Yes tons and tons. It's fairly easy to just imagine it as a sort of DFA/NFA. I personally like an SIR(D) model where from the "I" state you can either progress

Yes tons and tons. It's fairly easy to just imagine it as a sort of DFA/NFA. I personally like an SIR(D) model where from the "I" state you can either progress to recovered or dead. You can kinda take care of that in the regular "R" case with the SIR model, but the "D" state makes the fatality of the virus all the more visible. Additionally, you can add "Vital dynamics" to the model by basically adding a birth rate dependent on the total size of the reproducing population that feeds into the "S" state. You can also add a general baseline death rate (\mu) that takes away some population from each state in SIR. Lastly, like the sibling commenter mentioned, you can add an "E" state where there is a state between susceptible and infected where an individual is infected but not yet infectious. All that is added are more letters and computational time :)