Last I looked neither zinc nor silver had any anti-viral effect inside the body nor any preventative effect in a person not suffering a nutrient deficiency.
There are all sorts of drugs available but they just control the symptoms that result from your body fighting off your average cold. They can sometimes make you more comfortable and allow you to either sleep or work through the normal disease process but they don't change the timing or outcomes.
The old saying is that getting over a cold takes ten days if you do nothing and a week-and-a-half if you do everything right.
But if you feel better about it stuffing yourself with vitamins or chemicals have at it. Sometimes I down a couple of 500mg tablets of vitamin-C. makes me feel like I am doing something and, in return, I get Technicolor urine.
When I used to drink I chased the vitamin-C with a tall scotch. Followed by 16 hours of hugging my pet bedbug. Rinse and repeat as needed.