Derek Lowe has an interesting post on the three documented cases of reinfection so far:
"Immunity and Re-Infection"
He concludes, in part:
"So the situation, for now, seems to be that yes, re-infection is possible. But it’s also quite rare. There are surely cases that we’ve missed, but it’s clearly not something that is happening much."
I posted the following as a (currently unapproved) comment on that post:
So three people have been reinfected, as covered above.
However, Covid-19 is similar enough to other CoV that up to 81% of people have been found to have some level of immunity against the "new" version.
"Cross-reactive SARS-CoV-2 peptides revealed pre-existing T cell responses in 81% of unexposed individuals and validated similarity with common cold coronaviruses, providing a functional basis for heterologous immunity in SARS-CoV-2 infection."
So what about reinfection?
It would be reasonable to expect that this virus would also be similar to the other versions, as it's similar in almost every other way.
Unfortunately, reinfection is common in the other CoV.
See here:
"In OC43 infection, a serum hemagglutination inhibition anti-body titer of >=40 was associated with protection... against reinfection. Nevertheless, 43 per cent of men who became infected with OC43 virus had initial serum antibody titers of >= 20 and < 80."
43% is pretty high. And here:
"Researchers have studied four species of these seasonal coronaviruses across the past 35 years, and found reinfection occurred frequently, around a year after the first bout."
And here:
"Neutralizing antibody to 229E was commonly present in the sera of the students. The level... did not appear to influence the occurrence of, or likelihood of illness with, reinfection... underscores the natural occurrence of reinfection with this virus."
So yes, we've only seen three people get reinfected with the new virus.
But there's no reason to think this will be rare. It should be quite common, especially as this virus becomes endemic, as I expect it will.
It will be interesting to see if a vaccine is possible...