It's in the "spit" of the mosquito. malaria is caused in an infectious protozoa. It's a nasty bug that does not affect the vector (the mosquito), but "hangs out" in the insect and is injected into the wound when the mosquito bites. The mosquito drinks blood, as you know, but it injects an anticoagulant into a bite to insure good "blood flow" to enable it to dine. And the unfortunate individual who is bitten is infected in the process. A link can be found below.
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Malaria parasite lives in your liver. The plasmodium vivax can live in your liver for years together. They are called aptly as 'Hypnozoites' in that case. One US citizen developed malaria after twenty years of leaving the malaria zone. The plasmodium falciparum can not live in the liver like that. They say that either you or we will survive. But then, when the patient dies, the parasites are also are killed. They are foolish parasites. Wise parasites do not kill the host!
Mosquitoes get the protozoa that cause malaria from animals they bite that are infected with the protozoa.