SIR Model for Epidemiology, Ordinary Differential Equations 

Let's look at the SIR model, a basic framework to understand the spread of a disease within a population through a set of ordinary differential equations. We begin by outlining the model's fundamental assumptions: a constant-sized, homogeneously mixing population without births, deaths, or migrations unrelated to the disease. The disease has some level of transmissivity, affects people for a certain fixed amount of time, and cannot cause reinfection. Then the model categorizes individuals into three groups: Susceptible (S), Infectious (I), and Removed (R), with transitions from S to I to R based on fixed disease transmission and recovery rates (involving parameters beta and gamma, respectively).
We mathematically derive the flow of individuals between these compartments, leading to a set of ordinary differential equations. By adjusting parameters, we simulate various scenarios, such as the impact of social distancing and mask wearing on disease spread. The analysis highlights the model's ability to predict the epidemic's peak and duration, and the importance of beta in controlling the spread.
Lastly we introduce an alternative formulation using proportions of the population, removing the total population size from the equations.
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