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Police plan to step up their presence around Medway Maritime Hospital following reports on social media of a pre-planned protest.
Officers have briefed staff, including security teams at the Gillingham hospital, after the Medway NHS Foundation Trust is said to have received a "credible" threat.
A warning has been issued by the force to anyone attending such an event that they should abide by government guidelines to prevent the possible risk of Covid-19 cases.
The matter has apparently come to light after a man died on Saturday night, although neither the police nor the hospital could confirm the nature of the protest.
A police spokesman said: "Kent Police has received no formal notification of a pre-planned demonstration taking place at Medway Maritime Hospital, however the force is aware of posts advertising a possible event on a date to be confirmed.
"Officers have reviewed all available information and liaised with hospital staff to provide advice.
"The situation will continue to be monitored and members of the public may see an increased policing presence in the area.
"Anyone attending such an event is also reminded to ensure they adhere to the current government guidelines to prevent the possible spread of Covid-19."
It is understood that health unions are not aware of any kind of demonstration around the hospital, which is in a residential area in Windmill Road.
The last time there was a protest at the hospital was at the beginning of the month when unions and staff came together to show their dismay at the government's pay offer for NHS workers.