Overtraining, probably increases the chances of illness the greatest, specifically longer endurnace work or greater volumes due to parasympathetic activiation and suppressed immune response. This coupled with lowered bodyfat after a pre-season certainly leads to greater likelihood of colds & flu.
The only way I know possible to avoid it is using an Omegawave, but they are incredibly expensive, time consuming and not many are willing to invest.
Rest is paramount, but some training specific to the type of illness is possible and certainly effective in encouraging the body to respond. Vitamin C, Zinc, Ginger, Garlic & Lemon are all helpful also.
It may also be well worth looking at the timing of a cold and it’s appearance as you have some control over when it hits … if you know it’s coming (with OW or symptoms) you can speed up it’s arrival or delay it to just avoid major games. It is also possible to have a great performance just before it hits big time as the body, by mobilising defences and the CNS can allow greater energy expenditures … of course in the days post game the player is usually knocked for six. The Russians used these techniques in Olympic games with athletes.