Traveling Wave Solutions for a Class of Discrete Diffusive SIR Epidemic Model

Zhang, Ran; Wang, Jinliang; Liu, Shengqiang

doi:10.1007/s00332-020-09656-3

Cited by 30 publications

(9 citation statements)

References 45 publications

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“…Recently, such open problem was solved by constructing suitable Lyapunov functional. 26,27 One of the most important idea to show that the persistence of traveling wave solutions was proposed in Chen et al, 25 which will use a fundamental theory proposed in Chen and Guo. 36 But for our model has coupled linearized subsystem, we did not find that we can apply the theory in Chen and Guo 36 directly.…”

Section: Discussionmentioning

confidence: 99%

“…Note that system (1.4) lacks of external supply, Chen et al 25 studied the traveling wave solutions of a more complex situation of system (1.4) by introducing constant recruitment; they found that such kind of solutions is connecting the disease-free steady state to another positive steady state, but whether the solutions converge to the positive equilibrium at +∞ is left as open problem. Recently, the above open problem was solved by constructing a suitable Lyapunov functional in Zhang et al 26,27 Based on the above facts, we focus on the undermentioned discrete diffusive epidemic model with exposed class:…”

Section: Introductionmentioning

confidence: 99%

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Traveling waves for a four‐compartment lattice epidemic system with exposed class and standard incidence

Zhang

2021

Math Methods in App Sciences

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This paper is considering the problem of traveling wave solutions (TWS) for a susceptible-exposed-infectious-recovered (SEIR) epidemic model with discrete diffusion. The threshold condition for the existence and nonexistence of TWS is obtained. More specifically, such kind of solutions are governed by the threshold number ℜ 0 . We can find a critical wave speed c * if ℜ 0 > 1, by employing the Schauder's fixed point theorem, limiting argument and two-sided Laplace transform, we confirm that there exists TWS for c > c * , while there exists no TWS for c < c * . We also obtain the nonexistence of TWS for ℜ 0 ≤ 1. At last, we give some biological explanations from the epidemiological perspective.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Traveling waves for a four‐compartment lattice epidemic system with exposed class and standard incidence

Zhang

2021

Math Methods in App Sciences

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“…With the help of Lemma 3.2 and following from the standard arguments in [22], we can conclude that (S, V, I) is solution for system (2.4) with…”

Section: Existence Of Traveling Wave Solutionsmentioning

confidence: 83%

“…Chen et al shown that system (1.2) admits TWS when ℜ 0 > 1 and c ≥ c * . More recently, the TWS for (1.2) was proved to be converged to the endemic equilibrium by Zhang et al [21,22]. Model (1.2) is an SIR model with constant recruitment (i.e.…”

Section: Introductionmentioning

confidence: 99%

Wave propagation for a discrete diffusive vaccination epidemic model with bilinear incidence

Zhang¹,

Liu²

2022

Preprint

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The aim of the current paper is to study the existence of traveling wave solutions (TWS) for a vaccination epidemic model with bilinear incidence. The existence result is determined by the basic reproduction number ℜ 0 . More specifically, the system admits a nontrivial TWS when ℜ 0 > 1 and c ≥ c * , where c * is the critical wave speed. We also found that the TWS is connecting two different equilibria by constructing Lyapunov functional. Lastly, we give some biological explanations from the perspective of epidemiology.

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“…The importance of TWS makes many people devote themselves to the study of its existence with the help of various methods. Some well-known methods include the Schauder's fixed-point theorem [11,22,25,27,34,35,39,41,40], monotone iteration technique [36,37,42], shooting method [12,13,14,23], Conley index [19,20] and so on. However, there is another useful but little noticed approach, namely, the geometric singular perturbation (GSP) theory [18,24], which is mainly suitable for systems with small parameters [31].…”

mentioning

confidence: 99%

Geometric singular perturbation of a nonlocal partially degenerate model for Aedes aegypti

Wang

Zhao

Wang

2023

DCDS-B

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This paper is devoted to investigate the existence of traveling wave solutions for a partially degenerate Aedes aegypti model with nonlocal effects. By taking specific kernel forms and time scale transformation, we transform the nonlocal model into a singularly perturbed system with small parameters. A locally invariant manifold for wave profile system is obtained with the aid of the geometric singular perturbation theory, and then the existence of traveling wave solutions is proved provided that the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_0>1 $\end{document}</tex-math></inline-formula> through utilizing the Fredholm orthogonal method. Furthermore, we study the asymptotic behaviors of traveling wave solution with the help of asymptotic theory. The methods used in this work can help us overcome the difficulty that the solution map associated with the system is not compact. Numerically, we perform simulations to demonstrate the theoretical results.

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Traveling Wave Solutions for a Class of Discrete Diffusive SIR Epidemic Model

Cited by 30 publications

References 45 publications

Traveling waves for a four‐compartment lattice epidemic system with exposed class and standard incidence

Traveling waves for a four‐compartment lattice epidemic system with exposed class and standard incidence

Wave propagation for a discrete diffusive vaccination epidemic model with bilinear incidence

Geometric singular perturbation of a nonlocal partially degenerate model for <i>Aedes aegypti</i>

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