Introduction

The main task is to calculate the epidemiological parameters by comparing the output of mathematical models to data and investigate what kinds of interventions would be required in order to control the outbreak.

Background

One of the main goals in modelling of infectious diseases is to be able to understand the Basic Reproduction Number or $R_0$, but what actually is it? The exact definition is:

The average number of secondary cases for every primary case in a completely susceptible population

This may sound complicated, but it's fairly easy to understand. As an example, imagine a person is infected and passes the infection on to two people. Then one of those people passes the infection on to one other person and the other passes it on to three people. On average each person infected two other people, so we would estimate the $R_0$ of that disease to be 2.

Use the tool below to explore the consequences of different $R_0$ values. An epidemic is simulated for three generations and who infects who are connected by lines.

Notice that if $R_0$ is below one then the epidemic is not likely to get to the third generation. If it's above one then this is possible. It turns out after a long time, if the $R_0$ is less than one then the infection is guaranteed to die out, however if it is bigger than one then there's a probability it will persist and continue to become an epidemic. Estimating the $R_0$ is therefore vitally important for public health intervention.

Model fitting

In order to calculate the $R_0$ and other values of interest, we need some way of comparing a model to data. We can do this by looking at the mean squared error.

Look at the example below. Here we have a simple model that has two parameters: one that controls how steep the gradient of the output is and another that controls where the output intersects at zero. The mean squared error can be calculated by comparing each data point to the model and squaring the difference. All these are averaged (and a square root is also taken) to produce the mean squared error. The important thing is that the best model parameters are the ones that provides the smallest error.

Model comparisons

Diseases have many different modes of spreading either directly or through another organism like a mosquito. Diseases also have quite varied biological characteristics such as how long it can remain in a host before they become infectious. We use models to explore the consequences of these different phenomena and compare them to data in order to provide evidence as to which one is correct.

A list of likely models for the pathogen are given below.

Direct contact model

The simplest model we can consider is where the pathogen is spread through direct contact with people who are infected. Also someone who gets infected immediately becomes infectious themselves. Here's what the model looks like:

It turns out from this model if we know the infectivity (how many people on average one person infects someone else in a day) and the rate of recovery (how fast it takes to no longer be infectious) we can derive the $R_0$ as

$$R_0 = \frac{\text{infectivity}}{\text{rate of recovery}}$$

Many interventions exist at reducing the effective size of the $R_0$ such as the use of antibiotics, taking preventative medicines (prophylaxis) or hand washing. For the zombie outbreak we can ask if a vaccine were introduced then how many people would need the vaccine before the outbreak is contained. For this and similar models there is a threshold and if we can vaccinate above this then the pathogen is guaranteed to be eliminated:

$$1- \frac{1}{R_0}$$

As an example measles has an $R_0$ of about 13, which means optimal vaccination coverage is 92%. This means that in order for there to not be an outbreak of measles, the vaccination coverage needs to be very high.

Vector-borne model

The vector borne model is split into two compartments: humans and vectors (mosquitos, fleas, birds etc.). Humans aren't infected directly, but rather through the vector. If the vector was a mosquito then a susceptible vector bites an infected person and incubates the pathogen becoming infected itself. The vector then bites a susceptible person transmitting the pathogen.

Latent model

Some pathogens may have a latent stage before an individual becomes Infectious. Typically this is called the $E$ or exposed stage

The progression of the pathogen would look something like this:

The $R_0$ is the same as for the direct-contact model, but the latent stage changes how quickly the epidemic advances

Use the menu above to navigate to the different sections. Use the fitting section to compare the models to data and work out what the best parameters are that match the observed epidemic. Then use these parameters to simulate what might happen if an outbreak occurred in North America and what interventions might be necessary to stop the outbreak.

The math

If you're curious about the equations used for each of these models then they are given below.

Direct-contact model

$$\frac{dS}{dt} = -\beta SI$$ $$\frac{dI}{dt} = \beta SI - \gamma I$$

This produces an $R_0$ of

$$R_0 = \frac{\beta}{\gamma}$$

Vector-borne model

$$\frac{dx}{dt} = mabz(1-x) - rx$$ $$\frac{dz}{dt} = acx(e^{-g\nu} - z) - gz$$

This produces an $R_0$ of

$$R_0 = \frac{ma^2bc}{gr}e^{-g\nu}$$

Latent-stage model

$$\frac{dS}{dt} = -\beta SI$$ $$\frac{dE}{dt} = \beta SI - \alpha E$$ $$\frac{dI}{dt} = \alpha E - \gamma I$$

This produces an $R_0$ of

$$R_0 = \frac{\beta}{\gamma}$$