One of the biggest challenges for modern medical science will be dealing with diseases as the human population grows. On the one hand, we have lived in a golden age, with vaccines for the most common viral diseases and antibiotics for bacterial infections. However, bacteria have evolved to resist antibiotics, and recent estimates indicate that the most resistant forms of bacteria (tuberculosis and E. Coli variants) respond to only one antibiotic in the medical inventory. Perhaps even worse, as human populations increase humans come into contact with reservoirs of viral agents which we have no natural immunities to (HIV, Ebola, Marburg, Hantavirus...). With modern capitals all less than a day from one another in travel time, catastrophic amplification and spread of disease is one of the most realistic `doomsday' scenarios in the modern world.
The mathematical modelling of diseases began with Kermack and McKendrick studying the advent of plague in Bombay in 1927, and was put on firm theoretical setting by Anderson and May in the 70's.