Propagation direction of traveling waves for a class of bistable epidemic models
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Mathematical Biology
Propagation direction of traveling waves for a class of bistable epidemic models Je-Chiang Tsai1,2
· Yu-Yu Weng1
Received: 28 November 2019 / Revised: 5 September 2020 / Accepted: 16 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Traveling waves of a reaction–diffusion (RD) system connecting two spatially uniform stable equilibria are termed as bistable waves. Due to the uniqueness of a bistable wave in RD systems, it is difficult to determine its propagation direction, and there are very few analytical results on this subject. In this study, we propose an approach to give a complete characterization of the propagation direction of bistable waves for a class of bistable epidemic models arising from the spread of a cholera epidemic. Moreover, this characterization also gives a parameter threshold above which the epidemic disease eventually tends to extinction, and below which the epidemic outbreak happens. Keywords Bistable traveling waves · Wave speed · Propagation direction Mathematics Subject Classification 34A34 · 34A12 · 35K57
1 Introduction Traveling waves have been the subject of mathematical studies for many years, and have a number of applications in physical systems, chemical reactions, and biological processes (Murray 2004a, b). For instance, since the works by Fisher (1937) and Kolmogorov et al. (1937), waves of the Fisher equation, which was originally proposed to describe the spread of advantageous genes, have received continuous attention in population dynamics.
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Je-Chiang Tsai [email protected] Yu-Yu Weng [email protected]
1
Department of Mathematics, National Tsing Hua University, No. 101, Sec. 2, Kuang-Fu Road, Hsinchu 300, Taiwan
2
National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4 Roosevelt Rd., Taipei 106, Taiwan
123
J.-C. Tsai, Y.-Y. Weng
From a practical point of view, a good understanding of the propagation direction of traveling waves is necessary. For example, during the disease spread, if the propagation direction of waves of the infectious agent could be reversed, then the disease will decline and eventually die out. A model of wave propagations is often formularized as a reaction–diffusion (RD) system. RD systems which possess traveling waves fall into two major categories: monostable RD systems, and bistable RD systems. Monostable RD systems have two spatially uniform equilibria: one is stable and the other is unstable. Hence waves of monostable RD systems always propagate in the way that the unstable equilibrium is ahead of the wave front and the system tends to the stable equilibrium. One typical example of such monostable RD systems is the Fisher equation (Fisher 1937; Kolmogorov et al. 1937). On the other hand, bistable RD systems in general have three spatially uniform equilibria: two of which are stable and the other is unstable. The typical example is the Nagumo equation which arises in nerve conduction (Fife and McLeod 1977). Waves of bistable RD systems connecting two
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