lowered immunity is a significant factor in the infection's growth. Furthermore, time delays might drastically reduce R0, which would prevent HIV-1 from replicating. The information provided by our research in this work can improve our comprehension of HIV-1 dynamics within-host and provide guidance for the creation of novel pharmacological treatments.

A Cancer Subpopulation Competition Model Reveals Optimal Levels of Immune Response that Minimize Tumor Size.

Breast cancer is a complex disease with significant phenotypic heterogeneity of cells, even within a single breast tumor. Emerging evidence underscores the significance of intratumoral competition, which can serve as a key contributor to cancer drug resistance, imparting substantial clinical implications. Understanding the competitive dynamics is paramount as it can significantly influence disease progression and treatment outcomes. In the present work, a mathematical model was developed using a system of differential equations to describe the dynamic interactions between two cancer subtypes (each further classified into cancer stem cells and tumor cells) and innate immune cells. The purpose of the model is to comprehensively understand the competitive interactions between the heterogeneous subpopulations. The equilibrium points and stability analysis for each equilibrium point were established. Model simulations showed that the competition between two cancer subtypes directly affects the number of both species. When competition between two cancer subtypes is strong, increasing the immune response rate specific to the more competitive species effectively reduces the tumor size. However, if the competition is relatively weak, an optimal immune response rate is required to minimize the total number of tumor cells. Rates below the optimal level fail to reduce the population of the stronger species, whereas rates above the optimal level can lead to the recurrence of the weaker species. Overall, this model provides insights into breast cancer dynamics and guides the development of effective treatment strategies.

Fractional mathematical model for the transmission dynamics and control of Lassa fever

This paper aims to investigate various epidemiological aspects of Lassa fever viral infection using a fractional-order mathematical model so as to assess the impacts of treatment and vaccination on the spread of Lassa fever transmission dynamics. Initially, the model employs integer-order nonlinear differential equations, incorporating imperfect vaccination and treatment as control measures for the human population. Subsequently, the model is redefined using a fractional order derivative with power law to enhance understanding of disease dynamics. The paper establishes conditions ensuring the existence and uniqueness of the model’s solution in the fractional case, alongside presenting stability analysis of the endemic equilibrium using the Lyapunov function approach. Numerical simulations are performed using the fractional Adams–Bashforth–Moulton method, shedding light on the influence of model parameters and fractional order values on Lassa fever dynamics and control. Additional numerical simulations conducted utilizing surface and contour plots revealed that elevating the contact rates and diminishing the efficacy of vaccines would escalate the prevalence of Lassa fever among the human populace. It was also observed that optimizing treatment and enhancing vaccination strategies would ultimately mitigate the prevalence of Lassa fever within the human population.

Wiener and Lévy processes to prevent disease outbreaks: Predictable vs stochastic analysis

The study considers stochastic behavior along with modeling, mathematical analysis, theory development, and numerical simulation of the COVID-19 virus. To evaluate current trends and make future projections regarding the basic reproduction number and infection case, we have taken the modified five-compartment SEIRD mathematical model. Since the basic reproduction number R0 is not sufficient to predict the outbreak, we applied the Itô stochastic differential equations (SDEs) along with the Weiner process and Lévy jump to investigate the nature of the disease outbreak. Since COVID-19 is an infectious disease caused by the severe acute respiratory syndrome coronavirus 2 “(SARS-CoV-2)”, that has common symptoms including fever, cough, and difficulty breathing, we have illustrated the boundedness and positivity of solutions and investigated the stability of endemic and disease-free equilibrium points. The findings show that the dynamics of disease epidemics are influenced by contact patterns. For all the models, the threshold quantity, R0 is calculated which is the key reason to prove the global and local stability analysis of disease-free equilibrium and endemic equilibrium points. Using the least squares method for data fitting, we have performed a case study on COVID-19 data in Italy for this article. A sensitivity analysis is part of our investigation to find important factors. This paper investigates a stochastic epidemic model (SDE) using Lévy jumps and Weiner processes. Local and global stability in a stochastic setting are examined in the analysis. To view the analytical study, a multitude of numerical results are achieved. The study is important to comprehend the stochastic behavior of the virus in the contagious model in order to prevent similar disease outbreaks in the future.

Coherent Phase Change in Interstitial Solutions: A Hierarchy of Instabilities.

Metal hydrides or lithium ion battery electrodes can take the form of interstitial solid solutions with a miscibility gap. This work discusses theory approaches for locating, in temperature-composition space, coherent phase transformations during the charging/discharging of such systems and for identifying the associated transformation mechanisms. The focus is on the simplest scenario, where instabilities derive from the thermodynamics of the bulk phase alone, considering strain energy as the foremost consequence of coherency and admitting for stress relaxation at free surfaces. The extension of the approach to include capillarity is demonstrated by an example. The analysis rests on constrained equilibrium phase diagrams that are informed by geometry- and dimensionality-specific mechanical boundary conditions and on elastic instabilities-again geometry-specific-as implied by the theory of open-system elasticity. It is demonstrated that some scenarios afford the analysis of chemical stability to be based entirely on a linear stability analysis of the mechanical equilibrium, which provides closed-form solutions in a straightforward manner. Attention is on the impact of the system geometry (infinitely extended or of finite size) and on the chemical (closed or open system) and mechanical (incoherent or coherent) boundary conditions. Transformation mechanism maps are suggested for documenting the findings. The maps reveal a hierarchy of instabilities, which depend strongly on each of the above characteristics. Specifically, realistic, finite-sized systems differ qualitatively from idealized systems of infinite extension. Among the transformation mechanisms exposed by the analysis are a uniform switchover to the other phase when the open system reaches its chemical spinodal, practical coherent nucleation, as well as chemo-elastically coupled spontaneous buckling modes, which may take the form of either, single-phase or dual-phasestates.

From Chingford to Carsington: impact of soil mechanics on dam construction, 1937–1987

In 1937 the investigation of the failure during construction of Chingford number 2 embankment dam brought the concepts of modern soil mechanics to bear on the construction of British embankment dams. Effective stress theory highlighted the crucial significance of pore-water pressure in understanding embankment stability and it became standard practice to monitor pore pressures during embankment construction using twin-tube hydraulic piezometers. By the 1980s, embankment stability might have seemed assured, with embankment design utilising limit equilibrium stability analyses that incorporated strength parameters of fill and foundation materials determined by laboratory testing. However, in 1984 the upstream slope of Carsington dam suffered a major slip in the final stage of construction. Recently developed advanced finite-element analyses were able to demonstrate that progressive failure was a major factor in the slope instability, but the failure was also a reminder of the continuing need for experience-based judgement in design.

Mathematical modelling of COVID-19 dynamics using SVEAIQHR model

Abstract In this study, we formulate an eight-compartment mathematical model with vaccination as one of the compartments to analyze the dynamics of COVID-19 transmission. We examine the model’s qualitative properties, such as positivity and boundedness of solutions, and stability analysis of the illness-free equilibrium with respect to the basic reproduction number. We estimate ten significant parameters and also compute the magnitude of the basic reproduction number for India by fitting the proposed model to daily confirmed and cumulative confirmed COVID-19 cases in India. Sensitivity analysis with respect to basic reproduction number is conducted, and the main parameters that impact the widespread of disease are determined. We further extend this model to an optimal control problem by including four non-pharmaceutical and pharmaceutical intervention measures as control functions. Our numerical results show that the four control strategy has greater impact than the three control strategies, two control strategies, and single control strategies on reducing the dynamics of COVID-19 transmission.

Digital tools used on the teaching-learning process in geotechnical engineering

This article presents the incorporation of information and communication technologies on the teaching-learning process of geotechnical engineering to improve the quality of education and provide practical knowledge to civil engineering students. The content of this paper is divided into three main modules, which are: Treatment of laboratory tests results, Slope stability and Rock mass stability. For this, some software were used, including 2D limit equilibrium stability analysis, stability analysis of rock wedges and a dynamic mathematics, to obtain and analyze soil shear strength parameters, among others. Furthermore, a digital tool was developed to analyze the results gathered in compaction, one-dimensional consolidation, and direct soil shear tests, in order to clarify the relationship between theoretical concepts and practical results of the tests and analyses and to help students on doubts, in addition to increase their interest and motivation to perform the complete interpretation of the collected data. The activities were conducted in classes of geotechnical disciplines of the undergraduate course in Civil Engineering at the Federal University of Viçosa, aiming to promote active learning and improve teaching quality. Based on the results of an applied feedback questionnaire, it was observed that most students were satisfied with the resources used in the classroom, demonstrating that the implemented digital tools work as a didactic instrument that facilitates learning and comprehension of practical problems, in addition to enabling the resolution of several geotechnical engineering problems much more quickly and efficiently.

Turing Instabilities are Not Enough to Ensure Pattern Formation.

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

Mathematical modeling of the effects of vector control, treatment and mass awareness on the transmission dynamics of dengue fever

Dengue fever is a vital public health concern that affects about 40% of the world’s population. To address the dynamics of dengue disease, a mathematical model was formulated by incorporating three control strategies: vector control, treatment, and mass awareness. A stability analysis of the disease-free equilibrium (DFE) was conducted using the Jacobian matrix. The DFE was found to be locally and globally asymptotically stable when the effective reproductive number was less than one; otherwise, it was unstable. Additionally, an endemic equilibrium point (EEP) was identified. The global stability analysis of the EEP, performed using the Lyapunov method, showed that it is globally asymptotically stable whenever Re>1; otherwise, it is unstable. Bifurcation analysis revealed that the model system exhibits a forward bifurcation. Furthermore, sensitivity analysis of the effective reproduction number revealed that the most sensitive parameters are the biting rate (b) and insecticide efficacy (δ). Therefore, the results suggest that, in order to reduce new dengue cases, intervention strategies that decrease the biting rate, such as mosquito repellents and the use of insecticides to kill mosquitoes, should be implemented. Moreover, simulations were conducted for the extended model with vector control, treatment, and mass awareness. The results showed that the combination of vector control, treatment, and mass awareness has a more positive impact on the control of dengue fever than any single or paired intervention. Thus, for effective control of dengue fever, the three control measures should be implemented simultaneously, especially in endemic areas.

Stability Analysis of Equilibrium of DC Microgrid Under Distributed Control

Constant power loads (CPLs) often lead to system instability and voltage collapse. This paper analyzes the existence and stability of equilibrium of meshed DC microgrids with multiple CPLs under distributed control, which aims to realize current sharing and voltage regulation. Firstly, the power-flow equation of the system is derived. To analyze the solvability of power-flow equation, we transform the problem of the multidimensional quadratic equation solvability into the existence of a fixed-point for a contraction mapping. Then, the sufficient solvability condition is derived based on Banach's fixed-point theorem. Secondly, we build the small-signal model to determine the system qualitative behavior around equilibrium. By analyzing the eigenvalue of system Jacobian matrix based on inertia theorem, the analytical stability conditions are obtained. Further, under the load uncertainty, a robust stability condition which only depends on the maximum load information instead of the real-time load information is derived. Simulation results verify the feasibility of the proposed theorems.

Analyzing the Asymptotic Behavior of an Extended SEIR Model with Vaccination for COVID-19

Several research papers have attempted to describe the dynamics of COVID-19 based on systems of differential equations. These systems have taken into account quarantined or isolated cases, vaccinations, control measures, and demographic parameters, presenting propositions regarding theoretical results that often investigate the asymptotic behavior of the system. In this paper, we discuss issues that concern the theoretical results proposed in the paper “An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter”. We propose detailed explanations regarding the resolution of these issues. Additionally, this paper focuses on extending the local stability analysis of the disease-free equilibrium, as presented in the aforementioned paper, while emphasizing the derivation of theorems that validate the global stability of both epidemic equilibria. Emphasis is placed on the basic reproduction number R0, which determines the asymptotic behavior of the system. This index represents the expected number of secondary infections that are generated from an already infected case in a population where almost all individuals are susceptible. The derived propositions can inform health authorities about the long-term behavior of the phenomenon, potentially leading to more precise and efficient public measures. Finally, it is worth noting that the examined paper still presents an interesting epidemiological scheme, and the utilization of the Kalman filtering approach remains one of the state-of-the-art methods for modeling epidemic phenomena.

A cell-level dynamical model for malaria parasite infection with antimalarial drug treatment

Malaria is an infectious disease caused by intracellular parasites of the genus Plasmodium. It is a major health problem around the world. In this study, a cell-level mathematical model of malaria parasites with antimalarial drug treatments is formulated and analyzed. The model consists of seven compartments for cell populations. We analyzed the qualitative behavior of the model using various techniques. The stability analysis of the parasite-free equilibrium is obtained, whereas it is locally and globally stable if the basic reproduction number R0<1. The parasite persistence equilibrium point exists, and it is locally asymptotically stable if R0>1. The sensitivity analysis of the basic reproduction number is computed, and the results show that the infection rate of the erythrocyte by merozoites, the average number of merozoites per ruptured infected erythrocyte cells, the natural death rate of merozoites, and the requirement rate of the uninfected erythrocyte are the most influential parameters within-host dynamics of malaria infection. Different numerical simulations are performed to supplement our analytical findings. The effect of primary tissue schizontocides, blood schizontocides, and gametocytocides on infected hepatocytes, infected erythrocytes, and gametocytes have been investigated, respectively. Finally, some counterplots are presented in order to investigate the impact of parameters on the basic reproduction number. The in-host basic reproduction number decreases as the antimalarial treatment administration increases. Therefore, increasing antimalarial treatment administration is the best way to mitigate the in-host malaria infection.

Existence and influence of mixed states in a model of vegetation patterns

Abstract. The Rietkerk vegetation model is a system of partial differential equations, which has been used to understand the formation and dynamics of spatial patterns in vegetation ecosystems, including desertification and biodiversity loss. Here, we provide an in-depth bifurcation analysis of the vegetation patterns produced by Rietkerk's model, based on a linear stability analysis of the homogeneous equilibrium of the system. Specifically, using a continuation method based on the Newton–Raphson algorithm, we obtain all the main heterogeneous equilibria for a given size of the domain. We confirm that inhomogeneous vegetated states can exist and be stable, even for a value of rainfall for which no vegetation exists in the non-spatialized system. In addition, we evidence the existence of a new type of equilibrium, which we call “mixed state”, in which the equilibria are always unstable and take the form of a mix of two equilibria from the main branches. Although these equilibria are unstable, they influence the dynamics of the transitions between distinct stable states by slowing down the evolution of the system when it passes close to it. Our approach proves to be a helpful way to assess the existence of tipping points in spatially extended systems and disentangle the fate of the system in the Busse balloon. Overall, our findings represent a significant step forward in understanding the behaviour of the Rietkerk model and the broader dynamics of vegetation patterns.

Mathematical Model for COVID-19 pandemic with implementation of intervention strategies and Cost-Effectiveness Analysis

In this study, a novel mathematical model for the dissemination dynamics of COVID-19 is developed. The main objective is to analyze the effectiveness of pharmaceutical interventions in lowering the COVID-19 contagions. We first demonstrate the positivity and boundedness of the solutions of the model and then compute the fundamental reproduction number. The stability analysis of contagion-free equilibrium is performed. The model is fitted to the COVID-19 reported data for a period of twelve months in India and estimate three parameters. The sensitivity analysis is conducted to identify the significant factors which impact the COVID-19 disease prevalence. We then define an optimum control problem using pharmaceutical interventions vaccination and treatment as the control functions to minimize the dissemination of COVID-19 contagions and disease-related mortality. Cost-effectiveness analysis is employed to determine the most effective and least costly strategy. The results are determined that the combination of vaccination and treatment is the most effective and least costly strategy in mitigating the spread of COVID-19 contagions. Furthermore, the impact of different levels of vaccine efficacy on contagion trajectories is examined, and it is shown that COVID-19 contagions and disease related fatalities would decrease as vaccination efficacy increases. The outcomes would assist administrators in developing efficient strategies to reduce the scope of the COVID-19 pandemic.

Analysis of a non-integer order mathematical model for double strains of dengue and COVID-19 co-circulation using an efficient finite-difference method

An efficient finite difference approach is adopted to analyze the solution of a novel fractional-order mathematical model to control the co-circulation of double strains of dengue and COVID-19. The model is primarily built on a non-integer Caputo fractional derivative. The famous fixed-point theorem developed by Banach is employed to ensure that the solution of the formulated model exists and is ultimately unique. The model is examined for stability around the infection-free equilibrium point analysis, and it was observed that it is stable (asymptotically) when the maximum reproduction number is strictly below unity. Furthermore, global stability analysis of the disease-present equilibrium is conducted via the direct Lyapunov method. The non-standard finite difference (NSFD) approach is adopted to solve the formulated model. Furthermore, numerical experiments on the model reveal that the trajectories of the infected compartments converge to the disease-present equilibrium when the basic reproduction number ({mathbb {R}}_0) is greater than one and disease-free equilibrium when the basic reproduction number is less than one respectively. This convergence is independent of the fractional orders and assumed initial conditions. The paper equally emphasized the outcome of altering the fractional orders, infection and recovery rates on the disease patterns. Similarly, we also remarked the importance of some key control measures to curtail the co-spread of double strains of dengue and COVID-19.

Modeling economic growth with spatial migration: A stability analysis of the long-run equilibrium based on semigroup theory

We develop a new model of economic growth that fully accounts for the spatial dependence of the flows of workers and capital on salaries and returns. Considering a rather general setting in which we do not require the knowledge of the exact expressions of the production function and the population growth rate, we allow for a strictly positive steady state in which labor, capital, wages, and returns on capital are constant in space. Then, we establish the stability of such an equilibrium using the theory of abstract non-linear parabolic problems and analyzing the spectral properties of the derivative of the operator that describes the coupled dynamics of labor and capital. We present numerical simulations which agree with our theoretical investigation and show that the proposed model allows us to detect interesting transitional dynamics that the standard Solow model does not capture. In particular, we see how the migration of labor can slow down the process of wage convergence at some spatial locations and how the flow of workers, interacting with population dynamics, can reduce economic growth in countries where both developed and underdeveloped regions are present.

A Novel Mathematical Model of the Dynamics of COVID-19

The severity of the COVID-19 pandemic requires a better understanding of the spread of SARS-COV2. As of December 2019, several mathematical models have been developed to explain how SARS-COV2 spreads within populations, and proposed models have evolved as more is learned about the dynamics of the outbreak. In this study, we propose a new mathematical model that includes demographic characteristics of the population. Social isolation and vaccination are also taken into account in the model. Besides transmission arising from intercourse with undiagnosed infected persons, we also consider transmission by contact with the exposed group. In this study, after the model is established, the basic reproduction number is calculated and local stability analysis of disease-free equilibrium is given. Finally, we give numerical simulations for the proposed model.

Fractional Modeling of Cancer with Mixed Therapies.

Cancer is the biggest cause of mortality globally, with approximately 10 million fatalities expected by 2020, or about one in every six deaths. Breast, lung, colon, rectum, and prostate cancers are the most prevalent types of cancer. In this work, fractional modeling is presented which describes the dynamics of cancer treatment with mixed therapies (immunotherapy and chemotherapy). Mathematical models of cancer treatment are important to understand the dynamical behavior of the disease. Fractional models are studied considering immunotherapy and chemotherapy to control cancer growth at the level of cell populations. The models consist of the system of fractional differential equations (FDEs). Fractional term is defined by Caputo fractional derivative. The models are solved numerically by using Adams-Bashforth-Moulton method. For all fractional models the reasonable range of fractional order is between β = 0.6 and β = 0.9. The equilibrium points and stability analysis are presented. Moreover, positivity and boundedness of the solution are proved. Furthermore, a graphical representation of cancerous cells, immunotherapy and chemotherapy is presented to understand the behaviour of cancer treatment. At the end, a curve fitting procedure is presented which may help medical practitioners to treat cancer patients.

Local and global stability of an HCV viral dynamics model with two routes of infection and adaptive immunity

The aim of this article is to formulate and study a mathematical model describing hepatitis C virus (HCV) infection dynamics. The model includes two essential modes of infection transmission, namely, virus-to-cell and cell-to-cell. The effect of therapy and adaptive immunity are incorporated in the suggested model. The adaptive immunity is represented by its two categories, namely, the humoral and cellular immune responses. Our article begins by establishing some mathematical results through proving the model’s well-posedness in terms of existence, positivity and boundedness of solutions. We present all the steady states of the problem that depend on specific reproduction numbers. It moves then to the theoretical investigation of the local and global stability analysis of the free disease equilibrium and the four disease equilibria. The local and global stability analysis of the HCV mathematical model are established via the Routh-Hurwitz criteria and Lyapunov-LaSalle invariance principle, respectively. Finally, our article presents some numerical simulations to validate the analytical study of the global stability. Numerical simulations have shown the effect of the drug therapies on the system’s dynamical behavior.