Global Asymptotic Stability Of Equilibrium Research Articles

Stability and backward bifurcation for an HIV model with macrophages and CD4+T cells with latent reservoirs

Human immunodeficiency virus (HIV) primarily attacks CD4+ T cells but can also attack macrophage and dendritic cells leading to AIDS disease. Antiretroviral therapies are unavoidable to treat patients with HIV. However, the existing therapies failed in eliminating viral particles in infected patients. The establishment of a latency state in these patients could play a major role in this failure. To better understand the impact of different drug therapies on HIV dynamics, we propose an in-host model describing the dynamics of HIV and its interaction with both CD4+T and macrophage cells. This model incorporates the CD4+T latent reservoir as well. Results on the positivity, boundedness, and stability of solutions are shown. Furthermore, the basic reproduction number R0 is obtained using the next generation-matrix method. It is shown that the model is locally asymptotically stable at disease-free equilibrium (DFE) when the basic reproduction number R0<1. Moreover, the model exhibits either forward or backward bifurcation when R0=1. The global asymptotic stability of equilibria is investigated using suitable Lyapunov functions. To study the impact of our model parameters on the model dynamics, a sensitivity analysis is performed using the partial rank correlation coefficient method and the Latin hypercube sampling. Finally, numerical simulations are carried out to assess several drug therapy strategies to reduce HIV infection and improve health outcomes for HIV infected patients.

Global dynamics of a diffusive SARS-CoV-2 model with antiviral treatment and fractional Laplacian operator

In this paper, we propose and investigate the global dynamics of a SARS-CoV-2 infection model with diffusion and antiviral treatment. The proposed model takes into account the two modes of transmission (virus-to-cell and cell-to-cell), the lytic and nonlytic immune responses. The diffusion into the model is formulated by the regional fractional Laplacian operator. Furthermore, the global asymptotic stability of equilibria is rigorously established by means of a new proposed method constructing Lyapunov functions for a class of partial differential equations (PDEs) with regional fractional Laplacian operator. The proposed method is applied to the classical reaction-diffusion equations with normal diffusion.

Dynamics analysis of a diffusive SIRI epidemic system under logistic source and general incidence rate

This study presents and examines a diffusive SIRI epidemic model incorporating logistic source and a general incidence rate. Differing from existing works, the system incorporates two factors: the general incidence rate and the logistic source. We first consider the well-posedness of the system. Then, utilizing the construction of four Lyapunov functions, we thoroughly examine the global asymptotic stability of equilibria in both specific and general scenarios, assuming all coefficients remain constant. In addition, we establish the basic reproduction number, denoted as R0, and subsequently derive the correlation between R0 and the local basic reproduction number. Furthermore, we provide a detailed discussion of the persistence and extinction of the infective population. In particular, in the case where R0 equals one and certain assumptions are met, we demonstrate the global asymptotic stability of the disease-free equilibrium. Lastly, we substantiate the validity of our theoretical findings through the presentation of five illustrative examples.

Global stability and optimal control of epidemics in heterogeneously structured populations exhibiting adaptive behavior

We present a novel SIRS model on scale-free networks that takes into account behavioral memory and time delay to depict an adaptive behavioral feedback mechanism, which can better characterize the actual spread of epidemics. We conduct rigorous analysis on the dynamics of the model, including the basic reproduction number R0, uniform persistence and the global asymptotic stability of equilibria. The model has the sharp threshold property, namely, if R0 is less than 1 then the disease-free equilibrium is globally asymptotically stable while if R0 is larger than 1 then the endemic equilibrium is globally asymptotically stable. We further perform an optimal control study for the model to seek effective vaccination and treatment strategies. The existence and uniqueness of these optimal control strategies are demonstrated. Finally, we perform some stochastic network simulations that yield quantitative agreement with the deterministic mean-field approach. Our findings indicate that time delay does not affect R0, but behavioral memory does.

GENERAL THEORY FOR SIGNIFICANCE OF CULLING IN TWO-WAY DISEASE TRANSMISSION BETWEEN HUMANS AND ANIMALS

An epidemic model is proposed to comprehend the disease dynamics between humans and animals and back to humans with a culling intervention strategy. The proposed model is separated into two cases with two different culling rates: (1) at a per-capita constant rate and (2) constant population being culled. The global asymptotic stability of equilibria is determined in terms of the basic reproduction numbers. Further, we find that the culling rate (2) considered in the model could change the dynamics by having multiple positive equilibria. Sensitivity analysis recommends developing a strategy that promotes animals’ natural and disease-related death rates. By ranking the efficacies of various intervention strategies, we obtain that vaccination in the human population, isolation and public awareness are the largely effective control interventions. Our general theory raises concerns about both human and animal populations becoming reservoirs of the disease and affecting each other dynamically.

Stability and bifurcation in a two-patch model with additive Allee effect

&lt;abstract&gt;&lt;p&gt;A two-patch model with additive Allee effect is proposed and studied in this paper. Our objective is to investigate how dispersal and additive Allee effect have an impact on the above model's dynamical behaviours. We discuss the local and global asymptotic stability of equilibria and the existence of the saddle-node bifurcation. Complete qualitative analysis on the model demonstrates that dispersal and Allee effect may lead to persistence or extinction in both patches. Also, combining mathematical analysis with numerical simulation, we verify that the total population abundance will increase when the Allee effect constant $ a $ increases or $ m $ decreases. And the total population density increases when the dispersal rate $ D_{1} $ increases or the dispersal rate $ D_{2} $ decreases.&lt;/p&gt;&lt;/abstract&gt;

Stability analysis of a delayed sir epidemic model with diffusion and saturated incidence rate

In this paper, we investigate the effect of spatial diffusion and delay on the dynamical behavior of the SIR epidemic model. The introduction of the delay in this model makes it more realistic and modelizes the latency period. In addition, the consideration of an SIR model with diffusion aims to better understand the impact of the spatial heterogeneity of the environment and the movement of individuals on the persistence and extinction of disease. First, we determined a threshold value $$R_0$$ of the delayed SIR model with diffusion. Next, By using the theory of partial functional differential equations, we have shown that if $$R_0<1$$ , the unique disease-free equilibrium is asymptotically stable and there is no endemic equilibrium. Moreover, if $$R_0>1$$ , the disease-free equilibrium is unstable and there is a unique, asymptotically stable endemic equilibrium. Next, by constructing an appropriate Lyapunov function and using upper–lower solution method, we determine the threshold parameters which ensure the the global asymptotic stability of equilibria. Finally, we presented some numerical simulations to illustrate the theoretical results.

Spatiotemporal Dynamics of a Generalized Viral Infection Model with Distributed Delays and CTL Immune Response

In this paper, we propose and investigate a diffusive viral infection model with distributed delays and cytotoxic T lymphocyte (CTL) immune response. Also, both routes of infection that are virus-to-cell infection and cell-to-cell transmission are modeled by two general nonlinear incidence functions. The well-posedness of the proposed model is also proved by establishing the global existence, uniqueness, nonnegativity and boundedness of solutions. Moreover, the threshold parameters and the global asymptotic stability of equilibria are obtained. Furthermore, diffusive and delayed virus dynamics models presented in many previous studies are improved and generalized.

Dynamical Analysis of a Phytoplankton–Zooplankton System with Harvesting Term and Holling III Functional Response

A toxin-producing phytoplankton and zooplankton system is investigated. Considering that zooplankton can be harvested for food in some bodies of water, the harvesting term is introduced to zooplankton population. Firstly, from the ordinary differential equation (ODE) system, we obtain the global asymptotic stability of equilibrium and optimal capture problem. Secondly, based on the ODE system, the diffusion term is introduced and the global asymptotic stability of the steady state solution is obtained. As a result, the diffusion cannot affect the global asymptotic stability of equilibrium, and Turing instability cannot occur. Once again, a delayed differential equation (DDE) system is put forward. The global asymptotic stability of boundary equilibrium and the existence of local Hopf bifurcation at positive equilibrium are discussed. Furthermore, it is proved that there exists at least one positive periodic solution as delay varies in some region by using the global Hopf result of Wu for functional differential equations. Lastly, some numerical simulations are carried out for supporting the theoretical analyses and the positive impacts of harvesting effort, and the release rate of toxin is given. The unstable interval of the positive equilibrium becomes smaller and smaller with the increase of harvesting effort or the release rate of toxin.

Global dynamics of an SIR epidemic model with nonlocal diffusion

In this paper, we are concerned with the global asymptotic stability of each equilibrium of an SIR epidemic model with nonlocal diffusion. Under the assumption of Lipschitz continuity of parameters, the eigenvalue problem associated with the linearized system around the disease-free equilibrium has a principal eigenvalue corresponding to a strictly positive eigenfunction. By setting the eigenfunction as the integral kernel of a Lyapunov function, we prove the global asymptotic stability of the disease-free equilibrium when the basic reproduction number R0 is less than one. We also prove the uniform persistence of the system when R0>1 by using the persistent theory for dynamical systems. Furthermore, in a special case where the dispersal rate for susceptible individuals is equal to zero, we prove the existence, uniqueness and global asymptotic stability of the endemic equilibrium when R0>1 by constructing a suitable Lyapunov function.

Transmission Dynamics of an Epidemic Model with Vaccination, Treatment and Isolation

This paper focuses on the global stability of an epidemic model with vaccination, treatment and isolation. The basic reproduction number $${\mathcal {R}}_0$$ is derived. By constructing suitable Lyapunov functions, sufficient conditions for the global asymptotic stability of equilibria are obtained. Numerical simulations are performed to verify and complement the theoretical results. Furthermore, we consider the uncertainty and sensitivity analysis of the basic reproduction number $${\mathcal {R}}_0$$ . The results show that the transmission rate, the fraction of infected receives treatment, vaccination rate, the isolation rate are crucial to prevent the spread of infectious diseases. These suggest that public health workers design the control strategies of disease should consider the influence of vaccination, treatment and isolation.

Global stability of a two-mediums rumor spreading model with media coverage

Rumor spreading is a typical form of social communication and plays a significant role in social life, and media coverage has a great influence on the spread of rumor. In this paper, we present a new model with two media coverage to investigate the impact of the different mediums on rumor spreading. Then, we calculate the equilibria of the model and construct the reproduction number ℜ0. And we prove the global asymptotic stability of equilibria by using Lyapunov functions. Finally, we can conclude that the transition rate of the ignorants between two mediums has a direct effect on the scale of spreaders, and different media coverage has significant effects on the dynamics behaviors of rumor spreading.

On global asymptotic stability of the equilibrium of “predator–prey” system in varying environment

This paper considers a predator–prey system of differential equations. This ecological system is a model of Lotka–Volterra type whose prey population receives time-variation of the environment. It is not assumed that the time-varying coefficient is weakly integrally positive. We obtain sufficient conditions of global asymptotic stability of the unique interior equilibrium if the time-variation is bounded.

A metapopulation model for the population dynamics of anopheles mosquito

A more robust assessment of malaria control will come from a better understanding of the distribution and connectivity of breeding and blood feeding sites. Spatial heterogeneity of mosquito resources, such as hosts and breeding sites, affects mosquito dispersal behavior. This paper analyzes and simulates the spreading of anopheles mosquito on a complex metapopulation, that is, networks of populations connected by migratory flows whose configurations are described in terms of connectivity distribution of nodes (patches) and the conditional probabilities of connections between nodes. We examine the impacts of vector dispersal on the persistence and extinction of a mosquito population in both homogeneous and heterogeneous landscapes. For uncorrelated networks in a homogeneous landscape, we derive an explicit formula of the basic offspring number R0(m). Using the theory of monotone operators, we obtain sufficient conditions for the global asymptotic stability of equilibria. Precisely, the value 1 of the basic offspring number is a forward bifurcation for the dynamics of anopheles mosquito, with the trivial (mosquito-free) equilibrium point being globally asymptotically stable (GAS) when R0(m)≤1, and one stable nontrivial (mosquito-persistent) equilibrium point being born with well determined basins of attraction when R0(m)>1. Theoretical results are numerically supported and the impact of the migration of mosquitoes are discussed through global sensitivity analysis and numerical simulations.

Geometric approach to global asymptotic stability for the SEIRS models in epidemiology

In this paper, we present a more general criterion for the global asymptotic stability of equilibria for nonlinear autonomous differential equations based on the geometric criterion developed by Li and Muldowney. By applying this criterion, we obtain some results for the global asymptotic stability of SEIRS models with constant recruitment and varying total population size. Based on these results, we give a complete affirmative answer to Liu–Hethcote–Levin conjecture. Furthermore, an affirmative answer to Li–Graef–Wang–Karsai’s problem for SEIR model with permanent immunity and varying total population size is given.

Asymptotic stability analysis of autonomous systems by applying the method of localization of compact invariant sets

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.

On the non-linear stability of a continuous duopoly model with constant conjectural variation

The paper concerns a continuous model governed by a ODE system originated by a discrete duopoly model with bounded rationality, based on constant conjectural variation. The aim of this paper is to show (i) the existence of an absorbing set in the phase space; (ii) linear stability analysis of the critical points of the system; (iii) non-linear, global asymptotic stability of equilibrium of constant conjectural variation.

Global stability of a two-patch cholera model with fast and slow transmissions

A two-patch model for a waterborne disease, such as cholera, is considered, with the aim of investigating the impact of human population movements between two cities (patches). We derive the reproduction number R0, which depends on human movement rates. It is shown that the disease-free equilibrium is globally asymptotically stable whenever R0≤1. Three types of equilibria are explored: boundary endemic equilibria (patch-1 disease-free equilibrium and patch-2 disease-free equilibrium); interior endemic equilibrium (both patches endemic). They depend on four threshold parameters. The global asymptotic stability of equilibria is established using Lyapunov functions that combine quadratic, Volterra-type and linear functions. The theory is supported by numerical simulations, which further suggest that the human movement can increase or reduce the spread of the disease in one patch.

Using an analog of the Lyapunov function with sign-alternating derivative in the study of global asymptotic stability of equilibria

The paper deals with the problem of global asymptotic stability of the zero solution of a system of autonomous differential equations. A proposed method for studying this problem is based on the use of an auxiliary positive definite function whose derivative can be sign-alternating along solutions of the system.

A SIMPLE DISCRETE-TIME ANALOGUE PRESERVING THE GLOBAL STABILITY OF A CONTINUOUS SIRS EPIDEMIC MODEL

In this paper, we consider the backward Euler discretization derived from a continuous SIRS epidemic model, which contains a remaining problem that our discrete model has two solutions for infected population; one is positive and the other is negative. Under an additional positiveness condition on infected population, we show that the backward Euler discretization is one of simple discrete-time analogue which preserves the global asymptotic stability of equilibria of the corresponding continuous model.