No, it would never occur to me to sterilize my toothbrush. If I do decide to, Listerine now makes a Whitening rince made from sweetened hydrogen peroxide. Tastes much better than bleach or regular hydrogen peroxide.
Since research says most illnesses are transmitted through mucus membranes, like the nose & eyes (tear ducts,) I've always heard it is much more important to wash hands often & keep from touching the eyes & nose after touching things other people have handled. I wash my hands immediately after I get home from being out & about.
I would think it would be more beneficial to go around with a Clorox wipe (or bleach) and keep counters & the things that the whole family touches: remote controls, refidgerator knob, door handles, telephone, etc., constantly wiped down.
If you need a constant supply of new, FREE toothbrushes, look for Rite Aid's "Free After Rebate" sales. Almost every month I get a free toothbrush.

You can now enter the receipts online and get immediate confirmation that your rebate has gone through. No more cutting UPC labels & mailing in forms, wondering if you are really going to get your money back.
Yesterday I received a check for $15 back in FREE items!

Last month, I got a check for $18. These are only for the things I bought. I don't get everything offered, just because it's free.
The rebates are one per address, so if DH or you works at a different address, you can get another toothbrush & have another rebate check sent there.
Walgreens does something similar to the Free After Rebates, but you do have to mail in the rebates.
There are also coupons in the Sunday newspaper inserts. You can get Oral-B & GUM toothbrushes down to about 39 cents at both Rite Aid & Walgreens during a toothbrush sale each month.
Last idea is to look online & see if you can get a carton of assorted colored toothbrushes at a wholesale price.