scholarly journals Stability Analysis of a Multigroup Epidemic Model with General Exposed Distribution and Nonlinear Incidence Rates

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Ling Zhang ◽  
Jingmei Pang ◽  
Jinliang Wang
Keyword(s):  
Death Rate ◽  
Sufficient Conditions ◽  
Incidence Rates ◽  
Global Dynamics ◽  
Lyapunov Functionals ◽  
Natural Death ◽  
Nonlinear Incidence ◽  
Nonlinear Incidence Rates ◽  
Basic Production ◽  
Derivatives Of

We investigate a class of multigroup epidemic models with general exposed distribution and nonlinear incidence rates. For a simpler case that assumes an identical natural death rate for all groups, and with a gamma distribution for exposed distribution is considered. Some sufficient conditions are obtained to ensure that the global dynamics are completely determined by the basic production numberR0. The proofs of the main results exploit the method of constructing Lyapunov functionals and a graph-theoretical technique in estimating the derivatives of Lyapunov functionals.

GLOBAL DYNAMICS OF A MULTI-GROUP EPIDEMIC MODEL WITH GENERAL RELAPSE DISTRIBUTION AND NONLINEAR INCIDENCE RATE

2012 ◽  
Vol 20 (03) ◽  
pp. 235-258 ◽  
Author(s):  
JINLIANG WANG ◽  
JIAN ZU ◽  
XIANNING LIU ◽  
GANG HUANG ◽  
JIMIN ZHANG
Keyword(s):  
Incidence Rate ◽  
Host Population ◽  
Global Dynamics ◽  
Lyapunov Functionals ◽  
Nonlinear Incidence Rate ◽  
Oscillatory Solutions ◽  
Nonlinear Incidence ◽  
Persistence Theory ◽  
Graph Theoretical Approach ◽  
Derivatives Of

In this paper, we investigate a class of multi-group epidemic models allowing heterogeneity of the host population and that has taken into consideration with general relapse distribution and nonlinear incidence rate. We establish that the global dynamics are completely determined by the basic reproduction number R0. The proofs of the main results utilize the persistence theory in dynamical systems, Lyapunov functionals and a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach. Biologically, the disease (with any initial inoculation) will persist in all groups of the population and will eventually settle at a constant level in each group. Furthermore, our results demonstrate that heterogeneity and nonlinear incidence rate do not alter the dynamical behaviors of the basic SIR model. On the other hand, the global dynamics exclude the existence of Hopf bifurcation leading to sustained oscillatory solutions.

scholarly journals Asymptotic Dynamics of a Stochastic SIR Epidemic System Affected by Mixed Nonlinear Incidence Rates

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Ping Han ◽  
Zhengbo Chang ◽  
Xinzhu Meng
Keyword(s):  
Stochastic System ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Incidence Rates ◽  
Markov Semigroup ◽  
Nonlinear Incidence ◽  
Sir Epidemic ◽  
Nonlinear Incidence Rates ◽  
Asymptotic Dynamics ◽  
Stochastic Sir

This paper considers a stochastic SIR epidemic system affected by mixed nonlinear incidence rates. Using Markov semigroup theory and the Fokker–Planck equation, we explore the asymptotic dynamics of the stochastic system. We first investigate the existence of a positive solution and its uniqueness. Furthermore, we prove that the stochastic system has an asymptotically stable stationary distribution. In addition, the sufficient conditions for disease extinction are also obtained, which imply that the white noise can suppress and control the spread of infectious diseases. Finally, in order to illustrate the analytical results, we give some numerical simulations.

scholarly journals Global dynamics of a delayed SIRS epidemic model with a wide class of nonlinear incidence rates

2011 ◽  
Vol 39 (1-2) ◽  
pp. 15-34 ◽  
Author(s):  
Yoichi Enatsu ◽  
Eleonora Messina ◽  
Yukihiko Nakata ◽  
Yoshiaki Muroya ◽  
Elvira Russo ◽  
...  
Keyword(s):  
Wide Class ◽  
Epidemic Model ◽  
Incidence Rates ◽  
Global Dynamics ◽  
Nonlinear Incidence ◽  
Sirs Epidemic Model ◽  
Nonlinear Incidence Rates

scholarly journals Analysis of Pine Wilt Disease Model with Nonlinear Incidence and Horizontal Transmission

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Muhammad Ozair
Keyword(s):  
Global Analysis ◽  
Horizontal Transmission ◽  
Disease Model ◽  
Pine Wilt Disease ◽  
Incidence Rates ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Pine Wilt ◽  
Nonlinear Incidence Rates

The deterministic pine wilt model with vital dynamics to determine the equilibria and their stability by considering nonlinear incidence rates with horizontal transmission is analyzed. The complete global analysis for the equilibria of the model is discussed. The explicit formula for the reproductive number is obtained and it is shown that the “disease-free” equilibrium always exists and is globally asymptotically stable wheneverR0≤1. Furthermore, the disease persists at an “endemic” level when the reproductive number exceeds unity.

scholarly journals Asymptotic Behavior of Multigroup SEIR Model with Nonlinear Incidence Rates under Stochastic Perturbations

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Feng Wang ◽  
Shan Wang ◽  
Youhua Peng
Keyword(s):  
Asymptotic Behavior ◽  
Stochastic Perturbation ◽  
Incidence Rates ◽  
Nonlinear Incidence Rate ◽  
Stochastic Perturbations ◽  
Seir Model ◽  
Nonlinear Incidence ◽  
Nonlinear Incidence Rates ◽  
Uniqueness Of The Solution ◽  
Rate Functions

In this paper, the asymptotic behavior of a multigroup SEIR model with stochastic perturbations and nonlinear incidence rate functions is studied. First, the existence and uniqueness of the solution to the model we discuss are given. Then, the global asymptotical stability in probability of the model with R0<1 is established by constructing Lyapunov functions. Next, we prove that the disease can die out exponentially under certain stochastic perturbation while it is persistent in the deterministic case when R0>1. Finally, several examples and numerical simulations are provided to illustrate the dynamic behavior of the model and verify our analytical results.

Dynamics of a delayed epidemic model with varying immunity period and nonlinear transmission

2015 ◽  
Vol 08 (02) ◽  
pp. 1550027 ◽  
Author(s):  
Aadil Lahrouz
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
Incidence Rates ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
The Basic Reproduction Number

An epidemic model with a class of nonlinear incidence rates and distributed delay is analyzed. The nonlinear incidence is used to describe the saturated or the psychological effect of certain serious epidemics on the community when the number of infectives is getting larger. The distributed delay is derived to describe the dynamics of infectious diseases with varying immunity. Lyapunov functionals are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one. Moreover, it is shown that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions under which the endemic equilibrium is locally and globally asymptotically stable are obtained.

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