Short-term disability is covered in the Agreement under the heading of Serious Illness. The language reads:

A House Officer who is unable to work for at least one (1) week due to a) a single incident of serious disabling illness or injury, b) intermittent absences to receive multiple, on-going treatments resulting from and following a single disabling illness or injury (such as chemotherapy, radiation therapy, physical therapy or dialysis), or c) severe complications from pregnancy that prevent attendance at work, or (d) a medically necessary period of physical recovery from childbirth that exceeds the amount of paid Maternity Leave provided pursuant to Section F shall receive up to a maximum of six (6) months, inclusive of weekends, of full pay in each three (3) calendar year period following their date of hire. Renewal of this six (6) months sick time benefit will occur on the first of the month in which the three-year anniversary occurs. However, if a House Officer is on paid sick leave under this provision on the renewal date, the renewal will be deferred until the House Officer returns to active employment for 30 days. The Employer will require verification of the House Officer's inability to work, including intermittent absences, before granting paid time off. In the event the House Officer applies for and is approved for Long-Term Disability benefits pursuant to Article IV of this Agreement, this six (6) month period of paid Serious Illness Leave qualifies as the elimination period for the Long-Term Disability benefit.

After the six-month elimination period, your Long-term Disability insurance is activated after going through Work Connections and providing the required documentation.  

An employee who has a serious negative reaction to any employer-mandated vaccine would be covered under the employer's Worker's Compensation policy.  In Michigan, it's part of the law that your medical bills are included in this coverage.