Well, the only instance I know is of Pompeius Strabo (father of Gnaeus Pompeius, you know, the Magnus, the Great) having a disease broke up in his camp when he besieged Cinna and others. This must mean that other earlier armies were always good at avoiding it. The Persian, Macedonian, Theban, Spartan, Athenian, none to my memory. So protocols were that wastes and such were thrown far from the camp? Just some help please!
Ancient armies definitely suffered from disease, but anonymous soldiers dying usually gets less mention than one famous dude. Those wounded in battle were extremely susceptible to bacterial infection, particularly tetanus, gangrene, and septicemia. But your question about waste indicates that you're thinking more about hygiene and diseases that might spread to otherwise uninjured soldiers.
The Egyptians and the Romans had established rules about setting up camps that involved keeping food and water away from waste which reduced the risk of diseases from contaminated water supply. However, with that many people in a constrained place there is still risk. The Antonine Plague, for example, of the second century was likely smallpox brought to Rome by legions stationed in the east. By some estimates 8-10% of Roman legionaries died from wound or infection while about 35% died from another health related clause, including epidemics. (Numbers from Gabriel's The Ancient World 145). Dysentery was probably the big killer. Records of it go back to 1550 BC and Greek and Roman medical texts describe the disease in detail and its prevention. Even with a relatively low mortality rate for most types, dysentery would slow an army down by immobilizing a large number of people for 2-3 weeks. Typhoid, typhus, and smallpox were other dangerous diseases with higher mortality rates and lasting effects. Sieges were particularly dangerous. It's likely that the infamous Athenian plague which killed about a third of that city's population during the Peloponnesian War was one or a combination of those three (since getting one often left the victim weakened and susceptible to another).
There are plenty of accounts in histories that make passing references to diseases spreading in armies, but unless someone particularly important died or the disease was particularly bad it's more of an expectation than something worth mentioning. Also ancient cities had staggeringly high mortality rates from disease so this was in no way a characteristic of military life that was considered unusual.