Before addressing the specific case of the smallpox epidemic among the Aztecs, it is important to introduce the basic epidemiological model known as SIR (Susceptible – Infected – Recovered). This model helps explain how an infection spreads through a population over time by classifying people into three main categories:

• Susceptible (S): Individuals who have not yet been infected but are at risk.
• Infected (I): Individuals who have contracted the disease and can transmit it.
• Recovered (R): Individuals who are no longer contagious, either because they have developed immunity or have died.

Predator

dy/dt = δxy - γy

The predator population growth δxy depends on successful kills and the reproduction rate; however, delta is likely be different from beta. The loss rate, an exponential decay, of the predators {\displaystyle \displaystyle \gamma y}γy represents either natural death or emigration