Rapid spread of disease through a population in a relatively short time (defined by host lifespan)
Different from endemic diseases, which occur at some equilibrial level in the population (e.g., malaria, HIV)
Emergence: Previously unknown disease to a given population/region/country
Re-emergence: Known disease that is coming back in a given population/region/country after a period of absence
Outbreak: Known disease in a given population/region/country
What are examples of each of these?
SARS-COV2: $11-16 trillion
Influenza: $60-90 billion
But tough to say. These estimates consider missed work in these estimates, among other things. Perhaps it’s better to think about the cost of epidemics in terms of disability adjusted life years (1 DALY = 1 year of healthy life lost).
Mitigation: we want to lessen disease burden
Understanding: modeling allows us to understand when, why, and how epidemics emerge
Blame: some models can be used to identify key locations or persons responsible for epidemic initiation
Compartmental models of infectious disease (will definitely cover)
Individual-based or network models (potentially will have time to cover)
Fitting statistical distributions to time series data (may not have time to cover)
S : susceptible individuals
I : infectious individuals
R : recovered (and immune) individuals
Costs of parasitism do not have to involve death
\[ \begin{aligned} \frac{dS}{dt} &= -\beta SI \\ \frac{dI}{dt} &= \beta SI - dI \\ \frac{dR}{dt} &= dI \end{aligned} \]
\[ \begin{aligned} \frac{dS}{dt} &= -\beta SI \\ \frac{dI}{dt} &= \beta SI - dI \\ \frac{dR}{dt} &= dI \end{aligned} \]
\(R_0\): the number of secondary infections generated by a single infected individual in a wholly susceptible population
\(\frac{dI}{dt}>0\) only if \(\beta SI > dI\)
\(R_0 = \frac{\beta SI}{dI} = \frac{\beta S}{d}\)
\(\frac{1}{d}\): the average duration of infection
Ives and Buzzuto, 2021 https://www.nature.com/articles/s42003-020-01609-6
\[R_0 = \frac{\beta S}{d}\]
\(R_0 = \frac{\beta S (1 - c)}{d} < 1\)
\(c > 1 - \frac{d}{\beta S} = 1 - \frac{1}{R_0}\)
https://covid.cdc.gov/covid-data-tracker/#vaccinations_vacc-people-booster-percent-pop5
\(R_0\) gives us a nice criteria for whether a pathogen will invade a particular host population.
Invasion potential is often driven by host density, but also on many aspects of the infection process which are not considered in \(R_0\), like …
Think-pair-share: 5 minutes to talk with your neighbors about what you think drives pathogen invasion potential that is not in \(R_0\).
Dallas, T. A., Krkošek, M., & Drake, J. M. (2018). Experimental evidence of a pathogen invasion threshold. Royal Society open science, 5(1), 171975.
Clif et al. 1981
\(R_0\) estimation is dependent on the model used, right?
Diseases with \(R_0 < 1\) may persist (HOW!?)
Makes a fair number of assumptions (examples?)
https://web.stanford.edu/~jhj1/teachingdocs/Jones-on-R0.pdf, https://www.hindawi.com/journals/cmmm/2011/527610/, etc. etc.
The goal is to reduce transmission or move people out of the susceptible class, right?
How do we do these things?
https://marcoapintoo.shinyapps.io/sir-model/
The SIR model from our reading this week is different.
\[
\begin{aligned}
\frac{dS}{dt} &= -\beta SI \\
\frac{dI}{dt} &= \beta SI - dI \\
\frac{dR}{dt} &= dI
\end{aligned}
\]
\[ \begin{aligned} \frac{dS}{dt} &= S\lambda - \beta SI \\ \frac{dI}{dt} &= \beta SI - dI \\ \frac{dR}{dt} &= dI \end{aligned} \]
What’s going on here?
Typically epidemics are on a timescale where might ignore demographic changes.
What if we don’t?
\[ \begin{aligned} \frac{dS}{dt} &= S\lambda - \beta SI \\ \frac{dI}{dt} &= \beta SI - dI \\ \frac{dR}{dt} &= dI \end{aligned} \]
What happens in this model to the host population size in the absence of disease?
Consider \(\lambda\) to be birth - death. But infected and recovered individuals can die too, right?
\[ \begin{aligned} \frac{dS}{dt} &= S\lambda - \beta SI \\ \frac{dI}{dt} &= \beta SI - dI - \alpha I \\ \frac{dR}{dt} &= dI - \alpha R \end{aligned} \]
What might this do to population dynamics?
What might this do to infection dynamics?
\(\beta SI\), where \(\beta\) is encounter * susceptibility
We went over:
The SIR model is incredibly simple , but also incredibly extensible
\[ \begin{align} \frac{dS}{dt} & = -\beta SI \\ \frac{dE}{dt} & = \beta SI - \omega E\\ \frac{dI}{dt} & = \omega E - \gamma I \\ \frac{dR}{dt} & = \gamma I \end{align} \]
\(R_0\) is not affected by the addition of the exposed class, as all exposed individuals will get infected.
Since all exposed individuals get infected, the number of secondary infections from 1 infected individual in a wholly susceptible population (\(R_0\)) is the same as the SIR.
\[ R_0 = \frac{\beta SI}{dI} = \frac{\beta S}{d} \]
\[ \frac{d(E+I)}{dt} = (\beta S - \gamma)I \]
The above will become negative if
\[ \beta S_0 - \gamma < 0 \]
Then we can solve for the threshold condition
\[ R_0 = \frac{\beta S_0}{\gamma} \]
The well-mixed assumption means that everyone interacts with everyone else, such that every individual has the same risk of infection.
Density-dependent transmission: \(\beta SI\)
Frequency-dependent transmission: \(\beta S \frac{I}{N}\)
WHAT? WHY?
Begon et al. 2002 Epidemiology and Infection
Classic density dependent transmission in SIR
Classic frequency dependent transmission in SIR
For frequency-dependent transmission, population size does not matter.
For density-dependent transmission, it definitely does.
In the SIR model, the pathogen is already there, and it is going through a closed population.
What happens when we consider spatial spread of an emerging pathogen?
In 1988, Harbor seals were having massive die-offs (~60% of the colony) across much of this area
The cause of these declines was a virus (morbillivirus) related to measles and canine distemper.
spread through respiratory aerosol
affects lungs of seals causing starvation (can’t dive)
recovered individuals become immune
Each location experienced it’s own epidemic, separated in time.
Because it’s not meant for that
…but within host dynamics are important and have consequences for population-level spread.
unusually contagious organism infected with a disease
This can happen a number of ways, but all essentially influence \(\beta\).
A superspreader can:
A cook for multiple families (lots of contacts)
Had no (or very minor) symptoms (asymptomatic)
22 people presented signs of infection and many died
Quarantined at a hospital for 2 years, but freed in 1910
Went on to infect 25 more people (2 died)
Quarantined again until her death
Mary Mallon (1869-1938) and the history of typhoid fever: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3959940/
Goyal et al. 2021 eLife
Possible to incorporate into SIR models
Fitting these models to data is still one of the best tools we have to understand and mitigate epidemics
Sometimes it’s the best we can do
Superspreaders
We won’t until a couple weeks from now, but useful to think about how you might do it.
Individuals don’t have to have the same \(\beta\)
a <- rnorm(1000, 3, 0.5)/6 a2 <- sample(a, 10)
## [[1]] ## NULL ## ## [[2]] ## NULL ## ## [[3]] ## NULL ## ## [[4]] ## NULL ## ## [[5]] ## NULL ## ## [[6]] ## NULL ## ## [[7]] ## NULL ## ## [[8]] ## NULL ## ## [[9]] ## NULL ## ## [[10]] ## NULL ## ## [[11]] ## NULL ## ## [[12]] ## NULL ## ## [[13]] ## NULL ## ## [[14]] ## NULL ## ## [[15]] ## NULL ## ## [[16]] ## NULL ## ## [[17]] ## NULL ## ## [[18]] ## NULL ## ## [[19]] ## NULL ## ## [[20]] ## NULL ## ## [[21]] ## NULL ## ## [[22]] ## NULL ## ## [[23]] ## NULL ## ## [[24]] ## NULL ## ## [[25]] ## NULL ## ## [[26]] ## NULL ## ## [[27]] ## NULL ## ## [[28]] ## NULL ## ## [[29]] ## NULL ## ## [[30]] ## NULL ## ## [[31]] ## NULL ## ## [[32]] ## NULL ## ## [[33]] ## NULL ## ## [[34]] ## NULL ## ## [[35]] ## NULL ## ## [[36]] ## NULL ## ## [[37]] ## NULL ## ## [[38]] ## NULL ## ## [[39]] ## NULL ## ## [[40]] ## NULL ## ## [[41]] ## NULL ## ## [[42]] ## NULL ## ## [[43]] ## NULL ## ## [[44]] ## NULL ## ## [[45]] ## NULL ## ## [[46]] ## NULL ## ## [[47]] ## NULL ## ## [[48]] ## NULL ## ## [[49]] ## NULL ## ## [[50]] ## NULL ## ## [[51]] ## NULL ## ## [[52]] ## NULL ## ## [[53]] ## NULL ## ## [[54]] ## NULL ## ## [[55]] ## NULL ## ## [[56]] ## NULL ## ## [[57]] ## NULL ## ## [[58]] ## NULL ## ## [[59]] ## NULL ## ## [[60]] ## NULL ## ## [[61]] ## NULL ## ## [[62]] ## NULL ## ## [[63]] ## NULL ## ## [[64]] ## NULL ## ## [[65]] ## NULL ## ## [[66]] ## NULL ## ## [[67]] ## NULL ## ## [[68]] ## NULL ## ## [[69]] ## NULL ## ## [[70]] ## NULL ## ## [[71]] ## NULL ## ## [[72]] ## NULL ## ## [[73]] ## NULL ## ## [[74]] ## NULL ## ## [[75]] ## NULL ## ## [[76]] ## NULL ## ## [[77]] ## NULL ## ## [[78]] ## NULL ## ## [[79]] ## NULL ## ## [[80]] ## NULL ## ## [[81]] ## NULL ## ## [[82]] ## NULL ## ## [[83]] ## NULL ## ## [[84]] ## NULL ## ## [[85]] ## NULL ## ## [[86]] ## NULL ## ## [[87]] ## NULL ## ## [[88]] ## NULL ## ## [[89]] ## NULL ## ## [[90]] ## NULL ## ## [[91]] ## NULL ## ## [[92]] ## NULL ## ## [[93]] ## NULL ## ## [[94]] ## NULL ## ## [[95]] ## NULL ## ## [[96]] ## NULL ## ## [[97]] ## NULL ## ## [[98]] ## NULL ## ## [[99]] ## NULL ## ## [[100]] ## NULL
## [[1]] ## NULL ## ## [[2]] ## NULL ## ## [[3]] ## NULL ## ## [[4]] ## NULL ## ## [[5]] ## NULL ## ## [[6]] ## NULL ## ## [[7]] ## NULL ## ## [[8]] ## NULL ## ## [[9]] ## NULL ## ## [[10]] ## NULL ## ## [[11]] ## NULL ## ## [[12]] ## NULL ## ## [[13]] ## NULL ## ## [[14]] ## NULL ## ## [[15]] ## NULL ## ## [[16]] ## NULL ## ## [[17]] ## NULL ## ## [[18]] ## NULL ## ## [[19]] ## NULL ## ## [[20]] ## NULL ## ## [[21]] ## NULL ## ## [[22]] ## NULL ## ## [[23]] ## NULL ## ## [[24]] ## NULL ## ## [[25]] ## NULL ## ## [[26]] ## NULL ## ## [[27]] ## NULL ## ## [[28]] ## NULL ## ## [[29]] ## NULL ## ## [[30]] ## NULL ## ## [[31]] ## NULL ## ## [[32]] ## NULL ## ## [[33]] ## NULL ## ## [[34]] ## NULL ## ## [[35]] ## NULL ## ## [[36]] ## NULL ## ## [[37]] ## NULL ## ## [[38]] ## NULL ## ## [[39]] ## NULL ## ## [[40]] ## NULL ## ## [[41]] ## NULL ## ## [[42]] ## NULL ## ## [[43]] ## NULL ## ## [[44]] ## NULL ## ## [[45]] ## NULL ## ## [[46]] ## NULL ## ## [[47]] ## NULL ## ## [[48]] ## NULL ## ## [[49]] ## NULL ## ## [[50]] ## NULL ## ## [[51]] ## NULL ## ## [[52]] ## NULL ## ## [[53]] ## NULL ## ## [[54]] ## NULL ## ## [[55]] ## NULL ## ## [[56]] ## NULL ## ## [[57]] ## NULL ## ## [[58]] ## NULL ## ## [[59]] ## NULL ## ## [[60]] ## NULL ## ## [[61]] ## NULL ## ## [[62]] ## NULL ## ## [[63]] ## NULL ## ## [[64]] ## NULL ## ## [[65]] ## NULL ## ## [[66]] ## NULL ## ## [[67]] ## NULL ## ## [[68]] ## NULL ## ## [[69]] ## NULL ## ## [[70]] ## NULL ## ## [[71]] ## NULL ## ## [[72]] ## NULL ## ## [[73]] ## NULL ## ## [[74]] ## NULL ## ## [[75]] ## NULL ## ## [[76]] ## NULL ## ## [[77]] ## NULL ## ## [[78]] ## NULL ## ## [[79]] ## NULL ## ## [[80]] ## NULL ## ## [[81]] ## NULL ## ## [[82]] ## NULL ## ## [[83]] ## NULL ## ## [[84]] ## NULL ## ## [[85]] ## NULL ## ## [[86]] ## NULL ## ## [[87]] ## NULL ## ## [[88]] ## NULL ## ## [[89]] ## NULL ## ## [[90]] ## NULL ## ## [[91]] ## NULL ## ## [[92]] ## NULL ## ## [[93]] ## NULL ## ## [[94]] ## NULL ## ## [[95]] ## NULL ## ## [[96]] ## NULL ## ## [[97]] ## NULL ## ## [[98]] ## NULL ## ## [[99]] ## NULL ## ## [[100]] ## NULL
We talked about vaccination efforts and herd immunity
What about the timing of vaccination efforts?
What is the optimal timing of vaccine rollout? (think-pair-share 3 minutes)
