There should be no modern or practical anti-virus software (or need for any) in an old OS X
such as Leopard 10.5.8; you may need to attempt or troubleshoot the system and hardware.
Try restarting in Safe mode? Shift key, held through startup, perhaps you may need to login
again depending on if your system is set up that way, then proceed to desktop. Some items
will be disabled in Safe boot mode, you can check system files, look into Console logs to
see if there are entries by date, day, hour & minute you noticed a problem. You can also
repair disk permissions when booted in Safe mode, and restart normal to exit it.
If you have an alternative saved bootable copy of the same OS X (a clone on another HDD)
that could be used to test the hardware; especially if the system on the clone has not seen
any odd update or additional add-ons or impulsive changes, it can help troubleshoot or see
if the problem could be resolved by a new fresh installation in the computer. Or a reclone.
The computer model you have could see benefit of a replacement better spec hard drive
if the original is what's inside it now. They wear out and fail. Be sure to backup and then
also have another backup of your files. A system clone (use clone utility) in addition is also
recommended. Actually the best OS X for that computer would be Snow Leopard 10.6.8
and the retail DVD is worth it. That is the finished version where old Leopard was fixed.
For hardware upgrade to best run OS X, the RAM or memory chip total may need more.
If you have the install restore DVD, you could see if you can start up from that, it can be
helpful to see if there is any indication of hardware failure but that test in itself is not a
certainty, it may not show a failing drive as such. Sometimes a fresh installation over a
totally erased & reformatted HDD can bring the computer back to run well, but if the old
hard disk drive is still in there, a failure could happen without much notice. They wear out.
With a wired USB keyboard and mouse, you could also troubleshoot issues and use several
keyboard commands on startup easier than with touchpad and built-in keyboard. Sometimes
the built-in version can develop issues and the workaround is to have wired alternatives. I do.
In any event...
Good luck & happy computing! 🙂