With any non-negligible immunity rate (for example, 70% would be typical for immunity gained through an influenza vaccine [0]), we will still see exponential decay once a critical mass (R0 < 1) is reached. I believe immunity rates are higher after recovering from real infections, but still not perfect.

Therefore a headline reading "Japanese man gets reinfected" is not of particular concern while "50% of survivors get reinfected" would be highly concerning.

[0] https://www.ecdc.europa.eu/en/seasonal-influenza/prevention-...