scholarly journals On Equilibria Stability in an Epidemiological SIR Model with Recovery-dependent Infection Rate

2020 ◽  
Vol 21 (3) ◽  
pp. 409
Author(s):  
A. D. Baez Sanchez ◽  
N. Bobko
Keyword(s):  
Infection Rate ◽  
Saddle Points ◽  
Sufficient Conditions ◽  
Sir Model ◽  
Stable Equilibria ◽  
The Stability ◽  
Disease Free

We consider an epidemiological SIR model with an infection rate depending on the recovered population. We establish sufficient conditions for existence, uniqueness, and stability (local and global) of endemic equilibria and consider also the stability of the disease-free equilibrium. We show that, in contrast with classical SIR models, a system with a recovery-dependent infection rate can have multiple endemic stable equilibria (multistability) and multiple stable and unstable saddle points of equilibria. We establish conditions for the occurrence of these phenomena and illustrate the results with some examples.

Stability Analysis of Computer Virus Model System in Networks

2013 ◽  
Vol 278-280 ◽  
pp. 2033-2038 ◽  
Author(s):  
Shao Ting Ge ◽  
Gong You Tang ◽  
Xue Yang ◽  
Qi Lei Xu ◽  
Hao Yu ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Model System ◽  
Virus Model ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Mathematical Model ◽  
Disease Free

This paper considers stability analysis of a discrete-time computer virus model in networks. The disease-free equilibrium and the disease equilibrium are first derived from the mathematical model. Then the sufficient condition of stability for the disease-free equilibrium is obtained by the first Lyapunov method. And the sufficient conditions of stability for the disease equilibrium are given by disc theorem. Simulation results demonstrate the effectiveness of the stability conditions.

scholarly journals Qualitative and Bifurcation Analysis of an SIR Epidemic Model with Saturated Treatment Function and Nonlinear Pulse Vaccination

2016 ◽  
Vol 2016 ◽  
pp. 1-18
Author(s):  
Xiangsen Liu ◽  
Binxiang Dai
Keyword(s):  
Epidemic Model ◽  
Sufficient Conditions ◽  
Sir Epidemic Model ◽  
Pulse Vaccination ◽  
Sir Epidemic ◽  
Treatment Function ◽  
Existence And Stability ◽  
The Stability ◽  
Saturated Treatment ◽  
Disease Free

An SIR epidemic model with saturated treatment function and nonlinear pulse vaccination is studied. The existence and stability of the disease-free periodic solution are investigated. The sufficient conditions for the persistence of the disease are obtained. The existence of the transcritical and flip bifurcations is considered by means of the bifurcation theory. The stability of epidemic periodic solutions is discussed. Furthermore, some numerical simulations are given to illustrate our results.

scholarly journals Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics

2020 ◽  
Vol 21 (1) ◽  
Author(s):  
Mahmood Parsamanesh ◽  
Majid Erfanian ◽  
Saeed Mehrshad
Keyword(s):  
Discrete Time ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Euler Method ◽  
Continuous Version ◽  
Sis Epidemic Model ◽  
The Stability ◽  
Disease Free ◽  
Discretized Model

Abstract Background The spread of infectious diseases is so important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and to investigate under which conditions it will be wiped out or continued. Results A discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number $$\mathcal {R}_0$$ R 0 . Then the stability of the equilibria is given in terms of $$\mathcal {R}_0$$ R 0 , and the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of the model, a discretized model is obtained and analyzed. Conclusion It is proven that the disease-free equilibrium and endemic equilibrium are stable if $$\mathcal {R}_0<1$$ R 0 < 1 and $$\mathcal {R}_0>1$$ R 0 > 1 , respectively. Also, the disease-free equilibrium is globally stable when $$\mathcal {R}_0\le 1$$ R 0 ≤ 1 . The system has a transcritical bifurcation when $$\mathcal {R}_0=1$$ R 0 = 1 and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.

scholarly journals Mathematical modeling of COVID-19 spreading with asymptomatic infected and interacting peoples

2020 ◽  
Author(s):  
Mustapha SERHANI ◽  
Hanane Labbardi
Keyword(s):  
Mathematical Modeling ◽  
Reproduction Number ◽  
Sir Model ◽  
Control Coefficient ◽  
Infected Person ◽  
Containment Control ◽  
Disease Spreading ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

Abstract In this article we propose a modified compartmental (SIR) model describing the transmission of COVID-19 in Morocco. It takes account on the asymptomatic people and the strategies involving hospital isolation of the confirmed infected person, quarantine of people contacting them, and the home containment of all population to restrict mobility. We establish a relationship between the containment control coefficient $c_0$ and the basic reproduction number $\mathcal{R}_0$. Different scenarios are tested with different values of $c_0$, for which the stability of a Disease Free Equilibrium (DFE) point is correlated with the condition linking $\mathcal{R}_0$ and $c_0$. A worst scenario in which the containment is not respected in the same way during the period of confinement leads to several peaks of pandemic. It is shown that the home containment, if lived well, played a crucial role in controlling the disease spreading.

Stability and Bifurcation Analysis of a Stage-Structured SIR Model with Time Delays

2012 ◽  
Vol 468-471 ◽  
pp. 1070-1073
Author(s):  
Shan Wen Yan ◽  
Li Zhang
Keyword(s):  
Infectious Disease ◽  
Time Delays ◽  
Disease Model ◽  
Sufficient Conditions ◽  
Sir Model ◽  
Constant Time Delay ◽  
Infectious Disease Model ◽  
Stability And Bifurcation Analysis ◽  
The Stability ◽  
Two Stages

In this paper, we considered a SIR infectious disease model with two stages, immature and mature, with the time to maturity represented by a constant time delay. Positivity and boundness of solutions and sufficient conditions of the stability of equilibria are obtained.

scholarly journals A Discrete Numerical Solution of The SIR Model with Horizontal and Vertical Transmission

CAUCHY ◽  
2019 ◽  
Vol 6 (1) ◽  
pp. 1
Author(s):  
Trija Fayeldi
Keyword(s):  
Numerical Solution ◽  
Vertical Transmission ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Sir Model ◽  
Euler Method ◽  
Discrete Version ◽  
Step Size ◽  
The Stability ◽  
Disease Free

The aim of this paper is to is to generalize the SIR model with horizontal and vertical transmission. In this paper, we develop the discrete version of the model. We use Euler method to approximate numerical solution of the model. We found two equilibrium points, that is disease free and endemic equilibrium points. The existence of these points depend on basic reproduction number <em>R</em><sub>0</sub>. We found that if <em>R</em><sub>0</sub> <span style="text-decoration-line: underline;">&lt;</span> 1 then only disease free equilibrium points exists, while both points exists when <em>R</em><sub>0</sub> &gt; 1. We also found that the stability of these equilibrium points depend on the value of step-size <em>h</em>. Some numerical experiments were presented as illustration.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

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