If the internet connection begins OK and then slows down with one computer connected directly to the modem, it's probably the modem going bad. Only way I can think to make sure is by borrowing another modem.
If the problem doesn't occur with just the modem in the picture, then it is probably the router going bad. Try resetting it to factory default settings and then reconfigure. If still problem, look for a firmware update.
If the problem doesn't occur with just the modem in the picture, then it is probably the router going bad. Try resetting it to factory default settings and then reconfigure. If still problem, look for a firmware update.