A Diffusive Sveir Epidemic Model with Time Delay and General Incidence

A delayed epidemic model with nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals is considered. By analyzing the corresponding characteristic equations, the effects of time delay on the stability of the equilibria are studied. By choosing time delay as bifurcation parameter, the critical value of time delay at which a Hopf bifurcation occurs is obtained. In order to derive the normal form of the Hopf bifurcation, an extended method of multiple scales is developed and used. Then, the amplitude of bifurcating periodic solution and the conditions which determine the stability of the bifurcating periodic solution are obtained. The validity of analytical results is shown by their consistency with numerical simulations.

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A reaction–diffusion epidemic model with incubation period in almost periodic environments

European Journal of Applied Mathematics ◽

10.1017/s0956792520000303 ◽

2020 ◽

pp. 1-24

Author(s):

LIZHONG QIANG ◽

BIN-GUO WANG ◽

ZHI-CHENG WANG

Keyword(s):

Time Delay ◽

Spatial Heterogeneity ◽

Incubation Period ◽

Epidemic Model ◽

Reaction Diffusion ◽

Fixed Time ◽

Reaction Diffusion Equations ◽

Threshold Dynamics ◽

Almost Periodic ◽

Periodic Reaction

In this paper, we propose and study an almost periodic reaction–diffusion epidemic model in which disease latency, spatial heterogeneity and general seasonal fluctuations are incorporated. The model is given by a spatially nonlocal reaction–diffusion system with a fixed time delay. We first characterise the upper Lyapunov exponent $${\lambda ^*}$$ for a class of almost periodic reaction–diffusion equations with a fixed time delay and provide a numerical method to compute it. On this basis, the global threshold dynamics of this model is established in terms of $${\lambda ^*}$$ . It is shown that the disease-free almost periodic solution is globally attractive if $${\lambda ^*} < 0$$ , while the disease is persistent if $${\lambda ^*} < 0$$ . By virtue of numerical simulations, we investigate the effects of diffusion rate, incubation period and spatial heterogeneity on disease transmission.

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Effect of time delay on pattern dynamics in a spatial epidemic model

Physica A Statistical Mechanics and its Applications ◽

10.1016/j.physa.2014.06.038 ◽

2014 ◽

Vol 412 ◽

pp. 137-148 ◽

Cited By ~ 16

Author(s):

Yi Wang ◽

Jinde Cao ◽

Gui-Quan Sun ◽

Jing Li

Keyword(s):

Time Delay ◽

Epidemic Model ◽

Pattern Dynamics ◽

Spatial Epidemic Model ◽

Spatial Epidemic

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An Epidemic Model with a Time Delay in Transmission

Applications of Mathematics ◽

10.1023/a:1026002429257 ◽

2003 ◽

Vol 48 (3) ◽

pp. 193-203 ◽

Cited By ~ 12

Author(s):

Q. J. A. Khan ◽

E. V. Krishnan

Keyword(s):

Time Delay ◽

Epidemic Model

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Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay

Journal of Biomedicine and Biotechnology ◽

10.1155/2007/64870 ◽

2007 ◽

Vol 2007 ◽

pp. 1-10 ◽

Cited By ~ 38

Author(s):

Shujing Gao ◽

Zhidong Teng ◽

Juan J. Nieto ◽

Angela Torres

Keyword(s):

Periodic Solution ◽

Time Delay ◽

Epidemic Model ◽

Discrete Dynamical System ◽

Vaccination Rate ◽

Sir Epidemic Model ◽

Pulse Vaccination ◽

Sir Epidemic ◽

Distributed Time Delay ◽

Large Pulse

Pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as an effective strategy for the elimination of infectious diseases. An SIR epidemic model with pulse vaccination and distributed time delay is proposed in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the impulsive epidemic system and prove that the infection-free periodic solution is globally attractive if the vaccination rate is larger enough. Moreover, we show that the disease is uniformly persistent if the vaccination rate is less than some critical value. The permanence of the model is investigated analytically. Our results indicate that a large pulse vaccination rate is sufficient for the eradication of the disease.

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