# Infectious Disease Epidemiology

Infectious disease epidemiology is the study of how infectious diseases spread in a population.

## SIR model

An epidemic is when the number of people infected with a disease is increasing. A simple model of an epidemic is called the SIR model. S,I, and R stand for:

 S - susceptible. These are people that are not infected with the disease yet. However, they are not immune to it either and so they can become infected with the disease in the future. I - infected or infectious. These are people that are infected with the disease and can transmit the disease to susceptible people. R - recovered. These are people who have recovered from the disease and are immune, so they can no longer be infected with the disease.

Below is a graph of an SIR model. The bottom (x) axis is time. The left (y) axis is population (number of people.)

Blue represents the number of susceptible people. At the beginning of the epidemic, the number of susceptible (blue) people decreases as the number of infected (green) people increases. Gradually the number of recovered (red) people increases. The point at which the maximum number of people are infected at one time is called the height of the epidemic, after which the number of infected people falls, because there are fewer susceptible people to infect. In this particular model, after the epidemic is over, all of the people have been infected and recovered. This is not always the case; sometimes some susceptible people remain uninfected.

This model is also missing some elements, such as the creation of new susceptible people by being born, and the removal of susceptible, infected, and recovered people who die.

In addition, this model does not work well for every disease. It works well where people recover from a disease and become immune, such as measles. Not all infectious disease follow this pattern. For instance, people who are infected with HIV never recover and there is no vaccine, so there is no group of recovered people, only susceptible and infected people. There is also the case where people recover from a disease but do not become immune or lose their immunity over time. For instance, when people get the flu, they are generally only immune for a short time before they become susceptible again.

There are also different kinds of models when a disease is spread through the water, such as in the case of cholera, or by a vector like a mosquito, such as malaria.

### Quiz

1

 At the height of the epidemic, how many people are infected with the disease?

2

 Time = at the height of the epidemic.

3

 At time = 60, how many people have been infected with the disease in total?

4

 At time = 10, approximately how many people are infected with the disease?

## Basic Reproduction Number and Herd Immunity

When enough people are immune to a disease, either from vaccination or from having recovered from infection, this reduces the number of susceptible people in the population to levels so low than an epidemic is less likely to happen.

A very important number for determining whether an epidemic occurs is something called R0, pronounced "R naught." It refers to how many people any given person will infect, on average. It is called a basic reproduction number.

Look at the figure on the right. In the figure, the R0 of Ebola is 2. Each infected person infects 2 more people. Below that the R0 of SARS 4. Each infected person infects 4 more people.

If R0 > 1, an epidemic will occur. If R0 < 1, an epidemic won't occur. All known infectious diseases have R0 > 1, because otherwise they wouldn't be able to cause epidemics. However, we can use vaccination to make enough people immune in a population to stop epidemics from happening. We can also use other measures to make the effective reproduction rate (Re) lower than the basic reproduction rate (R0).

An example of vaccination working really well is the smallpox vaccine, which stopped smallpox virus from spreading so well that it longer exists, except in laboratories. An example of making the effective reproduction rate lower than the basic reproduction rate is using condoms to stop sexually transmitted diseases, or social distancing to stop respiratory diseases.

### Herd immunity threshold

Vaccination or recovery from a natural case of the disease can make people immune to it. In the SIR model above, herd immunity is what causes the number of infected people to drop. If enough people are immune, this can stop further epidemics from occurring.

You can calculate how many people must be immune for future epidemics to not occur, with the following equation:

qc = 1 − 1/R0

qc is the ratio of people that must be immune for future epidemics to not occur.

### Quiz

1

 At time = 0, 1 person is infected with a disease. If R0 is 2, and patient 0 does not recover by time = 1, how many people are infected at time = 1?

2

 At time = 1, 4 people are infected with a disease. If R0 is 2, and no patients recover by time = 2, how many people are infected at time = 2?

3

 If R0 is 4, what ratio of the population must be immune for there to be herd immunity?

4

 If R0 is 2, what percentage of the population must be immune for there to be herd immunity?