scholarly journals Successive Vaccination and Difference in Immunity of a Delay SIR Model with a General Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Yongzhen Pei ◽  
Li Changguo ◽  
Qianyong Wu ◽  
Yunfei Lv
Keyword(s):  
Dynamic Behavior ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Vaccination Rate ◽  
Disease Spread ◽  
Computational Techniques ◽  
Threshold Condition ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
The Basic Reproduction Number

A delay SIR epidemic model with difference in immunity and successive vaccination is proposed to understand their effects on the disease spread. From theorems, it is obtained that the basic reproduction number governs the dynamic behavior of the system. The existence and stability of the possible equilibria are examined in terms of a certain threshold condition about the basic reproduction number. By use of new computational techniques for delay differential equations, we prove that the system is permanent. Our results indicate that the recovery rate and the vaccination rate are two factors for the dynamic behavior of the system. Numerical simulations are carried out to investigate the influence of the key parameters on the spread of the disease, to support the analytical conclusion, and to illustrate possible behavioral scenarios of the model.

scholarly journals Stability Analysis of a Delayed SIR Epidemic Model with Stage Structure and Nonlinear Incidence

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Xiaohong Tian ◽  
Rui Xu
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
The Stability ◽  
The Basic Reproduction Number

We investigate the stability of an SIR epidemic model with stage structure and time delay. By analyzing the eigenvalues of the corresponding characteristic equation, the local stability of each feasible equilibrium of the model is established. By using comparison arguments, it is proved when the basic reproduction number is less than unity, the disease free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, sufficient conditions are derived for the global stability of an endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals Threshold Dynamics of an SIR Model with Nonlinear Incidence Rate and Age-Dependent Susceptibility

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-15
Author(s):  
Junyuan Yang ◽  
Xiaoyan Wang
Keyword(s):  
Incidence Rate ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Threshold Dynamics ◽  
Sir Epidemic Model ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Sir Epidemic ◽  
Age Dependent ◽  
The Basic Reproduction Number

We propose an SIR epidemic model with different susceptibilities and nonlinear incidence rate. First, we obtain the existence and uniqueness of the system and the regularity of the solution semiflow based on some assumptions for the parameters. Then, we calculate the basic reproduction number, which is the spectral radius of the next-generation operator. Second, we investigate the existence and local stability of the steady states. Finally, we construct suitable Lyapunov functionals to strictly prove the global stability of the system, which are determined by the basic reproduction number ℛ0 and some assumptions for the incidence rate.

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 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 Impact of early detection and vaccination strategy in COVID-19 eradication program in Jakarta, Indonesia

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Dipo Aldila ◽  
Brenda M. Samiadji ◽  
Gracia M. Simorangkir ◽  
Sarbaz H. A. Khosnaw ◽  
Muhammad Shahzad
Keyword(s):  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Vaccination Strategy ◽  
Vaccination Rate ◽  
Rapid Testing ◽  
Social Distancing ◽  
Rapid Spread ◽  
Vaccine Potential ◽  
The Basic Reproduction Number

Abstract Objective Several essential factors have played a crucial role in the spreading mechanism of COVID-19 (Coronavirus disease 2019) in the human population. These factors include undetected cases, asymptomatic cases, and several non-pharmaceutical interventions. Because of the rapid spread of COVID-19 worldwide, understanding the significance of these factors is crucial in determining whether COVID-19 will be eradicated or persist in the population. Hence, in this study, we establish a new mathematical model to predict the spread of COVID-19 considering mentioned factors. Results Infection detection and vaccination have the potential to eradicate COVID-19 from Jakarta. From the sensitivity analysis, we find that rapid testing is crucial in reducing the basic reproduction number when COVID-19 is endemic in the population rather than contact trace. Furthermore, our results indicate that a vaccination strategy has the potential to relax social distancing rules, while maintaining the basic reproduction number at the minimum possible, and also eradicate COVID-19 from the population with a higher vaccination rate. In conclusion, our model proposed a mathematical model that can be used by Jakarta’s government to relax social distancing policy by relying on future COVID-19 vaccine potential.

scholarly journals Analysis of COVID-19 based on SEIR epidemic models in a multi-patch environment

2021 ◽  
Author(s):  
Lan Meng ◽  
Wei Zhu
Keyword(s):  
Numerical Simulations ◽  
Dynamic Behavior ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Migration Rate ◽  
Reproduction Number ◽  
Population Migration ◽  
Theoretical Results ◽  
Disease Free ◽  
The Basic Reproduction Number

Abstract In this paper, an n-patch SEIR epidemic model for the coronavirus disease 2019 (COVID-19) is presented. It is shown that there is unique disease-free equilibrium for this model. Then, the dynamic behavior is studied by the basic reproduction number. Some numerical simulations with three patches are given to validate the effectiveness of the theoretical results. The influence of quarantined rate and population migration rate on the basic reproduction number is also discussed by simulation.

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 Rich Dynamics of an Epidemic Model with Saturation Recovery

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Hui Wan ◽  
Jing-an Cui
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Medical Resource ◽  
Sir Epidemic Model ◽  
The Impact ◽  
The Basic Reproduction Number

A SIR epidemic model is proposed to understand the impact of limited medical resource on infectious disease transmission. The basic reproduction number is identified. Existence and stability of equilibria are obtained under different conditions. Bifurcations, including backward bifurcation and Hopf bifurcation, are analyzed. Our results suggest that the model considering the impact of limited medical resource may exhibit vital dynamics, such as bistability and periodicity when the basic reproduction numberℝ0is less than unity, which implies that the basic reproductive number itself is not enough to describe whether the disease will prevail or not and a subthreshold number is needed. It is also shown that a sufficient number of sickbeds and other medical resources are very important for disease control and eradication. Considering the costs, we provide a method to estimate a suitable treatment capacity for a disease in a region.

PERMANENCE OF SOME DISCRETE EPIDEMIC MODELS

2009 ◽  
Vol 02 (04) ◽  
pp. 443-461 ◽  
Author(s):  
MASAKI SEKIGUCHI
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Epidemic Models ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Seirs Epidemic Model ◽  
Two Delays ◽  
Appl Math ◽  
The Basic Reproduction Number

In this paper, we discuss some discrete epidemic models, that is, discrete SIR epidemic model with no delay, discrete SIR epidemic model with one delay and discrete SEIRS epidemic model with two delays. By applying the method given in Wang, Appl. Math. Lett. 15(2002) 423–428, we prove the permanence of these discrete epidemic models. These sufficient conditions are similar to the continuous epidemic models, that is, the basic reproduction number of each model is larger than one.

scholarly journals Stability analysis in a delayed SIR epidemic model with a saturated incidence rate

2010 ◽  
Vol 15 (3) ◽  
pp. 299-306 ◽  
Author(s):  
A. Kaddar
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Basic Reproduction Number

We formulate a delayed SIR epidemic model by introducing a latent period into susceptible, and infectious individuals in incidence rate. This new reformulation provides a reasonable role of incubation period on the dynamics of SIR epidemic model. We show that if the basic reproduction number, denoted, R0, is less than unity, the diseasefree equilibrium is locally asymptotically stable. Moreover, we prove that if R0 > 1, the endemic equilibrium is locally asymptotically stable. In the end some numerical simulations are given to compare our model with existing model.

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