What Cougie said.... you can't get any fitter in a week anyway. Even without the sickness bug you would be all trained up now and just doing a few shorter easy runs this week to ensure you are rested and fresh for the race.
In this case you are exhausted... therefore I'd say don't run. Just eat right and try to recover your energy as best you can. If you still feel exhausted come race day, you won't be able to run. However even if you feel better it may still be best if you are not 100% back to normal energy levels but are determined to run, to use a run/walk method.
So I'd be doing nothing if you are exhausted, and if you are recovered, only a few or less gentle short runs. Then come race day either don't race or just run/walk. Sickness happens. It has to me and it's rubbish. But your family/friends understand that too.
Yeah I know, I often think you probably are but then again, I also know when my body is sore or tired or at least needs a rest from say running but not necessarily a complete rest, meaning I can still cycle or swim or go to gym, or climb... you get my point. I think if you are new to exercising regularly then certainly you need rest days, and training should be built up gradually but I have always been active and find sitting about quite difficult. I do take days off, just not all the time. Sometimes it is forced upon me by work or other commitments but 90% of the time I take them when I feel the need.
It can vary person to person and I am certainly no expert but I would say go on feel which is what I do. I tend not to have too structured a plan due to working shifts which have no routine on a week to week basis so sometimes I do have to shift things about. I also know what you mean about being keen to train every day.. I am one of those who hates a rest day, unless I am actually knackered.
Some training plans (IronMan) for example don't have definite rest days written in as there is so much to train for but they might have easy days where there is the option for a "recovery swim" or a rest.
I'd say if you are really keen on training most days make sure you at least have recovery days where maybe as above you just have an easy swim or similar. That way your body gets a rest from the heavy stuff and your legs can recover but you still feel you have done something. Otherwise you will eventually knacker yourself.
In general though if you are marathon training I'd say 1 or 2 rest days... and some other days will be easier as well...