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Public Health, Epidemiology, and Models (Day 1)

Simple Models�(Day 1)

Foundations of Dynamic Modeling (Day 1)

(Hidden) Assumptions of Simple ODE’s (Day 2)

Breaking Assumptions!

Consequences of Heterogeneity (Day 6)

Introduction stochastic simulation models (Day 3)

Heterogeneity tutorial (Day 6)

Introduction to Infectious Disease Data (Day 1)

Thinking about Data (Day 2)

Data management and cleaning (Day 9)

Creating a Model World (Day 4)

Study design and analysis in epidemiology (Day 3)

Introduction to Statistical Philosophy (Day 4)

Variability, Sampling Distributions, & Simulation (Day 10)

HIV in Harare tutorial (Day 3)

Integration!

Introduction to Likelihood (Day 4)

Fitting Dynamic Models I – III (Day 5, 8, & 9)

Modeling for Policy (Day 11)

Model Assessment (Day 10)

MCMC Lab (Day 9)

MLE Fitting SIR model to prevalence data (Day 5)

Likelihood Lab (Day 4)

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Simple Models

Carl�MMED 2024

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Goals

develop general model world perspective
think about what a model needs
introduce the Reed-Frost Model

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What is the simplest, formal model of transmission?

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Simplest?

shortest that can encapsulate …

Formal Model?

rule-based, assumptions, …

Transmission?

spread of infectious disease

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One element of your groups answer?

???

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One element of your groups answer?

people who aren’t infected
people are infected
people are recovered
people who are infectious
control measures
people who are vaccinated

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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?

2 hosts
2 states
time / steps
interaction => state change

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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; possible starts?

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

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5 minutes:

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

We have a simple rule: if a sick host interacts with a with a not sick host, the not sick host becomes sick in a fixed amount of time.

This is a really simple model. Is it *too* simple? For many purposes, yes, but it’s just right for grasping the point of modelling: different initial conditions lead to different outcomes. No one sick? No gets sick. Everyone sick? Also no new infections. One person sick? Other gets infected, but *it doesn’t matter who is sick first*.

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.

We’re also making very explicit assumptions here about everyone has the same incubation period / latent period

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Less Minimal Models

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More host states

Interaction may or may not lead to infection

More than two hosts

Also:

Time passes in different durations
Some hosts interact each step
Becoming sick takes multiple steps
Et cetera...

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A Particular Less Minimal Model

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Hosts are either susceptible, infectious, or removed

Interaction may or may not lead to infection

Many hosts

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Let’s try it!

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Outcomes!

total:

duration:

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Plotting!

comparison to Reed Frost simulations

in MTM package

github.com/pearsonca/MTM

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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

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Reed-Frost Model

There is a population of N hosts
Time proceeds in fixed steps
Hosts are either infectious, susceptible, or removed
Each time step, all hosts interact
When an infectious 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 infectious 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 infectious 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

Not easy to turn into intermediate / final size statistics

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