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Schools will stay open in the face of renewed calls to close ahead of the Christmas break.
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Alan Brookes, who's chair of the Kent Association of Headteachers, says closing schools early needs urgent consideration.
Medway Council's leader has hit out at Labour councillors who say pupils should stay home as the Towns' Covid rate remains one of the highest in the country.
Cllr Alan Jarrett was responding after the opposition made a second plea to education leaders at the council and the Department for Education in an open letter calling on a 'circuit break' before the end of term.
It follows the drastic move taken by Greenwich council last night to shut all schools in the London borough.
The Mayor of London Sadiq Khan wants all secondary schools and colleges in the capital to close from tonight and open later in January, citing a big spike in cases among children aged between 10 and 19.
Medway has the second worst rate of coronavirus infection in England, second only to neighbouring Swale. The latest government data shows a rate of 604.9 cases per 100,000 people in the Towns, an increase of 1.4% in new cases week-on-week.
Labour's education spokesman Cllr Clive Johnson said it was likely schools remaining open would "exacerbate" rising cases .
He expressed deep concerns about the high rate of transmission being seen and the increased pressure it is having on Medway Maritime Hospital, Gillingham.
He said: "I wrote to the Regional Schools Commissioner two weeks ago to urge a reconsideration of the need for a circuit break within Medway’s schools.
"Since then cases have increased, and Medway now has the highest transmission rates in the country.
"This is a public health matter, as SAGE [Scientific Advisory Group for Emergencies] has identified, primary and secondary schools as two of the top three vectors for transmission of the virus, and we must do all we can to ease pressure on our local hospitals.
"However, this is also about the welfare of our young people. Medway schools are a credit to our local community and have done an amazing job in delivering quality education amidst real adversity.
"However, their ability to sustain this is threatened as increasing numbers of pupil and teacher populations are forced to isolate.
"I absolutely urge the commissioner, and indeed the Conservative council leader, to urgently reassess the need to implement a circuit break."
The isolation period is now 10 days, meaning if children contract coronavirus from tomorrow they will be in isolation on Christmas Day or will risk spreading Covid-19.
In response, Cllr Jarrett criticised Labour's "abject refusal" to listen to the commissioner and government guidance.
He accused the group of "incessant politicking" and said the "most disappointing" was the opposition's "constant calls" to close schools.
He said: "Closures would result in the isolation and loss of educational opportunity for so many of Medway’s most vulnerable children.
"Indeed, senior educational figures have expressed to me their bitter disappointment at this lack of understanding and disrespect for the hard work of teachers and staff across Medway."
He went on to urge Labour leader Cllr Vince Maple and his colleagues to "stop and consider its conduct moving forwards".
Cllr Jarrett added: "I would like to take the opportunity to remind Cllr Maple and his colleagues that at the beginning of this pandemic, his Labour counterparts elsewhere in Kent were quick to encourage him not to oppose proposals for opposition’s sake, and to instead work together to ensure the very best outcomes and continuity of service for Medway residents. It appears that this advice fell on deaf ears."
As part of the council's asymptomatic testing programme which kicked off on December 5, members of the armed services visited schools in Medway last week at mobile pop-up testing sites.
Those aged over 16 will be offered a symptom-free test where they carry out a swab themselves and send off for analysis.
Speaking on Friday, the council's director of public health, James Williams, said: "We have already tested pupils and staff in five large secondary schools in Medway and have visits arranged for a further seven next week.
"Symptom-free testing will help identify those who unknowingly have the virus and stop the spread to their loved ones and wider school community.
"Primary school staff are also being encouraged to get tested at the pop-up sites and we have directly contacted them with details of how they can book.
"All education staff are able to visit our four community testing centres in Rochester, Chatham, Rainham and Chattenden. We all need to act now and play our part in the fight against coronavirus."
Schools commissioner Claire Burton said in a reply to Cllr Johnson that her team remains in close contact with education leaders in Medway.
She was "acutely aware" of the "levels of anxiety and uncertainty" relating to the rising incident rates in Medway, which currently sit at 604.9 per 100,000 and still among the highest in the country.
She wrote: "The government has made clear its commitment that education and childcare settings should continue to operate as normal during the Covid-19 outbreak even in areas like Medway, under Tier 3 restrictions.
"The Department has published its Contingency Framework guidance which outlines our approach to considering restrictions to full time, face to face education as a last resort in a local area.
"We are regularly reviewing regional data and working alongside the Joint Biosecurity Centre and Public Health England in the region to ensure that we have all the available evidence.
"We are not currently in a position where the scientific and public health advice supports implementation of the contingency framework in Medway."