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Spreading dynamics of a diffusive epidemic model with free boundaries and two delays

Part of: Functional-differential and differential-difference equations Parabolic equations and systems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  11 August 2023

Qiaoling Chen Open the ORCID record for Qiaoling Chen [Opens in a new window] ,
Sanyi Tang ,
Zhidong Teng  and
Feng Wang
Qiaoling Chen*
Affiliation:
School of Science, Xi’an Polytechnic University, Xi’an, PR China School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, PR China
Sanyi Tang
Affiliation:
School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, PR China
Zhidong Teng
Affiliation:
College of Medical Engineering and Technology, Xinjiang Medical University, Urumqi, PR China
Feng Wang
Affiliation:
School of Mathematics and Statistics, Xidian University, Xi’an, PR China
*
Corresponding author: Qiaoling Chen; Email: qiaolingf@126.com
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Abstract

A delayed reaction-diffusion system with free boundaries is investigated in this paper to understand how the bacteria spread spatially to larger area from the initial infected habitat. Under the assumptions that the nonlinearities are of monostable type and the initial values satisfy some compatible condition, we show that the free boundary problem is well-posed and discuss the long-time behaviour of solution (including spreading and vanishing) in terms of the spatial-temporal risk index. Furthermore, to determine the spreading speed of free boundaries when spreading occurs, we first study the distribution of roots of a transcendental equation containing a polynomial of degree four and then establish the existence and uniqueness of monotone solution to a delay-induced nonlocal semi-wave problem by employing the approximation method, lower-upper solutions technique and Schauder fixed point theorem. It is shown that time delays slow down the spreading of bacteria.

Keywords

Free boundarysemi-wavespreading speedtime delayepidemic model

MSC classification

Primary: 35K57: Reaction-diffusion equations 35R35: Free boundary problems
Secondary: 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations 34K30: Equations in abstract spaces
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Issue 6 , December 2023 , pp. 1133 - 1169
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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