Propagation dynamics of an influenza transmission model with nonlocal dispersal

Xuerui Li; Boyi Wang; Yuanyuan Wu; Xuerui Li; Boyi Wang; Yuanyuan Wu

doi:10.3934/math.2025952

AIMS Mathematics

2025, Volume 10, Issue 9: 21422-21451. doi: 10.3934/math.2025952

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Research article

Propagation dynamics of an influenza transmission model with nonlocal dispersal

1.
Department of Statistics, Beijing Normal University, Beijing 100875, China
2.
Department of Mathematics, Pennsylvania State University, 54 McAllister Street, Pennsylvania, USA
3.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received: 07 June 2025 Revised: 30 August 2025 Accepted: 08 September 2025 Published: 16 September 2025
MSC : 35K57, 35R20, 92D25

This paper investigated the existence and nonexistence of traveling wave solutions for a nonlocal dispersal influenza transmission model with human mobility. We established the existence of nonnegative entire solutions by combining the upper-lower solution method with Schauder's fixed point theorem. Appropriate Lyapunov functionals were constructed to determine the asymptotic behavior of solutions at $ +\infty $. Due to the influence of the nonlocal dispersal operator, the asymptotic behavior at $ -\infty $ for the critical wave speed could not be directly established via the Hartman-Grobman theorem. Through careful analysis of the wave equation, we overcame this difficulty and established the desired asymptotic properties. Finally, we used numerical simulations to verify the existence of the traveling wave solution, and compared the effects of the nonlocal dispersal pattern and the local dispersal pattern on the wave speed.
- influenza transmission models,
- nonlocal dispersal,
- traveling wave solutions,
- existence,
- fixed point theorem
Citation: Xuerui Li, Boyi Wang, Yuanyuan Wu. Propagation dynamics of an influenza transmission model with nonlocal dispersal[J]. AIMS Mathematics, 2025, 10(9): 21422-21451. doi: 10.3934/math.2025952

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Abstract

This paper investigated the existence and nonexistence of traveling wave solutions for a nonlocal dispersal influenza transmission model with human mobility. We established the existence of nonnegative entire solutions by combining the upper-lower solution method with Schauder's fixed point theorem. Appropriate Lyapunov functionals were constructed to determine the asymptotic behavior of solutions at $ +\infty $. Due to the influence of the nonlocal dispersal operator, the asymptotic behavior at $ -\infty $ for the critical wave speed could not be directly established via the Hartman-Grobman theorem. Through careful analysis of the wave equation, we overcame this difficulty and established the desired asymptotic properties. Finally, we used numerical simulations to verify the existence of the traveling wave solution, and compared the effects of the nonlocal dispersal pattern and the local dispersal pattern on the wave speed.

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