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Journal Articles Journal of Mathematical Analysis and Applications Year : 2022

Traveling waves of a differential-difference diffusive Kermack-McKendrick epidemic model with age-structured protection phase

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Abstract

We consider a general class of diffusive Kermack-McKendrick SIR epidemic models with an age-structured protection phase with limited duration, for example due to vaccination or drugs with temporary immunity. A saturated incidence rate is also considered which is more realistic than the bilinear rate. The characteristics method reduces the model to a coupled system of a reaction-diffusion equation and a continuous difference equation with a time-delay and a nonlocal spatial term caused by individuals moving during their protection phase. We study the existence and non-existence of non-trivial traveling wave solutions. We get almost complete information on the threshold and the minimal wave speed that describes the transition between the existence and non-existence of non-trivial traveling waves that indicate whether the epidemic can spread or not. We discuss how model parameters, such as protection rates, affect the minimal wave speed. The difficulty of our model is to combine a reaction-diffusion system with a continuous difference equation. We deal with our problem mainly by using Schauder's fixed point theorem. More precisely, we reduce the problem of the existence of non-trivial traveling wave solutions to the existence of an admissible pair of upper and lower solutions.
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Dates and versions

hal-03518604 , version 1 (10-01-2022)

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Mostafa Adimy, Abdennasser Chekroun, Toshikazu Kuniya. Traveling waves of a differential-difference diffusive Kermack-McKendrick epidemic model with age-structured protection phase. Journal of Mathematical Analysis and Applications, 2022, 505 (1), pp.125464. ⟨10.1016/j.jmaa.2021.125464⟩. ⟨hal-03518604⟩
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