Thresholds and travelling waves in an epidemic model for rabies

Källén, Anders

doi:10.1016/0362-546x(84)90107-x

Cited by 60 publications

(42 citation statements)

References 3 publications

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“…While there is a rich literature on spatial epidemic models with delays or integral forms (see [27] for a brief review), several previous works have studied traveling wave solutions of basic diffusive Kermack-McKendrick SIR models. Källén [15] and Hosono and Ilyas [11] considered the existence of a traveling wave with mass-action incidence.…”

Section: Kermack-mckendrick Sir Model With Standard Incidencementioning

confidence: 99%

Traveling Wave Phenomena in a Kermack–McKendrick SIR Model

Wang

2015

J Dyn Diff Equat

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We study the existence and nonexistence of traveling waves of general diffusive Kermack-McKendrick SIR models with standard incidence where the total population is not constant. The three classes, susceptible S, infected I and removed R, are all involved in the traveling wave solutions. We show that the minimum speed for the existence of traveling waves for this three-dimensional non-monotonic system can be derived from its linearizaion at the initial disease-free equilibrium The proof in this paper is based on Schauder fixed point theorem and Laplace transform and provides a promising method to deal with high dimensional epidemic models.

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Section: Kermack-mckendrick Sir Model With Standard Incidencementioning

confidence: 99%

Traveling Wave Phenomena in a Kermack–McKendrick SIR Model

Wang

2015

J Dyn Diff Equat

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“…We note that the reaction term f in the existing models [16,18,7,21,20,19] satisfies the assumptions (A1)-(A4). Hence our result establishes the existence and uniqueness of traveling wave solutions to the reaction-diffusion system associated with the reactions (1.2)-(1.3) (i.e., the system (1.1) with f (u) = k 1 u and K = k 2 ) for an isothermal autocatalytic chemical reaction of any order in which the autocatalyst is assumed to decay to the inert product at a rate of the same order.…”

mentioning

confidence: 99%

“…The existence of traveling wave solutions of (1.1) has been investigated by several authors. When d 1 = 0 and m = 1, the problem (1.4)-(1.5) can be reduced to a system of two first order ordinary differential equations, and Källén [16] and Kennedy [18] have used the phase-plane analysis to solve the existence and uniqueness of solutions of the problem (1.4)-(1.5) for the particular choice f (u) = u and f (u) = m 1 u m 2 /(m 3 + u m 2 ) with some positive constants m 1 , m 2 , and m 3 , respectively. For sufficiently small d 1 > 0 and m = 1, Smith and Zhao [25] have employed the so-called geometric singular perturbation theory to treat the case when f (u) = m 1 u/(m 2 + u) for some positive constants m 1 and m 2 .…”

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confidence: 99%

Traveling waves of two-component reaction-diffusion systems arising from higher order autocatalytic models

Guo

Tsai

2009

Quart. Appl. Math.

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Abstract. We study the existence and uniqueness of traveling wave solutions for a class of two-component reaction-diffusion systems with one species being immobile. Such a system has a variety of applications in epidemiology, bio-reactor models, and isothermal autocatalytic chemical reaction systems. Our result not only generalizes earlier results of Ai and Huang (Proceedings of the Royal Society of Edinburgh 2005; 135A:663-675), but also establishes the existence and uniqueness of traveling wave solutions to the reactiondiffusion system for an isothermal autocatalytic chemical reaction of any order in which the autocatalyst is assumed to decay to the inert product at a rate of the same order.

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“…The existence of traveling wave solutions of (1.1) for some special choices of functions f and K > 0 has been studied by numerous authors. When d 1 = 0 and m = 1, Kennedy and Aris [21] and Källén [19] have solved the problem (1.4)-(1.5) for the special cases f (u) = u and 3). Recently, Huang [17] (see also Ai and Huang [1]) has developed a method to deal with the problem (1.4)-(1.5) with arbitrary positive constants d 1 and d 2 for a general class of functions f which include all the reaction terms f mentioned above.…”

mentioning

confidence: 99%

“…We conclude the introduction with three remarks. The first remark is that the reaction terms f (u) in the above-mentioned models [22,3,21,19,26,27,29,4] are smooth and increasing in u, and f (0) = 0 and f (0) > 0. Hence f in these models satisfies the assumptions (A1)-(A3).…”

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confidence: 99%

Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

Tsai

2010

Quart. Appl. Math.

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Abstract. The reaction-diffusion systems which are based on an isothermal autocatalytic chemical reaction involving both an autocatalytic step of the (m + 1)th order (A + mB → (m + 1)B) and a decay step of the same order (B → C) have very rich and interesting dynamics. Previous studies in the literature indicate that traveling waves play a key role in understanding these interesting dynamical phenomena. However, there is a lack of rigorous proof of the existence of traveling waves to this system. Here we generalize this isothermal autocatalytic chemical reaction model and provide a rigorous proof of the existence of traveling waves for the resulting reaction-diffusion system which also includes the systems arising from epidemiology and the microbial growth in a flow reactor. Introduction.Since the pioneering works of Fisher [9] and of Kolmogorov, Petrovsky and Piskunov [23], traveling waves in a wide array of biological and chemical systems, based on the interaction between reaction and diffusion processes, have been extensively investigated. In many realistic systems, traveling waves can be initiated by a highly localized disturbance of a stable rest state (e.g., a highly localized increase in ion concentrations). A typical example is the Hodgkin-Huxley model [13] which describes the electrical activity across membranes of never cells. It is known that if a short, but sufficiently large, burst of simulating current is injected into a nerve axon, a so-called action potential wave can then be generated. In this paper, we will discuss a system which enjoys a similar phenomenon. Specifically, this system is governed by the following equations:

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Thresholds and travelling waves in an epidemic model for rabies

Cited by 60 publications

References 3 publications

Traveling Wave Phenomena in a Kermack–McKendrick SIR Model

Traveling Wave Phenomena in a Kermack–McKendrick SIR Model

Traveling waves of two-component reaction-diffusion systems arising from higher order autocatalytic models

Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

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