Compartmental Epidemiological Models: Discussing Fundamental Elements via the SIR Model

Abstract

We present a didactic treatment of compartmental disease spread modeling centered on the SIR framework, aimed at mathematics education. Our focus is on how modeling choices encode transmission mechanisms and data-collection practices. We begin by stating the structural hypotheses of compartmental models, including population partitioning, homogeneous mixing, flow closure, and Markovian transitions. We then derive the basic SIR dynamic equations in both continuous-time (differential equations) and discrete-time (difference equations) settings. A key goal is to clarify the distinction between density-dependent and frequency-dependent transmission. We do this by constructing the force of infection from the contact rate, prevalence, and per-contact transmissibility while tracking units and interpretations. We go beyond the usual mass-balance SIR model by introducing the discrete Kermack-McKendrick perspective through the cumulative force of infection. This naturally leads to discrete SIR updates with an exponential escape probability term. We also derive Bernoulli-based exponential formulations for both density- and frequency-dependent transmission, interpret these probabilistically over observation intervals, and obtain the associated discrete infectivity kernel. This highlights how geometric infectious survival emerges from per-step removal. The paper includes reproducible, introductory-level Python code to implement the basic discrete SIR model, the continuous SIR model, and the exponential discrete variants. We provide instructors with an example of parameter estimation from prevalence data using least squares, along with guidance on interpretation and limitations. This framework is designed to help teachers integrate model assumptions, units, simulation, and inference in a coherent introduction to infectious disease modeling.