Traveling waves in a nonlocal dispersal SIR epidemic model

Wave propagation in a diffusive SEIR epidemic model with nonlocal reaction and standard incidence rate

“…There exists a positive constant c * such that for any S 0 > 0, if R 0 ∶= + > 1 and c > c * , then there exist a constant S ∞ depending on S 0 and a nontrivial and nonnegative traveling wave solution (S, E, I, R) for the system (16) to (19)…”

“…Finally, it follows from (16) to (19) that S ′′ ( − ∞) = 0, E ′′ ( − ∞) = 0, I ′′ ( − ∞) = 0 and R ′′ ( − ∞) = 0. Now, we are ready to investigate the asymptotic behavior of S( ), E( ), I( ) and R( ) as → + ∞.…”

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

“…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…”

“…Moreover, other traditional methods such as the method of monotone iteration together with upper-lower solutions [45] and the shooting method [18] cannot be applied any more. To obtain the existence theorem for (1.5), we construct an invariant cone of initial functions defined on a large spatial domain and apply Schauder's fixed point theorem on this cone and a limiting argument (motivated by [30,42,49,50,54]) to establish the existence of a solution for the wave system. Note also that in [3], the authors investigated the nonexistence of the solutions for their model for the cases R 0 < 1 and R 0 = 1, simultaneously.…”

Traveling waves in an SEIR epidemic model with the variable total population

“…On the other hand, there is no traveling wave solution for (1.2) when βS 0 ≤ γ. There are substantial recent developments on the existence and non-existence of traveling wave solutions for Kermack-Mckendrick SIR model, see [1,2,7,12,17,19,20,24,25,27,28] and references therein. In particular, Wang and Wang [17] introduced the following general diffusive Kermack − γI(t, x) − δI(t, x), ∂ ∂t R(t, x) = d 3 ∆R(t, x) + γI(t, x), (1.3) in which N = S + I + R is the total population, d 1 , d 2 and d 3 are the diffusion rates of the susceptible (S), infective (I) and recovered (R) individuals, respectively.…”