The Epidemic Lab
Model Structure
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Predictive Models

Epidemiologists draw on a variety of models to predict the death rates from the COVID-19 Epidemic: Columbia University; Imperial College of London; MIT; Northeastern University; University of Texas at Austin; Institute for Health Metrics and Evaluation at University of Washington; Yougang Gu.

These models employ variations of the SEIR (Susceptible; Exposed; Infected; Removed) model structure along with various statistical methods for including exogenous data and projecting death rates. Like weather models, their results vary from each other and also from themselves as they evolve.

Simulation Models

A simulation model has a different function. It starts with a definition of a system structure (such as SEIR). This definition includes an enumeration of all the system elements and also how these elements inform each other. The simulation starts with initial values for all system elements and then, extrapolates new values for the elements after a very small time interval, dt.

Then, with the new element values, it makes another small-step extrapolation. And then another. The computation proceeds in this manner, generating the system behavior entirely from the system structure, endogenously, without incorporating exogenous data.

A simulation model does not purport to deliver accurate predictions. It aims to show the behavior that logically follows from a structure. This has various benefits:

    1. It presents the structural theory and the behavior as two parts of the same theory.

    2. It shows the evolution of all the system elements, including critically important ones, not just the ones for which we have real-world metrics.

    3. It does not depend on exogenous metrics, many of which suffer from bias and scarcity.

    4. It allows testing of different theories about how the structure operates.

    5. It allows testingt interventions, qualitatively.

Current Epidemic Model Structures

Epidemiologists who wish to study epidemics currently rely on the SIR and SEIR models.

The SIR Model

The SIR Model has three stages:

   Susceptible:     not yet infected, disease-free.
   Infected:     infected and contagious.
   Removed:     recovered with lifelong immunity.

The model also has equations for the flows between these states:

   Infection Rate = b * Susceptible * Infected     flow from Susceptible to Infected.
   Removal Rate = v * Infected     flow from Infected to Removed.

   b     the effective transmission rate, has units: (people/day)/people2 = (people-days)-1
   v     the removal rate, has units: (people/day)/people = (days)-1

The SIR model has Infected directly informing Infection Rate, in a positive feedback loop with little delay. This leads simulations showing a very rapid rise in both Infection Rate and Infected, quickly exhausting Susceptible. This behavior does not fit exactly with the more gradual course of actual epidemics.

The SEIR model addresses the Infected / Infection Rate feedback loop by inserting an additional element, Exposed, in between Susceptible and Infected. This puts a gestation delay in the loop and slows down the conversion of Susceptible.

The SEIR model employs four states:

   Susceptible     not yet infected, disease-free.
   Exposed     infected and not yet contagious; incubating.
   Infected     infected and contagious.
   Removed     recovered with lifelong immunity.

The Epidemic Lab Model

The Epidemic Lab Simulation Model further extends the SEIR model to include six stages.

The Epidemic Lab Model Structure

   Susceptible     people with no exposure
   Exposed     people who have the infection and show no symptoms
   Infected     people who have the infection and also show symptoms
   Bedridden     people who require care
   Deceased     people who die
   Immune     people who develop immunity

The Converter

The SIR model, the SEIR model and Epidemic Lab Simulation Model (S6) all revolve around a common method, namely the Converter.

A basic form of the Converter, from the SIR model, converts Susceptible to Infected.

The Converter

The converter has four elements:

   Susceptible     People with no infection and no immunity (people)
   Infection Rate     Flow from Susceptible to Infected (people / day)
   Infected Group     People with the infection who can infect Susceptible (people)
   Heat     Inherent propensity of the virus to propagate (1/people-days)

The connecting arrow lines indicate that Susceptible, Infected and Heat all inform Infection Rate. Specifically, in this model, Infection Rate = Susceptible * Infected * Heat.

From this model structure, we can simulate the behavior of the converter, using Euler’s Method step-by-step on a spreadsheet.

Spreadsheet Simulation of Converter Dynamics

The spreadsheet above shows the behavior of the Converter elements (Susceptible, Infected and Infection Rate) as they co-evolve step-by-step according to the Converter system structure.

The converter converts the Susceptible Group (green line) into the Infected (blue line) via the Infection rate (yellow line).

The Infection Rate (yellow line) grows, peaks and then declines in a bell-shape curve. Decreasing the Viral Heat tends to flatten the curve.

At first, Infection Rate increases exponentially as it engages positive feedback with Infected. Later, Infection Rate declines exponentially as it engages negative feedback with Susceptible. These loops appear in Figure 1 with (+) and (-) signs.

The converter has a logical structure, consistent units and also a behavior that approximates the types of behavior we find in actual viral epidemics. We may now extend the model to include other system elements.

The S6 Epidemic Lab Simulation Model Structure Extends the Converter

The Epidemic Lab Model Structure chart above shows the six stages and the flows between them. It does not show the connections through which the stages inform the flow rates. In one variation, we might have Infected and Bedridden informing Infection Rate; in another (SIR) we might set Exposed as the only informant.

In one variation we might set Immune to stay permanently immune; in another we might “recycle” some Immune back to start over as Susceptible, to begin another journey through the stages.

We could code this up, as before, in a spreadsheet and run simulations to see how it works. As the model grows in complexity, however, running simulations on spreadsheets gets cumbersome.

Alternatively, we might add a GUI (Graphical User Interface) to the model and then put it all on line. People could then use the model as a kind of video game, to see if they can beat the virus. In the process they could come learn about the dynamics of epidemics and to form their own opinions about how to protect themselves personally and what public policies to support.

Epidemic Lab with Behavior Graph Avove and Control Panel Below

(c) 2020 by Ed Seykota