scholarly journals Global Stability for Novel Complicated SIR Epidemic Models with the Nonlinear Recovery Rate and Transfer from Being Infectious to Being Susceptible to Analyze the Transmission of COVID-19

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Fehaid Salem Alshammari ◽  
F. Talay Akyildiz
Keyword(s):  
Saudi Arabia ◽  
Invariance Principle ◽  
Direct Method ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Lasalle’S Invariance Principle ◽  
Sir Epidemic ◽  
Lasalle's Invariance Principle

Epidemiological models play pivotal roles in predicting, anticipating, understanding, and controlling present and future epidemics. The dynamics of infectious diseases is complex, and therefore, researchers need to consider more complicated mathematical models. In this paper, we first describe the dynamics of a complex SIR epidemic model with nonstandard nonlinear incidence and recovery rates. In this model, we consider the rate at which individuals lose immunity. Rigorous mathematical results have been established from the point of view of stability and bifurcation. The basic reproduction number ( R 0 ) is determined. We then apply LaSalle’s invariance principle and Lyapunov’s direct method to prove that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 . The model has a unique endemic equilibrium when R 0 > 1 . A nonlinear Lyapunov function is used together with LaSalle’s invariance principle to show that the endemic equilibrium is globally asymptotically stable under some conditions. Further, for the case when   R 0 = 1 , we analyze the model and show a backward bifurcation under certain conditions. In the second part of this paper, we analyze a modified SIR model with a vaccination term, which must be a function of time. We show that the modified model agrees well with COVID-19 data in Saudi Arabia. We then investigate different future scenarios. Simulation results suggest that a two-pronged strategy is crucial to control the COVID-19 pandemic in Saudi Arabia.

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

scholarly journals Transmission Dynamics of a Two-City SIR Epidemic Model with Transport-Related Infections

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Yao Chen ◽  
Mei Yan ◽  
Zhongyi Xiang
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
The Basic Reproduction Number

A two-city SIR epidemic model with transport-related infections is proposed. Some good analytical results are given for this model. If the basic reproduction numberℜ0γ≤1, there exists a disease-free equilibrium which is globally asymptotically stable. There exists an endemic equilibrium which is locally asymptotically stable if the basic reproduction numberℜ0γ>1. We also show the permanence of this SIR model. In addition, sufficient conditions are established for global asymptotic stability of the endemic equilibrium.

scholarly journals Global stability for a discrete SIR epidemic model with delay in the general incidence function

2019 ◽  
Vol 8 (2) ◽  
pp. 32
Author(s):  
Guiro Aboudramane ◽  
Dramane Ouedraogo ◽  
Harouna Ouedraogo
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Step Size ◽  
Globally Asymptotically Stable ◽  
Incidence Function ◽  
Sir Epidemic ◽  
Disease Free ◽  
Boundedness Of Solution

In this paper, we construct a backward difference scheme for a class of general SIR epidemic model with general incidence function f. We use the step size h > 0, for the discretization. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, under the conditions that function f satisfies some assumptions. The global stabilities of equilibria are obtained. If the basic reproduction number R0<1, the disease-free equilibrium is globally asymptotically stable. If R0>1, the endemic equilibrium is globally asymptotically stable.

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.

scholarly journals Global dynamics for a class of reaction–diffusion multigroup SIR epidemic models with time fractional-order derivatives

2022 ◽  
Vol 27 (1) ◽  
pp. 142-162
Author(s):  
Zhenzhen Lu ◽  
Yongguang Yu ◽  
Guojian Ren ◽  
Conghui Xu ◽  
Xiangyun Meng
Keyword(s):  
Equilibrium Point ◽  
Fractional Order ◽  
Propagation Speed ◽  
Reaction Diffusion ◽  
Incidence Rates ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

This paper investigates the global dynamics for a class of multigroup SIR epidemic model with time fractional-order derivatives and reaction–diffusion. The fractional order considered in this paper is in (0; 1], which the propagation speed of this process is slower than Brownian motion leading to anomalous subdiffusion. Furthermore, the generalized incidence function is considered so that the data itself can flexibly determine the functional form of incidence rates in practice. Firstly, the existence, nonnegativity, and ultimate boundedness of the solution for the proposed system are studied. Moreover, the basic reproduction number R0 is calculated and shown as a threshold: the disease-free equilibrium point of the proposed system is globally asymptotically stable when R0 ≤ 1, while when R0 > 1, the proposed system is uniformly persistent, and the endemic equilibrium point is globally asymptotically stable. Finally, the theoretical results are verified by numerical simulation.

scholarly journals Mathematical Analysis of a Fractional-order "SIR" Epidemic Model with a General Nonlinear Saturated Incidence Rate in a Chemostat

2019 ◽  
pp. 1-12
Author(s):  
Miled El Hajji
Keyword(s):  
Incidence Rate ◽  
Fractional Order ◽  
Equilibrium Points ◽  
Continuous Reactor ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
Saturated Incidence Rate ◽  
Saturated Incidence

In the present work, a fractional-order differential equation based on the Susceptible-Infected- Recovered (SIR) model with nonlinear incidence rate in a continuous reactor is proposed. A profound qualitative analysis is given. The analysis of the local and global stability of equilibrium points is carried out. It is proved that if the basic reproduction number R > 1 then the disease-persistence (endemic) equilibrium is globally asymptotically stable. However, if R ≤ 1, then the disease-free equilibrium is globally asymptotically stable. Finally, some numerical tests are done in order to validate the obtained results.

scholarly journals Global stability analysis for a generalized delayed SIR model with vaccination and treatment

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
A. Elazzouzi ◽  
A. Lamrani Alaoui ◽  
M. Tilioua ◽  
A. Tridane
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Distributed Delay ◽  
Vaccination Strategy ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Susceptible Population ◽  
Sir Epidemic ◽  
The Basic Reproduction Number

AbstractIn this work, we investigate the stability of an SIR epidemic model with a generalized nonlinear incidence rate and distributed delay. The model also includes vaccination term and general treatment function, which are the two principal control measurements to reduce the disease burden. Using the Lyapunov functions, we show that the disease-free equilibrium state is globally asymptotically stable if $\mathcal{R}_{0} \leq 1 $R0≤1, where $\mathcal{R}_{0} $R0 is the basic reproduction number. On the other hand, the disease-endemic equilibrium is globally asymptotically stable when $\mathcal{R}_{0} > 1 $R0>1. For a specific type of treatment and incidence functions, our analysis shows the success of the vaccination strategy, as well as the treatment depends on the initial size of the susceptible population. Moreover, we discuss, numerically, the behavior of the basic reproduction number with respect to vaccination and treatment parameters.

scholarly journals Stability Analysis of an Age-Structured SIR Epidemic Model with a Reduction Method to ODEs

Mathematics ◽  
2018 ◽  
Vol 6 (9) ◽  
pp. 147 ◽  
Author(s):  
Toshikazu Kuniya
Keyword(s):  
Stability Analysis ◽  
Epidemic Model ◽  
Disease Transmission ◽  
Transmission Function ◽  
Endemic Equilibrium ◽  
Dimensional System ◽  
Relevant Parameter ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Age Structured

In this paper, we are concerned with the asymptotic stability of the nontrivial endemic equilibrium of an age-structured susceptible-infective-recovered (SIR) epidemic model. For a special form of the disease transmission function, we perform the reduction of the model into a four-dimensional system of ordinary differential equations (ODEs). We show that the unique endemic equilibrium of the reduced system exists if the basic reproduction number for the original system is greater than unity. Furthermore, we perform the stability analysis of the endemic equilibrium and obtain a fourth-order characteristic equation. By using the Routh–Hurwitz criterion, we numerically show that the endemic equilibrium is asymptotically stable in some epidemiologically relevant parameter settings.

scholarly journals Mathematical Modelling and Analysis of Corruption of Morals amongst Adolescents with Control Measures in Kenya

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Nathan Oigo Mokaya ◽  
Haileyesus Tessema Alemmeh ◽  
Cyrus Gitonga Ngari ◽  
Grace Gakii Muthuri
Keyword(s):  
Integrated Control ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Control Measures ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Appropriate ◽  
Manifold Theory ◽  
Necessary And Sufficient ◽  
Well Posed

In the present paper, we formulate a new mathematical model for the dynamics of moral corruption with comprehensive age-appropriate sexual information and provision of guidance and counselling. The population is subdivided into three (3) different compartments according to their level of information on sexual matters. The model is proved to be both epidemiologically and mathematically well posed. The existence of unique morally corrupt-free and endemic equilibrium points is investigated. The basic reproduction number with respect to morally corrupt-free equilibrium is obtained using next generation matrix approach to monitor the dynamics of corrupt morals and ascertain its level in order to suggest effective intervention strategies to control this problem. The local as well as global asymptotic stability of these equilibrium points is studied. The analysis reveals a globally asymptotically stable morally corrupt-free equilibrium whenever ℛ 0 ≤ 1 and a globally asymptotically stable endemic equilibrium if otherwise. Further analysis, using center manifold theory, shows that the model exhibits forward bifurcation insinuating that the classical epidemiological requirement of ℛ 0 ≤ 1 is necessary and sufficient for elimination of moral corruption. A brief discussion on the graphical results using the available numerical procedures is shown. From numerical simulations, it was ascertain that integrated control strategy is the best approach to fight against moral corruption transmission. Lastly, some key parameters that show significance in the moral corruption elimination from the society are also exploited.

Stability of a CD4+ T cell viral infection model with diffusion

2018 ◽  
Vol 11 (05) ◽  
pp. 1850071 ◽  
Author(s):  
Zhiting Xu ◽  
Youqing Xu
Keyword(s):  
T Cell ◽  
Viral Infection ◽  
Lyapunov Functions ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Viral Infection Model

This paper is devoted to the study of the stability of a CD[Formula: see text] T cell viral infection model with diffusion. First, we discuss the well-posedness of the model and the existence of endemic equilibrium. Second, by analyzing the roots of the characteristic equation, we establish the local stability of the virus-free equilibrium. Furthermore, by constructing suitable Lyapunov functions, we show that the virus-free equilibrium is globally asymptotically stable if the threshold value [Formula: see text]; the endemic equilibrium is globally asymptotically stable if [Formula: see text] and [Formula: see text]. Finally, we give an application and numerical simulations to illustrate the main results.

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