There is no scientifically proven way to improve the immune system, but lifestyle changes, diet, and others factors may in theory. One thing is an infection can set you back, especially with a compromised immune system, such as cancer, and rapidly progress, have a domino effect, which happened to me after 5 days of chemo only.
As far as Probiotics, there are positives and negatives aspects. In infants, and people with a compromised immune system, like cancer, they may not be advised, and can do more harm than good. In healthy people it may be ok as long as their stored properly, not expired, and has a certain amount of good bacteria.
http://ajcn.nutrition.org/content/83/6/1256.fullAs far as vitamins, you may want to check with your doctors before using. Some, like antioxidants, may interfere with treatment by protecting cancer cells, and not recommended in certain cancers like lung, prostrate, with vitamins A, E. Vitamins are either fat or water soluble, so put added stress on the already stressed organs and fat souluable vitamins can build up to dangerous levels with vitamins A,D,E, K. A healthy diet of natural foods, mainly fruits vegetsbles, and adequate protein, is better. One should work with the CCC dietician or nutritionist.
http://oralcancernews.org/wp/antioxidants-may-cause-more-harm-than-good-in-cancer-patients/Here is a laundry list of style changes, diet, and infection controls that may help.
Adequate sleep
Avoid stress
adequate diet, mainly fruits, vegetables
Light exercise. Too much weakens the immune system
Control blood pressure, diabetes other co-morbidities
Avoid Infection by avoiding public exposure, sickness, use PPE (personal protection equipment) mask, gloves, wash hands with soap water
Use liquid soap
Use PPE when doing yard work, cleaning Pets or avoid totally
Cook and store foods properly to specified temperature, storage regulations
Avoid raw fish, meats, Walsh hands after touching
Use caution with outside food sources, if using
Don't smoke, need we say that lol.
Don't drink alcohol
Change tooth brushes often
Don't share bath cloth, towels, soap, again use liquid soap
Use an electric shave or don't shave at all
I hope this helps.