scholarly journals Model Penyebaran Penyakit Menular dengan Transmisi Vertikal

CAUCHY ◽  
2009 ◽  
Vol 1 (1) ◽  
pp. 25
Author(s):  
Usman Pagalay
Keyword(s):  
Reproduction Number ◽  
Host Population ◽  
Endemic Equilibrium ◽  
Epidemic Models ◽  
Feasible Region ◽  
Globally Asymptotically Stable ◽  
Threshold Phenomenon ◽  
Single Host ◽  
Globally Stable ◽  
Disease Free

<p>The dynamics of many epidemic models for infectious diseases that spread in a single host population demonstrate a threshold phenomenon. If the basic reproduction number R0 is below unity, the disease free equilibrium P is globally stable in the feasible region and the disease always dies out. If R0 &gt; 1, a unique endemic equilibrium P is globally asymptotically stable in the interior of the feasible region and the disease will persist at the endemic equilibrium if it is initially present. In this paper this threshold phenomenon is established for two epidemic models of SEIR type using two recent approaches to the global-stability problem.</p>

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 Global Dynamics of Infectious Disease with Arbitrary Distributed Infectious Period on Complex Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Xiaoguang Zhang ◽  
Rui Song ◽  
Gui-Quan Sun ◽  
Zhen Jin
Keyword(s):  
Complex Networks ◽  
Reproduction Number ◽  
Epidemic Models ◽  
Infectious Period ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Sis Epidemic Model ◽  
Different Types ◽  
Theoretical Results ◽  
Disease Free

Most of the current epidemic models assume that the infectious period follows an exponential distribution. However, due to individual heterogeneity and epidemic diversity, these models fail to describe the distribution of infectious periods precisely. We establish a SIS epidemic model with multistaged progression of infectious periods on complex networks, which can be used to characterize arbitrary distributions of infectious periods of the individuals. By using mathematical analysis, the basic reproduction numberR0for the model is derived. We verify that theR0depends on the average distributions of infection periods for different types of infective individuals, which extend the general theory obtained from the single infectious period epidemic models. It is proved that ifR0<1, then the disease-free equilibrium is globally asymptotically stable; otherwise the unique endemic equilibrium exists such that it is globally asymptotically attractive. Finally numerical simulations hold for the validity of our theoretical results is given.

THE DYNAMICS OF A DISCRETE SEIT MODEL WITH AGE AND INFECTION AGE STRUCTURES

2012 ◽  
Vol 05 (03) ◽  
pp. 1260004 ◽  
Author(s):  
HUI CAO ◽  
YANNI XIAO ◽  
YICANG ZHOU
Keyword(s):  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Disease Spread ◽  
Threshold Parameter ◽  
Infection Age ◽  
The Stability ◽  
Globally Stable ◽  
Disease Free ◽  
The Basic Reproduction Number ◽  
Age Structures

Age and infection age have significant influence on the transmission of infectious diseases, such as HIV/AIDS and TB. A discrete SEIT model with age and infection age structures is formulated to investigate the dynamics of the disease spread. The basic reproduction number R0 is defined and used as the threshold parameter to characterize the disease extinction or persistence. It is shown that the disease-free equilibrium is globally stable if R0 < 1, and it is unstable if R0 > 1. When R0 > 1, there exists an endemic equilibrium, and the disease is uniformly persistent. The stability of the endemic equilibrium is investigated numerically.

scholarly journals Modes of transmission of COVID-19 outbreak- a mathematical study

2020 ◽  
Author(s):  
Chandan Maji
Keyword(s):  
Equilibrium Point ◽  
Reproduction Number ◽  
Mathematical Study ◽  
Globally Asymptotically Stable ◽  
Modes Of Transmission ◽  
Novel Coronavirus ◽  
Respiratory Droplets ◽  
Globally Stable ◽  
Disease Free ◽  
Close Contacts

The world has now paid a lot of attention to the outbreak of novel coronavirus (COVID-19). This virus mainly transmitted between humans through directly respiratory droplets and close contacts. However, there is currently some evidence where it has been claimed that it may be indirectly transmitted. In this work, we study the mode of transmission of COVID-19 epidemic system based on the susceptible-infected-recovered (SIR) model. We have calculated the basic reproduction number $R_0$ by next-generation matrix method. We observed that if $R_0<1$, then disease-free equilibrium point is locally as well as globally asymptotically stable but when $R_0>1$, the endemic equilibrium point exists and is globally stable. Finally, some numerical simulation is presented to validate our results.

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 Global Behaviors of a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Lei Wang ◽  
Zhidong Teng ◽  
Long Zhang
Keyword(s):  
Incidence Rate ◽  
Endemic Equilibrium ◽  
Epidemic Models ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
General Incidence Rate ◽  
Special Cases ◽  
Disease Free

We study a class of discrete SIRS epidemic models with nonlinear incidence rateF(S)G(I)and disease-induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease-free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction numberℛ0<1, then the disease-free equilibrium is globally asymptotically stable, and ifℛ0>1, then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only whenℛ0>1, the disease in the model is permanent. Some special cases ofF(S)G(I)are discussed. Particularly, whenF(S)G(I)=βSI/(1+λI), it is obtained that the endemic equilibrium is globally asymptotically stable if and only ifℛ0>1. Furthermore, the numerical simulations show that for general incidence rateF(S)G(I)the endemic equilibrium may be globally asymptotically stable only asℛ0>1.

scholarly journals Mathematical Analysis of a Cholera Model with Vaccination

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Jing'an Cui ◽  
Zhanmin Wu ◽  
Xueyong Zhou
Keyword(s):  
Disease Transmission ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Disease Spread ◽  
Relative Importance ◽  
Globally Asymptotically Stable ◽  
Perform Sensitivity Analysis ◽  
Compound Matrix ◽  
Disease Free

We consider aSVR-Bcholera model with imperfect vaccination. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. We calculate the certain threshold known as the control reproduction numberℛv. Ifℛv<1, we obtain sufficient conditions for the global asymptotic stability of the disease-free equilibrium; the diseases will be eliminated from the community. By comparison of arguments, it is proved that ifℛv>1, the disease persists and the unique endemic equilibrium is globally asymptotically stable, which is obtained by the second compound matrix techniques and autonomous convergence theorems. We perform sensitivity analysis ofℛvon the parameters in order to determine their relative importance to disease transmission and show that an imperfect vaccine is always beneficial in reducing disease spread within the community.

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.

THE IMPACT OF MEDIA COVERAGE ON THE DYNAMICS OF INFECTIOUS DISEASE

2008 ◽  
Vol 01 (01) ◽  
pp. 65-74 ◽  
Author(s):  
YIPING LIU ◽  
JING-AN CUI
Keyword(s):  
Infectious Disease ◽  
Media Coverage ◽  
Compartment Model ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Impact ◽  
Globally Stable ◽  
Disease Free

In this paper, we give a compartment model to discuss the influence of media coverage to the spreading and controlling of infectious disease in a given region. The model exhibits two equilibria: a disease-free and a unique endemic equilibrium. Stability analysis of the models shows that the disease-free equilibrium is globally asymptotically stable if the reproduction number (ℝ0), which depends on parameters, is less than unity. But if ℝ0 > 1, it is shown that a unique endemic equilibrium appears, which is asymptotically stable. On a special case, the endemic equilibrium is globally stable. We discuss the role of media coverage on the spreading based on the theory results.

scholarly journals Global Stability Analysis of Two-Stage Quarantine-Isolation Model with Holling Type II Incidence Function

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 350 ◽  
Author(s):  
Mohammad A. Safi
Keyword(s):  
Disease Transmission ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Control Measures ◽  
Asymptotically Stable ◽  
Two Stage ◽  
Globally Asymptotically Stable ◽  
Incidence Function ◽  
Special Case ◽  
Disease Free

A new two-stage model for assessing the effect of basic control measures, quarantine and isolation, on a general disease transmission dynamic in a population is designed and rigorously analyzed. The model uses the Holling II incidence function for the infection rate. First, the basic reproduction number ( R 0 ) is determined. The model has both locally and globally asymptotically stable disease-free equilibrium whenever R 0 < 1 . If R 0 > 1 , then the disease is shown to be uniformly persistent. The model has a unique endemic equilibrium when R 0 > 1 . A nonlinear Lyapunov function is used in conjunction with LaSalle Invariance Principle to show that the endemic equilibrium is globally asymptotically stable for a special case.

Sign in / Sign up

Export Citation Format

Share Document