Abstract
In this paper, we propose a nonlocal diffusion infectious disease model with nonlinear incidences and distributed delay to model the transmission of the epidemic. By a fixed point theorem and a limiting argument, we establish the existence of traveling wave solutions for the model. Meanwhile, we obtain the non-existence of traveling wave solutions for the model via two-sided Laplace transform. It is found that the threshold dynamics of traveling wave solutions are entirely determined by the basic reproduction number of the corresponding spatially-homogenous delayed differential system and the minimum wave speed. A typical example is given for supporting our abstract results. Moreover, the effect of the diffusive rate of the infected individuals on the minimum wave speed is discussed.
Highlights
With the rapid development of modern mathematical epidemiology, many mathematical models have been proposed to describe the transmission of communicable diseases
In the present paper we propose a nonlocal diffusion infectious disease model with nonlinear incidences and distributed delay
By introducing an auxiliary system and applying the Schauder’s fixed point theorem together with a limiting argument, Bai and Zhang [3] obtained the existence of a traveling wave solution for their model directly
Summary
With the rapid development of modern mathematical epidemiology, many mathematical models have been proposed to describe the transmission of communicable diseases. Model (1.5) with nonlinear incidences and distributed delay describes the spread of an infectious disease (involving only susceptible and infected individuals) transmitted by a vector (e.g., mosquitoes) after a latent period [39]. By introducing an auxiliary system and applying the Schauder’s fixed point theorem together with a limiting argument, Bai and Zhang [3] obtained the existence of a traveling wave solution for their model directly. 2.4 Existence of solutions of (1.7) on R Choose a positive increasing constant sequence {ln}∞ n=1 such that ln > ρ and limn→∞ ln = ∞. Integrating the second equation in (1.7) over R and applying Fubini theorem and i(±∞) = 0, we deduce β s(z) f (τ )g i(z – cτ ) dτ dz = γ i(z) dz
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