Infectanator is such a game. You get one shot on each screen to start the infection. Everyone who comes into contact with it becomes a zombie. The zombies then have a percentage chance to turning anyone they come into contact with also into a zombie. Each screen has a target number of infected you have to hit.
You can purchase upgrades between levels, thanks to the gold coins dropped by the infected. I pumped all my cash into increasing the chance of infection, but you might want to boost their lifespan (so they come into contact with more people) or up their damage resistance (useful when the zombie killing agents turn up).
At the time of writing the current high score is over 900,000. Which, er, leaves my first go score of 122,000 for dead.