JUST THROW THE DARNED THING AWAY!

That old sponge is a Petri dish!

We use two sponges in our kitchen. One, the little one with the rough and smooth sides, we reserve for  dishes. The other larger one has a less defined purpose. It gets used for cleaning the sink and performs occasional counter duty. 

The wet sponge is a perfect place for bacteria to live and prosper. The bacteria in the sponge get regular  feedings as we clean up the crumbs on the counter or the lasagne in the casserole, and so on. Bacteria love it!

It helps to  let the sponge dry between uses because bacteria do need moisture, but eventually the big one gets stinky. Washing it out  with Chlorox, abates the nasty smell left on my hands from the old sponge for a few days. Some microwave their sponge, some run it with the dishes or in the laundry. Actually, recent research has shown that attempts at cleaning household sponges can only make the matter worse. It turns out that none of these methods of cleaning is all that effective in getting rid of the most noxious bacteria. We might get rid of a lot of the bacteria in these cleanings, but those bacteria that remain are resistant and can prosper in the absence of  competition.

For most of us whose immune  systems are in good working  order a little bacterial contamination is not a problem. So we can hold off on replacing that sponge for a while longer. Overlook that funky smell. It's a different story for those whose immune systems are stressed, for example from a cancer or maybe somebody fighting a serious infection. At these times our bodies are not well defended and opportunistic bacteria, like some of those that can live in funky sponges, can be dangerous.

Economics aside, you would be better to think of using paper towels (if you are a gardener, paper composts at home very nicely), especially when cleaning up, for example, vomit or 'cat urp.' Restaurants use cloth towels that are frequently washed, and a restaurant must maintain a strict standard of cleanliness if it hopes to pass a health inspection. At a minimum, don't use the kitchen sink sponge to clean messes on the floor. 

The overarching recommendation is that you replace that sponge frequently, even weekly, or more often if somebody in the house is ill (think about cross contamination in the kitchen). 

There is no virtue or economy in keeping a kitchen sponge that fails the sniff test. It's a reservoir of bacteria -- your own kitchen Petri Dish. Ditch it!

Want to read more? Here are two articles from the New York Times for you:
https://nyti.ms/2uNnbBd and https://nyti.ms/2urCwap