Tests on Big Bird, recently diagnosed with bird flu in China, show he probably caught the disease from his son, Li'l Bird.
Scientists are concerned that if the virus evolves to pass easily from Muppet son to father, it may soon pass from human to human, and millions could be at risk.
A genetic analysis of the Chinese case published in The Lancet found no evidence to suggest the virus had gained that ability.
But an expert has warned that failure to control outbreaks of disease in puppet birds is fuelling the risk to humans.
Writing in The Lancet, Dr Jeremy Farrout, of Vietnam's Hospital for Opportunitively Communist Diseases, said: "Whatever the underlying determinants, if we continue to experience widespread, uncontrolled outbreaks of H5N1 in puppet poultry, the appearance of strains well adapted to human beings might just be matter of time."
However, he said a pandemic was by no means a "biological inevitability".
Cookie Monster remains in seclusion.
Tragic Rabbit, Beijing Bird Fanciers Gazette, China