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Public Health, Epidemiology, & 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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assumptions of simple compartmental ODE models

Elisha Are, PhD

Simon Fraser University

SACEMA, Stellenbosch University and MMED Alumnus

MMED 2024

(Hidden)

Adapted from Rebecca Borchering, 2023 slides

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Goals

Review the main uses of applied epidemiological modelling

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Introduce our conceptual framework for applied modelling

Review commonly overlooked assumptions that are inherent in the structure of simple compartmental ODE models

Discuss when these assumptions might be problematic, and when they may be desirable

Begin to explore some alternative model structures that relax the assumptions

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Applied Epi. Modelling

The use of simplification to represent the key components of something you’re trying to understand more clearly

Related to the distribution and determinants of health-related states and events

For application to the real world

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What are some uses of applied epidemiological modeling?

from:https://childdevelop.ca

Improve public health

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Applied Epi. Modelling

Improving understanding of the dynamics of health and disease
Translation of results into decision-making and communication tools

Insight

Improving measurement and interpretaiton of key health indicators at the population and individual levels

Estimation

Projection and forecasting of expected future trends

Prediction

Guiding study design and intervention roll-out
Informing decisions through analysis and comparison of policy scenarios

Planning

Evaluating the impact of public health interventions
Assessing risk of future public health events

Assessment

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Applied Epi. Modelling

Improving understanding of the dynamics of health and disease
Translation of results into decision-making and communication tools

Insight

Improving measurement and interpretation of key health indicators at the population and individual levels

Estimation

Projection and forecasting of expected future trends

Prediction

Guiding study design and intervention roll-out
Informing decisions through analysis and comparison of policy scenarios

Planning

Evaluating the impact of public health interventions
Assessing risk of future public health events

Assessment

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Applied Epi. Modelling

Improving understanding of the dynamics of health and disease
Translation of results into decision-making and communication tools

Insight

Improving measurement and interpretaiton of key health indicators at the population and individual levels

Estimation

Projection and forecasting of expected future trends

Prediction

Guiding study design and intervention roll-out
Informing decisions through analysis and comparison of policy scenarios

Planning

Evaluating the impact of public health interventions
Assessing risk of future public health events

Assessment

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Applied Epi. Modelling

Improving understanding of the dynamics of health and disease
Translation of results into decision-making and communication tools

Insight

Improving measurement and interpretaiton of key health indicators at the population and individual levels

Estimation

Projection and forecasting of expected future trends

Prediction

Guiding study design and intervention roll-out
Informing decisions through analysis and comparison of policy scenarios

Planning

Evaluating the impact of public health interventions
Assessing risk of future public health events

Assessment

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Applied Epi. Modelling

Improving understanding of the dynamics of health and disease
Translation of results into decision-making and communication tools

Insight

Improving measurement and interpretaiton of key health indicators at the population and individual levels

Estimation

Projection and forecasting of expected future trends

Prediction

Guiding study design and intervention roll-out
Informing decisions through analysis and comparison of policy scenarios

Planning

Evaluating the impact of public health interventions
Assessing risk of future public health events

Assessment

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Applied Epi. Modelling

Improving understanding of the dynamics of health and disease
Translation of results into decision-making and communication tools

Insight

Improving measurement and interpretation of key health indicators at the population and individual levels

Estimation

Projection and forecasting of expected future trends

Prediction

Guiding study design and intervention roll-out
Informing decisions through analysis and comparison of policy scenarios

Planning

Evaluating the impact of public health interventions
Assessing risk of future public health events

Assessment

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Goals

Review the main uses of applied epidemiological modelling
Introduce our conceptual framework for applied modelling
Review commonly overlooked assumptions that are inherent in the structure of simple compartmental ODE models
Discuss when these assumptions might be problematic, and when they may be desirable
Begin to explore some alternative model structures that relax the assumptions

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REAL WORLD

MODEL WORLD

MODEL

abstraction

implementation

understanding

interpretation

https://publichealth.jhu.edu

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Model Worlds

A model world is an abstraction of the world that is simple and fully specified, which we construct to help us understand particular aspects of the real world

by:Vildana Šuta

A mathematical model is a formal description of the assumptions that define a model world

We know exactly what assumptions we’ve made, and we can follow those assumptions to their logical conclusions to address research questions

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The SIR Model World

S

I

R

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SIR: ODE Model

S

I

R

λ

γ

N=S+I+R

FOI depends on: the number of contacts per unit time, probability of disease transmission per contact, and proportion of contacts that are infected

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SIR: Reed-Frost Model

S

I

R

N=S+I+R

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SIR: Stochastic Reed-Frost

S

I

R

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SIR: Chain Binomial Model

S

I

R

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The SIR Model Family

S

I

R

A mathematical model is formal description of the assumptions that define a model world

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Taxonomy of compartmental models

continuous time

Gillespie algorithm

discrete time

Chain binomial type models

(eg, Stochastic Reed-Frost models)

Stochastic

continuous time

Stochastic differential equations

discrete time

Stochastic difference equations

Discrete treatment of individuals

Deterministic

Continuous treatment of individuals

(averages, proportions, or population densities)

continuous time

Ordinary differential equations
Partial differential equations

discrete time

Difference equations

(eg, Reed-Frost type models)

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Taxonomy of compartmental models

continuous time

Gillespie algorithm

discrete time

Chain binomial type models

(eg, Stochastic Reed-Frost models)

Stochastic

continuous time

Stochastic differential equations

discrete time

Stochastic difference equations

Discrete treatment of individuals

Deterministic

Continuous treatment of individuals

(averages, proportions, or population densities)

continuous time

Ordinary differential equations
Partial differential equations

discrete time

Difference equations

(eg, Reed-Frost type models)

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Taxonomy of compartmental models

continuous time

Gillespie algorithm

discrete time

Chain binomial type models

(eg, Stochastic Reed-Frost models)

Stochastic

continuous time

Stochastic differential equations

discrete time

Stochastic difference equations

Discrete treatment of individuals

Deterministic

Continuous treatment of individuals

(averages, proportions, or population densities)

continuous time

Ordinary differential equations
Partial differential equations

discrete time

Difference equations

(eg, Reed-Frost type models)

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Taxonomy of compartmental models

continuous time

Gillespie algorithm

discrete time

Chain binomial type models

(eg, Stochastic Reed-Frost models)

Stochastic

continuous time

Stochastic differential equations

discrete time

Stochastic difference equations

Discrete treatment of individuals

Deterministic

Continuous treatment of individuals

(averages, proportions, or population densities)

continuous time

Ordinary differential equations
Partial differential equations

discrete time

Difference equations

(eg, Reed-Frost type models)

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Taxonomy of compartmental models

What is a compartmental model?

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I

R

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Goals

Review the main uses of applied epidemiological modelling
Introduce our conceptual framework for applied modelling
Review commonly overlooked assumptions that are inherent in the structure of simple compartmental ODE models
Discuss when these assumptions might be problematic, and when they may be desirable
Begin to explore some alternative model structures that relax the assumptions

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Taxonomy of compartmental models

Stochastic

Discrete treatment of individuals

Deterministic

Continuous treatment of individuals

(averages, proportions, or population densities)

large population size

continuous time

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time

Borchering & McKinley (2018) Multiscale Modeling and Simulation

DOI: 10.1137/17M1155259

I*

Demographic stochasticity

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Taxonomy of compartmental models

continuous time

Stochastic

Discrete treatment of individuals

Deterministic

Continuous treatment of individuals

(averages, proportions, or population densities)

large population size

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Environmental stochasticity

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Compartmental ODE models assume

Large population size
Deterministic progression

for a given set of initial conditions and parameter values, a deterministic model always gives the same outcome

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Benefit: Simplicity and consequences of assumptions facilitate quick assessments of what model outcomes are possible, and when

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Compartmental ODE models assume

Large population size
Deterministic progression

for a given set of initial conditions and parameter values, a deterministic model always gives the same outcome

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Breaking Assumptions!

Discrete Individuals and Finite Populations

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Taxonomy of compartmental models

Stochastic

Discrete treatment of individuals

Deterministic

Continuous treatment of individuals

(averages, proportions, or population densities)

continuous time

Ordinary differential equations

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Time proceeds in a continuous manner
Parameter values remain constant

continuous time

Ordinary differential equations

N

Simple ODE models assume

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Time proceeds in a continuous manner
Parameter values remain constant

continuous time

Ordinary differential equations

N

Simple ODE models assume

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continuous time

Ordinary differential equations

discrete time

Difference equations

N

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Simple compartmental ODE models assume

Homogeneity within compartments
Large population size
Deterministic progression
Time proceeds in a continuous manner
Parameter values remain constant
Memory-less processes

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Simple ODE models assume

Memory-less processes

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N

Proportion surviving to at least τ

τ

00

Again, we can understand the implications of this assumption by integrating the differential equation that corresponds to our very simplified model, this time from the initial time 0 to an arbitrary time we’ll call tau

We can now divide both sides by the initial population size to calculate the proportion of the population that survives at least to time tau

Looking at the shape of this curve, you can see that under this assumption, most individuals will leave the compartment very quickly and a small proportion of the population will remain in the population for a very long time. The distribution of times until an event occurs is known as a waiting time distribution, and the WTD for an event that occurs at a constant per capita rate, such as in this example, is known as an exponential WTD, for reasons that should be clear from the equations.

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Realistic waiting times

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Estimate based on data from Joshi et al. (2009) Transactions of the Royal Society of Tropical Medicine and Hygiene

A real-world example

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Again, this assumption is often violated in reality. Here, I’m showing example waiting time distributions based on data from a pneumonic plague outbreak in India.

In this case, we’re looking at the incubation period distribution. two types of probability distributions have been fitted to data available on the time from exposure to symptom onset for the 30 people who were identified as cases during this outbreak.

The grey curve shows the exponential fit to the data, and the green curve shows the best fit gamma distribution, which has a mean of 4.5 days and a shape parameter of 2.3. The gamma distribution is a much better representation of the actual distribution of the data.

Pneumonic plague in India

Mean is 4.5 days; shape is 2.3

For Gamma: 5.6 % > 10 days

For exponential >11% > 10 days

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Realistic waiting times

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Estimate based on data from Joshi et al. (2009) Transactions of the Royal Society of Tropical Medicine and Hygiene

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Realistic waiting times

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Estimate based on data from Joshi et al. (2009) Transactions of the Royal Society of Tropical Medicine and Hygiene

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Simple compartmental ODE models assume

Homogeneity within compartments
Large population size
Deterministic progression
Time proceeds in a continuous manner
Parameter values remain constant
Memory-less processes

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Breaking Assumptions!

Non-exponential waiting times

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Summary

Simple ODE models are important tools for building understanding

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Summary

Simple ODE models are important tools for building understanding
It’s important to recognize the assumptions built into these models

When populations are small, average behaviors can be misleading

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Summary

Simple ODE models are important tools for building understanding
It’s important to recognize the assumptions built into these models

When populations are small, average behaviors can be misleading
When rates vary, simple ODEs can fail to reproduce important (observed) dynamics

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Data from Earn et al. 2000 Science

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Summary

Simple ODE models are important tools for building understanding
It’s important to recognize the assumptions built into these models

When populations are small, average behaviors can be misleading
When rates vary, simple ODEs can fail to reproduce important (observed) dynamics

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Dushoff lecture on heterogeneity

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Summary

Simple ODE models are important tools for building understanding
It’s important to recognize the assumptions built into these models

When populations are small, average behaviors can be misleading
When rates vary, simple ODEs can fail to reproduce important (observed) dynamics

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Summary

The applied epidemiological modelling process requires

abstraction
specification and implementation
gaining an understanding of the dynamics
interpretation

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REAL WORLD

MODEL WORLD

MODEL

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https://figshare.com/articles/Mathematical_Assumptions_of_Simple_ODE_models/5044606

(Hidden) assumptions of simple compartmental ODE models. DOI: 10.6084/m9.figshare.5044606

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Juliet Pulliam & Rebecca Borchering

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This presentation is made available through a Creative Commons Attribution-Noncommercial license. Details of the license and permitted uses are available at� http://creativecommons.org/licenses/by/3.0/

Attribution:

Juliet Pulliam

Clinic on the Meaningful Modeling of Epidemiological Data

Source URL:

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