Two options: a) Indigenous people have a higher infection rate of HIV than non-indigenous people. b) There is a higher infection rate of HIV in indigenous people than in non-indigenous people.

It helps to think of it in stages. 1. Indigenous people [have a higher infection rate of HIV] than non-indigenous people [have an infection rate of HIV]. 2. Indigenous people [have a higher infection rate of HIV] than non-indigenous people [do]. (repetitive part becomes "do"). 3. Indigenous people [have a higher infection rate of HIV] than non-indigenous people. ("do" is repetitive too, so remove it). (If you think of it this way, you can see there is no reason to add "in" in the sentence) Similarly, 1. [There is a higher infection rate of HIV] among indigenous people than [there is an infection rate of HIV] among non-indigenous people. 2. [There is a higher infection rate of HIV] among indigenous people than [there is] among non-indigenous people. 3. [There is a higher infection rate of HIV] among indigenous people than among non-indigenous people. ("among" seems to be repetitive too, but it sounds better leaving it in, because it binds tightly with its object) In both cases, I think sentence (2) is also fine, as they only have a one- or two-word anchor phrases repeated. But the modern trend seems to be to reduce it to the shortest form possible.

Yes, I agree with Michael. "These people have a higher rate THAN those other people" or "There is a higher rate IN these people THAN IN those people."