Concerning the hedging, you can use so-called catastrophe bonds. They are often issued by insurers and reinsurers, development banks and I can imagine issue by a pharma company. Funds raised from these bonds are put on separated account where they waint until a catastrophe (specified in the bond documentation) occurs. When this happens, the issuer can use funds on the account to treat consequences of the catastrophe. Sometimes there is a provision that in case of the catastrophe, the issuer will return no or smaller nominal to an investor. In case the catastrophe does not occur, the issuer returns whole nominal plus interest to the investor.
If we look at a normal investor (e.g. central bank, mutual fund etc.), a pandemic is very similar to any financial crisis. So an approach to hedging depends on how much risk-averse the investor is. You can for example use negative correlation between high quality government bonds and equities - when a financial crisis break out, equities would lose but there would be risk-off behavior and fly to government bonds whose price would increase.