How long a COVID positive case can be infective?

To address this not so easy question we need to first understand the concept of Cycle Threshold (CT) in RT-PCR for COVID 

CT is the number of replication cycles required to produce a fluorescent signal, so lower CT values represent  higher viral RNA loads. A CT value less than 40 is clinically reported as PCR positive. This positivity starts to decline by week 3 and subsequently becomes undetectable. However, the CT values obtained in severely ill hospitalized patients are lower than the CT values of mild cases and PCR positivity may persist beyond 3 weeks after illness onset when most mild cases will yield a negative result. 

However, a “positive” PCR result reflects only the detection of viral RNA and does not necessarily indicate presence of viable virus.

In some cases, viral RNA has been detected by RT-PCR even beyond week 6 following the first positive test. Because isolation of the virus in culture were not successful beyond day 8 of illness onset, which correlates with the decline of infectivity beyond the first week.