Enforcement Rules for the Issuance of ROC Visas to Foreign-Passport Holders Article 4

A foreign-passport holder who enters the ROC by visa-exemption, or who obtains a visa upon arrival, shall leave the ROC upon or before the expiration of permitted legal period of stay. However, in the following circumstances, the person may submit supporting documents and apply to the Bureau of Consular Affairs or its subordinate agencies for a visitor visa before the expiration of permitted legal period of stay;1.The person is afflicted with an acute, severe disease, excluding infectious, mental or other diseases that may pose a threat to public health or social order;2.Natural disasters or forces majeur; and3.Other justifiable causes.
中文原文 外國護照簽證條例施行細則第4條
持外國護照以免簽證或抵我國時申請簽證入境者,於停留期限屆滿時,應即出境。
但有下列各款情形之一,並提出證明者,得於停留期限屆滿前,向外交部領事事務局或其分支機構申請適當期限之停留簽證:
一、罹患急性重病。
但不包括足以妨害公共衛生或社會安寧之傳染病、精神病或其他疾病。
二、遭遇天災或其他不可抗力事故。
三、其他正當理由。

English translation for reference only; the Chinese text prevails. Not legal advice.