The SIR model for the spread of an infectious disease

(Source: This example is from the Calculus in Context project.)

The scene: An epidemic disease has broken out in a small town with a population of 50,000. We want to understand how the disease spreads through the population, to the point where we can predict details of the spread and the effectiveness of various control measures. For simplicity we make the following assumptions:

1. Everyone who contracts the disease eventually recovers - no one dies.
2. It confers permanent immunity on those who have recovered.
3. The total population remains fixed at 50,000.

Strategy and notation: Split the population into 3 distinct groups: the susceptibles, the infected, the recovered.

Let's use the following units and notation:

• $t$ = time in days
• $S(t)$ = susceptible population at $t$ (in number of people)
• $I(t)$ = infected population (in number of people)
• $R(t)$ = recovered population (in number of people)

(Now, can you guess why it is called the SIR model!)

Since the population remains constant, and the recovered people never become susceptible again, there is a strictly one-way flow in a schematic diagram of the situation:

We can now start building a model for predicting how $S(t)$, $I(t)$ and $R(t)$ might evolve if we know their initial values. A good way to do that is to find suitable mathematical experessions that capture the behavior of $\displaystyle \frac{dS}{dt}$, $\displaystyle \frac{dI}{dt}$ and $\displaystyle \frac{dR}{dt}$. Here are some explorations to help you get started:

• The schematic diagram immediately tells us the shape (or, trend) of 2 of the 3 variables as a function of time. Which are those variables? Sketch graphs showing their qualitative behavior.
• Although the behavior of the 3rd variable is less clear, can you conjecture what it looks like on a graph, and explain your thinking?
• From the general trend in your graphs, what can you say about the sign of each derivative: $\displaystyle \frac{dS}{dt}$, $\displaystyle \frac{dI}{dt}$ and $\displaystyle \frac{dR}{dt}$?
• Next, we will try to model $\displaystyle \frac{dS}{dt}$, or $S^\prime(t)$. In other words, we want to find a mathematical experession $expr$ such that $S^\prime(t)= expr$. When modeling it is often best to start with the fewest terms, and to keep them simple. Accordingly, think about how many terms, and what sign ($+$ or $-$) they should have, to capture the qualitative behavior of $S^\prime$ seen in your graph.
• It turns out that $S^\prime = -aSI~$ (where $a$ is a positive constant) does reasonably well for modeling $S^\prime$. The term $SI$ represents the product of the population of susceptible and infected people. Can you suggest a physical interpretation, or reason why $S^\prime = -aSI~$ works, and what the parameter $a$ might represent?
• Modeling $R^\prime(t)$ is relatively straightforward, since we assume the rate at which people recover only depends on $I$. Thus, the model looks like: $R^\prime = b I$, (with some positive constant $b$). Conjecture why $R^\prime$ does not depend on $S$ or $R$. Also, how might we estimate the value of $b$?
• The model for $I^\prime$ follows directly from the models for $S^\prime$ and $R^\prime$, keeping in mind that the total population $(S+I+R)$ is constant. Put everything together and write the system of 3 equations that models $S^\prime$, $I^\prime$ and $R^\prime$.

Solve the SIR model Let's simulate the following SIR model using Sage:

$S^\prime(t) = - 0.000003 S I \\ I^\prime(t) = 0.000003 S I - I/14 \\ R^\prime(t) = I/14$

Carry out the simulation with this starting scenario
At $t=0$: $S=45400$, $I=2100$, $R=2500$.

Interpreting model results A key goal of modeling is to assist in planning and decision-making. For example, in the case of an infectious disease, public health officials and medical facilities must be prepared to cope with the demand for medical supplies, doctors/nurses, hospital beds, etc. Here are some related questions to consider:

• What proportion of the total population will become infected?
• When does the infection reach its peak, and what is the peak number of infected people? (E.g., The sign of which derivative tells you whether new infections are increasing or decreasing?)
• It is not hard to show there is a threshold value of $S$. When $S$ is above this threshold, the infection rate increases, and when it is below the threshold it decreases. Can you find this value?
• If a vaccine exists, it is possible (according to this model) to vaccinate enough of the susceptible population to stop the spread of the disease. What is the minimum number of susceptible people that must be vaccinated to stop the spread?
• Would a quarantine policy help significantly reduce the number of people who catch the infection? How would you model and simulate the effect of a quarantine?