"At that time infections in Osaka were increasing rapidly."
- "At that time infections in Osaka were increasing rapidly."
"at that time" 「そのころ」
"infections" 「感染者数」"the number of infections" も言えます。
"in Osaka" 「大阪で」
"were increasing rapidly" 「急激に〜増えてきていた」"were rapidly increasing" や "were suddenly increasing" も使えます。
"At that time infections in Osaka were increasing rapidly, so I had to refrain from playing tennis."
「そのころ大阪で急激に感染者数が増えてきていたのでテニスを自粛しなくてはならなかった」
At that time infections were spreading sharply in Osaka.
既に回答挙がっていますが、少しだけ別の言い方も紹介します(*^_^*)
At that time infections were spreading sharply in Osaka.
「その頃感染が大阪で急速に広がっていた」
spread「広がる」
sharply「急激に」
以上です。
少しでも参考になれば幸いです(*^_^*)