Can a virus stay in your body (after recovery) in a concentration that is sufficient to infect someone?
Is it known how long the Zika virus persists in the body?
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According to the CDC the zika virus persists in the blood for around a week.
Zika virus usually remains in the blood of an infected person for about a week but it can be found longer in some people. - CDC
I can't find much information on how long the Zika virus specifically can persist after recovery, it seems unknown just now. However, it is advised to be careful about sexual contact after infection, which suggests longer persistence is suspected in other fluids/tissues.
British health authorities have suggested using condoms for at least 28 days. - NYTimes
Some viruses, including Ebola, can linger for some time after recovery in parts of the body (for example, testes, eyes, spinal column fluid) not easily reached by the immune system. - CDC
These parts of the body have immune privilege. This means they can be infected by antigens without producing strong immune responses. This is thought to occur to protect vital tissues/areas which could be damaged by inflamation and the immune response, such as the brain, eyes, spinal column, testes, and fetuses.
Since I wrote this answer there have been several reported cases of sexually transmitted zika, highlighting the problems that can arise from immune privilege.
Some viruses can stay in your body for your entire life, for example Herpes Simplex Virus.