It starts with sporadic reports of animals dying or showing signs of sickness. Birds are a likely culprit, because they can travel and mix without restriction. The scientists might be straight on it, identifying the particular virus, and giving it a catchy name that sounds like a droid from star wars. At this point nobody is too concerned. It's just dead animals. We're at the World Health Organisation's level 1.
When people start to take notice is when humans start getting infected. The media starts whipping itself into a frenzy as people start dying; but so long as it's only the very young, the old, and the infirm, then it isn't too different from the standard flu that surfaces every year. When young healthy people start dying, and if it seems as though the disease is passing between people, then it's a different story. This is the point when politicians start telling you that everything is under control; which at the same time taking exotic vaccinations home for their close family.
Soon people are dying everywhere, the advice is to stay indoors and avoid contact with anyone else. Soldiers in bio-hazard suits are walking the streets torching pigeons with flame-throwers, there's a strange smell coming from your neighbours house, and there's only repeats and sweaty news-readers on TV. Rich people are throwing fist-fulls of cash at shady pharmaceutical salesmen, and labs are running 24/7. Eventually the pandemic starts to burn itself out; but what's left?
There have already been several pandemics, and in many regions there is seasonal flu. Over 39,000,000 people are currently living with HIV/AIDS, which in most cases will significantly reduce their lifespan. To be apocalyptic a pandemic has to walk a tight-rope of being fatal on one hand, but not so fatal that people die before they can pass it on. There is the possibility of a 'carrier', such as the suspected role of the flea in the bubonic plague, but a lot of people don't have the same exposure to animals that they used to.
It might not even be viral. Overuse and misuse of antibiotics is leading to increased resistance from bacteria. We could get to the point where minor infections become fatal.