Natural disasters cause the balance in nature to go awry. This could mean that feeding grounds for normal pests are disturbed so they seek new areas through migration. Locusts coming to areas they've never been historically reported. Small mammals who carry fleas and ticks with disease will look for the easiest food available. If hundreds of thousands of people die, and so quickly that they can't be buried properly, this now will spread disease from the decaying bodies, plus the wildlife that feeds on them. an increase in flies and other insects comes to mind which we know spread bacteria and disease. You also have no humans to clean up and keep rats and mice from getting into the grain supply. In medieval Europe and up to the plague of London in the 17th century, cats and dogs were killed as people thought they were dirty and spread disease. This makes the rat population boom. In that specific case, the plaque was spread from fleas on the rats. More rats, more plaque. Natural disasters have been spreading disease since we have existed.