Apparently we’re in the timeline before Kyle Reese gets sent back.

This may be true but sudden climactic shifts in the past have spurred ecological chaos that have fomented plagues. Plague of Justinian, for example, came in the wake of volcanic eruptions that darkened the skies and cooled the earth. That shift spurred rodents carrying the bacteria to go beyond their normal range in search of food, thus bringing the disease in contact with population centers.