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

Wei, Jingdong; Zhou, Jiangbo; Chen, Wenxia; Zhen, Zaili; Li, Tian

doi:10.3934/cpaa.2020125

Cited by 7 publications

(4 citation statements)

References 45 publications

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“…A maximum point is reached until a certain quantity of the population is infected or recovered, and subsequently the infection fraction begins to decline. The traveling wave solutions to the diffusive epidemic models have been theoretically analyzed in [6,15,24,[28][29][30][31]. In Figure 6, we illustrate the traveling wave profiles of susceptible, infected and recovered fractions along the central horizontal line of the domain at different times.…”

Section: 2mentioning

confidence: 99%

“…To describe the epidemic spreading in space accurately, the diffusive SIR models have been attracted more attentions of researchers in recent years [6,15,24,[28][29][30][31]. Diffusion can produce traveling waves, the existence of which has been proved rigorously in [6,15,24,[28][29][30][31]. In 2021, Ramaswamy, Oberai and Yortsos [18] proposed a comprehensive temporal-spatial infection model that takes into consideration the influence of advection on the spread of the infectious disease in addition to diffusion.…”

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confidence: 99%

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A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

Kou¹,

Chen²,

Wang³

et al. 2023

CPAA

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In this paper, we propose an efficient numerical method for a comprehensive infection model that is formulated by a system of nonlinear coupling advection-diffusion-reaction equations. Using some subtle mixed explicitimplicit treatments, we construct a linearized and decoupled discrete scheme. Moreover, the proposed scheme is capable of preserving the positivity of variables, which is an essential requirement of the model under consideration. The proposed scheme uses the cell-centered finite difference method for the spatial discretization, and thus, it is easy to implement. The diffusion terms are treated implicitly to improve the robustness of the scheme. A semi-implicit upwind approach is proposed to discretize the advection terms, and a distinctive feature of the resulting scheme is to preserve the positivity of variables without any restriction on the spatial mesh size and time step size. We rigorously prove the unique existence of discrete solutions and positivity-preserving property of the proposed scheme without requirements for the mesh size and time step size. It is worthwhile to note that these properties are proved using the discrete variational principles rather than the conventional approaches of matrix analysis. Numerical results are also provided to assess the performance of the proposed scheme.

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Section: 2mentioning

confidence: 99%

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confidence: 99%

A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

Kou¹,

Chen²,

Wang³

et al. 2023

CPAA

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“…When 0 < R 0 ≤ 1 or 0 < c < c * , system (1.4) has no nontrivial nonnegative traveling waves. Very recently, Wei et al [25] investigated a nonlocal delayed version of model (1.4) and derived the existence of nontrivial positive traveling waves with super-critical and critical speeds. For more study of nonlocal diffusion epidemic systems, we refer to [2, 3, 9, 15-19, 21, 33, 35, 41, 44].…”

Section: Introductionmentioning

confidence: 99%

Traveling Waves for a Discrete Diffusion Epidemic Model with Delay

et al. 2021

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This paper is concerned with traveling wave solutions in a discrete diffusion epidemic model with delayed transmission. Employing the way of contradictory discussions and the bilateral Laplace transform, we obtain the nonexistence of nontrivial positive bounded traveling wave solutions. Utilizing the super-/sub-solutions method and the fixed point theory, we derive the existence of nontrivial positive traveling wave solutions with both super-critical and critical speeds. Our results indicate that the critical speed is the minimal speed.

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“…To prove the main theorem, we first construct a pair of upper-and lower-solutions of system (6) in this section, cf. [1,3,4,9,13]. Inspired by the construction of [15], we define the following functions:…”

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confidence: 99%

Critical traveling wave solutions for a vaccination model with general incidence

Yu¹,

Zhou²,

Hsu³

2022

DCDS-B

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This paper is concerned with the existence of traveling wave solutions for a vaccination model with general incidence. The existence or nonexistence of traveling wave solutions for the model with specific incidence were proved recently when the wave speed is greater or smaller than a critical speed respectively. However, the existence of critical traveling wave solutions (with critical wave speed) was still open. In this paper, applying the Schauder's fixed point theorem via a pair of upper-and lower-solutions of the system, we show that the general vaccination model admits positive critical traveling wave solutions which connect the disease-free and endemic equilibria. Our result not only gives an affirmative answer to the open problem given in the previous specific work, but also to the model with general incidence. Furthermore, we extend our result to some nonlocal version of the considered model.

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

Cited by 7 publications

References 45 publications

A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

Traveling Waves for a Discrete Diffusion Epidemic Model with Delay

Critical traveling wave solutions for a vaccination model with general incidence

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