Not the probable answer. But the answer as to why neanderthals died out is still conjecture.
Neanderthals existed on the European and Asian continents for better than 400,000 years. They were ice-age hunters who followed, killed and ate big game-mammoths. Neanderthals ‘ prey included musk oxen and aurochs, the large ancestor of modern cattle, to name a few.
More probable was that rapid climate change as our climate warmed at the end of the last ice age along with increased hunting from modern man moving into their territory, saw the large scale demise of many of the species they hunted for food. What remained of those animals migrated north to cooler climates. Musk ox are an arctic mammal still found today.
You can read about that here:.
We do know that there was an overlap of many thousands of years when modern man co-existed with Neanderthals, and interbred- likely not always consensually. Modern non African populations still carry Neanderthal DNA.
Another interesting article is linked here.
We have witnessed pandemics and other disease related human die-offs such as plague in our history. We have no way to know what part that may have played in Neanderthals demise.
If Neanderthal populations became stressed due to competition from modern man, lack of new hunting technology such as the bow and arrow, brought to bear by growing populations of modern man, the competition for food along with reduced available game, might well have also made them more vulnerable to disease. We simply don’t really yet know.
A quote from the LiveScience article above also postulates:
“Neanderthals may not even have truly disappeared, but instead have been assimilated into modern human populations.”