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Simple Models

Carl�MMED 2025

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Goals

Be able to

Create a “model world” & rules to run it
Distinguish “state” versus “parameter” and identify these for a “model world”
Explain the S-I-R “model world” and the Reed-Frost representation of it

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What is the simplest, formal model of transmission?

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Simplest?

least parameters, easiest to explain/analyze, linear

Formal Model?

unambiguous, mathematical rules

Transmission?

disease from one person to another

Answer:

???
???
???
???

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Simplest?

Shortest-but-completist version of …

Formal Model?

unambiguous rules that …

Transmission?

can exhibit infection moving from one host to another

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One element of your groups answer?

initially infected person
states: susceptible, infectious, recovered (it has three compartments)
constant infection rate
there is an arrow of time
there are differential equations

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A Minimal Model

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All hosts interact each step

I

S

Hosts are either infectious or susceptible

t=0

t=1

t=2

Time passes in fixed steps

Simplest: 2 hosts

If infected host interacts with a susceptible host, the susceptible host is infected on the next time step

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should be no more than 20 mins in after this.

What were the key requirements? Transmission, formal, simple.

We have a simple rule: if an infected host interacts with a with a uninfected host, the uninfected host becomes infected on the next step.

Important tangent: I’ve described this as a “simple” model. In the sense of outlining the model, this is reasonable use of the word - I have given a very simple description. In another sense, however, this is a ridiculously complicated: when we try to connect this model to the real world, we have to be clear about all the assumptions that to translate between the two.

E.g., I’ve used a “infected” vs. “not infected” label - but we know lots of people may be infected without symptoms and not transmit infection onward. This model also has no time *units* - to start to be useful, we have to *add* a definition of time.

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Too simple?

Need to show transmission? Check.

But: models also need to show negatives - are there initial conditions where transmission DOESN’T occur?

Yes! Only see transmission with 1 I, 1 S - no transmission with 2 I or 2 S.

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Stepping back: what was different for your model vs mine?

Some had different rules, but about the same “thing”. Some were about a different “thing”

We call the “thing” the model world: the collection of states and processes

Distinctly, the rules–how those processes work–is the model representation.

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A Particular Less Minimal Model

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Hosts are either Susceptible, Infectious, or Recovered

Interaction may or may not lead to infection

Many hosts

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No more than 25 mins in after this.

Okay, but sometimes that model is actually too simply to capture the dynamics you want.

What are some other characteristic trends of infectious disease spread, aka outbreaks, epidemics, pandemics?

[elicit answers: fishing for 1) non-linear growth and 2) waves or turnover etc - susceptible depletion]

Okay - what do we need to make that happen?

To see a “wave” and non-linear growth, we need several hosts - also note that we now must have a way for not everyone to become instantly infected. We have only some people interact every step, or only some interactions lead to infection.

What do we need to see the wave turnover? Let’s imagine people don’t stay infectious forever, and they are immune after infection. Maybe that will do the trick?

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Quick check:

what’s new about the model world?

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Let’s try it!

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Get into groups of 4. You’re all Susceptible.

Initially, you all interact with me. Roll to see if you become Infected: a 1 or 2. I’m going to Recover.

On each subsequent round: if you are Susceptible, roll once for each Infected person in your group. After those rolls, all Infected people Recover.

Record how many people get Infected each round.

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Plotting!

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Question:

How many people did we expect to get infected in the first round?

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Get into groups of 8. You’re all Susceptible.

Initially, you all interact with me. Roll to see if you become Infected: a 1. I’m going to Recover.

On each subsequent round: if you are Susceptible, roll once for each Infected person in your group. After those rolls, all Infected people Recover.

Record how many people get Infected each round.

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Plotting!

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Summary

start simple with model worlds, then build
model implementation should be incremental
model-thinking requires both creativity and specificity
Reed-Frost: very simple, but still provides insights
Surprise! We also introduced the�Basic Reproductive number, R₀, and the�Effective Reproductive number, R_eff!

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Reed-Frost Model

Take a population of N hosts, one initially infected
Time proceeds in fixed steps
Hosts are either infected, susceptible, or removed
Each time step, all hosts interact
When an infected and susceptible host interact during a time step, there is a probability p that the susceptible host will become infectious on the next time step
All infected hosts at a time step become removed on the next time step�

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Reed-Frost Model Math

All interactions instantaneous, independent, universal
Host counts at time steps: I_t, S_t, R_t
R_t+1 = I_t + R_t -- i.e., all infected become immune
For a particular susceptible,�P(infected @ t+1 | I_t) = 1 - (1-p)^I = 1-q^I

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Therefore,�P(S_t+1 = S_t - x, I_t+l = x | S_t, I_t) = (1-q^I)^x(q^I)^S-x�

( )

S_t

x

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