Taiwan has recorded six new deaths from Covid-19, and several hundred more cases in the continuing outbreak.

It is the second day in a row of six recorded deaths from the outbreak, which has brought to an end Taiwan’s long run of living largely Covid-free since the pandemic began. On Monday the CECC reported 334 new local cases, and added 256 to the previous week’s total. Over the weekend hundreds of new cases were added to the previous week’s total, after authorities started to clear a backlog of thousands of tests.

Taiwan does not have mass testing set up, and the figures are coming from cases identified through targeted rapid testing stations and then confirmation tests. The CECC said the rapid testing stations were showing positivity rates of 5.9% in Taipei and about 4% in New Taipei.

Taiwan remains on a level 3 alert, which requires the closure of entertainment and recreation venues, public venues, and sporting grounds. People are encouraged to work from home and restaurants told to ensure social distancing.

There is consideration of extending level three, but the CECC has said level four – lockdown – won’t happen unless there are 14 consecutive days of 100 or more cases, with 50% untraced.

Some local governments appear to have wanted stricter measures than those ordered by the CECC. On the weekend Taipei’s mayor joined some other city governments in banning dine-in services at restaurants.

Taiwan is also under pressure over vaccines. Chairman of the Chinese vaccine manufacturer Fosun said the company has offered to provide doses of the German-developed BioNtech vaccine to Taiwan, state media reported over the weekend. Taiwan reportedly rejected a similar offer in March saying it was talking directly to BioNtech before the deal fell over, and has suggested China interfered – a charge China denies.

Dr Peter Chang, director general of the Global Taiwan Medical Alliance, and former national ombudsman, told the Guardian vaccinations were the leading solution to an outbreak, but an area where Taiwan – which has otherwise run a world-leading response to the Covid-19 pandemic – has faltered.

“I think the government started thinking seriously [about vaccines] too late. They thought we were very good with mask wearing, and people are very disciplined so social distancing, so it is OK,” he said.

“I personally think if the vaccine is good, wherever they come from we should take it,” he said, adding that he did not know the data about Chinese vaccines.

“I think we should be very neutral. We don’t need to put politics in front of the safety of our people.”