Avian H7N9 influenza viruses are evolutionarily constrained by stochastic processes during replication and transmission in mammals.

H7N9 avian influenza viruses (AIVs) have caused over 1,500 documented human infections since emerging in 2013. Although wild-type H7N9 AIVs can be transmitted by respiratory droplets in ferrets, they have not yet caused widespread outbreaks in humans. Previous studies have revealed molecular determinants of H7N9 AIV host switching, but little is known about potential evolutionary constraints on this process. Here, we compare patterns of sequence evolution for H7N9 AIV and mammalian H1N1 viruses during replication and transmission in ferrets. We show that three main factors-purifying selection, stochasticity, and very narrow transmission bottlenecks-combine to severely constrain the ability of H7N9 AIV to effectively adapt to mammalian hosts in isolated, acute spillover events. We find rare evidence of natural selection favoring new, potentially mammal-adapting mutations within ferrets but no evidence of natural selection acting during transmission. We conclude that human-adapted H7N9 viruses are unlikely to emerge during typical spillover infections. Our findings are instead consistent with a model in which the emergence of a human-transmissible virus would be a rare and unpredictable, though highly consequential, 'jackpot' event. Strategies to control the total number of spillover infections will limit opportunities for the virus to win this evolutionary lottery.