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This project contains three simple mathematical models for the spread of an epidemic: Standing Disease, Network Disease and Counter Plague. They are taken from the Disease Dynamics Schools Pack on the Motivate website, where you will find more activities and background material.

The purpose of a mathematical model is to simplify a real but complex situation so that individual features can be studied more easily. This project provides several models, which could be used for several sessions, to help students understand how epidemics grow and then die out. These models are not meant to be realistic, but give an introduction to mathematical modelling: criticising the models is really important - what features of reality do they help us to understand, what features do they miss?

Studying epidemics is an important aspect of public health. We expect our governments to plan ahead so that when an epidemic occurs, there are plans in place to provide for the needs of both individuals and society generally.

Some epidemics are annoying but not generally life-threatening, such as the common cold or childhood chicken pox. Other epidemics are more serious, such as measles, flu and HIV.

Infectious diseases are transmitted by a variety of methods:

- contaminated water, eg. cholera
- contaminated food, eg. Salmonella infection
- air contaminated by coughing and sneezing, eg. the common cold
- another living organism, eg. mosquitos spread malaria
- direct bodily transfer, eg. HIV

An important parameter in deciding how rapidly an epidemic will escalate and how long it will last is R_{0}, the reproductive ratio or transmission index (different names for the same thing). For the Standing Disease R_{0} is 2, since each infected person causes two new infections. Any disease where R_{0} is greater than 1 will escalate, whereas if
R_{0} is less than 1 it will die out. R_{0} will not remain constant throughout an epidemic, but is a good way of modelling the initial escalation phase.

An important issue in controlling epidemics is that of vaccination. The purpose of vaccination is not to immunise an entire vulnerable population, but to immunise enough people so that R_{0} is brought below 1. The Network Disease is one way of investigating what effect immunising key individuals would have - the less connected the network, the fewer routes for
infection.

It is straightforward to calculate what proportion of a population should be vaccinated. Suppose R_{0} is 5, meaning that on average each infected person will infect 5 more. Clearly we would expect this to lead to a rapid increase in the number of cases of the illness, and an escalation of an epidemic. However if 4 out of 5 people are immune to the illness, then there
will be 5 new cases for 5 existing cases. Vaccinating a little over 80% of the population would be enough to reduce R_{0} to below 1.

Mathematical modelling is an important way of tackling problems which are complex and unlikely to yield simple solutions. The point is not to be realistic, but to incorporate some significant features of reality. Too much realism makes a model complicated, too little makes it unhelpful. Modellers generally start with very simple models, then increase the complexity, until what they have is good enough to provide useful information.

The three models in this project are very simple, but do provide useful information.

Standing Disease illustrates how quickly a disease with R_{0} greater than 1 can take off. It would only take 33 steps for the whole world to be infected if R_{0} = 2, and there could be perfect transmission from one person to two people at each stage. However, experience tells us that epidemics generally self-limit or become stable in a population (as malaria has
in sub-Saharan Africa, for instance).

The Network Disease includes attempted transmission to people who are already ill (and this could also be seen as transmission to people who have become immune), and thus provides a mechanism which shows how transmission of a disease might be constrained.

Counter Plague is a model which incorporates a way of changing R_{0} in more subtle ways. The blue dice give R_{0} = 5/6, whereas the red dice give R_{0} = 7/6. Using the values on the blue dice will mean that in general epidemics die out quite quickly, whereas using the values on the red dice will mean they tend to escalate. These calculations depend
on the probability of getting a particular value on a die being 1/6. So for the blue dice:

R_{0} = 0 x 1/6 + 0 x 1/6 + 1 x 1/6 + 1 x 1/6 + 1 x 1/6 + 2 x 1/6 = 5/6

and for the red dice:

R_{0} = 0 x 1/6 + 0 x 1/6 + 1 x 1/6 + 1 x 1/6 + 2 x 1/6 + 3 x 1/6 = 7/6

How does this model help us to understand real epidemics?

What features of real epidemics does it miss?

Could we improve this model?

The background tab links to an introduction to the maths behind epidemics, part of the Disease Dynamics Schools' Pack on the Motivate website.

More ideas links to four related simulations which model how an epidemic might progress (or not) in a class or in two classes in a school.

More advanced ideas links to an simulation which allows students to input different probabilities of catching a disease and dying from it, and hence test the effects of policy decisions.

Read: science links to the Maths and Our Health packs on the Motivate website, where you will find videos and classroom resources in five different health science areas.

Health statistician links to an article on Plus about Florence Nightingale, the pioneer of using data to understand and address health issues.