The Susceptible-Infectious-Recovered (SIR) model stands as a cornerstone in epidemic modeling, providing a mathematical framework to describe the spread of infectious diseases within a population. This model not only encapsulates the transition of individuals through different stages of an infection but also aids in understanding the dynamics of disease spread and the potential impact of public health interventions. In this Epi Explained, let’s delve into the SIR model, breaking down its components, mathematical underpinnings, and real-world applications to ensure a comprehensive understanding.

### The Basics of the SIR Model

The SIR model segments the population into three compartments: Susceptible (**S**), Infectious (**I**), and Recovered (**R**). Each compartment represents a group of individuals sharing the same status regarding the disease:

**Susceptible (S):**Individuals who have not yet contracted the disease and are vulnerable to infection.**Infectious (I):**Individuals who are currently infected with the disease and can transmit it to susceptible individuals.**Recovered (R):**Individuals who have recovered from the disease, assumed to have gained immunity, and thus cannot become infected again or transmit the disease.

The transition from S to I to R is modeled through a set of differential equations that describe the rate at which individuals move between compartments over time.

### A quick word about Differential Equations

Before we get into the mathematics of the SIR model itself, let’s take a quick detour to discuss Differential Equations, of which the SIR model is a set. Simply put, differential equations are mathematical tools used to describe the rate at which something changes over time, relative to other things. So for the SIR model, the S, I, and R variables are all changing in relation to each other as they represent the 3 stages any 1 person can be in. There are also equations called Ordinary Differential Equations (which the SIR model fits into) where most of the factors are static in relation to one changing variable (in our case time), and Partial Differential Equations, which will be covered sometime later as they are fundamentally more complex and aren’t needed right now.

### Mathematical Framework

The dynamics of the SIR model are governed by the following set of ordinary differential equations (ODEs)

- [math] \frac{dS}{dt} = -\beta \frac{SI}{N} [/math]
- [math] \frac{dI}{dt} = \beta \frac{SI}{N} – \gamma I [/math]
- [math] \frac{dR}{dt} = \gamma I[/math]

Where:

- [math] N [/math]is the total population size (assumed constant; [math] N = S + I + R [/math]).
- [math] \beta [/math] (Beta) is the effective contact rate, representing the rate at which an infectious individual infects susceptible individuals.
- [math] \gamma [/math] (Gamma) is the recovery rate, indicating the fraction of infectious individuals recovering from the disease per unit time.

### Breaking Down the Equations

**Susceptible Equation ([math] \frac{dS}{dt} [/math]****):**This equation depicts the rate of decrease in the susceptible population over time. The term [math] -\beta \frac{SI}{N} [/math] captures the process of susceptible individuals becoming infected after interacting with infectious individuals. The rate of this transition is proportional to both the number of encounters between susceptible and infectious individuals and the probability of transmission per encounter.**Infectious Equation ([math] \frac{dI}{dt} [/math]****):**This equation shows the rate of change in the infectious population. The increase in infectious individuals ([math] \beta \frac{SI}{N} [/math]) is offset by the recovery rate ([math] -\gamma I [/math] ), reflecting the number of individuals who recover and move into the recovered compartment per unit time.**Recovered Equation****([math] \frac{dR}{dt} [/math]****)**: This equation represents the rate at which the recovered population increases, directly correlated with the number of individuals recovering from the disease ([math]\gamma I [/math]).

### Understanding Parameters

**Effective Contact Rate ([math] \beta [/math]****):**This parameter is crucial in determining the disease’s ability to spread. A higher Beta signifies a faster spreading of the disease. Public health measures, such as social distancing and wearing masks, aim to reduce Beta.**Recovery Rate ([math] \gamma [/math]****):**This parameter indicates the proportion of infectious individuals who recover each day. The inverse of [math] \gamma [/math] ( [math] 1/ \gamma [/math] ) gives the average infectious period of the disease.

### Real-World Applications

The SIR model has been extensively applied in public health to simulate outbreaks, estimate disease parameters, and evaluate intervention strategies. For instance, during the COVID-19 pandemic, variations of the SIR model were employed to predict the course of the outbreak under different scenarios, guiding policymakers in implementing timely and effective public health measures.

### Example Problem

Say you are at a resort with 29 other people who all arrived yesterday (N = 30). Due to wild winter weather, as soon as you all arrived you were cut off from travel in or out of the resort. At the start of the first day, 1 person is infected with a fairly contagious cold, and given close proximity, this cold can be spread to 20% of the resort per day. Say the infectious period lasts 3 days, and there is an incubation period of 1 day. How long would it be until 10 people are likely infectious?

A) 1 Day

B) 4 days

C) 7 days

D) There are never 10 concurrent infections.

## Answer Key, click to reveal

### Solution Process

**Day 1:**

**Initially infectious:**1 person, who infects others.**New infections:**20% of 29 = 5.8, approximately 6 new infections. These individuals are now in the incubation period and not yet infectious.

**Day 2:**

- The initially infected person continues to be infectious.
- The 6 individuals infected on Day 1 are in their incubation period.
**Becoming infectious:**The initially infected person from Day 0 remains infectious, but no one else joins them in being infectious because the 6 are still in their incubation period.

**Day 3:**

- The 6 people infected on Day 1 become infectious.
**Total infectious:**Now, we have the initial person plus the 6 newly infectious individuals, totaling 7 people who are infectious.

**Day 4:**

- On Day 3, the group of 7 infectious individuals can infect more people. Given the infection rate and the number of susceptible individuals left, it’s very likely that additional infections occurred on Day 3, which would become infectious on Day 4.
- Considering the exponential growth pattern and the high likelihood of each infectious individual infecting around 20% of the remaining susceptible population, reaching a total of 10 infectious individuals by Day 4 is plausible.

### Correct Answer

The correct answer is **C) 4 days**.

### Explanation

By Day 3, we have a total of 7 individuals who are infectious. Given the dynamics of the disease spread and the daily infection rate, it’s highly likely that the necessary additional infections to bring the total number of infectious individuals to at least 10 occur on Day 3, with these individuals becoming infectious by Day 4. This is because the infectious individuals from Day 1 and Day 2 are actively spreading the infection, and the incubation period ensures that new infections from Day 3 become infectious by Day 4.

### Conclusion

The SIR model is a fundamental tool in epidemiology, providing insights into the dynamics of disease transmission and the effectiveness of control measures. By mathematically representing the flow of individuals through different stages of an infection, the SIR model helps public health scientists understand the potential trajectory of an outbreak and the impact of interventions. While incredibly simplified, the model’s strength lies in its flexibility and the foundational understanding it offers, making it a vital component in the arsenal against infectious diseases.

## Humanities Moment

The featured image for this Epi Explained was “Female – Head Study from an Italian Model” by

Sir Edward Coley Burne-Jones (English, 1833 – 1898). Sir Edward Coley Burne-Jones was a pivotal British artist and designer within the Pre-Raphaelite movement, known for reviving stained glass art in Britain and contributing to the decorative arts through partnership with William Morris. His works spanned various media, significantly influencing the Aesthetic Movement and leaving a lasting legacy in stained glass across notable British churches and institutions.