Compartmental Frameworks The foundation of epidemic theory is the SIR model, formulated by Kermack and McKendrick, which divides a population into three compartments: Susceptible, Infectious, and Recovered (or Removed). Transitions between these states are governed by differential equations where the rate of new infections is proportional to the contact between susceptible and infectious individuals. Variations include the SEIR model (adding an "Exposed" latent period) and SIS model (where individuals return to susceptibility after recovery).
Deterministic vs. Stochastic Models
• Deterministic Models: Utilizing ordinary differential equations (ODEs), these models describe the average behavior of large populations. They assume fixed rates and continuous variables, predicting that if transmission conditions are met, the disease will reach a stable endemic equilibrium.
• Stochastic Models: Essential for small populations or the early stages of an outbreak, these models (e.g., Markov chains) incorporate randomness. Unlike deterministic models, stochastic simulations show that a pathogen can go extinct purely by chance even if conditions favor spread, because the "disease-free" state is an absorbing state from which the system cannot escape.
Key Metrics and Thresholds
• Basic Reproduction Number (R0): Defined as the average number of secondary infections produced by a single infected individual in a totally susceptible population. It acts as a sharp threshold: if R0>1, an epidemic can occur; if R0<1, the disease dies out.
• Effective Reproduction Number (Re): This tracks transmission potential as the susceptible population declines or interventions are applied. An epidemic ends when Re drops below 1.
• Herd Immunity: The population fraction that must be immune to prevent epidemic growth is calculated as 1−1/R0. Vaccination strategies aim to reach this threshold to break the chain of transmission.
