Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay

Zhou, Jiangbo; Song, Liyuan; Wei, Jingdong

doi:10.1016/j.jde.2019.10.034

Cited by 29 publications

(16 citation statements)

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“…As a matter of fact, we only obtained the existence of traveling wave solution for c > c * . Generally, for the case c = c * , Schauder's fixed point theorem with upper-lower-solutions is an appropriate method [4,8,22,21]. In the meantime, approximating argument is also a useful method to prove the existence of critical traveling waves, referring to [3,15,17].…”

Section: )mentioning

confidence: 99%

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Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment

San¹,

He²

2021

CPAA

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<p style='text-indent:20px;'>In this paper, we consider a two-group SIR epidemic model with bilinear incidence in a patchy environment. It is assumed that the infectious disease has a fixed latent period and spreads between two groups. Firstly, when the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_{0}>1 $\end{document}</tex-math></inline-formula> and speed <inline-formula><tex-math id="M2">\begin{document}$ c>c^{\ast} $\end{document}</tex-math></inline-formula>, we prove that the system admits a nontrivial traveling wave solution, where <inline-formula><tex-math id="M3">\begin{document}$ c^{\ast} $\end{document}</tex-math></inline-formula> is the minimal wave speed. Next, when <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_{0}\leq1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ c>0 $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{R}_{0}>1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ c\in(0,c^{*}) $\end{document}</tex-math></inline-formula>, we also show that there is no positive traveling wave solution, where <inline-formula><tex-math id="M8">\begin{document}$ k = 1,2 $\end{document}</tex-math></inline-formula>. Finally, we discuss and simulate the dependence of the minimum speed <inline-formula><tex-math id="M9">\begin{document}$ c^{\ast} $\end{document}</tex-math></inline-formula> on the parameters.</p>

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Section: )mentioning

confidence: 99%

“…For discrete version, we can look through e.g. [3,9,14,15,18,21]. This paper is organized as follows.…”

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Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment

San¹,

He²

2021

CPAA

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“…If β ≤ γ or c < c * , then (1.5) has no nontrivial positive bounded traveling wave solutions. For other progress of discrete diffusion epidemic models, see [5,11,24,30,36,42].…”

Section: Introductionmentioning

confidence: 99%

“…We should point out that the difference-differential epidemic models in the existing references [5,11,24,26,30,36,42] are two-component systems, while (1.1) is indeed a threecomponent system and we need to overcome some difficulties. Due to the deficiency of monotonicity for (1.1), it is hard to obtain the exact boundaries of S-component and R-component at plus infinity.…”

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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“…In contrast to the non-critical traveling wave solutions, the existence problem of critical traveling wave solutions is more difficult in various models. In past years, using limiting arguments or Schauder's fixed point theorem, there were some literature devoting to investigate the existence of critical traveling wave solutions of different models, see e.g., [2,3,7,8,10,13,14]. Applying the limiting arguments, it is sufficient to consider the convergence problem for a sequence of traveling wave solutions with wave speed greater than the critical speed.…”

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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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Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay

Cited by 29 publications

References 50 publications

Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment

Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment

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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