It comes just days after Prime Minister Boris Johnson urged people to follow social distancing guidelines and asked Britons to only go to work if they fell into the key workers category.
Bin collections will be continuing in East Suffolk.
Carron McDiarmid, executive chief officer for communities and place, said: "While we are doing our best to maintain our regular waste and recycling collections, we need to be realistic and plan for staff shortages".
The council's bulky waste service is still operating as normal for the removal of large items from your home, as are green waste collections.
"In order to protect members of the public and employees, all 16 recycling sites will be closed until they can be operated safely".
As of 24 March 2020, Auckland's kerbside rubbish, recycling and food scraps collections are considered essential services and will continue as usual.
If your collection has been missed, please check our service disruption page on the coronavirus section of the Dorset Council website before reporting.
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"By postponing the garden waste collections, we are able to use these vehicles, drivers and loaders to support general rubbish, recycling and clinical collections, which must be prioritised".
They are following the lead of local authorities including Cambridge City Council, Cheshire West and Chester and Derby City Council, which had already suspended services such as garden and food waste pickups.
Do not put your glass box out for collection until it is full. But we now need to make some service adjustments.
These bags should be placed outside for at least 72 hours before being put in the rubbish bin for collection.
The GMCA has said over 46,000 people visited the 20 centres across the region over the weekend, an average of around 2,300 people at each site.
'Any disruption to recycling services will be a last resort and, if this is unavoidable, we must not allow it to break our national recycling habit in the long term.
Anyone with garden waste is asked to take it to their nearest household waste recycling centre or consider home composting.