scholarly journals Dynamics of a Diffusive Multigroup SVIR Model with Nonlinear Incidence

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Jinhu Xu ◽  
Yan Geng
Keyword(s):  
Reproduction Number ◽  
Classical Method ◽  
Reaction Diffusion ◽  
Theoretic Approach ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Graph Theoretic ◽  
Graph Theoretic Approach ◽  
Special Case ◽  
Disease Free

In this paper, a multigroup SVIR epidemic model with reaction-diffusion and nonlinear incidence is investigated. We first establish the well-posedness of the model. Then, the basic reproduction number ℜ 0 is established and shown as a threshold: the disease-free steady state is globally asymptotically stable if ℜ 0 < 1 , while the disease will be persistent when ℜ 0 > 1 . Moreover, applying the classical method of Lyapunov and a recently developed graph-theoretic approach, we established the global stability of the endemic equilibria for a special case.

scholarly journals Global Dynamic Behavior of a Multigroup Cholera Model with Indirect Transmission

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Ming-Tao Li ◽  
Gui-Quan Sun ◽  
Juan Zhang ◽  
Zhen Jin
Keyword(s):  
Classical Method ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Theoretic Approach ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Graph Theoretic ◽  
Indirect Transmission ◽  
Global Dynamic ◽  
Disease Free

For a multigroup cholera model with indirect transmission, the infection for a susceptible person is almost invariably transmitted by drinking contaminated water in which pathogens,V. cholerae, are present. The basic reproduction numberℛ0is identified and global dynamics are completely determined byℛ0. It shows thatℛ0is a globally threshold parameter in the sense that if it is less than one, the disease-free equilibrium is globally asymptotically stable; whereas if it is larger than one, there is a unique endemic equilibrium which is global asymptotically stable. For the proof of global stability with the disease-free equilibrium, we use the comparison principle; and for the endemic equilibrium we use the classical method of Lyapunov function and the graph-theoretic approach.

scholarly journals Stability Analysis of a Multigroup SEIR Epidemic Model with General Latency Distributions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Nan Wang ◽  
Jingmei Pang ◽  
Jinliang Wang
Keyword(s):  
Epidemic Model ◽  
Classical Method ◽  
Theoretic Approach ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Graph Theoretic ◽  
Graph Theoretic Approach ◽  
Latency Distributions ◽  
The Given

The global stability of a multigroup SEIR epidemic model with general latency distribution and general incidence rate is investigated. Under the given assumptions, the basic reproduction numberℜ0is defined and proved as the role of a threshold; that is, the disease-free equilibriumP0is globally asymptotically stable ifℜ0≤1, while an endemic equilibriumP*exists uniquely and is globally asymptotically stable ifℜ0>1. For the proofs, we apply the classical method of Lyapunov functionals and a recently developed graph-theoretic approach.

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 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.

Global dynamics of two heterogeneous SIR models with nonlinear incidence and delays

2016 ◽  
Vol 09 (03) ◽  
pp. 1650046 ◽  
Author(s):  
Haitao Song ◽  
Weihua Jiang ◽  
Shengqiang Liu
Keyword(s):  
Incidence Rate ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Group Model ◽  
Theoretic Approach ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
The Basic Reproduction Number

To investigate the effect of heterogeneity on the global dynamics of two SIR epidemic models with general nonlinear incidence rate and infection delays, we formulate a multi-group model corresponding to the heterogeneity in the host population and a multi-stage model corresponding to heterogeneous stages of infection. Under biologically motivated considerations, we establish that the global dynamics for each of the two models is determined completely by the corresponding basic reproduction number: if the basic reproduction number is less than or equal to one, then the disease-free equilibrium is globally asymptotically stable and the disease dies out in all groups or stages; if the basic reproduction number is larger than one, then the disease will persist in all groups or stages, and there is a unique endemic equilibrium which is globally asymptotically stable. Then we conclude that the heterogeneity does not change the global dynamics of the SIR model when the incidence rate is a general nonlinear function. Our results extend a class of previous results and can be applied to the other epidemiological models. The proofs of the main results use Lyapunov functional and graph-theoretic approach.

DYNAMICS ANALYSIS OF A VACCINATION MODEL FOR HPV TRANSMISSION

2014 ◽  
Vol 22 (04) ◽  
pp. 555-599 ◽  
Author(s):  
ALIYA A. ALSALEH ◽  
ABBA B. GUMEL
Keyword(s):  
Deterministic Model ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Dynamics Analysis ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Effective Community ◽  
Hpv Types ◽  
Special Case ◽  
Disease Free

A new deterministic model for the transmission dynamics of human papillomavirus (HPV) and related cancers, in the presence of the Gardasil vaccine (which targets four HPV types), is presented. In the absence of routine vaccination in the community, the model is shown to undergo the phenomenon of backward bifurcation. This phenomenon, which has important consequences on the feasibility of effective disease control in the community, arises due to the re-infection of recovered individuals. For the special case when backward bifurcation does not occur, the disease-free equilibrium (DFE) of the model is shown to be globally-asymptotically stable (GAS) if the associated reproduction number is less than unity. The model with vaccination is also rigorously analyzed. Numerical simulations of the model with vaccination show that, with the assumed 90% efficacy of the Gardasil vaccine, the effective community-wide control of the four Gardasil-preventable HPV types is feasible if the Gardasil coverage rate is high enough (in the range 78–88%).

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