### Mathematical Biosciences and Engineering

2019, Issue 4: 2391-2410. doi: 10.3934/mbe.2019120
Research article

# Traveling waves of di usive disease models with time delay and degeneracy

• Received: 05 February 2018 Accepted: 12 December 2018 Published: 22 March 2019
• In this paper, we propose a diffusive epidemic model with a standard incidence rate and distributed delays in disease transmission. We also consider the degenerate case when one of the diffusion coe cients vanishes. By establishing existence theory of traveling wave solutions and providing sharp lower bound for the wave speeds, we prove linear determinacy of the proposed model system. Sensitivity analysis suggests that disease propagation is slowed down by transmission delay but fastened by spatial diffusion. The existence proof is based on the construction of a suitable convex set which is invariant under the integral map of traveling wave equations. An innovative argument is formulated to study the boundary value problems of nonlinear elliptic equations satisfied by the traveling wave solutions, which enables us to prove that there does not exist a positive traveling wave connecting two nontrivial equilibria.

Citation: Xiao-Min Huang, Xiang-ShengWang. Traveling waves of di usive disease models with time delay and degeneracy[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2391-2410. doi: 10.3934/mbe.2019120

### Related Papers:

• In this paper, we propose a diffusive epidemic model with a standard incidence rate and distributed delays in disease transmission. We also consider the degenerate case when one of the diffusion coe cients vanishes. By establishing existence theory of traveling wave solutions and providing sharp lower bound for the wave speeds, we prove linear determinacy of the proposed model system. Sensitivity analysis suggests that disease propagation is slowed down by transmission delay but fastened by spatial diffusion. The existence proof is based on the construction of a suitable convex set which is invariant under the integral map of traveling wave equations. An innovative argument is formulated to study the boundary value problems of nonlinear elliptic equations satisfied by the traveling wave solutions, which enables us to prove that there does not exist a positive traveling wave connecting two nontrivial equilibria.

###### 通讯作者: 陈斌, bchen63@163.com
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沈阳化工大学材料科学与工程学院 沈阳 110142

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