Those five have all returned negative Covid test results and will be subject to more testing in the coming days.
Of the 15 people to test positive on the ship, 12 are regarded as being in the early stages of their infection, and one may be a historical infection.
More testing is planned today to help provide more information, the ministry said.
Plans are now in place for the management of the five people who remain on the Viking Bay should any of them start showing symptoms or are confirmed to have Covid.
Health officials said just over half of the crew have been at sea since February.
Another nine crew members joined them after arriving in New Zealand last Monday, July 5, as part of a scheduled crew change.
"No link has been shown between this case and any other cases previously identified in New Zealand."
Genome sequencing of the 13 new cases will start today.
Last week public health staff identified three contacts who were isolating, including two people who worked at the port and the driver who transported the mariners from Auckland to New Plymouth where they boarded the fishing vessel.
The rapid spread of the virus on the ship reinforces the importance of bringing it back to port to transfer crew to a quarantine facility, health officials said.
"This allows the public health risks to be managed and the crew to be kept in isolation until they have completed their 14 days isolation and recovered."
The Viking Bay is a Spanish-flagged fishing vessel.
Officials have previously described the infection on the ship as being "low risk" to the public because of standard infection prevention controls in place.
A second fishing vessel has this morning docked at Port Taranaki with crew members experiencing a "flu-like" illness.
The Ministry of Health said those on board the Playa Zahara would be tested for Covid today before returning to sea.
He expected the operation would take four hours.
"Our pilot will be onboard our launch, which will lead the vessel into port, so no one will go aboard. In addition, those tying up the vessel will wear full PPE and will have no contact with crew," said Port Taranaki chief executive Guy Roper.
"After the crew has been tested by the Public Health Unit on the wharf beside the vessel, the crew will return to the vessel and head offshore while they await the results."
Roper said he was satisfied all safety precautions were in place, and the vessel posed a low risk.
The vessel had been at sea for more than three weeks and there had been no contact with others or crew changes, he said.
