Both the WHO (World Health Organization) and the CDC (Centers for Disease Control) agree that there is no credible evidence that pets can catch or spread COVID-19, the novel coronavirus currently wreaking havoc around the world. There was a flurry of concern when two dogs in Hong Kong tested “weak positive” for the virus and were quarantined along with their infected owners. Neither showed any signs of illness and other pets in the homes did not test positive. There have been no other cases reported anywhere else in the world, even as the virus has spread extensively among the human population of every continent except Antarctica. There is actually a test for covid-19 in animals (not the same as the human test) and it has been given to thousands of animals around the world, with all other companion animals testing negative.
Since companion animals can neither get Covid-19 or spread it to humans, they pose no medical danger to humans, according to the best evidence at this time. There is no reason to abandon animals or to turn them in to shelters because of fears regarding the coronavirus. The advice given by scientists and medical professionals is to interact with your animals normally if you are not sick yourself. If you become sick, it is best to have another family member care for your pets, maintaining distance from your animals just as you would your human family members, out of an abundance of caution.
Excellent information about pets and the coronavirus can be found in this FAQ from the veterinary college at the University of Illinois at Urbana-Champaign: https://vetmed.illinois.edu/pet_column/coronavirus-pets/
The following page from the AVMA (American Veterinary Medical Association), which is the leading organization of American veterinarians, also addresses these issues with the best information available from these medical professionals: https://www.avma.org/resources-tools/animal-health-and-welfare/covid-19
There is another piece of misinformation going around the internet that needs to be debunked— that hand sanitizer includes an ingredient found in antifreeze that can poison a pet who licks the hand sanitizer applied to a human. That is NOT true. The ingredient in question is ethylene glycol, which is indeed extremely poisonous to pets. However, hand sanitizer contains either ethyl or isopropyl alcohol, NOT ethylene glycol. There is not a danger if your pet licks your sanitized hand.