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If you haven't watched CGP Grey's video Americapox: The Missing Plague, I suggest you watch it. It's an excellent video.

One assertion he makes is that "germs jumping species is extraordinarily rare [...] being the patient zero of a new animal-to-human plague is winning a terrible lottery."

Why is this true? Why is it so rare for a germ to move from a cow's body to a human's? Is it because it can't travel that distance while remaining alive? Is it because the human's immune system will kill it? More details will be appreciated.

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The main reason is the biochemistry of two unrelated species are different. The more closely related two species the more similar things like the hormones and signaling molecules will be and inversely the more distantly related they are the more different those things will be. Even highly constrained molecules like hemoglobin may still pick of mutation that get carried on, and things like an immune system actually favor variation evolutionary speaking. Even within the same species these things vary, and this is often cited as a possible reason sexual reproduction exists at all, to make each individual slightly different, so there is less chance of catching disease or parasites from others.

Animals have many many barriers to infection, getting pass all of them is hard and tends to only occur with organisms that evolve alongside that animal and thus have been selected to get pass that specific combination of barriers. Getting past them all if you have not been evolving along side it is incredibly unlikely, some diseases manage it be cause there are billions of them, and if you run billion upon billion of random keys through a lock some may fit just due to chance. this is also why disease transfer is more likely the more closely related species are. SIV/HIV easily jumps between great apes because they are very closely related (and the insist on eating each other) the chances of a dog catching it are astronomically unlikely because the chances of the viruses surface proteins matching receptor proteins of a dog is basically only chance as there has been no evolutionary pressure to alter it.

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