scholarly journals Dynamical Analysis of a Plateau Pika with Cross-Diffusion under Contraception Control

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xiaoyan Wang ◽  
Junyuan Yang ◽  
Fengqin Zhang
Keyword(s):  
Steady State ◽  
Local Stability ◽  
Degree Theory ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Plateau Pika ◽  
Globally Asymptotically Stable ◽  
Cross Diffusion ◽  
Contraception Control ◽  
Diffusion Rates

A plateau pika model with spatial cross-diffusion is investigated. By analyzing the corresponding characteristic equations, the local stability of an coexistence steady state is discussed whend21is small enough. However, whend21is large enough, the model shows Turing bifurcation ifB2 -4AC > 0. Furthermore, it is proved that if,R > R0, βK > dand cross-diffusion rates are zero, the positive coexistence steady state is globally asymptotically stable. A nonconstant positive solution bifurcates from the coexistent steady state by the Leray-Schauder degree theory. Numerical simulations are carried out to illustrate the main results.

Dynamical analysis of a reaction–diffusion SEI epidemic model with nonlinear incidence rate

2021 ◽  
pp. 2150041
Author(s):  
Jianpeng Wang ◽  
Binxiang Dai
Keyword(s):  
Steady State ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Reaction Diffusion ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Positive Steady State

In this paper, a reaction–diffusion SEI epidemic model with nonlinear incidence rate is proposed. The well-posedness of solutions is studied, including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions. The basic reproduction numbers are given in both heterogeneous and homogeneous environments. For spatially heterogeneous environment, by the comparison principle of the diffusion system, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] if [Formula: see text], the system will be persistent and admit at least one positive steady state. For spatially homogenous environment, by constructing a Lyapunov function, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if [Formula: see text]. Finally, two examples are given via numerical simulations, and then some control strategies are also presented by the sensitive analysis.

Dynamics of a diffusion-driven HBV infection model with capsids and time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750062 ◽  
Author(s):  
Kalyan Manna
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Dynamical Behavior ◽  
Hbv Infection ◽  
Infection Model ◽  
Direct Lyapunov Method ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Discrete Time Delay ◽  
Number Formula

In this paper, a diffusive hepatitis B virus (HBV) infection model with a discrete time delay is presented and analyzed, where the spatial mobility of both intracellular capsid covered HBV DNA and HBV and the intracellular delay in the reproduction of infected hepatocytes are taken into account. We define the basic reproduction number [Formula: see text] that determines the dynamical behavior of the model. The local and global stability of the spatially homogeneous steady states are analyzed by using the linearization technique and the direct Lyapunov method, respectively. It is shown that the susceptible uninfected steady state is globally asymptotically stable whenever [Formula: see text] and is unstable whenever [Formula: see text]. Also, the infected steady state is globally asymptotically stable when [Formula: see text]. Finally, numerical simulations are carried out to illustrate the results obtained.

scholarly journals Global Dynamics of an HIV Infection Model with Two Classes of Target Cells and Distributed Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
A. M. Elaiw
Keyword(s):  
T Cells ◽  
Steady State ◽  
Hiv Infection ◽  
Infection Model ◽  
Target Cells ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Infection Model ◽  
Nonlinear Delay

We investigate the global dynamics of an HIV infection model with two classes of target cells and multiple distributed intracellular delays. The model is a 5-dimensional nonlinear delay ODEs that describes the interaction of the HIV with two classes of target cells, CD4+T cells and macrophages. The incidence rate of infection is given by saturation functional response. The model has two types of distributed time delays describing time needed for infection of target cell and virus replication. This model can be seen as a generalization of several models given in the literature describing the interaction of the HIV with one class of target cells, CD4+T cells. Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states of the model. We have proven that if the basic reproduction numberR0is less than unity then the uninfected steady state is globally asymptotically stable, and ifR0>1then the infected steady state exists and it is globally asymptotically stable.

scholarly journals Dynamical Analysis of a Nitrogen-Phosphorus-Phytoplankton Model

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Yunli Deng ◽  
Min Zhao ◽  
Hengguo Yu ◽  
Yi Wang
Keyword(s):  
Nitrogen Concentration ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Threshold Values ◽  
Globally Asymptotically Stable ◽  
Global System ◽  
Water Ecosystem ◽  
Existence And Stability ◽  
Nitrogen Phosphorus ◽  
Coexistence Equilibrium

This paper presents a nitrogen-phosphorus-phytoplankton model in a water ecosystem. The main aim of this research is to analyze the global system dynamics and to study the existence and stability of equilibria. It is shown that the phytoplankton-eradication equilibrium is globally asymptotically stable if the input nitrogen concentration is less than a certain threshold. However, the coexistence equilibrium is globally asymptotically stable as long as it exists. The system is uniformly persistent within threshold values of certain key parameters. Finally, to verify the results, numerical simulations are provided.

Global dynamics for a live-virus-blood model with distributed time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750067 ◽  
Author(s):  
Ding-Yu Zou ◽  
Shi-Fei Wang ◽  
Xue-Zhi Li
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Live Virus ◽  
Number Formula ◽  
Distributed Time Delay ◽  
The Basic Reproduction Number

In this paper, the global properties of a mathematical modeling of hepatitis C virus (HCV) with distributed time delays is studied. Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states. It is shown that if the basic reproduction number [Formula: see text] is less than unity, then the uninfected steady state is globally asymptotically stable. If the basic reproduction number [Formula: see text] is larger than unity, then the infected steady state is globally asymptotically stable.

Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response

2017 ◽  
Vol 10 (05) ◽  
pp. 1750070 ◽  
Author(s):  
A. M. Ełaiw ◽  
A. A. Raezah ◽  
Khalid Hattaf
Keyword(s):  
Immune Response ◽  
Steady State ◽  
Dynamical Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
State Formula ◽  
Number Formula ◽  
Saturated Incidence ◽  
Ctl Immune Response ◽  
Hiv 1

This paper studies the dynamical behavior of an HIV-1 infection model with saturated virus-target and infected-target incidences with Cytotoxic T Lymphocyte (CTL) immune response. The model is incorporated by two types of intracellular distributed time delays. The model generalizes all the existing HIV-1 infection models with cell-to-cell transmission presented in the literature by considering saturated incidence rate and the effect of CTL immune response. The existence and global stability of all steady states of the model are determined by two parameters, the basic reproduction number ([Formula: see text]) and the CTL immune response activation number ([Formula: see text]). By using suitable Lyapunov functionals, we show that if [Formula: see text], then the infection-free steady state [Formula: see text] is globally asymptotically stable; if [Formula: see text] [Formula: see text], then the CTL-inactivated infection steady state [Formula: see text] is globally asymptotically stable; if [Formula: see text], then the CTL-activated infection steady state [Formula: see text] is globally asymptotically stable. Using MATLAB we conduct some numerical simulations to confirm our results. The effect of the saturated incidence of the HIV-1 dynamics is shown.

An age-structured epidemic model with waning immunity and general nonlinear incidence rate

2018 ◽  
Vol 11 (05) ◽  
pp. 1850069 ◽  
Author(s):  
Xia Wang ◽  
Ying Zhang ◽  
Xinyu Song
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Threshold Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Waning Immunity ◽  
Age Structured ◽  
The Basic Reproduction Number

In this paper, a susceptible-vaccinated-exposed-infectious-recovered epidemic model with waning immunity and continuous age structures in vaccinated, exposed and infectious classes has been formulated. By using the Fluctuation lemma and the approach of Lyapunov functionals, we establish a threshold dynamics completely determined by the basic reproduction number. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable, and otherwise the endemic steady state is globally asymptotically stable.

scholarly journals Stability of Virus Infection Models with Antibodies and Chronically Infected Cells

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Mustafa A. Obaid ◽  
A. M. Elaiw
Keyword(s):  
Immune Response ◽  
Steady State ◽  
Virus Infection ◽  
Incidence Rate ◽  
Lyapunov Functions ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Infection Models ◽  
Infected Cells

Two virus infection models with antibody immune response and chronically infected cells are proposed and analyzed. Bilinear incidence rate is considered in the first model, while the incidence rate is given by a saturated functional response in the second one. One main feature of these models is that it includes both short-lived infected cells and chronically infected cells. The chronically infected cells produce much smaller amounts of virus than the short-lived infected cells and die at a much slower rate. Our mathematical analysis establishes that the global dynamics of the two models are determined by two threshold parametersR0andR1. By constructing Lyapunov functions and using LaSalle's invariance principle, we have established the global asymptotic stability of all steady states of the models. We have proven that, the uninfected steady state is globally asymptotically stable (GAS) ifR0<1, the infected steady state without antibody immune response exists and it is GAS ifR1<1<R0, and the infected steady state with antibody immune response exists and it is GAS ifR1>1. We check our theorems with numerical simulation in the end.

Mathematical analysis and optimal control for Chikungunya virus with two routes of infection with nonlinear incidence rate

2021 ◽  
pp. 2150088 ◽  
Author(s):  
Miled El Hajji ◽  
Abdelhamid Zaghdani ◽  
Sayed Sayari
Keyword(s):  
Optimal Control ◽  
Steady State ◽  
Chikungunya Virus ◽  
Optimal Solution ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
State Formula ◽  
Number Formula ◽  
Global Threat

Chikungunya fever, caused by Chikungunya virus (CHIKV) and transmitted to humans by infected Aedes mosquitoes, has posed a global threat in several countries. In this paper, we investigated a modified within-host Chikungunya virus (CHIKV) infection model with antibodies where two routes of infection are considered. In a first step, the basic reproduction number [Formula: see text] was calculated and the local and global stability analysis of the steady states is carried out using the local linearization and the Lyapunov method. It is proven that the CHIKV-free steady-state [Formula: see text] is globally asymptotically stable when [Formula: see text], and the infected steady-state [Formula: see text] is globally asymptotically stable when [Formula: see text]. In a second step, we applied an optimal strategy in order to optimize the infected compartment and to maximize the uninfected one. For this, we formulated a nonlinear optimal control problem. Existence of the optimal solution was discussed and characterized using some adjoint variables. Thus, an algorithm based on competitive Gauss–Seidel-like implicit difference method was applied in order to resolve the optimality system. The theoretical results are confirmed by some numerical simulations.

scholarly journals On Oscillatory Pattern of Malaria Dynamics in a Population with Temporary Immunity

2007 ◽  
Vol 8 (3) ◽  
pp. 191-203 ◽  
Author(s):  
J. Tumwiine ◽  
J. Y. T. Mugisha ◽  
L. S. Luboobi
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Oscillatory Pattern ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number ◽  
Temporary Immunity

We use a model to study the dynamics of malaria in the human and mosquito population to explain the stability patterns of malaria. The model results show that the disease-free equilibrium is globally asymptotically stable and occurs whenever the basic reproduction number,R0is less than unity. We also note that whenR0>1, the disease-free equilibrium is unstable and the endemic equilibrium is stable. Numerical simulations show that recoveries and temporary immunity keep the populations at oscillation patterns and eventually converge to a steady state.

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