It comes after similar concerns about pressure on bed availability at the Borders General.
NHS Dumfries and Galloway said an upsurge in demand for services was being seen across Scotland.
It said work was ongoing to improve "patient flow" to deal with the increase in hospital admissions.
Infection control manager Elaine Ross said flu and other respiratory viruses were circulating in communities across the region.
She urged everyone to play their part in tackling the issue.
"Currently, the demand on our local hospital is extreme," she said.
"Patients are arriving at our emergency department and via GP referrals with flu-like symptoms and persistent coughs that may be compounded by underlying conditions.
"The flu vaccine is effective but won't protect against other viruses that are going round."
She asked anyone with symptoms to take lots of fluid and paracetamol and to stay indoors and rest until they felt better.
"We are also urging people to not visit relatives in the hospital if they are showing any symptoms," she said.
"It is vital that we stop the spread of infection."
Interim general manager Gail Meier said staff were "working round the clock" to assess patients and determine the need for admissions.
She said they aimed to ensure that people were "seen, treated and either discharged or admitted in a timely manner".
"It is a very demanding time for the health service and I take the opportunity to urge everyone to consider the use of emergency services as it is a finite resource," she said.
"Community pharmacies and NHS 24 offer help and advice as an alternative to attending hospital.
"I would like to thank everyone for their commitment over the past few weeks and 365 days of the year for that matter. It does not go unnoticed."

Agents arrested more than 150 people on sex-trafficking charges in dozens of cities including Atlanta, Denver and Seattle.
The youngest victim found was 12 years old, the FBI said.
The FBI targeted hotels, casinos and lorry stops to find underage victims.
The majority of the teens found were girls. Agents also found three boys and three transgender teens.
The FBI worked with local police departments and the National Center for Missing and Exploited Children to help the teenage victims to find shelter and get medical attention.
The agencies want to ensure the teenagers do not return to prostitution.
"When kids are treated as a commodity in seedy hotels and on dark roadsides, we must rescue them from their nightmare and severely punish those responsible for that horror," FBI Director James Comey said in a statement.
The raids involved about 500 local police departments from across the country. A similar FBI-led campaign carried out last year uncovered about 160 teenagers working as prostitutes.

Unite union members based at plants in Cowley, Goodwood, near Chichester, Hams Hall in the West Midlands and Swindon, turned down the offer by 56.6%.
Workers have already staged four walkouts during the dispute, which is over the closure of a final-salary pension scheme.
BMW said the offer was "fair" and it was disappointed at the ballot result.
Unite representatives will meet on Tuesday to discuss the next steps in the dispute, which has seen the first-ever strikes by the company's British employees.
Three 24-hour strikes scheduled in May were suspended while workers considered the offer, which the union did not recommend to its members.
Unite national officer Fred Hanna said: "It is clear that it did not go far enough or deal with the concerns many of our members have over BMW's pension plans.
"We would urge BMW bosses to reflect on the result and listen to the workforce by further engaging in meaningful talks with Unite."
A BMW spokesman said: "We are now considering the implications of the ballot result and we will be meeting with the union in due course to discuss next steps.
"We believe the offer, which resulted from lengthy negotiations with the union since September last year, was fair.
"It was designed to improve competitiveness, which is in the long-term interests of all our employees in the future."
Health chiefs have warned Dumfries Infirmary is facing "extreme" demand from people with flu-like symptoms and persistent coughs.
[NEXT_CONCEPT]
The FBI has broken up several sex trafficking rings across the US, uncovering more than 100 sexually exploited teenagers in a nationwide operation.
[NEXT_CONCEPT]
Workers at car giant BMW have voted to reject an offer aimed at resolving a long-running dispute over pensions.