Natural disasters are VERY different from what you propose in that they are essentially an interaction-free mechanism: just pure random events that may or may not occur over the course of the game and that you, player, have almost no influence in or ways to prevent (off the top of my head: governors and dams?). For everything else, it works through the normal tile improvement system, pillaging, destroying and rebuilding tiles, the same as barbarian attacks and the rest.
There's no trying to prevent the disease from spreading or finding a cure or any of those sorts of things. You do not send Emergency Response teams to look for survivors, you do have workers building sandbag defenses on tiles, you do not have firefighter units to combat the forest fire.
Really, by most standard, I'd qualify natural disasters as much more of a passive system (as in: you the player have a passive role in regard to them) than an active one. Whereas what you propose is firmly on the active side, as in it requires constant actions by the players that are specific to this mechanism.