Introduction
I need to start with a disclaimer: Epidemiological Models isn’t my field of study. Rather, I am a retired physicist who has done extensive work doing mathematical simulations of physical systems on computers. When the COVID-19 outbreak started I naturally got interested in what was going to happen. So I started reading the technical literature on the subject. I wanted an introduction to Epidemiological models. Some of the models for pandemics are quite sophisticated and there is no way I would even attempt to get to that level. Leave that to the experts. I started exploring the models that rely on coupled differential equations, something I am very familiar and comfortable with. These are classic models that are very straight forward, but lend themselves to considerable refinement. I have also explored Monte-Carlo techniques and methods based on Markov Chains. But more on that later.
The Nature of Modeling: Including Epidemiological Models.
No model includes every effect done to the last detail. If it did, it wouldn’t be a model. A model’s success or failure depends on the modeler’s ability to discern what can be left out. Simple models leave out a lot! However, simpler models are easier to interpret. Complicated models include a lot of different effects. As a result it is not always clear what exactly is causing a given unexpected result.
All models have numbers we have to supply to evaluate the model. We call these parameters of the model. Some parameters can be estimated from existing data. Others are not easily obtainable. A good epidemiologist would be able to guess at these numbers far better than I can. For example, in Part II we introduce a number that represents how many possible infectious contacts an infectious person will have in a day. Not surprisingly, this is a fundamental parameter for most epidemiological models.
This infection rate depends both on the disease, as well as human responses. perhaps at the beginning of the disease people are unaware and so their behavior is unaffected. Observing friends getting sick is likely to change people’s behavior, modifying the infection rate. In fact this is the desired result of social distancing. We can attempt to measure the infection rate by examining past data. But how accurate are such determinations? What happens as peoples perception change the infection rate?
The effects of any inaccuracy can be determined by running a model with different values to determine how sensitive the results are to inaccuracies this parameter. But will a given social policy have the effect on the model parameters required to achieve a stated goal? Leave that question for the epidemiologists and public health officials. A good epidemiologist, familiar with models, would be a better judge than I of the limitations of these models.
Why Epidemiological Models?
Modeling by itself is never sufficient to answer our questions. Modeling should be a tool of epidemiology. Epidemiology should be more than interpreting the results of computer models. I deeply believe this is true of models in any discipline. Beware of the modeler who simply fits the model to data and thinks they have proved something.
One of my early reads (1970’s) in numerical methods was a book by R. W. Hamming Numerical Methods for Scientists and Engineers. The book opens with the following.
“The purpose of computing is insight, not numbers.”
This is why we model, for insight. I hope that these notes provide insight into what is happening. That insight should help you discern the fake news, the uninformed news, the “I haven’t a clue but I want to get on the news” reports, from reality. I pray these notes help you avoid both false hope and unnecessary despair.
What can we expect from Epidemiological Models?
The manager of a hospital in crisis certainly would like definitive numbers like, “how many ICU beds and how many respirators do I need to get through the next month.” I believe that the future is not fully determined by the past so long as we are open to changing our collective decisions and behavior. The future is also determined by the decisions the leaders make and when they make them. This is not to say that the future is arbitrary, especially in the near term. Those that are sick today remain sick regardless of our decisions and actions today. However, the farther we look into the future, the more open it is to modification. Models should help us understand these limitations.
I am taking the position that the models facilitates exploring “What if” questions. For instance:
- If present trends trends continue, how many people will be infected?
- What if we adopt a different strategy, can we improve those results?
- Which strategies are most effective in the short run?
- What happens in the long run?
When models suggest answers to these questions they can guide policy. However, current data is always tell us about the past. Consequenctly. the faster things change, the less the current statistics tell us about the future. When we base decisions about the future by using the data alone, without modeling, it is like driving your car by ONLY looking in the rear view mirror.
Plan for these notes
I am primarily putting these notes together for my own edification. As a retired professor of physics, I miss the students, but not grading, Writing computer codes that solve real world problems, in other words models, is something that I have always enjoyed. I am making them available in the hope that others will benefit.
I plan to cover:
- Exponential growth and its limitations
- Susceptible-infections models (SI), where everyone gets infected
- Susceptible-infections-susceptible (SIS) models, where people recover but can be infected again (think the common cold).
- Susceptible – infection – recovered (SIR) models. In these model you only get the disease once, then you become immune (or dead).
- Elaborated (SIR models) – here we add back in some of the effects we missed.
- Elaborate transition matrix models.
The models will grow in detail as we progress down the list. This will allow for me to explore the limitations of one model as we move to the next.
What’s next
Here is a link to the next section.
References
If your are familiar with differential equations the following three articles will give you a good introduction to the basic models. These are the source of much of what I am covering when I talk about the differential equations. (Especially Three basic Epidemiological Models). I’ll give more references as I go.
Hethcote H.W. (1989) Three Basic Epidemiological Models. In: Levin S.A., Hallam T.G., Gross L.J. (eds) Applied Mathematical Ecology. Biomathematics, vol 18. Springer, Berlin, Heidelberg (link)
Hethcote, Herbert W. The Mathematics of Infectious Diseases SIAM Review, Vol. 42, No. 4. (Dec., 2000), pp. 599-653. (link)
Marathe, Madhan and Vullikanti, Anil Kumir S. Computational Epidemiology, Communications of the ACM 56(7):88-96 (link)
We will visit Monte Carlo simulations later in the notes.