A model is an abstraction often represented using mathematics that describe some pattern and that pattern can be static, like a snapshot, or dynamic, like a movie. Models are used everywhere in our lives, most commonly to predict the weather. And are invaluable tools in the study of infectious disease biology. One of the goals of modeling, when trying to describe a complex process like an epidemic. Is to pair down the number of processes, that you represent to only those that are necessary to adequately represent the pattern. Of course how far to pair down that list, and what constitutes an adequate representation of the pattern Isn't necessarily objective. The golden rule for developing models was best stated by Albert Einstein when he said that everything should be made as simple as possible, but not simpler. There are an infinite number of possible ways to develop models of epidemic processes, but we can come up with some broad dichotomies. One of the major dichotomies in infectious disease modeling is whether to represent the population as many individuals or as broad compartments that describe groups of individuals. Perhaps the most famous class of models in epidemiology are of this latter type. The so called susceptible, infectious, recovered. Or, SIR, class of compartment models. Here we divide the population into three classes of individuals. Those that are susceptible to infection, represented by the box S. Those that are infectious, represented by the box I. And, those that are recovered and immune to infection, represented by the box R. Though each of these boxes contains many individuals, we can still represent the change in infection through time, as the way the number of individuals in each compartment changes. Susceptible individuals move to the infectious class when transmission occurs. And infectious individuals move to the recovered class when they cease to be infectious. Thus taking a large population and simplifying it down to only three components, which can be represented by only three equations. Though very simple, this allows us to calculate simple analytical summaries of how such models will behave. Such as calculating the basic reproductive number or not, or the vaccination threshold necessary to achieve herd immunity. The downside to this is that we must treat each individual in each compartment as though they are the same, which is sometimes too simple. In that case, we can still sub-divide the population into any arbitrary number of compartments that represent the set of divisions that we think are relevant to the process. For example, perhaps we think that children transmit at a different rate than adults. So each class of S, I, and R, should be subdivided based on age. So long as the number of subdivisions isn't too great, we can still calculate all the simple analytical summaries, albeit using some more complicated mathematics. Of course, if you subdivide the population into enough compartments, such that each contains only one person, then you're in the realm of so called individual based models. Representing each individual in a population is computationally intensive. But allows us to move away from the simplifying assumption that all individuals do the same thing. And represent the kind of individual variability that we know exists in real populations. In general, these models are too complex to solve analytically, and we often need to use computers to simulate epidemics in these models in order to see the patterns that emerge from the set of rules that we put in. Another important dichotomy in infectious disease modeling is whether or not the outcomes are deterministic, that is predetermined by the rules of the model or stochastic, that is variable each time the model is evaluated. The simplest way to think of this is in accounting for the change of infectious individuals through transmission. If I have 100 susceptible individuals and they each get infected at a rate of 10% per day, then, if I constrain my model to be deterministic, I know that after one day, I'll have 90 susceptible individuals and 10 infected individuals. And the neext day, I'll know that there are 10 individuals that can susequently infect others and so on. So even at the start, I know what will happen in the future. However, if I say that the probability of any individual getting infected after one day is 0.1, then it's like each of those 100 individuals flipping a coin that comes up heads 10% of the time. On average, 10 of them will get infected, but sometimes only 7, and sometimes 13 will. This has important consequences, for what would then happen the following day. If only 7 got infected, then there are fewer individuals to be sources on infection the next day, and the outbreak might grow slower. If 13 got infected the outbreak might grow faster. The coin-flipping analogy makes sense for the way we see the world work. Lots of things, whether they're up to chance or not, behave as though they are. However, interpreting the outcomes of models that are stochastic is difficult. If I run the model forward and only seven people get infected the first day, I might think that the epidemic would always tend to grow slower. It's only if I evaluate many outcomes of the model that I would see that there is a range of outcomes that might happened on the next day and a range of outcomes that might happen the day after that. and the farther I go out, the harder it is to predict. Deterministic models can be useful because they can characterize what the average outcome might be in a very simple and interpretable way. Sarcastic models, though more challenging to interpret, Are useful when we want to represent the uncertainty in potential outcomes, where the prediction at any time is a range rather than a point. This is important when we consider how to evaluate models. The observation of an epidemic through time might follow a generic shape, but with lots of fluctuation in the number of infections from time to time. A deterministic model might get the average shape of the pattern well, but never accurately predict any one outbreak. A stochastic model might provide a range at each time that captures all the observed points, but only provides a fuzzy answer. There are two major uses of models. Models are often used to describe the patterns that we've already observed in the past and models are commonly used to try to forecast patterns that might occur in the future. In the best case scenario models are developed and evaluated based on their ability to replicate the patterns of the past. Then used to try to predict the future. And we can express our confidence in our prediction of the future based on their ability to replicate the past. We can then use these models to project things, like how fast might an epidemic spread in a new population, or which vaccination strategy would limit cases the most? Invariably the projections of a model are likely to be wrong. The future is inherently challenging to predict. But the utility of models lies in their ability to help us pair down a complicated world. To a relatively simple and tractable set of explanations. And in the extent to which they can at least provide some picture however hazy of what the future may hold.