No. Mosquitoes only ever remove blood from a person. They do not inject it or circulate it. This is due to the design of their unique proboscis. Made up of two tubes, one sends saliva into the host while the other sucks up blood.Â
This two-tube system is why mosquitoes cannot transmit HIV, which spreads through infected blood. Any HIV-positive blood ingested by a mosquito is thus impossible to transmit to another host.