Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold

Li, Wan–Tong; Lin, Guo; Ma, Cong; Yang, Fei-Ying

doi:10.3934/dcdsb.2014.19.467

Cited by 36 publications

(27 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…considered the diffusive SIR epidemic model with the standard incidence rate

\{\begin{matrix} \frac{\partial S (x, t)}{\partial t} = d_{1} \frac{\partial^{2} S (x, t)}{\partial x^{2}} - \frac{β S (x, t) I (x, t)}{S (x, t) + I (x, t)}, \\ \frac{\partial I (x, t)}{\partial t} = d_{2} \frac{\partial^{2} I (x, t)}{\partial x^{2}} + \frac{β S (x, t) I (x, t)}{S (x, t) + I (x, t)} - γ I (x, t), \\ \frac{\partial R (x, t)}{\partial t} = d_{3} \frac{\partial^{2} R (x, t)}{\partial x^{2}} + γ I (x, t) . \end{matrix}

They also obtained the existence and non‐existence results of traveling wave solutions, which is related to the basic reproduction number. Li et al . further studied the traveling wave solutions for nonlocal delayed and nonlocal dispersal SIR model with this incidence rate.…”

Section: Introductionmentioning

confidence: 99%

Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

Zhou

Han

Wang

2016

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we study the traveling wave solutions of a delayed diffusive SIR epidemic model with nonlinear incidence rate and constant external supplies. We find that the existence of traveling wave solutions is determined by the basic reproduction number of the corresponding spatial‐homogenous delay differential system and the minimal wave speed. The existence is proved by applying Schauder's fixed point theorem and Lyapunov functional method. The non‐existence of traveling waves is obtained by two‐sided Laplace transform. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

“…considered the diffusive SIR epidemic model with the standard incidence rate

\{\begin{matrix} \frac{\partial S (x, t)}{\partial t} = d_{1} \frac{\partial^{2} S (x, t)}{\partial x^{2}} - \frac{β S (x, t) I (x, t)}{S (x, t) + I (x, t)}, \\ \frac{\partial I (x, t)}{\partial t} = d_{2} \frac{\partial^{2} I (x, t)}{\partial x^{2}} + \frac{β S (x, t) I (x, t)}{S (x, t) + I (x, t)} - γ I (x, t), \\ \frac{\partial R (x, t)}{\partial t} = d_{3} \frac{\partial^{2} R (x, t)}{\partial x^{2}} + γ I (x, t) . \end{matrix}

Section: Introductionmentioning

confidence: 99%

Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

Zhou

Han

Wang

2016

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…As we known that the application of the Laplace transform requires the prior estimate of the exponential decay of the traveling wave solutions [3,10,18,19]. However, it seems that the analytical method in [3,10,18,19] cannot give the prior estimate for the model because of the four dimensional system (3.1) again. Instead, we approve the approach recently introduced by [25] to get the prior estimate.…”

Section: Accepted Manuscriptmentioning

confidence: 96%

“…To prove the existence theorem (Theorem 4.1), we first introduce an auxiliary system (3.1) and employ Schauder fixed point theorem [8,9,12] to prove the existence of traveling wave solutions for the auxiliary system, and then by the limit arguments and Arzelà-Asocli's theorem, we extend the results for (3.1) to (1.2). Here, in order to apply Schauder fixed point theorem, we also use the idea of the iteration process [1,10,18,19] to construct the upper-lower solutions. One important feature of our method, which is different from the ones [1,10,18,19], is that we need to construct the vector-value upper-lower solutions for (3.1) (Section 3.1) since system (3.1) consists of four equations.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

“…Here, in order to apply Schauder fixed point theorem, we also use the idea of the iteration process [1,10,18,19] to construct the upper-lower solutions. One important feature of our method, which is different from the ones [1,10,18,19], is that we need to construct the vector-value upper-lower solutions for (3.1) (Section 3.1) since system (3.1) consists of four equations. The idea of such a construction is motivated by Weng and Zhao [21], see also [17] for the related works.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

See 1 more Smart Citation

Traveling waves in a diffusive influenza epidemic model with vaccination

2016

Applied Mathematical Modelling

View full text Add to dashboard Cite

Highlights• We derive a mathematical model of diffusive influenza disease with vaccination• We obtain the explicit formula of the basic reproduction number R 0 for the model.• We establish the existence of the traveling wave solutions for the model if R 0 > 1 and c > c * .• We further study the non-existence of the traveling wave solutions for the model when R 0 < 1 and c ≥ 0 or R 0 > 1 and c (0, c * ) .Abstract: In this paper, a mathematical model of influenza disease with vaccination is formulated to incorporate a spatially homogeneous structure. The explicit formula of the basic reproduction number R 0 for the model is obtained. It is shown that the existence and non-existence of the traveling wave solutions for the model are determined completely by the threshold value R 0 . Here, by introducing an auxiliary system and applying Schauder fixed theorem, we show that the auxiliary system admits a nontrivial traveling wave solution. And then, by the limit arguments and Arzelà-Ascli's theorem, we establish the existence of the traveling wave solutions for the model if R 0 > 1 and c > c * . Furthermore, by the two-sided Laplace transform, we show that the model has no nontrivial traveling wave solutions when R 0 > 1 and c ∈ [0, c * ). In addition, we further study the non-existence of the traveling wave solutions for the model when R 0 < 1 and c ≥ 0.

show abstract

“…As we know, since Bartlett [1] in 1956 predicted a wave of infection moving out from the initial source of infection with a constant speed, epidemic waves have been extensively studied, see for example [10,29,22,45,15,38,40,42,20,12,19,11,23,46]. See also the monograph [30], the survey papers [32,33] and the references therein for more details.…”

Section: Introductionmentioning

confidence: 98%

Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold

Cited by 36 publications

References 45 publications

Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

Traveling waves in a diffusive influenza epidemic model with vaccination

Traveling waves for a nonlocal dispersal SIR model with delay and external supplies

Contact Info

Product

Resources

About