Sounds like a common reaction to pesticides as well. Make sure you have not just used flea spray on your pet and had them lay near your computer, nor sprayed bug spray or have a pest strip near your computer.
This problem is far more common than you would expect, and many more people today are becoming sensitive to the chemicals in their environment. After being poisoned by a "sick building", I now need to have my computer in a box vented by a fan to the outdoors, and had to switch to LCD for less fumes off the monitor. The fumes off electronics make me very sick.
Constant exposure to many things can also cause a sensitivity over time. Some woodworkers eventually become sensitized to wood dust. What makes it worse, is that while it may start with only Cedar, it eventually broadens it's scope until one is sensitive to quite a few species, if not all wood dust.
To track down the real culprit, move parts of your computer, one at a time, to a new location and sit with it a while. See if the reaction starts, and try to isolate the part(s) that cause the reaction. You can also remove everything from your computer area other than the actual hardware and see if the reaction still happens. Maybe a book or papers with something on it causes your problem.
When they sprayed for fleas at my wifes office, she broke out with a similar rash. When they moved her to a new location, the rash would occasionally re-appear. She finally had to have every piece of paper copied, and all new office supplies before the rash stopped. Everything that was at her old desk was contaminated by the flea spray. Worst part is, now she has become sensitized to a similar class of chemicals and gets wooly headed and headaches when around many perfumes (most of which contain many toxic chemicals).
There are many sites on the web about Multiple Chemical Sensitivity, Gulf War Syndrome, and Environmental Illness. There are just way too many chemicals out there that cause long term problems for a large segment of the population. I wish I had learned more about it before I got so bad that my sensitivity became permanent.