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Asymptotic boundary and nonexistence of traveling waves in a discrete diffusive epidemic model. (English) Zbl 1475.37091

The author considers the lattice dynamical system given by \[ \begin{aligned} \frac{dS_j(t)}{dt}=d\left[S_{j+1}(t)-2S_j(t)+S_{j-1}(t)\right]-\beta S_j(t)I_j(t), \quad j\in\mathbb{Z}\\ \frac{dI_j(t)}{dt}=I_{j+1}(t)-2I_j(t)+I_{j-1}(t)+\left[\beta S_j(t)-\gamma\right]I_j(t),\quad j\in\mathbb{Z}. \end{aligned} \] First a set of conditions under which the system has a traveling wave solution is determined. Then the asymptotic boundary of the solutions is studied. The author also establishes the conditions that ensure the existence of nontrivial, positive and bounded traveling wave solutions. Moreover, the nonexistence of traveling wave solutions with non-positive wave speed is proved.
Reviewer: Fei Xue (Hartford)

MSC:

37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
37N25 Dynamical systems in biology
34A33 Ordinary lattice differential equations
46N60 Applications of functional analysis in biology and other sciences
92D30 Epidemiology
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