Mayor Walsh said Wednesday that "we are very close into moving into the red category" on the state's Covid-19 map, possibly within the next few days, because Boston is now approaching an infection rate of 8 new daily cases per 100,000 residents, the state's threshold to label a city as high risk.

At a City Hall press conference, Walsh and Health and Human Services Chief Marty Martinez pointed to Dorchester, East Boston, Roslindale and Hyde Park as areas of particular concern as infection numbers slowly rise.

They said about 5 percent of Dorchester residents getting Covid-19 tests are getting "positive" results, compared to a citywide rate of 2.7 percent and a statewide rate of 0.8 percent. East Boston continues to have the highest positivity rate in the city - about 6 percent, although that's down from 11 percent a few weeks ago, while Roslindale and Hyde Park are around 4 percent.

Walsh added the city has seen an uptick in the number of people hospitalized with Covid-19.

Walsh and Martinez said the city will be ramping up testing and educational efforts in the four neighborhoods. Walsh pointed to a a pop-up testing site on Geneva Avenue in Grove Hall, which runs through Oct. 3, as an example (other testing sites).

"We're not going to set a panic off right now with the numbers," Walsh said, adding that the city is nowhere near an increase that would make Walsh consider rolling back any of the re-opening efforts of the last few months.

He said he would only start thinking about something like that should the test positivity rate begin to approach 9 or 10 percent.

Instead, Walsh said residents can help keep the numbers low by continuing to socially distance, wear masks outdoor, stay away from large gatherings, especially indoors and wash their hands and surfaces frequently.

Martinez said college students have been responsible for a part of the increase in numbers, But with few college students living in East Boston and Roslindale he and Walsh also pointed to permanent residents getting out more in the warm weather and congregating. Martinez praised local colleges for doing a lot of testing, which lets officials track people who might have been exposed and get them to quarantine for two weeks.

Walsh said young people in particular have a role to play - to use their social networks to get their friends to follow the same precautions as older residents.

Mayor's press conference, which also discusses Breonna Taylor and the re-opening of Boston schools: