This can be a tough situation. Having worked both in a spay/neuter clinic and in the field trapping cats, I can see both sides. I deeply respect veterinarians, but sometimes they aren't aware of changes in the field of veterinary medicine.
The science on retroviruses, such as FIV, has changed a lot in recent years. The leading feline and shelter veterinary organizations recommend against testing healthy cats in TNR programs. FIV is also no longer considered a reason for euthanasia, as FIV+ cats can live long, normal lives. Another change based on newer science is that FIV+ cats can live with FIV- cats without transmitting the disease. Mating and fighting (deep bite wounds) are the main way FIV is passed on, thus neutering the cats greatly reduces opportunity for transmission. Also consider that an intact FIV+ cat has already exposed the other cats in his neighborhood, so returning him after neutering him isn't going to make a difference.
Here are some documents that may be of interest to your veterinary team:
2020 AAFP Feline Retrovirus Management
https://catvets.com/guidelines/practice-guidelines/retrovirus-management-guidelines
What is recommended for TNR programs and management of community cats?
https://www.sheltermedicine.com/library/resources/?r=what-is-recommended-for-tnr-programs-and-management-of-community-cats
It is unfortunate that the trappers you work with are sending negative emails. They often don't recognize the emotional toll this takes on the vet staff. But understand they are doing it because they are also having to deal with the emotion of loosing a cats. And more. When you work really hard to develop a relationship with the people feeding cats in a community, convince them to let you take their cats away for surgery, and promise they will come back....and then they don't. That's not good. While the feeders may not be the legal "owners" of the cats, they are invested. The trapper may not be able to regain their trust to be able to trap the rest of the cats in that area for TNR. The population grows, more kittens come into the shelter, and no one wins.
My advice is to have a discussion with your veterinary team about the latest guidelines for FIV testing, especially with regards to TNR programs. Make sure any policy you have is based on current science. If your vet continues to insist on testing and euthanizing FIV+ cats, make this policy clear to your trappers so they can decide if they want to bring those cats to your clinic or find an alternative.
If you'd like more resources on this, please don't hesitate to reach out. I'll do my best to point you in the right direction.
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DanielleBays
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