Traveling wave solutions for a nonlocal dispersal SIR model with delayed transmission

Ashley Evette Embry; William Kyle Barker; Ashley Evette Embry; William Kyle Barker

doi:10.3934/math.2026527

AIMS Mathematics

2026, Volume 11, Issue 5: 12795-12824. doi: 10.3934/math.2026527

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Traveling wave solutions for a nonlocal dispersal SIR model with delayed transmission

Ashley Evette Embry ,
William Kyle Barker ^,

Department of Mathematics and Statistics, University of Arkansas at Little Rock, 2801 S. University Ave, Little Rock, AR 72204, USA

Received: 31 December 2025 Revised: 24 March 2026 Accepted: 30 March 2026 Published: 08 May 2026
MSC : 35C07, 35K57

We study the existence of traveling wave solutions for a spatial susceptible-infected-recover (SIR) epidemic model with nonlocal dispersal and delayed transmission. The model incorporates convolution-type dispersal operators and a nonlocal time-delay incidence mechanism, which together lead to a non-cooperative and non-monotone traveling wave system. To overcome these difficulties, we construct an invariant cone on a large bounded interval and define a suitable integral operator associated with the traveling wave equations. Uniform a priori bounds and regularity estimates are established independently of the truncation parameter. By applying Schauder's fixed point theorem and a limiting argument, we obtain the existence of nontrivial traveling wave solutions connecting the disease-free equilibrium to the endemic equilibrium when the basic reproduction number exceeds one. Explicit upper and lower solutions are constructed to illustrate the applicability of the approach.
- traveling waves,
- reaction-diffusion equations,
- nonlocal dispersal,
- delay,
- SIR model
Citation: Ashley Evette Embry, William Kyle Barker. Traveling wave solutions for a nonlocal dispersal SIR model with delayed transmission[J]. AIMS Mathematics, 2026, 11(5): 12795-12824. doi: 10.3934/math.2026527

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Abstract

We study the existence of traveling wave solutions for a spatial susceptible-infected-recover (SIR) epidemic model with nonlocal dispersal and delayed transmission. The model incorporates convolution-type dispersal operators and a nonlocal time-delay incidence mechanism, which together lead to a non-cooperative and non-monotone traveling wave system. To overcome these difficulties, we construct an invariant cone on a large bounded interval and define a suitable integral operator associated with the traveling wave equations. Uniform a priori bounds and regularity estimates are established independently of the truncation parameter. By applying Schauder's fixed point theorem and a limiting argument, we obtain the existence of nontrivial traveling wave solutions connecting the disease-free equilibrium to the endemic equilibrium when the basic reproduction number exceeds one. Explicit upper and lower solutions are constructed to illustrate the applicability of the approach.

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