Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate

Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay

“…To derive the existence of S(∞), we'll apply the method of reduction to absurdity. Assume that lim sup Utilizing the similar arguments in [40,46,49], one can deduce that S(ξ n + y) → ρ 1 and S(η n + y) → ρ 2 as n → ∞ for arbitrary y ∈ Together with (1.8)-(1.10), we conclude that S (±∞) = I (±∞) = R (±∞) = 0. The proof is completed.…”

“…In diffusive epidemic models, traveling waves can describe the state that a disease spreads geographically with a constant speed. The existence of traveling waves in these models has become one of the important issues in mathematical epidemiology [1,5,[7][8][9]11,11,12,14,20,[27][28][29][30][31][32][33]35,36,38,38,[40][41][42][43][44][45][46][48][49][50]. For example, Wang et al [30] where S(x, t), I(x, t) and R(x, t) refer to the densities of susceptible, infected and recovered individuals at location x and time t, respectively.…”

Wave propagation in a infectious disease model with non-local diffusion

“…Applying Theorems 2.3, 3.1 and 3.2, we obtain the following result. It is known that traveling wave solutions of nonlinear partial differential equations have been extensively studied due to significant applications in many fields [1,2,33,34,37,[55][56][57]. In the present paper, we have studied the traveling wave solutions of a nonlocal diffusion epidemic model with nonlinear incidences and distributed delay.…”

“…to model the propagation of the epidemic. In (1.5), the integral operators J * s(x, t)s(x, t) and J * i(x, t)i(x, t) mean that the rate of susceptible and infected individuals in position x and at time t depend on the influence of neighboring s(x, t) and i(x, t) in all other positions y [56,57]. 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].…”

“…Throughout this paper, we always assume that f (τ ) satisfies (1.3), the nonlinear function g and the kernel function J satisfy (A1) g(0) = 0, g (i) > 0 and g (i) ≤ 0 for i ≥ 0; (A2) g(i)/i is continuously differentiable, nonincreasing for i > 0 and lim i→∞ g(i)/i = 0; (A3) J ∈ C 1 (R), J(y) = J(-y) ≥ 0, R J(y) dy = 1, J is compactly supported; (A4) lim λ→∞ λ -1 R J(y)e -λy dy = ∞ and R J(x)e λx dx < +∞ for all λ > 0. Assumptions (A1)-(A4) have been used in the literature, one can refer to [3,17,26,49,50,56,57]. The aim of the current paper is to study the existence and nonexistence of the nontrivial and nonnegative traveling wave solutions of the form…”