scholarly journals Global stability analysis of a delayed HIV model with saturated infection rate

2018 ◽  
Vol 241 ◽  
pp. 01007
Author(s):  
jaouad Danane ◽  
Karam Allali
Keyword(s):  
Global Stability ◽  
Equilibrium Point ◽  
Infection Rate ◽  
Asymptotically Stable ◽  
Delay Model ◽  
Globally Asymptotically Stable ◽  
Hiv Model ◽  
Global Properties ◽  
Discrete Delays ◽  
Infected Cells

In this paper, the global stability of a delayed HIV model with saturated infection rate infection is investigated. We incorporate two discrete delays into the model; the first describes the intracellular delay in the production of the infected cells, while the second describes the needed time for virions production. We also derive the global properties of this two-delay model as function of the basic reproduction number R0. By using some suitable Lyapunov functions, it is proved that the free-equilibrium point is globally asymptotically stable when R0 ≤ 1, and the endemic equilibrium point is globally asymptotically stable when R0 ≥ 1. Finally, in order to support our theoretical findings we have illustrate some numerical simulations.

scholarly journals Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Qilin Sun ◽  
Lequan Min
Keyword(s):  
Drug Resistance ◽  
Antiretroviral Therapy ◽  
Hiv Infection ◽  
Equilibrium Point ◽  
Infection Rate ◽  
Reproductive Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Rna

This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive numberR0of the model is less than one, then the infection-free equilibrium point of the model is globally asymptotically stable; ifR0of the model is more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of the two groups of patients’ anti-HIV infection treatment. The numerical simulation results are in agreement with the evolutions of the patients’ HIV RNA levels. It can be assumed that if an HIV infected individual’s basic virus reproductive numberR0<1then this person will recover automatically; if an antiretroviral therapy makes an HIV infected individual’sR0<1, this person will be cured eventually; if an antiretroviral therapy fails to suppress an HIV infected individual’s HIV RNA load to be of unpredictable level, the time that the patient’s HIV RNA level has achieved the minimum value may be the starting time that drug resistance has appeared.

scholarly journals Threshold dynamics of a general delayed HIV model with double transmission modes and latent viral infection

2021 ◽  
Vol 7 (2) ◽  
pp. 2456-2478
Author(s):  
Xin Jiang ◽  
Keyword(s):  
Time Delay ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Model ◽  
Transmission Modes ◽  
Latently Infected ◽  
Vital Effect ◽  
Infected Cells

<abstract><p>In this paper, a general HIV model incorporating intracellular time delay is investigated. Taking the latent virus infection, both virus-to-cell and cell-to-cell transmissions into consideration, the model exhibits threshold dynamics with respect to the basic reproduction number $ \mathfrak{R}_0 $. If $ \mathfrak{R}_0 &lt; 1 $, then there exists a unique infection-free equilibrium $ E_0 $, which is globally asymptotically stable. If $ \mathfrak{R}_0 &gt; 1 $, then there exists $ E_0 $ and a globally asymptotically stable infected equilibrium $ E^* $. When $ \mathfrak{R}_0 = 1 $, $ E_0 $ is linearly neutrally stable and a forward bifurcation takes place without time delay around $ \mathfrak{R}_0 = 1 $. The theoretical results and corresponding numerical simulations show that the existence of latently infected cells and the intracellular time delay have vital effect on the global dynamics of the general virus model.</p></abstract>

scholarly journals Mathematical Modeling of Typhoid Fever Disease Incorporating Unprotected Humans in the Spread Dynamics

2019 ◽  
pp. 1-11
Author(s):  
Julia Wanjiku Karunditu ◽  
George Kimathi ◽  
Shaibu Osman
Keyword(s):  
Typhoid Fever ◽  
Global Stability ◽  
Equilibrium Point ◽  
Local Stability ◽  
Jacobian Matrix ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Spread Dynamics ◽  
Disease Free

A deterministic mathematical model of typhoid fever incorporating unprotected humans is formulated in this study and employed to study local and global stability of equilibrium points. The model incorporating Susceptible, unprotected, Infectious and Recovered humans which are analyzed mathematically and also result into a system of ordinary differential equations which are used for interpretations and comparison to the qualitative solutions in studying the spread dynamics of typhoid fever. Jacobian matrix was considered in the study of local stability of disease free equilibrium point and Castillo-Chavez approach used to determine global stability of disease free equilibrium point. Lyapunov function was used to study global stability of endemic equilibrium point. Both equilibrium points (DFE and EE) were found to be local and globally asymptotically stable. This means that the disease will be dependent on numbers of unprotected humans and other factors who contributes positively to the transmission dynamics.

scholarly journals Global Stability and Optimal Control of Dengue with Two Coexisting Virus Serotypes

MATEMATIKA ◽  
2019 ◽  
Vol 35 (4) ◽  
pp. 149-170
Author(s):  
Afeez Abidemi ◽  
Rohanin Ahmad ◽  
Nur Arina Bazilah Aziz
Keyword(s):  
Optimal Control ◽  
Global Stability ◽  
Disease Transmission ◽  
Deterministic Model ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Effective Control ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Boundary Equilibrium

This study presents a two-strain deterministic model which incorporates Dengvaxia vaccine and insecticide (adulticide) control strategies to forecast the dynamics of transmission and control of dengue in Madeira Island if there is a new outbreak with a different virus serotypes after the first outbreak in 2012. We construct suitable Lyapunov functions to investigate the global stability of the disease-free and boundary equilibrium points. Qualitative analysis of the model which incorporates time-varying controls with the specific goal of minimizing dengue disease transmission and the costs related to the control implementation by employing the optimal control theory is carried out. Three strategies, namely the use of Dengvaxia vaccine only, application of adulticide only, and the combination of Dengvaxia vaccine and adulticide are considered for the controls implementation. The necessary conditions are derived for the optimal control of dengue. We examine the impacts of the control strategies on the dynamics of infected humans and mosquito population by simulating the optimality system. The disease-freeequilibrium is found to be globally asymptotically stable whenever the basic reproduction numbers associated with virus serotypes 1 and j (j 2 {2, 3, 4}), respectively, satisfy R01,R0j 1, and the boundary equilibrium is globally asymptotically stable when the related R0i (i = 1, j) is above one. It is shown that the strategy based on the combination of Dengvaxia vaccine and adulticide helps in an effective control of dengue spread in the Island.

scholarly journals Global Stability of Malaria Transmission Dynamics Model with Logistic Growth

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Abadi Abay Gebremeskel
Keyword(s):  
Sensitivity Analysis ◽  
Global Stability ◽  
Malaria Transmission ◽  
Equilibrium Points ◽  
Transmission Dynamics ◽  
Logistic Growth ◽  
Asymptotically Stable ◽  
Dynamics Model ◽  
Globally Asymptotically Stable ◽  
The Impact

Mathematical models become an important and popular tools to understand the dynamics of the disease and give an insight to reduce the impact of malaria burden within the community. Thus, this paper aims to apply a mathematical model to study global stability of malaria transmission dynamics model with logistic growth. Analysis of the model applies scaling and sensitivity analysis and sensitivity analysis of the model applied to understand the important parameters in transmission and prevalence of malaria disease. We derive the equilibrium points of the model and investigated their stabilities. The results of our analysis have shown that if R0≤1, then the disease-free equilibrium is globally asymptotically stable, and the disease dies out; if R0>1, then the unique endemic equilibrium point is globally asymptotically stable and the disease persists within the population. Furthermore, numerical simulations in the application of the model showed the abrupt and periodic variations.

Global properties for an age-structured within-host model with Crowley–Martin functional response

2017 ◽  
Vol 10 (02) ◽  
pp. 1750030 ◽  
Author(s):  
Shaoli Wang ◽  
Xinyu Song
Keyword(s):  
Functional Response ◽  
Viral Infections ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Multi Scale ◽  
Global Properties ◽  
Age Structured

Based on a multi-scale view, in this paper, we study an age-structured within-host model with Crowley–Martin functional response for the control of viral infections. By means of semigroup and Lyapunov function, the global asymptotical property of infected steady state of the model is obtained. The results show that when the basic reproductive number falls below unity, the infection dies out. However, when the basic reproductive number exceeds unity, there exists a unique positive equilibrium which is globally asymptotically stable. This model can be deduced to different viral models with or without time delay.

scholarly journals Global Properties of Latent Virus Dynamics Models with Immune Impairment and Two Routes of Infection

2019 ◽  
Vol 8 (2) ◽  
pp. 16
Author(s):  
Aeshah A. Raezah ◽  
Ahmed M. Elaiw ◽  
Badria S. Alofi
Keyword(s):  
Global Stability ◽  
Lyapunov Method ◽  
Incidence Rates ◽  
Steady States ◽  
Global Properties ◽  
Infection Models ◽  
Latently Infected ◽  
Immune Impairment ◽  
Infected Cells ◽  
Dynamics Models

This paper studies the global stability of viral infection models with CTL immune impairment. We incorporate both productively and latently infected cells. The models integrate two routes of transmission, cell-to-cell and virus-to-cell. In the second model, saturated virus–cell and cell–cell incidence rates are considered. The basic reproduction number is derived and two steady states are calculated. We first establish the nonnegativity and boundedness of the solutions of the system, then we investigate the global stability of the steady states. We utilize the Lyapunov method to prove the global stability of the two steady states. We support our theorems by numerical simulations.

scholarly journals Global Stability of Humoral Immunity HIV Infection Models with Chronically Infected Cells and Discrete Delays

2015 ◽  
Vol 2015 ◽  
pp. 1-25
Author(s):  
A. M. Elaiw ◽  
N. A. Alghamdi
Keyword(s):  
Hiv Infection ◽  
Global Stability ◽  
Incidence Rate ◽  
Sufficient Conditions ◽  
Humoral Immune ◽  
Infection Models ◽  
Existence And Stability ◽  
Discrete Delays ◽  
Infected Cells ◽  
Theoretical Results

We study the global stability of three HIV infection models with humoral immune response. We consider two types of infected cells: the first type is the short-lived infected cells and the second one is the long-lived chronically infected cells. In the three HIV infection models, we modeled the incidence rate by bilinear, saturation, and general forms. The models take into account two types of discrete-time delays to describe the time between the virus entering into an uninfected CD4+T cell and the emission of new active viruses. The existence and stability of all equilibria are completely established by two bifurcation parameters,R0andR1. The global asymptotic stability of the steady states has been proven using Lyapunov method. In case of the general incidence rate, we have presented a set of sufficient conditions which guarantee the global stability of model. We have presented an example and performed numerical simulations to confirm our theoretical results.

Non Linear Difference Equation of Orlicz Type

2017 ◽  
Vol 14 (1) ◽  
pp. 306-313
Author(s):  
Awad. A Bakery ◽  
Afaf. R. Abou Elmatty
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Solution ◽  
Orlicz Function ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
The Difference

We give here the sufficient conditions on the positive solutions of the difference equation xn+1 = α+M((xn−1)/xn), n = 0, 1, …, where M is an Orlicz function, α∈ (0, ∞) with arbitrary positive initials x−1, x0 to be bounded, α-convergent and the equilibrium point to be globally asymptotically stable. Finally we present the condition for which every positive solution converges to a prime two periodic solution. Our results coincide with that known for the cases M(x) = x in Ref. [3] and M(x) = xk, where k ∈ (0, ∞) in Ref. [7]. We have given the solution of open problem proposed in Ref. [7] about the existence of the positive solution which eventually alternates above and below equilibrium and converges to the equilibrium point. Some numerical examples with figures will be given to show our results.

GLOBAL PROPERTIES OF A MODEL OF IMMUNE EFFECTOR RESPONSES TO VIRAL INFECTIONS

2007 ◽  
Vol 10 (04) ◽  
pp. 495-503 ◽  
Author(s):  
XIA WANG ◽  
XINYU SONG
Keyword(s):  
Immune Response ◽  
Immune Responses ◽  
Lyapunov Functions ◽  
Viral Infections ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Immune Effector ◽  
Global Properties ◽  
Ctl Immune Response

This article proposes a mathematical model which has been used to investigate the importance of lytic and non-lytic immune responses for the control of viral infections. By means of Lyapunov functions, the global properties of the model are obtained. The virus is cleared if the basic reproduction number R0 ≤ 1 and the virus persists in the host if R0 > 1. Furthermore, if R0 > 1 and other conditions hold, the immune-free equilibrium E0 is globally asymptotically stable. The equilibrium E1 exists and is globally asymptotically stale if the CTL immune response reproductive number R1 < 1 and the antibody immune response reproductive number R2 > 1. The equilibrium E2 exists and is globally asymptotically stable if R1 > 1 and R2 < 1. Finally, the endemic equilibrium E3 exists and is globally asymptotically stable if R1 > 1 and R2 > 1.

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