A time-periodic reaction–diffusion epidemic model with infection period

2016 ◽  
Vol 67 (5) ◽  
Author(s):  
Liang Zhang ◽  
Zhi-Cheng Wang
Keyword(s):  
Epidemic Model ◽  
Reaction Diffusion ◽  
Infection Period ◽  
Periodic Reaction ◽  
Time Periodic

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

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.

Cylindrically symmetric travelling fronts in a periodic reaction—diffusion equation with bistable nonlinearity

2015 ◽  
Vol 145 (5) ◽  
pp. 1053-1090 ◽  
Author(s):  
Zhi-Cheng Wang
Keyword(s):  
Wave Speed ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Qualitative Properties ◽  
One Dimensional ◽  
Cylindrically Symmetric ◽  
Periodic Reaction ◽  
Bistable Nonlinearity ◽  
Travelling Fronts ◽  
Time Periodic

This paper is concerned with the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts for time-periodic reaction–diffusion equations with bistable nonlinearity in ℝm with m ≥ 2. It should be mentioned that the existence and stability of two-dimensional time-periodic V-shaped travelling fronts and three-dimensional time-periodic pyramidal travelling fronts have been studied previously. In this paper we consider two cases: the first is that the wave speed of a one-dimensional travelling front is positive and the second is that the one-dimensional wave speed is zero. For both cases we establish the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts. In particular, for the first case we furthermore show the asymptotic behaviours of level sets of the cylindrically symmetric travelling fronts.

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