Mathematical Modeling: Seeing the World through Mathematical Lenses
Dr. Jose Maria Balmaceda
When a typhoon approaches the country, weather forecasters are able to make predictions about the typhoon’s path. These predictions rely on a process called mathematical modeling. Today, I want to talk to you about this method and how it has become a powerful tool to help understand our world and solve many real-life problems.
Mathematical modeling refers to the process of describing mathematically the behavior of some real-life system or phenomenon. The mathematical description is called a mathematical model. A model may help to explain a system and to study the effects of different components. It may also make predictions about the system’s behavior. Mathematical modeling has emerged as a powerful tool for studying a variety of problems in research. It’s used in seismology, in climate modeling, economics, business, in drug research and manufacturing, in medicine, in the law, and even the social sciences. Mathematical modeling is an interplay of various disciplines. Of course, you have the mathematics behind it, but they also need technology, and for that, computer science is very important. And when mathematical modelers study a problem, they actually study a problem—a problem from a particular area of science or of the social sciences. So it’s really an interplay of various subjects.
Real-life problems can be studied using mathematical models—problems like the recent landslide, in Itogon, Benguet, typhoon Ompong, traffic, flooding… These are big problems of our society and mathematics can help understand and solve some of these problems using this process. [Modelers] for instance can study climate and make forecasts and predictions about weather and climate. Our floods, which are big problems not just in the city but also in other places of the country, require a lot of this modeling process. This processes is a very scientific approach that is necessary to understand and analyze these huge problems. Our population problems due to overpopulation can be studied mathematically and we need mathematics because populations really have to be observed over long periods of time, which is physically impossible to do. And therefore you need to do a lot of these theoretical computations and predictions scheme.
What is mathematical modeling? Well, it starts with a problem—a real-life problem which is usually difficult to solve and requires a real-life solution. What a mathematician does is to model the problem. Converts the real-life problem into a mathematical problem. The mathematical problem hopefully, is easier to solve. The mathematical solution can then be interpreted and that might help you provide or obtain the solution to the real-life problem. Mathematical modelling does not replace experimentation or theory but is used in complement with those two things. Sometimes it is necessary to do modeling. It is useful because sometimes the problem is too large to study. If you want to study the universe and the galaxies in astrophysics then you need models because it’s physically impossible to visit all those far stars and planets. Sometimes, the problem being studied that is too small. We encounter that when we do nanotechnology or when we do quantum physics. Sometimes, it’s too expensive. For instance in creating a new aircraft, we can’t just build prototypes and models of the actual aircraft because that would be expensive and time-consuming. So, we use computers instead to make the science about these rockets and aircrafts. Sometimes, the problem is too dangerous or destructive. When we work with nuclear weapons, when we study pathogens, when you study disease. And, as I said, it could be time consuming. For instance population studies or geological studies require long periods of time. When the Boeing 777 was first launched, well, it was important because it was the first jetliner to be digitally designed. So it was really preassembled on a computer, eliminating the need for costly full-scale production. So, computational modeling improves the quality of work and reduces errors and changes, and reduces expenses.
Archaeologists can study the past, reconstructing artifacts and even the entire communities using just a few missing pieces of pottery or implements. Aside from problems, natural phenomena can also be studied using models. You ever wonder why birds can fly in a particular pattern without colliding? Well, this can be explained by mathematical models. These models are really just a set of equations using laws of physics. When we study our environment or the ecology, we use a lot of models. The environment or communities are very difficult to understand. They are very complex and would require really all parts of mathematics to simplify and analyze this problem. In the social sciences, we also use a lot of math, in particular a lot of modeling—modeling to simulate social behavior, social systems, to study large populations, to study other problems in the social sciences, problems during voting, during elections, in apportioning property or land… So, these are done in using math models.
When we study supply chains in business, when we do manufacturing studies, we need a lot of these models.
And also, we can understand our culture and heritage by using a lot of math. Using mathematical models, we can analyze kinship relations, indigenous music, languages, dialects, weaving patterns.
I want to discuss what a model is. A model has certain elements. They are usually composed of variables and relations—between these variables. What are variables? Variables are really abstractions of the parameters of the system that are of interest and these variables usually need to be quantified. Relationships on the other hand are described by various functions operators. These operators can be algebraic. They can use a lot of calculus, but can also use discrete tools like graphs and networks. In the physical sciences, a typical model consists of a set of governing equations usually from the laws of physics. These laws will probably require some supplementary models or laws, and then you will need assumptions—assumptions that will impose on your model. These assumptions are usually constrained by many factors. For instance, if the assumptions involve rates of change of a variable, the variable “time” denoted by t is often involved. In this case, we say that a solution of the model is obtained when we can describe the state of the system at a particular time.
I’ll give you an example of a very important model.
And these are Maxwell’s equations of electromagnetism—four equations involving derivatives, partial derivatives. There are probably enough to describe all phenomena in classical electricity and magnetism. So in Maxwell’s equations, the variables and the relationships are described by differential equations. The same differential equations are another set of differential equations, for instance, can describe the dynamics of an infectious-disease. You can write equations that relate how people infect others, how people receive the infection, how this is spread around.
But there’s a type of modeling that does not use calculus or differential equations. And that’s when we use objects like graphs. Graphs and networks are very useful tools in modeling. We can provide information, for instance, of the epidemic by studying how it moves from one infected person to the other using graphs. So through these graphs, we can study the spread of the disease. What are graphs? Graphs are very useful objects because they’re very simple. A graph consists of a set of vertices, sometimes we call them nodes. And a set of edges which connect these vertices. And the important point here is that these objects can be used in very many situations. They’re very general because the edges which join vertices can stand for any relation, even relationships among people like friends, friendships, enemies, collaboration…
Graph theory [is an] important tool that molecular biologists studied them, for instance, to look at interactions in the gene between proteins and genes, between proteins and proteins, and even the biochemical reactions between these materials. This is a very complicated problem because the graphs can be so complicated. A useful graph would be a subway map. If you take a subway map, you see edges which connect subway stations. The map itself doesn’t look like the physical layout of a city but it tells you how to get from one point to the other by just following the connections in the graph. Facebook, Instagram, all these social networks, really, rely on this concept of graphs, connecting you to your friends and your friends among your friends.
How does one build a model? Well, it’s very difficult actually, because a model is a description that hopefully preserves important characteristics and properties of the system or phenomenon being modeled. So, a model is not perfect; it just describes with some degree of accuracy what you want to study. When you want to study something, you can’t put all properties in your model. So, part of the challenge of [modeling] is to decide which characteristics are variables may be excluded. This is important because although you may want your model to be very realistic, that will only make the product the model very complicated. Which makes it mathematically harder to understand or to analyze. So, a good model should be accurate—should be accurate enough to be useful. On the other hand, it should be simple and elegant enough to generate realistic and interesting mathematical problems.
Occam’s Razor is a principle particularly relevant to modeling. The essential idea behind Occam’s Razor is that among models with roughly equal predictive power, the simplest one is the most desirable. Of course, systems tend to be very complex. The biologist Francis Crick, co-discoverer of the double helix structure of the DNA, cautions “While Occam’s razor is a useful tool in the physical sciences it can be a very dangerous implement in biology,” he says, “it is thus very rash to use simplicity and elegance as a guide in biological research”. Well, we have to admit it; we can’t simplify things all the time. How does the mathematical modeling process work? Well, it begins with a problem, a question, a research question, which you want to find the solution of. So, starting with the real-world problem, you want to simplify it. Try to write a working model, and at this point you try to perform your research and conduct investigations so that you can set up the necessary equations to describe your real-life situation. When you do that, you have to simplify these properties. You have to identify and select factors so that only the important aspects of your problem are included in your model. This means that you need to state your simplifying assumptions and you have to determine the factors that can be neglected. This is a challenging part of the modeling process. And then you use principles of physics, physical laws, governing principles that relate all these factors. You identify which things are known and which things you want to find out. You want to find the [identified] set up relationships between those variables. Now, once you’ve set up your working model you want to transform it into a mathematical model, so, this means expressing your working equations and relationships in mathematical terms. You may have to write down a set of equations, like the differential equations I showed you earlier. And you want to find algorithms that allow you to understand and analyze these equations and find solutions if necessary. In general, the successful model depends on how easy it is to use and how accurately it predicts. Once you have the mathematical model, [the] next step would be to even transform it into a computation model.
Today, we have a lot of software that will provide algorithms to solve your mathematical model. It can be programming languages—it can be software that will provide your solutions. Once you’ve run and solved the computational model, you have to look at your results. How do you analyze your results? You have to compare them with old ones. You can prepare graphs or charts to visualize your results. You have to go back and look at your computer program. And you want to check if your program [runs] correctly. Now, once you’ve understood the results of your computer or computational programs and resolutions, you want to check whether your model really validates or confirms what you expect in the real world problem. Hopefully, the model will predict the real-world problem’s behavior and this is a very hard part of the modeling process because, one, your results may not agree with what you’ve observed from experimental data. On the other hand, your results may seem to agree but the challenge is to determine why this is so? It could just be a coincidence and this is where a lot of statistical, mathematical error analysis is performed. Once you’ve check your answers and you can make some conclusions, well, the modeling process doesn’t end there because you want to refine your results. You want to check whether this can be replicated. You want to see whether you can remove some factors or add some variables that you missed out.
So the cycle goes on. You do the same steps from transforming into math, and then transform into something that you can compute, and then analyzing the results. So, this makes mathematical modeling a very, very challenging process. Sometimes, you don’t know when to end.
Let me give you an example for a very simple modeling project. Suppose we want to study the movement of a car run by an internal combustion engine. From physics, we know that velocity can be calculated using the principle of conservation of energy. So we can set up an equation: the Initial Potential Energy stored in the fuel equals the result on the Resultant Kinetic Energy. Well, that’s one equation which says something about the velocity. But if your model consists only of that single equation, then it’s a poor model, because other factors that affect the velocity or movement of the car are neglected. What are these? Some of these are air resistance, heat loss, friction, road roughness among others. So when we do a model we have to make sure that, really, the important properties are included in our equations in our model. Sometimes, that already is a big challenge.
Another equally difficult part of the modeling process is evaluating its usefulness or its correctness. You want to know whether your model really describes a system accurately. An easier part of this evaluation is to check whether your model fits experimental measurements or other data, but I’ve told you earlier that sometimes this can just be coincidences and you want to make sure that that’s not the case. A more difficult part of evaluating as a model is to determine what situations the model can be applied to. In particular, does the model describe properties between the measurement data, in which case, clearly we do interpolation, or does it describe properties outside the data, and therefore, we can do extrapolation? One type of problem in modeling is called an inverse problem. What are these? These are problems where, well, the answer is known but not the question, or you can think of it as where the results or consequences are known but not the cost. I’ll give an example. Suppose you want to study the specific heat capacity for liquid. There are two types of problems you can setup. One is the direct problem. If you know the specific heat, you can actually calculate how long it will take to boil a given quantity of liquid, so, this is usually easy. So, different liquids require different specific heats—have different specific heats—but once you know that, you know how long a certain volume will boil.
The inverse problem is sometimes harder. Suppose that you know that a certain liquid takes a certain amount of time to boil, say, five minutes. The inverse problem asks “can you tell me what liquid that is?” So, that’s the inverse problem. And we see that the problem becomes more complex, especially if more than one liquid shares the same specific heat capacity.
Let me give another example. Suppose I know that three streams join to form a river and that four factories are putting known amounts of pollutants into the streams, then we can actually calculate the resultant pollutant in the river. So this is the forward or direct problem. But a more likely problem is that we only know the pollutants in the river and the problem is to determine which factory is putting what into what stream, and that is the inverse problem.
So inverse problems are types of modeling problems which are challenging and many scientists do a lot of these inverse problems.
Let me end by saying no model is perfect but a good model often enables us to distill the essential characteristics of natural phenomena and real-world problems, helping lead to their solutions. Although mathematical models are approximations of reality, through these mathematical lenses we can better see and understand our world.
Thank you very much.