scholarly journals Global Stability of a SLIT TB Model with Staged Progression

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Yakui Xue ◽  
Xiaohong Wang
Keyword(s):  
Mathematical Model ◽  
Latent Period ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Infectious Period ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
N Stage ◽  
Disease Free ◽  
The Basic Reproduction Number

Because the latent period and the infectious period of tuberculosis (TB) are very long, it is not reasonable to consider the time as constant. So this paper formulates a mathematical model that divides the latent period and the infectious period into n-stages. For a general n-stage stage progression (SP) model with bilinear incidence, we analyze its dynamic behavior. First, we give the basic reproduction numberR0. Moreover, ifR0≤1, the disease-free equilibriumP0is globally asymptotically stable and the disease always dies out. IfR0>1, the unique endemic equilibriumP∗is globally asymptotically stable and the disease persists at the endemic equilibrium.

Dynamical system of a SEIQV epidemic model with nonlinear generalized incidence rate arising in biology

2017 ◽  
Vol 10 (07) ◽  
pp. 1750096 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Yasir Khan ◽  
Taj Wali Khan ◽  
Saeed Islam
Keyword(s):  
Dynamical System ◽  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, a dynamical system of a SEIQV mathematical model with nonlinear generalized incidence arising in biology is investigated. The stability of the disease-free and endemic equilibrium is discussed. The basic reproduction number of the model is obtained. We found that the disease-free and endemic equilibrium is stable locally as well as globally asymptotically stable. For [Formula: see text], the disease-free equilibrium is stable both locally and globally and for [Formula: see text], the endemic equilibrium is stable globally asymptotically. Finally, some numerical results are presented.

scholarly journals Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 328 ◽  
Author(s):  
Yanli Ma ◽  
Jia-Bao Liu ◽  
Haixia Li
Keyword(s):  
Lyapunov Function ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

scholarly journals A Mathematical Model of COVID-19 Pandemic: A Case Study of Bangkok, Thailand

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Pakwan Riyapan ◽  
Sherif Eneye Shuaib ◽  
Arthit Intarasit
Keyword(s):  
Mathematical Model ◽  
Reproduction Number ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Proposed Model ◽  
Next Generation Matrix ◽  
Disease Free ◽  
The Basic Reproduction Number

In this study, we propose a new mathematical model and analyze it to understand the transmission dynamics of the COVID-19 pandemic in Bangkok, Thailand. It is divided into seven compartmental classes, namely, susceptible S , exposed E , symptomatically infected I s , asymptomatically infected I a , quarantined Q , recovered R , and death D , respectively. The next-generation matrix approach was used to compute the basic reproduction number denoted as R cvd 19 of the proposed model. The results show that the disease-free equilibrium is globally asymptotically stable if R cvd 19 < 1 . On the other hand, the global asymptotic stability of the endemic equilibrium occurs if R cvd 19 > 1 . The mathematical analysis of the model is supported using numerical simulations. Moreover, the model’s analysis and numerical results prove that the consistent use of face masks would go on a long way in reducing the COVID-19 pandemic.

scholarly journals Stability of a Mathematical Model of Malaria Transmission with Relapse

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Hai-Feng Huo ◽  
Guang-Ming Qiu
Keyword(s):  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Recovery Rate ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Next Generation Matrix ◽  
The Government ◽  
Disease Free ◽  
The Basic Reproduction Number

A more realistic mathematical model of malaria is introduced, in which we not only consider the recovered humans return to the susceptible class, but also consider the recovered humans return to the infectious class. The basic reproduction numberR0is calculated by next generation matrix method. It is shown that the disease-free equilibrium is globally asymptotically stable ifR0≤1, and the system is uniformly persistence ifR0>1. Some numerical simulations are also given to explain our analytical results. Our results show that to control and eradicate the malaria, it is very necessary for the government to decrease the relapse rate and increase the recovery rate.

STABILITY OF AN AGE-STRUCTURED SEIS EPIDEMIC MODEL WITH INFECTIVITY IN INCUBATIVE PERIOD

2010 ◽  
Vol 03 (03) ◽  
pp. 299-312 ◽  
Author(s):  
SHU-MIN GUO ◽  
XUE-ZHI LI ◽  
XIN-YU SONG
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Stability Conditions ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an age-structured SEIS epidemic model with infectivity in incubative period is formulated and studied. The explicit expression of the basic reproduction number R0 is obtained. It is shown that the disease-free equilibrium is globally asymptotically stable if R0 < 1, at least one endemic equilibrium exists if R0 > 1. The stability conditions of endemic equilibrium are also given.

scholarly journals Partial Differential Equations of an Epidemic Model with Spatial Diffusion

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
El Mehdi Lotfi ◽  
Mehdi Maziane ◽  
Khalid Hattaf ◽  
Noura Yousfi
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.

GLOBAL STABILITY IN A TUBERCULOSIS MODEL INCORPORATING TWO LATENT PERIODS

2009 ◽  
Vol 02 (03) ◽  
pp. 357-362 ◽  
Author(s):  
LUJU LIU
Keyword(s):  
Latent Period ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Short Term ◽  
Globally Asymptotically Stable ◽  
Tuberculosis Model ◽  
The Stability ◽  
Disease Free

A tuberculosis (TB) model with two latent periods, short-term latent period (E1) and long-term latent period (E2), and fast and slow progressions is analyzed. The stability of the unique endemic equilibrium of the model is proved. It turns out that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number R0 ≤ 1, and the endemic equilibrium is globally asymptotically stable if R0 > 1.

Global stability of SEIVR epidemic model with generalized incidence and preventive vaccination

2015 ◽  
Vol 08 (06) ◽  
pp. 1550082 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Yasir Khan ◽  
Qaiser Badshah ◽  
Saeed Islam
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Preventive Vaccination ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an SEIVR epidemic model with generalized incidence and preventive vaccination is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number [Formula: see text]. If the basic reproduction number [Formula: see text], the disease-free equilibrium is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number [Formula: see text], the disease is uniformly persistent and the unique endemic equilibrium of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.

scholarly journals Dynamics of A Multi-stage Epidemic Model with and without Treatment

2020 ◽  
Vol 8 (5) ◽  
pp. 5293-5300
Keyword(s):  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Numerical Solutions ◽  
Reproduction Number ◽  
Dynamical Behaviour ◽  
Model Parameters ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, a non-linear mathematical model is proposed with the thought of treatment to depict the spread of infectious illness and assessed with three contamination stages. We talk about the dynamical behaviour and analytical study of the framework for the mathematical model which shows that it has two non-negative equilibrium points i.e., disease-free equilibrium (DFE) and interior(endemic) equilibrium. The outcomes show that the dynamical behaviour of the model is totally determined by the basic reproduction number. For the basic reproduction number , the disease-free equilibrium is locally as well as globally asymptotically stable under a particular parameter set. In case , the model at the interior equilibrium is locally as well as globally asymptotically stable. Finally, numerical solutions of the model corroborate the analytical results and facilitate a sensitivity analysis of the model parameters.

scholarly journals Global dynamics of a delayed epidemic model with latency and relapse

2013 ◽  
Vol 18 (2) ◽  
pp. 250-263 ◽  
Author(s):  
Rui Xu
Keyword(s):  
Basic Reproduction Number ◽  
Local Stability ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Lyapunov Functionals ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

A mathematical model describing the transmission dynamics of an infectious disease with an exposed (latent) period, relapse and a saturation incidence rate is investigated. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. By using suitable Lyapunov functionals and LaSalle’s invariance principle, it is proven that if the basic reproduction number is less than unity, the diseasefree equilibrium is globally asymptotically stable and therefore the disease fades out; and if the basic reproduction number is greater than unity, the endemic equilibrium is globally asymptotically stable and the disease becomes endemic.

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