Key Epidemiological Models

Why This Matters

Epidemiological models are the backbone of how public health officials predict, respond to, and control disease outbreaks. You're being tested on your ability to understand why different models exist, when each is most appropriate, and how they inform real-world interventions—from vaccination campaigns to social distancing policies. These models connect directly to core epidemiology concepts: disease transmission dynamics, population immunity, outbreak prediction, and intervention effectiveness.

Don't just memorize model names and their compartments. Know what assumptions each model makes, what types of diseases each best represents, and how changing parameters like $R_0$ affects predictions. Exam questions often ask you to select the appropriate model for a given scenario or explain why one model captures certain disease dynamics better than another. Master the underlying logic, and you'll be ready for anything.

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Compartmental Models: The Foundation

Compartmental models divide populations into distinct groups based on disease status and track how individuals move between these groups over time. The key insight is that disease dynamics depend on the relative sizes of each compartment and the rates of transition between them.

SIR Model (Susceptible-Infectious-Recovered)

• Three compartments (S, I, R) form the simplest framework for modeling infectious disease spread through direct contact
• Complete immunity assumption—recovered individuals cannot be reinfected, making this ideal for diseases like measles or mumps
• Foundation for $R_0$ calculations—the basic reproduction number emerges naturally from SIR dynamics and transition rates

SEIR Model (Susceptible-Exposed-Infectious-Recovered)

• Exposed compartment (E) captures the incubation period when individuals are infected but not yet infectious
• Essential for diseases with latent periods—COVID-19, influenza, and Ebola all require this additional compartment for accurate modeling
• More realistic timing predictions—helps public health officials anticipate when exposed individuals will become contagious

SIS Model (Susceptible-Infectious-Susceptible)

• No lasting immunity—recovered individuals return directly to the susceptible pool, enabling repeated infections
• Models endemic diseases like the common cold, gonorrhea, and bacterial infections where reinfection is common
• Continuous transmission dynamics—disease can persist indefinitely in a population without external reintroduction

Compartmental Models (General Framework)

• Flexible architecture—compartments can be added or modified to match specific disease characteristics (e.g., adding vaccination status or age groups)
• Transition rates between compartments represent biological and behavioral parameters like infection probability and recovery time
• Intervention analysis—changing transition rates simulates effects of treatments, vaccines, or behavioral changes on disease spread

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Modeling Approaches: Deterministic vs. Stochastic

The choice between deterministic and stochastic approaches depends on population size, the questions you're asking, and how much real-world variability matters. Deterministic models give you the average expected outcome; stochastic models show you the range of possible outcomes.

Deterministic Models

• Fixed parameters and equations produce the same output every time, assuming no randomness in disease transmission
• Best for large populations—when numbers are big enough, random variation averages out and deterministic predictions hold
• Differential equations (like $\frac{dS}{dt} = -\beta SI$) describe how compartment sizes change continuously over time

Stochastic Models

• Incorporate randomness to reflect real-world uncertainty in who infects whom and when transmission events occur
• Critical for small populations—random chance can determine whether an outbreak dies out or explodes when case numbers are low
• Probability distributions replace fixed values, generating a range of possible epidemic trajectories rather than a single prediction

Reed-Frost Model

• Classic stochastic model that simulates infection spread using random processes and probability calculations
• Contact-based transmission—focuses on the probability of infection during each interaction between susceptible and infectious individuals
• Ideal for small, closed populations—schools, nursing homes, or households where individual contacts matter

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Individual-Level Models: Capturing Complexity

When population-level averages aren't enough, individual-level models simulate how heterogeneity in behavior, social connections, and decision-making affects disease spread. These models sacrifice simplicity for realism.

Agent-Based Models

• Simulate individual agents with unique characteristics, behaviors, and decision-making rules that affect disease transmission
• Capture heterogeneity—differences in age, occupation, risk behavior, and social networks influence who gets infected and when
• Scenario testing—explore "what if" questions about interventions targeting specific groups or behaviors

Network Models

• Nodes and edges represent individuals and their connections, mapping the actual structure of social contact patterns
• Network topology matters—highly connected individuals (hubs) can become superspreaders, while clustering affects local outbreak dynamics
• Reveals hidden vulnerabilities—identifies which connections to target for maximum intervention impact

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Key Metrics: Reproduction Numbers

Reproduction numbers quantify transmission potential and guide intervention thresholds. Understanding the difference between $R_0$ and $R_t$ is essential for interpreting outbreak data and evaluating control measures.

Basic Reproduction Number ($R_0$)

• Average secondary infections from one case in a completely susceptible population—the theoretical maximum transmission potential
• Threshold value of 1—when $R_0 > 1$, outbreaks can grow exponentially; when $R_0 < 1$, disease dies out
• Determines herd immunity threshold—the proportion needing immunity to stop transmission is approximately $1 - \frac{1}{R_0}$

Effective Reproduction Number ($R_t$)

• Real-time transmission metric that accounts for current immunity levels, interventions, and behavioral changes
• Dynamic value—changes throughout an outbreak as more people recover, get vaccinated, or modify their behavior
• Intervention effectiveness measure—if $R_t$ drops below 1 after implementing controls, the outbreak is declining

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Quick Reference Table

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Self-Check Questions

1. A disease allows reinfection after recovery and persists in the population indefinitely. Which model best captures this dynamic, and why would SIR be inappropriate?

2. Compare $R_0$ and $R_t$: If a disease has $R_0 = 3$ but current $R_t = 0.8$, what does this tell you about the state of the outbreak and population immunity?

3. You're modeling a potential outbreak in a nursing home with 50 residents. Would you choose a deterministic or stochastic approach? Explain your reasoning.

4. Both SEIR and SIR models assume permanent immunity. What distinguishes them, and for which type of disease would choosing SIR over SEIR lead to inaccurate predictions?

5. An FRQ asks you to evaluate an intervention targeting "superspreaders." Which modeling approach would best capture this strategy's effectiveness—compartmental, agent-based, or network models? Justify your choice.