webinar: the role of applied math in real-time pandemic ...webinar: the role of applied math in...
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Webinar: The Role of Applied Math in Real-time Pandemic Response: How Basic Disease Models Work
Presented by: Professor Nina Fefferman
National Institute for Mathematical and Biological Synthesis, University of Tennessee, Knoxville
With support from the National Science Foundation (DBI-1300426)
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MEET YOUR MODERATOR
Louis J. Gross, PhD
Director, National Institute for Mathematical and Biological Synthesis (NIMBioS)
Director,, The Institute for Environmental Modeling, University of Tennessee
Chancellor’s Professor of Ecology and Evolutionary Biology and Mathematics, University of Tennessee
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HOW TO INTERACT TODAY
Type here
Question appears here
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NIMBioS.org/Webinars A recording of each webinar will be posted
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MEET YOUR PRESENTER
Nina Fefferman, PhD
Professor of Ecology and Evolutionary Biology and Mathematics, University of Tennessee
Director, The Mathematical Modeling Consulting Center, University of Tennessee
Associate Director, One Health Initiative, University of Tennessee
March 31, 2020
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Webinar Objectives
Understand how math models help us analyze and predict outbreaks
Gain familiarity with concepts in the news about pandemics: 𝑅𝑅0, "Flatten the Curve", etc.
See how we can use models to design public health policy
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The role of applied math in real-time pandemic response:
How basic disease models work
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What Do Math Modelers Do?
Start with some problem in the world
• Complicated Interactions• Zillions of possible factors• Lots of different possible
measurements / observations• Lots of things we can’t
measure
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Start with some problem in the world• Complicated Interactions• Zillions of possible factors• Lots of different possible measurements /
observations• Lots of things we can’t measure
Distill the world to abstract logic
We can use math to explain!
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Turn logic into equations
Lets us turn conceptual logic into quantitative calculations and
predictions: Numbers
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Lots of types of quantitative model methods
Linear Algebra and Matrices
Difference Equations
Differential Equations
Game Theory
Networks
Cellular Automata
Agent Based Models
Statistical Models
The details of what you do in each of these are
different, but the basic idea is the same:
Logic → Equations → Predictions
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Let’s Use Models to Understand Outbreaks
Measles (Gastañaduy et al. 2016) 1918 Flu (Hatchett et al. 2007)
West Nile virus (TX Dept of SHS, 2012)
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Let’s try to explain the pattern: Why do outbreaks end? (even if we don’t have a vaccine)
Measles (Gastañaduy et al. 2016) 1918 Flu (Hatchett et al. 2007)
West Nile virus (TX Dept of SHS, 2012)
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Let’s Abstract the Logic of Infections Together
Healthy, Susceptible
People
What do we know about infectious diseases and how they work?
Catch Infections From
Sick, Infectious People
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What happens to Susceptible People?
ContactA couple of steps:
1)
2)
Transmit Germs
3)
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Mathy things about what happens to Susceptible People
Contact
Transmit Germs
We need both Together this gives a rate of becoming
Let’s call it Rate 1 and keep it for later
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What happens to Infectious People?
Simpler:
Recover
Die
Good
Not Good
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Mathy things about what happens to Infectious People?
Still simpler:
Rate
Rate
Good
Not Good
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Mathy things about what happens to Infectious People?
Still simpler:
Rate
Rate
Good
Not Good
This option is depressing – let’s
ignore it
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Mathy things about what happens to Infectious People?
Still simpler:
RateGood
Let’s call this one Rate 2
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Let’s Put it All Together
S I RRate 1 Rate 2
Susceptible People become Infectious at Rate1 and Recover at Rate 2
Abstract logic of our system is:
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Now let’s make our logic into equations
S I RRate 1 Rate 2
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How many people will still be Susceptible tomorrow?
S I R
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
Rate 1 Rate 2
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How many people will be Infectious tomorrow?
S I R
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
Rate 1 Rate 2
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S I R
𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
Rate 1 Rate 2
How many people will be Recovered tomorrow?
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S I R
𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
Rate 1 Rate 2
The Complete Math!
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S I R
𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
Rate 1 Rate 2
This is really simple – Is it useful?YES
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Let’s go back to the question:Why do outbreaks end?
West Nile virus (TX Dept of SHS, 2012)
What is the pattern to explain?• First the number of Infectious people
goes up• Then it goes down Why?
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Why do outbreaks end?
• First the number of Infectious people goes up
• Then it goes down
Notice: This is a question just about the numbers of Infectious People over time
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When will the number of Infectious people fit that pattern?
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅)𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 is increasing when
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅)𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 > 0
the parts we add are bigger than the parts we subtract
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Why you actually took algebra:
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅)𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
This means:
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 > 0
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 > 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 > 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅
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Aha!
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 > 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅)𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
So what happens? Remember:
I is increasing when
S can only decrease!
We HAVE to get this picture for I over time
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This is also how we compare outbreaks
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 > 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅I is increasing when
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅
> 𝑅We can re-write as a ratio
This ratio at the start of an outbreak (when most people are
Susceptible) is called 𝑅𝑅0 for this model
The bigger the 𝑅𝑅0 for a disease, the worse we expect the outbreak to be
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Some 𝑅𝑅0 values for our favorite diseases
Disease 𝑹𝑹0Measles 12–18Smallpox 3.5–6HIV/AIDS 2–5COVID-19 1.4–3.9 ???Influenza 1918 1.4–2.8Ebola 2014 1.5–2.5Influenza (seasonal) 0.9–2.1
Shamelessly copied from Wikipedia.org for this talk
We can estimate 𝑅𝑅0from equations or
from observations of outbreak curves
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But wait, we know diseases don’t just go away permanently – What’s going on?
Measles in China: Chao Ma et al. 2014
Global Flu Pandemics: Potter 2008
Annual Flu Outbreaks: Potter 2008
Rotavirus in Africa: Platts-Mills et al. 2017
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Babies!
Measles in China: Chao Ma et al. 2014 Annual Flu Outbreaks: Potter 2008
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In reality, the number of Susceptibles ISN’T only decreasing
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 =𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑆𝑆𝑏𝑏𝑡𝑡𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅)𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
How much disease we can keep around the
population is directly tied to the birth rate
(and if people lose immunity over time)
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This is also why Vaccines can protect people who aren’t even vaccinated
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 =
Taking out enough Susceptiblesso that 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ≯
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 is the goal!
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑆𝑆𝑣𝑣𝑡𝑡𝑣𝑣𝑣𝑣 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅)𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
Called the “herd immunity threshold”
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S I R
𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑅𝑅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
Rate 1 Rate 2
This really simple model is so powerful!
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𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
Let’s use it to understand what we mean by “flatten the curve”
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𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
It’s all about Rate 1
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𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡− (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) ∗ 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
If we can slow down Rate 1
• People move out of S more slowly• I increases more slowly• It takes longer to run out of S, but we never build up
as many I at onceKeeps people alive – never run out of hospital beds
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Remember how we got Rate 1?
Contact
Transmit Germs
It was a combination of:
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This is how to slow down Rate 1
Contact
Transmit Germs
Social Distancing
Better Hygiene
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This is how to slow down Rate 1
Contact
Transmit Germs
Social Distancing
Better Hygiene
Using just this, we can predict: • How effective is shelter-in-place?
• How long do we need to continue sheltering?
• What percent of people have to shelter to shut down the outbreak?
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Not all the answers can be obtained this easily…
• We usually use continuous averages instead • Many diseases are more logically complicated
• Different ways of being transmitted• Insect bites• Sexual contact• Contamination of an environment (fomite)• Mother to fetus• etc.
• Delay between being infected and becoming infectious• Immunity can go away over time• Age-based differences• Seasonal/Climatological differences• Opportunistic infection• etc.
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The logic and the math can look complicated
The logic of a Tuberculosis and HIV/AIDS model my students and I just published:
DeNegre, Myers, Fefferman. Antibiotics 2020
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The equations were worse:
The first 12 equations (out of 40) from a model to understand how
different strains of TB would circulate in parts of Africa where
there is already a high prevalence of AIDS in the population
But the basic ideas are EXACTLY THE SAME
The logic and the math can look complicated
DeNegre, Myers, Fefferman. Antibiotics 2020
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The predictions our models make allow us to compare interventions
Model outcomes comparing mosquito control
strategies to combat Zika
virus in different cities
Stone, Schwab, Fonseca, Fefferman. PLoS NTD 2020
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Sometimes we use predicted curves, sometimes we use theoretical thresholds
𝑅𝑅0 =𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅
Together, these are the critical tools of mathematical epidemiology
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Moral: Math Models Keep Us All Safer
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