# Mathematical Epidemiology

by Jillian Byrnes, La’Ziyah Henry, Delphine Liu, Sophie Ussery, and Elizabeth Whetzel.

## What is Mathematical Epidemiology?

What is mathematical epidemiology? Well, mathematical epidemiology is when mathematicians use math to predict outcomes in various statistical problems. These problems include growth in infectious bacteria, change in population, and even the effects of climate change. Why is this used? It is used because it doesn’t need a complete set of data to figure out a solution, as long as you can create an equation and plug in the values.

Who uses it? Mathematicians and scientists use it in fields such as biotechnology, medical science, civil engineering, and as public health professionals.

## Mathematical Models

Mathematical models are used to predict the outcome of a real-life situation, and are shown by one or more equations. To create a model, you would need to collect and examine data, consider the effects on your scenario, and make predictions about future data amounts. Let’s say that you work in Biotechnology, and you’re trying to figuring out how much Ciprofloxacin to put in an antibiotic. You can use mathematical models to predict the amount of Ciprofloxacin you’ll need in one pill based off of how fast it kills bacteria, as well as the starting amount of bacteria in the antibody recipient.

## Linear Population Models

Linear population models are models where you study existing population models to predict future population, but without calculating the possibility of a change in the rate at which it is increasing. You’d use a type of conversion called dimensional analysis, which is converting units, such as hours to days, and days to weeks. That way, if you only had the data set for a day, you could multiply it by 365 to predict the population change in a year. This is very useful as a civil engineer, because you need to predict how many people will drive on certain roads so that you know how wide to make them.

## Nonlinear Population Models

The difference between linear and nonlinear population models are the equations used. This means that linear models end up being lines and nonlinear models end up as curves on a graph. Here are some key terms: recursive relations, discrete models, geometric models, exponential models, and logistics models. Discrete models are only useful for estimating population in small intervals of time, exponential models are used to represent specific formulas, and they are also known as geometric models. Recursive relations are used to define populations one unit of time later than the given; and logistics models are another common type of nonlinear population model.

## Mathematical Modeling

For the modeling of the population change in the event of a virus, we would need to separate the population into 3 main groups: Susceptible, as in the population that has not contracted the disease; Infected, the population who has contracted the disease; and Removed, people who have either recovered from the disease or died from it. If we assume that once you recover from the disease you can’t get it again, then we can put the recovered and the diseased in the same group. We would need to come up with three equations, one for how the Susceptible move to the Infected group, how the Infected group gains and loses population to and from the Susceptible and Removed groups, and how the Removed group gains from the Infected population. We could also show this in what’s called a SIR flowchart (Susceptible, Infected, Removed). It would look like S->I->R, and shows the direction the population flows between the groups. But it only shows the direction, so we need the equations to show the amount, and the rate of change. These equations would need to be recursive, so, for example, if you wanted the value for the Susceptible group after 2 years, you’d need the value for the first year to find it.

## Conclusion

Mathematical Epidemiology is a very important skill, it is useful for many jobs and professions. It’s difficult to use because of the many different factors you have to take into account and also it’s not a perfect solution because it’s just an estimate and will never be perfect since it is predicting the future