Nonlocal Dispersal Epidemic Model Research Articles

Traveling wave solutions in a nonlocal dispersal SIR epidemic model with nonlocal time-delay and general nonlinear incidences

This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed [Formula: see text] and the basic reproduction number [Formula: see text] are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder’s fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when [Formula: see text], for every speed [Formula: see text] there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any [Formula: see text] when [Formula: see text] or [Formula: see text] when [Formula: see text]. Finally, we analyze the effects of nonlocal time delay on the minimum wave speed.

Dynamics and calculation of the basic reproduction number for a nonlocal dispersal epidemic model with air pollution.

In order to reflect the dispersal of pollutants in non-adjacent areas and the large-scale movement of individuals, this paper proposes an epidemic model of nonlocal dispersal with air pollution, where the transmission rate is related to the concentration of pollutants. This paper checks the uniqueness and existence of the global positive solution and defines the basic reproduction number, . We simultaneously explore the global dynamics: when , the disease-free stable point is global asymptotic stability; when , the disease is uniformly persistent. Additionally, in order to approximate , a numerical method has been introduced. Illustrative examples are used to verify the theoretical outcomes and show the effect of the dispersal rate on the basic reproduction number .

Existence of traveling waves for a nonlocal dispersal SIR epidemic model with treatment

This paper is concerned with the existence and nonexistence of traveling wave solutions for a nonlocal dispersal epidemic model with treatment. The existence of traveling wave solutions is established by Schauder's fixed point theorem, while the nonexistence of traveling wave solutions is proved by two-sided Laplace transform. From the results, we conclude the minimal wave speed, which is an important threshold to predict how fast the disease invades. Compared with the work in [35], we obtain more accurate results about the existence and nonexistence of nontrivial traveling wave solutions. We prove that when the basic reproduction number R0>1, there exists a critical number c1⁎>0 such that for each c>c1⁎, the system has a nontrivial traveling wave solution with speed c, while for 0<c<c1⁎ the system admits no nontrivial traveling wave solution. When R0<1, we show that there exists no nontrivial traveling wave solution. In addition, based on [24], we obtain the existence of traveling waves with the critical speed c=c1⁎ under certain conditions.

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

In this paper, we investigate the existence and nonexistence of traveling wave solutions in a nonlocal dispersal epidemic model with spatio-temporal delay. It is shown that this model admits a nontrivial positive traveling wave solution when the basic reproduction number $ R_0&gt;1 $ and the wave speed $ c\geq c^* $ ($ c^* $ is the critical speed) and this model has no traveling wave solutions when $ R_0\leq1 $ or $ c&lt;c^* $. This indicates that $ c^* $ is the minimal wave speed.

Entire solutions originating from multiple fronts of an epidemic model with nonlocal dispersal and bistable nonlinearity

This paper is concerned with the entire solutions of a nonlocal dispersal epidemic model which arises from the spread of fecally–orally transmitted diseases. Under bistable assumptions, it is well-known that this model has three different types of traveling wave fronts. The annihilating-front and merging-front entire solutions originating from two fronts of the system have also been constructed in [38]. We first prove the uniqueness, Liapunov stability and continuous dependence on shift parameters of annihilating-front entire solutions obtained in [38]. A positive time-derivative estimate for such entire solution is also obtained. Then, we establish the existence of two different types of entire solutions merging three different fronts. Furthermore, we show that these entire solutions are global Lipschitz continuous with respect to the spatial variable x. To the best of our knowledge, it is the first time that the entire solutions originating from three fronts of diffusion systems have been constructed.

Exponential stability of traveling waves in a nonlocal dispersal epidemic model with delay

In this paper, we are concerned with the stability of traveling waves of a nonlocal dispersal epidemic model with delay. In the quasi-monotone case, we prove the exponential stability of traveling wavefronts by the weighted-energy method and the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as x → − ∞ , but can be arbitrarily large in other locations. In the non-quasi-monotone case, we investigate the exponential stability of traveling waves when the initial perturbation around the traveling wave is properly small in a weighted norm.

Multi-type Entire Solutions in a Nonlocal Dispersal Epidemic Model

This paper deals with entire solutions of a nonlocal dispersal epidemic model. Unlike local (random) dispersal problems, a nonlocal dispersal operator is not compact and the solutions of nonlocal dispersal system studied here lack regularity in suitable spaces, which affects the uniform convergence of the solution sequences and the technique details in constructing the entire solutions. In the monostable case, some new types of entire solutions are constructed by combining leftward and rightward traveling fronts with different speeds and a spatially independent solution. In the bistable case, the existence of many different entire solutions with merging fronts are proved by constructing different sub- and super-solutions. Various qualitative features of the entire solutions are also investigated. A key idea is to characterize the asymptotic behaviors of the traveling wave solutions at infinite in terms of appropriate sub- and super-solutions. Finally, we also obtain the smoothness of the entire solutions in space, i.e., the solutions established in our paper are global Lipschitz continuous in space.