The real world is full of complex biological relationships. A rabbit population might fluctuate depending on the fox population, the amount of edible plants, and the spread of disease. Mathematical biologists like Associate Professor of Mathematics and Statistics Rebecca Everett make sense of these relationships using mathematical models. She explains that the mathematical models she creates represent biological systems much like a model airplane represents a real airplane. They are not perfect copies of reality, but they can be used to understand and predict what happens in the real world.
Professor Everett primarily uses differential equations to create her models. A differential equation is an equation containing a derivative, or a rate of change of a function. If P (t) is the population of squirrels on Haverford’s campus at a given time t, the derivative of P (t), which we denote dP/dt, is the rate of change of the population of squirrels over time. Consider this assumption: the rate of change of the squirrel population is proportional to the current squirrel population. This makes sense considering that the greater the number of squirrels, the more they can reproduce. The phrase “is proportional to” tells us that the rate of change of the squirrel population is equal to the current squirrel population multiplied by a proportionality constant, which we will call k. We can now write the differential equation,
dP/dt = kP.
This differential equation is an example of exponential growth. The differential equations Professor Everett uses are more complex than our dP/dt, but they often include an exponential growth term in addition to a death term and terms that depend on other differential equations in the model. For example, the squirrel population will decrease if the population of predators increases, so a differential equation for the squirrel population might include a term that depends on the hawk population. When considered together, these interrelated differential equations represent a complex biological system.
Professor Everett works in close collaboration with researchers who are experts in the subject of the model she is creating. These collaborators come to Professor Everett with a research question, and they discuss what should go into the model. There must be a balance between simplicity and complexity in these models: if a model is too simple, it does not accurately represent reality, but biological complexity makes mathematical analysis of the model more difficult. Professor Everett and her collaborators generally start with a simple model, and build up the complexity by gradually incorporating more factors.
One of Professor Everett’s current areas of research is linking the currently separate fields of ecosystem and disease ecology. When the nutrients available to plants increases, they are better able to defend themselves against disease, but pathogens that cause disease in plants also benefit from an increase in nutrients. The question Professor Everett’s ecologist collaborators want to answer is what happens in the long term when the amount of nutrients changes. The models she creates for this system are continuous, meaning that each equation has a value for every point in time. In a recent paper, Professor Everett and her collaborators use nine differential equations to model a deciduous forest. These nine equations correspond with nine variables, including the amounts of carbon, nitrogen, and phosphorus in the soil.
Professor Everett also collaborates with psychologists to model the alcohol usage and craving of individuals with mild to moderate alcohol use disorder. The model Professor Everett and her collaborators created is discrete, so the equations are defined for only distinct timesteps. They use difference equations, the discrete version of differential equations. There are two equations in their model: one for alcohol use and one for craving. The model shows how these two factors relate, and Professor Everett and her collaborators plan to add complexity by adding equations for factors such as confidence in being able to reduce alcohol use. The ultimate goal of this work is to help psychologists individualize treatment for alcohol use disorder by shedding light on what factors cause individuals with the disorder to change their behavior.
Professor Everett emphasizes that her collaboration with psychologists is unconventional. Generally psychologists use statistics to gain information from data, not differential equations, so Professor Everett is excited that this research expands the range of mathematical methods used in psychology.
After creating a mathematical model, Professor Everett uses real-world data to evaluate its accuracy. For the model to be considered accurate, its predictions must match the expected results. Mathematical analysis can also show that the model makes sense. In most cases the solutions should be positive and bounded. For our squirrel example, this means that the number of squirrels should not be negative or approach infinity because these values cannot represent reality.
Once the model is created and shown to be accurate, it can be used to “experiment.” Testing biologically can be expensive, impractical, or unethical, and a model offers a way to run tests mathematically. Researchers can observe long-term behavior and test how changing parameters affects the system.
Professor Everett’s research is just one example of how mathematicians use math to understand the real world. Mathematical models are powerful tools that can be applied to a wide array of disciplines.
This article was edited by Anagha Aneesh and Ashley Schefler.