Fitting the Curve: Methods of Reasoning and Coronavirus Modeling
By: Zahid P
Tags: Data, Data Science, Coronavirus

Before I get started, I want to reiterate that I am a software engineer and not an epidemiologist.

I broke down 2 widely cited models, their critiques, and how they relate to various ways of reasoning and predicting outcomes we have never seen.

The goal here is to walk through the process of learning variables, making assumptions, and building reasoning structures that can turn those values into predictions.

If you are interested in infectious disease modeling I have linked every paper I reference and also there is (of course) a Wikipedia page on it.

And remember, the usual George Box disclaimer:

All models are wrong but some are useful.

Contents

Reasoning and Modeling Techniques

Let’s say I ask you to predict the 1980 population of LA, Chicago, and Kansas City based only on the first chart. How would you go about doing it?

Only Cleveland and Omaha are shown through 1980.

(The second chart shows the actual data)

Based on only that chart, it would be extremely hard to predict.

Now if I told you: LA borders an ocean, Kansas City and Omaha are landlocked, Chicago and Cleveland are on a lake. Also if I mentioned LA had no nearby competitors, and that Chicago had already been established as the hub of the Midwest. That might help your prediction.

What about the reduction in Western birth rates as we grew wealthier? Then there are totally unpredictable events like LA becoming the world’s film capital in the 1920s. On top of that, what if the data is unreliable? Should suburbs count? Did a city recently incorporate a new area?

Dynamics like these are near impossible to predict. While this is not a perfect analogy. It’s something to consider when looking at the epidemiological models later.

Reasoning Definitions

There are a few different ways of reasoning towards a conclusion. One of the key comparisons is described in this paper:

The deductive method requires expert knowledge to build a mechanistic-based model and depends on a first-principles understanding of the mechanisms acting within the ecological system. In contrast, the inductive method only uses the information content of the available empirical output data of the ecological system to construct a predictive model.

In other words

Nowadays, when we hear about statistical modeling we are typically talking about a 3(ish) step process.

Model training/fitting Prediction Evaluation

We then improve our models over time by re-training with better models, better data, or both.

Fitting and training can be associated with rule-learning. This learning typically occurs with many examples used to derive an association. More akin to inductive reasoning.

Predictions or calculations can be associated with applying these learned rules. More akin to (though not exactly) deductive reasoning.

Every model does some combination of these steps. They do so at different levels of granularity, with different learning techniques, but they do these steps.

Epidemic Modeling for Coronavirus

We will look at three approaches.

Compartmental Models

The IHME / UW approach

The Imperial College Approach

Compartmental Models

Compartmental models look to place segments of the population into different states. Many epidemic models use compartmental approaches. I want to mention these because they have a history of use in epidemiology. The SEIR model includes these main states,

Susceptible, Exposed , Infected / Infectious, Removed

Differential equations are then set up based on how each of these states relate to each other. These flow states look akin to this MIT DELPHI model,

The equations produced look like this,

Equations for Susceptible, Exposed and Infectious

Note that these models attempt to capture ebbs and flows of infections as people go from susceptible, to infectious, to recovered.

We see how the change in infectious population over time (dI/dt) is governed by the number of exposed individual (E(t)) minus the number of already infected individuals. Individuals can move between states depending on how much time has elapsed. This can result in situations where we see some sort of dampening curve or oscillation that moves toward a stable state.

I would consider this as more of a deductive approach, because we set up rules for the states to follow, and through these rules we arrive at the solution. Some of the parameter values that modulate these variables, however, are learned in an inductive manner.