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Before you catch or even approach a stray cat, you need to observe them for signs of disease and injury. For your own safety and the safety as a cat, it's important to evaluate the health of the animal and come up with a viable plan for determining when and how to help. If a cat has been frequenting a particular area for a few days, spend some time watching it closely. If it's a friendly cat, your job will be a lot easier. If not, you've got some work to do.  If a cat behaves erratically, breathes heavily, drools excessively, or generally acts lethargic and unusual, call animal control. Do not attempt to approach cats exhibiting signs of disease. There are some diseases that cats carry that are zoonotic—or transferred to humans from animals and vice versa. A very frightening, and virtually untreatable disease is rabies transmitted by saliva through bites or cuts in the skin. Another danger is the dangerous infections that can set in from cat bites. Professionals who work at animal control have the necessary equipment and protection to safely capture sick stray cats. Not every cat needs to be caught. Don't try to catch well-fed looking cats with collars. Call your neighbors and inquire about whether or not anyone’s missing a stray, instead. Live traps are extremely safe and simple mechanisms that catch cats easily and humanely. You bait the trap with food, and then the doors of the cage will close when an animal wanders inside, trapping it safely. After trapping a stray cat, leave it in the live trap and transport it to the vet. Do not remove the animal from the trap.  Local vets and animal shelters will lend you live traps for catching strays. You don’t have to buy one, though it might be a good tool to have on hand if you live in a rural area and commonly encounter strays and other animals in need of relocation. If you absolutely can't find or use a trap, use a cat carrier or a box to bait with food and trap the cat inside.  Check with your vet before using a carrier to catch a cat some vets won't accept cats that are brought to them in anything except a live trap. A live trap is safer and more effective, but this can do in a pinch. Don’t attempt to catch a stray by picking it up or by using a pillowcase or other type of bag. These methods can anger and agitate strays, as well as injure them, and risk injuring yourself. Never, under any circumstances, should you handle a stray cat with your bare hands. Treat stray cats as you would a wild animal, even if you hope to domesticate the cat in the long run. Give it time. You'll need to have a suitable place for keeping the cat, even if you just plan on taking it to the vet and then re-releasing it. Ideally, you want to catch the cat as near as possible to the spay or neuter date, so you can take the cat immediately to the vet's office. If you need to house the cat for a little while before, though, you need to prepare a quiet room in the house for the cat to stay.  Strays should be kept in out-of-the-way areas of your house that you can keep quiet so the cat will calm down and feel safe. Basements, spare bedrooms, and other temperature controlled areas that you can keep dark will help to calm cats and keep them feeling safe. If you're taking the cat to the vet in less than 12 hours, don't worry about feeding the animal. It'll be fine and it'll be much safer to avoid opening the cage and risking having to wrangle the cat again. Provide some clean water and let the cat stay in the live trap. The ASPCA now promotes a policy called Trap Neuter Return (TNR). Whatever you intend to do with the animal after you’ve caught it, this needs to be the first step after catching it.
Observe the cat for a while. Get a live trap. Avoid unreliable methods for catching a cat. Prepare an area in which to keep the cat. Set up an appointment to have the cat neutered and medically checked out.