Influence of latent period and nonlinear
incidence rate on the dynamics of SIRS epidemiological models

Yu, Yang; Xiao, Dongmei

doi:10.3934/dcdsb.2010.13.195

Cited by 19 publications

(11 citation statements)

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“…Mathematical epidemiology describing the population dynamics of infectious diseases has been made a significant progress in better understanding of disease transmissions and behavior of epidemics. Many epidemic models have been described by ordinary differential equations [1][2][3][4][5][6][7][8][9][10][11]. These important and useful deterministic investigations offer a great insight into the effects of infectious disease, but in the real world, epidemic dynamics is inevitably affected by the environmental noise, which is an important component in the epidemic systems.…”

Section: Introductionmentioning

confidence: 99%

Survival and Stationary Distribution in a Stochastic SIS Model

Zhou

Zhang

Yuan

2013

Discrete Dynamics in Nature and Society

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The dynamics of a stochastic SIS epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: whenR0≤1, we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions; whenR0>1, we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. Finally, numerical simulations are carried out to illustrate the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Survival and Stationary Distribution in a Stochastic SIS Model

Zhou

Zhang

Yuan

2013

Discrete Dynamics in Nature and Society

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“…Here we will use the saturating incidence rate denoted as βSI 1+aI (a > 0) [3] other than the bilinear incidence rate βSI. As reported in the literature, the saturating incidence rate is more realistic than the bilinear rate and can cause some interesting dynamic behaviors of infectious diseases, such as limit cycle, heteroclinic orbit, saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation, see, [7,14,18,28,29,31], for example.…”

Section: Zhiting Xumentioning

confidence: 99%

Traveling waves for a diffusive SEIR epidemic model

Xu¹

2016

CPAA

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In this paper, we propose a diffusive SEIR epidemic model with saturating incidence rate. We first study the well posedness of the model, and give the explicit formula of the basic reproduction number R 0. And hence, we show that if R 0 > 1, then there exists a positive constant c * > 0 such that for each c > c * , the model admits a nontrivial traveling wave solution, and if R 0 ≤ 1 and c ≥ 0 (or, R 0 > 1 and c ∈ [0, c *)), then the model has no nontrivial traveling wave solutions. Consequently, we confirm that the constant c * is indeed the minimal wave speed. The proof of the main results is mainly based on Schauder fixed theorem and Laplace transform.

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“…In their work, they considered the inhibition effect from the behavioral change of the healthy population or to consider the crowding effect of the infected individuals due to large amount of infected individuals presented in the system as in this type of disease transmission. Recently, using this type of incidence rate can be seen in previous studies, for example. Here, we consider the saturated incidence rates for systems and , individuals pass from the susceptible classes to the infective classes in the same population group with a saturating incidence rate defined by

f_{1} false(x false) = \frac{x}{1 + κ_{1} x}

, κ 1 > 0, while individuals pass from the susceptible classes to the opposite infective classes with another saturating incidence rate defined by

f_{2} false(x false) = \frac{x}{1 + κ_{2} x}

, κ 2 > 0, the positive constants κ i , i = 1,2, determine the saturation level when the infectious population is large.…”

Section: Introductionmentioning

confidence: 99%

Traveling waves in a diffusive epidemic model with criss‐cross mechanism

Guo

2019

Math Methods in App Sciences

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In this paper, we propose a reaction‐diffusion system to describe the spread of infectious diseases within two population groups by self and criss‐cross infection mechanism. Firstly, based on the eigenvalues, we give two methods for the calculation of the critical wave speed c∗. Secondly, by constructing a pair of upper‐lower solutions and using the Schauder fixed‐point theorem, we prove that the system admits positive traveling wave solutions, which connect the initial disease‐free equilibrium false(u10,0,u30,0false) at t = −∞, but the traveling waves need not connect the final disease‐free equilibrium false(u1∗,0,u3∗,0false) at t = +∞. Hence, we study the asymptotic behaviors of the traveling wave solutions to show that the traveling wave solutions converge to false(u1∗,0,u3∗,0false) at t = +∞. Finally, by the two‐sided Laplace transform, we establish the nonexistence of traveling waves for the model. The approach in this paper provides an effective method to deal with the existence of traveling wave solutions for the nonmonotone reaction‐diffusion systems consisting of four equations.

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Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models

Cited by 19 publications

References 0 publications

Survival and Stationary Distribution in a Stochastic SIS Model

Survival and Stationary Distribution in a Stochastic SIS Model

Traveling waves for a diffusive SEIR epidemic model

Traveling waves in a diffusive epidemic model with criss‐cross mechanism

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