When pigs are regionally slaughtered due to diseases concerns, prices will generally be tempered by associated drop in demand for pork, but if there is an imbalance either the demand reduction will exceed the culls resulting in cheap pork for a while, or demand reduction won’t exceed it, meaning people sufficiently keen to eat pork will pay a little more and everyone else will eat chicken and fish for a while. It’s really not that big a deal.
The **** will hit the fan when an actual serious disaster means there isn’t enough food to feed everyone. That will occur, but probably not in a gradual way. We are still in a situation where significantly more food is produced than required, meaning that a significant percentage is wasted, largely due to indifference (why bother trying to sell the least pretty 20% of the crop when no one will buy it? Just dump it. Why bother eating leftovers from 2 days ago when we can just get something fresh?).
Food will probably become more expensive as the spare capacity is reduced, and at some stage, some actually significant issue will disrupt food production or distribution. A freak weather event stopping the monsoon season from happening in Asia, an all out war stopping distribution networks, a major state carrying out an attack on production and/or distribution, and then things will get interesting. Alternatively, if some major event wipes out a significant percentage of the world’s human population, the food shortage may never happen.