The most professional thing you can do is plan ahead - if you know certain situations are more likely to make you ill, then avoid finding yourself there in the first place.
For example, say after a long trip you feel so tired you can barely walk. In this case, avoid scheduling meetings the next day. Or, avoid traveling before a deadline. Simply put: if you want to be a professional, be professional with yourself first. It's likely that you have accumulated a lot of "mental" data about your illness, to the point where you should have a model that can help you work around what your work schedule should be. If you don't have one, try writing down your work pattern, see if you can spot what circumstances allow you to work undisturbed, and what hinders you.
Secondly, have somebody trusted back you up. Consider this a "plan B": your boss, a colleague, anybody who could catch the bullets in these emergency situations where you can't deal with them. Be honest with this person: let them understand they are helping you, let them know about your illness, and never ever abuse their patience. Consider this a "business deal" and treat is as such. Don't mix friendship with professionalism.
Everything else is very much dependent on your illness and the nature of your job.