Seven to eight COVID-19 epidemic waves have been observed in Japan as of December 2022. By performing a wavelet analysis using publicly available data on the number of COVID-19 positive individuals reported and calculating the phase difference between the infection waves, the spatial and temporal characteristics of the spread of infection were examined. In Japan's prefectural analysis, the spread of infection from the large cities of Tokyo, and Osaka to the surrounding areas and to the Sanyo and Kyushu regions was indicated. On the other hand, it was found that the standard deviation of the phase difference in other prefectures was large across zero phase difference. The former is considered to represent the regularity involving with the spread of COVID-19, while the latter is considered to represent its stochastic aspect.

We present a minimal version of Landscape Evolution Models (LEMs) to capture the essence of tree-shaped channel network formation on the assumption of two mechanisms, soil erosion and diffusion. The continuous tectonic uplift is not required, which affects landscape evolution only at the starting point. We refer to the mathematical model as the Erosion-Diffusion Model (EDM). No steady state exists in the EDM except for the perfectly flat plane, which is realized at the ultimate final stage of the landform transition. It is suggested that generated channel patterns are temporal and transient creatures during long-term gravitational and erosional processes on the earth.