Quasi traveling waves with quenching in a reaction-diffusion equation in the presence of negative powers nonlinearity

On global behavior of a some SIR epidemic model based on the Poincaré compactification

“…In this section, we briefly introduce the Poincaré compactification. Here Section 2 of [5,6,8] are reproduced. Also, it should be noted that we refer [3,9,10] for more details.…”

“…An overview of Poincaré compactification is given later in Section 2, which is one of the compactifications of the phase space (the embedding of R n into the unit upper hemisphere of R n+1 ) (see, e.g. [3,5,6,8,9,10]). The most important feature of applying this method is that it allows us to investigate the global behavior of the system of ODEs of interest by revealing all its dynamics including infinity.…”

“…The most important feature of applying this method is that it allows us to investigate the global behavior of the system of ODEs of interest by revealing all its dynamics including infinity. This method has been used, for instance, in the analysis of the Liénard equation ( [3] and references therein), in the classification of phase portraits of ODE systems derived from the Gray-Scotte model ( [2]), in the reconsideration of blowup solutions of systems of ODEs in the view of dynamical systems ( [9,10]), and in the analysis of the behavior of characteristic and typical solutions of certain partial differential equations ( [5,6,8]). To the best of the author's knowledge, there are no studies of its application to investigate the global behavior of solutions to ODEs related to infectious disease epidemics, such as the susceptible-infectious-recovered (SIR) model.…”

Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations

“…Since this system of ODEs has φ −1 , it turns out to have a singularity at φ = 0, which is not easy to analyze about its dynamics. However, as shown in [13,14,15,20,21], it is possible to study the dynamics of these ODEs to infinity in the framework that combines Poincaré compactification (for instance, see Section 2 of [13] and [8,14,15,20,21] for the details of it) and classical dynamical systems theory. Note that, unlike the literature cited above, we do not use blow-up technique in our analysis, and therefore do not include it in this framework.…”

“…Remark 1. In the above definition, see Definition 1 in [13] and Definition 2.1 in [11] for the definition of a quasi traveling wave of (1) on a semi-infinite interval. Furthermore, note that although this definition is similar to a quasi traveling wave with quenching as defined in these papers, the meaning of singularity is different.…”