Can p. acne bacteria multiply inside facial creams?

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fiftyeight
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Can p. acne bacteria multiply inside facial creams?

I apply acne creams every day and afraid that when I apply the product,
my hand touches my face and then the cream again to take more of the cream,
usually I'd take 3 or 4 times from the cream.

I wanted to know if I might transfer acne bacteria from my face to the cream
and than the acne bacteria will continue multiplying in the cream and eventually
applying the cream will do more harm than good, is this possible?

I'll type the ingredients of one of the products I'm concerned about
it is called "Active Gel":

Ingredients:
Water, Artemisia Vulgaris Extract, Hexylene Glycol, Porpyridium Cruentum Extract,
Coleus Forskohili Extract, Camellis Sinensis Ext, Withania Somnifera,
Melia Azadirachta Seed Extract, Guar Gum, Carbomer, Polysorbate-20, Benzoic Acid,
Retynil Palmitate, Citric Acid.

Also concerned about moisturizer if it can happen

Thanx in advance