The changes will come into effect at Royal Shrewsbury Hospital and Princess Royal Hospital in Telford from 6pm on Monday until further notice, in the wake of national guidance.
The Shrewsbury and Telford Hospital NHS Trust, which runs the hospitals, is asking anyone with a fever or a new persistent cough to stay away.
Children under the age of 16, over 70s and anyone with existing heart disease, lung disease, cancer, diabetes or any condition that weakens their immune system, will not be able to visit the hospitals.
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The rules were introduced as Boris Johnson said all people in the UK should avoid pubs, clubs and theatres, stop all non-essential contact and travel, and work from home if they can.
Neither RSH or PRH currently has any confirmed cases of coronavirus, but the trust says it is taking steps to limit the risk to vulnerable patients.
Patients who are undergoing testing for possible coronavirus will not be allowed visitors until they are given the all clear.
The other restrictions are:
Visitors will not be allowed in x-ray or radiology.
Wards at both hospitals will also have restrictions on visiting times.
People will be able to visit their loved ones between 10am and 8pm, but will only be able to visit for two hours between those times.
Only one visitor per patient will be allowed to visit at any one time.
Families and wards are being encouraged to nominate the same person to visit regularly, in order to minimise the number of visitors to the hospitals.
Visitors are asked to only visit the ward or area where their loved one is being treated and refrain from walking into the canteens, cafes and, especially, other clinical areas if this can be avoided.
There will be exceptions in maternity and end of life care where the two-hour visiting window does not apply.
For births, only one of two named birthing partners will be allowed at any one time to support the mother through delivery.
All other maternity areas, including postnatal and antenatal wards, will operate one named visitor only.
Children under the age of 16, including siblings of the newborn baby, are not permitted to visit any maternity area.
In paediatric and neonatal services, one designated carer can visit for the duration of the child’s stay.
There will also be an exception where end of life care is concerned, with discussions to be held on an individual basis.
Certain areas in the hospitals might be cordoned off or closed to visitors for deep cleaning or certain treatments, but the trust says it does not mean that there is a positively tested coronavirus patient in that area.
All visitors will be asked to use alcohol gel to disinfect their hands on entering the hospital, and again on entering the clinical area where their relative is treated.
They are also asked to use the gel again when leaving the clinical area and the hospital.
Visitors who are picking up a discharged patient to take them home, or are providing transport for a patient who is attending as an outpatient or for a short procedure, are asked to stay in their car, to avoid direct contact with other people and pick up their relative wherever this is possible.
Dr Arne Rose, medical director at SaTH, said: “Friends and relatives are encouraged to stay in touch with patients via other means, such as video messaging, text messaging and social media. Mobile phones can be used in almost all areas of the hospital.
“We are asking the public and friends and families of patients to please restrict the number of phone calls to our hospitals in what is a highly-pressured time for our teams.
“Clinical teams will give short updates to families of patients via telephone; in particular where there is a change of treatment or a change in the course of an illness.
“We appreciate that these measures may cause some anxiety and inconvenience, and we are sorry for that, but this is a highly unusual situation and our priority needs to be in protecting our vulnerable patients. I want to thank the public for their understanding and our teams for their hard work at this time.”