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disease.txt
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UNIT 8: Infectious disease
----------------------------------------------------------------------
EXTRA
Could flesh out more
A bit more detail about reciprocal control/replenishment
More on pathogen aggressiveness biology
As much more heterogeneity as you want; at least the fragile-heterogeneity lines
Case fatality
Incidence and prevalence
----------------------------------------------------------------------
SEC Introduction
----------------------------------------------------------------------
Infectious disease
Extremely common
Huge impacts on ecological interactions
A form of exploitation, but doesn't fit well into our previous
modeling framework
How many people are there?
How many influenza viruses are there?
How do they find each other?
----------------------------------------------------------------------
Disease agents
POLL Name an infectious agent that causes disease in humans.
Disease agents vary tremendously:
Most \textbf{viruses} have just a handful of genes that allow
them to hijack a cell and get it to make virus copies
ANS \scv, influenza virus, HIV, measles
\textbf{Bacteria} are independent, free-living cells with
hundreds or thousands of chemical pathways
ANS Tuberculosis, anthrax, pertussis
\textbf{Eukaryotic} pathogens are nucleated cells who are more
closely related to you than they are to bacteria
ANS Malaria, intestinal worms, trichomoniasis
----------------------------------------------------------------------
PSLIDE Influenza virus
FIG webpix/flu_virus.jpg
----------------------------------------------------------------------
PSLIDE Tuberculosis bacilli
FIG webpix/tb.jpg
----------------------------------------------------------------------
PSLIDE Malaria sporozoite
FIG webpix/sporozoite.jpg
----------------------------------------------------------------------
Microparasites
For infections with small pathogens (viruses and bacteria), we don't
attempt to count pathogens, but instead divide disease into stages
Latently infected
Productively infected
Recovered
----------------------------------------------------------------------
Microparasite models
We model microparasites by counting the number of hosts in various
\textbf{states}:
\textbf{Susceptible} individuals can become infected
\textbf{Infectious} individuals are infected and can infect others
\textbf{Resistant} individuals are not infected and cannot become
infected
More complicated models might include other states, such as
latently infected hosts who are infected with the pathogen but
cannot yet infect others
----------------------------------------------------------------------
Models as tools
BC
Models are the tools that we use to connect scales:
individuals to populations
single actions to trends through time
NC
SIDEFIG boxes/trans.jpg
SIDEFIG dd/ewmeas.Rout.pdf
EC
----------------------------------------------------------------------
TSEC Rate of spread
POLL For many diseases, especially new diseases, we can \emph{observe}
and \emph{estimate} $r$. | What is r?
ANS Instantaneous rate of increase (per capita)
ANS Units of 1/t
ANS Gives the exponential rate of spread
POLL Want to know what factors contribute to that, and how it relates to
\R. | What is R in a disease model?
ANS number of new cases per case
ANS Unitless
----------------------------------------------------------------------
Basic reproductive number
People in the disease field love to talk specifically about \Ro
But they don't always mean the same thing when they say \Ro:
Actual value of \R\ before an epidemic
Hypothetical value assuming no immunity
Hypothetical value assuming no immunity and no control efforts whatsoever
Often easier to talk simply about \R.
----------------------------------------------------------------------
Example: the West African Ebola epidemic
DOUBLEPDF WA_Ebola_Outbreak/liberia.npc.tsplot.Rout
----------------------------------------------------------------------
PRESLIDE COVID in Ontario
FIG covidCA/ON_class.Rout-0.pdf
----------------------------------------------------------------------
PRESLIDE COVID in Ontario
FIG covidCA/ON_class.Rout-1.pdf
----------------------------------------------------------------------
COVID in Ontario
DOUBLEFIG covidCA/ON_class.Rout-0.pdf covidCA/ON_class.Rout-1.pdf
----------------------------------------------------------------------
OFFSLIDE REPSLIDE COVID in Ontario
FIG covidCA/ON_class.Rout-2.pdf
Blowup likely intended to show current conditions in 2022.
----------------------------------------------------------------------
OFFSLIDE REPSLIDE COVID in Ontario
FIG covidCA/ON_class.Rout-3.pdf
----------------------------------------------------------------------
Scales
Which scale should we look at?
ANS Log scale is better for looking at trends
ANS Linear scale is better for looking at impacts
----------------------------------------------------------------------
Population biology
What quantities do we want to look at?
ANS Speed of exponential growth $r$
ANS Finite rate of increase $\lambda$
OFF ANS Skipped this year
ANS Lifetime reproduction, \Rx
----------------------------------------------------------------------
Instantaneous rate of growth $r$
What are the components?
ANS Birth rate
ANS Instantaneous rate of a case producing new cases
ANS [case/(case $\cdot$ time]
ANS Death rate
ANS Virus-centered!
ANS Rate of death, recovery, or effective quarantine
How do you think we estimate?
ANS We estimate $r$ from the population-level
increase in disease
ANS Then we use that to estimate $b = d+r$
ANS Models go both directions!
Individuals $\leftrightarrow$ Populations
----------------------------------------------------------------------
ONSLIDE Finite rate of growth $\lambda$
Why do we want this?
ANS to communicate with policy-makers or the public
ANS maybe to make concrete predictions, though we could use $r$
How do we calculate it?
ANS Pick a time step (week? year?)
ANS Use a formula $\lambda = \exp(r\Delta t)$
----------------------------------------------------------------------
ONSLIDE Example
$r \approx 0.14/\dd$ for early COVID spread
What is $\lambda$?
At a time scale of a day?
At a time scale of a week?
----------------------------------------------------------------------
Reproductive number $\R$
What is it?
ANS Expected number of new cases per case over the lifetime of a case
Why do we want this?
ANS An important measure of how hard the epidemic will be to stop
How do we calculate it?
ANS $\R = b/d$; if we can estimate those
----------------------------------------------------------------------
Example
$r \approx 0.14/\dd$
What is our estimate of \R?
When average length of infection $L = 5\dd$?
ANS $d = 1/(5\dd) = 0.2/\dd$
ANS $b = 0.14\dd + 0.2\dd = 0.34/\dd$
ANS $\R = 0.34/0.2 = 1.7$
When average length of infection $L = 10\dd$?
ANS $d = 1/(10\dd) = 0.1/\dd$
ANS $b = 0.14\dd + 0.1\dd = 0.24/\dd$
ANS $\R = 0.24/0.1 = 2.4$
----------------------------------------------------------------------
ONSLIDE Generation intervals
BC
Researchers try to estimate the \emph{proportion} of transmission
that happens for different \textbf{ages of infection}
How long from the time you are \emph{infected} to the time you
\emph{infect someone else}?
Analogous to a life table
The effective generation time $\hat G$ has units of time
COMMENT $\hat G$ is fairly deep; we'll skip the details
NC
SIDEFIG boxes/generationTime.Rout-0.pdf
EC
ADD We could have more about compound interest and variation
----------------------------------------------------------------------
ONSLIDE Generation intervals
DOUBLEFIG boxes/generationTime.Rout-0.pdf boxes/generationTime.Rout-1.pdf
----------------------------------------------------------------------
ONSLIDE Speed and risk
BC
Which is more dangerous, a fast disease, or a slow disease?
How are we measuring speed?
How are we measuring danger?
\emph{What do we already know?}
NC
SIDEFIG boxes/generationTime.Rout-0.pdf
EC
----------------------------------------------------------------------
ON PSLIDE Fighting Ebola
FIG webpix/burial_team.jpg
----------------------------------------------------------------------
ONSLIDE Generation time and risk
If we know $\R$, what does the generation time tell us about $r$?
ANS The faster the generations (small $\hat G$), the faster the
exponential growth (large $r$)
If we know $r$, what does the generation time tell us about $\R$?
ANS The faster the generations (small $\hat G$), the
\emph{smaller} the strength of the epidemic (small reproductive
number $\R$)
$\R = \exp(r \hat G)$
----------------------------------------------------------------------
ON REPSLIDE Generation time and risk
FIG boxes/steps.Rout-0.pdf
----------------------------------------------------------------------
ON REPSLIDE Generation time and risk
FIG boxes/steps.Rout-1.pdf
----------------------------------------------------------------------
ONSLIDE Generation time and risk
DOUBLEPDF boxes/steps.Rout
----------------------------------------------------------------------
ONSLIDE Generation time and risk
$\R = \exp(r \hat G)$
An intuitive view:
Epidemic speed = Generation strength $\times$ Generation speed
COMMENT Mathematically: $r = \log(\R) * (1/\hat G)$
If we know generation speed, then a faster epidemic speed means:
ANS More strength required (greater $\R$)
If we know epidemic speed, a faster generation speed means
ANS Less strength required (smaller $\R$)
----------------------------------------------------------------------
TSEC Single-epidemic model
PRESENT WIDEFIG boxes/sir.np.three.pdf
Susceptible $\to$ Infectious $\to$ Recovered
We also use $N$ to mean the total population
----------------------------------------------------------------------
Transition rates
WIDEFIG boxes/sir.three.pdf
What factors govern movement through the boxes?
People get better independently
People get infected by infectious people
----------------------------------------------------------------------
Conceptual modeling
BC
POLL What happens in the long term if we introduce an infectious individual?
ANS There \emph{may be} an \textbf{epidemic} -- an outbreak of
disease
ANS Disease burns out
ANS Everyone winds up recovered
ANS … or susceptible
ANS Or, there may not be an outbreak
NC
SIDEFIG boxes/sir.three.pdf
EC
----------------------------------------------------------------------
Interpreting
Why might there not be an epidemic?
ANS If the disease can't spread well enough in the population
ANS Could depend on season, or immunity \ldots
ANS Demographic stochasticity: if we only start with one
individual, we expect an element of chance
Why doesn't everyone get infected?
NOANS
----------------------------------------------------------------------
ONSLIDE Implementing the model
BC
The {simplest} way to implement this conceptual
model is with differential equations:
$$\frac{dS}{dt} = - \beta \frac{SI}{N} $$
$$\frac{dI}{dt} = \beta \frac{SI}{N}- \gamma I $$
$$\frac{dR}{dt} = \gamma I $$
$$ N = S+I+R $$
NC
SIDEFIG boxes/sir.three.pdf
EC
----------------------------------------------------------------------
Quantities
CLASS WIDEFIG boxes/sir.three.pdf
\textbf{State variables}
$S, I, R, N$: [people] or [people/ha]
----------------------------------------------------------------------
CONT Quantities
\textbf{Parameters}
Susceptible people have \textbf{potentially effective} contacts at rate
$\beta$ (units [1/time])
These are contacts that would lead to infection if the person
contacted is infectious
Total infection rate is $\beta I/N$, because $I/N$ is the
proportion of the population infectious
Infectious people recover at \emph{per capita}
rate $\gamma$ (units [1/time])
Total recovery rate is $\gamma I$
Mean time infectious is $D = 1/\gamma$ (units [time])
----------------------------------------------------------------------
APRIL REPSLIDE Implementing the model
$$\frac{dS}{dt} = - \beta \frac{SI}{N} $$
$$\frac{dI}{dt} = \beta \frac{SI}{N}- \gamma I $$
$$\frac{dR}{dt} = \gamma I $$
$$ N = S+I+R $$
This is a Hamiltonian dynamical system
ANS Apply Principle of Least Action
----------------------------------------------------------------------
APRIL Principle of Least Action
If:
$$ \mathbf{q} : \mathbf{R} \to \mathbf{R}^N $$
$$ \mathcal{S}[\mathbf{q}, t_1, t_2]
= \int_{t_1}^{t_2} L(\mathbf{q}(t),\mathbf{\dot{q}}(t), t) dt $$
Then:
ANS $$ \delta \mathcal{S} = 0 $$
----------------------------------------------------------------------
APRIL Hamilton's equation of motion
Write down the Lagrangian energy:
$$ E_{\cal L}(\boldsymbol{q},\boldsymbol{\dot q},t)\, \stackrel{\text{def}}{=}\, \sum^n_{i=1} \dot q^i \frac{\partial {\cal L}}{\partial \dot q^i} - {\cal L} $$
The inverse Legendre transform gives the Hamiltonian:
$$ {\cal H}\left(\frac{\partial {\cal L}}{\partial \boldsymbol{\dot q}},\boldsymbol{q},t\right) = E_{\cal L}(\boldsymbol{q},\boldsymbol{\dot q},t) $$
Dynamics conserve these level sets!
----------------------------------------------------------------------
APRIL The Lorenz attractor
BCC
CFIG webpix/butterfly.png
NCC
Calculate the (maximal) Lyapunov exponent as:
$$ \lambda = \lim_{t \to \infty} \lim_{|\delta \mathbf{Z}_0| \to 0}
\frac{1}{t} \ln\frac{| \delta\mathbf{Z}(t)|}{|\delta \mathbf{Z}_0|} $$
EC
----------------------------------------------------------------------
REPSLIDE Quantities
CLASS WIDEFIG boxes/sir.three.pdf
State variables
$S, I, R, N$: [people] or [people/ha]
----------------------------------------------------------------------
REPSLIDE Simulating the model
DOUBLEPDF sims/burnout.plots.Rout
----------------------------------------------------------------------
Simulating the model
DOUBLEFIG sims/burnouts.plots.Rout-0.pdf sims/burnouts.plots.Rout-4.pdf
----------------------------------------------------------------------
Basic reproductive number
POLL What \emph{unitless} parameter can you make from the model above?
How can you make a unitless parameter reflecting disease spread?
ANS $\Ro = \beta D = \beta/\gamma$ is the \textbf{basic
reproductive number}
ANS The \emph{potential} number of infections caused by an
average infectious individual
ANS That is: the number they would cause on average if
everyone else were susceptible
ANS The product of the rate $\beta$ (units [1/t]) and the
duration $D$ ([t])
----------------------------------------------------------------------
Basic reproductive number implications
POLL What happens early in the epidemic if $\Ro>1$?
What happens early in the epidemic if Ro>1?
ANS Number of infected individuals grows exponentially
What happens early in the epidemic if $\Ro<1$?
ANS Number of infected individuals does not grow (disease cannot
invade)
----------------------------------------------------------------------
Effective reproductive number
The effective reproductive number gives the number of new infections
per infectious individual in a partially susceptible population:
ANS $\Reff = \Ro S/N$
Is the disease increasing or decreasing?
ANS It will increase when $\Reff > 1$ (more than one case per case)
ANS This happens when $S/N > 1/\Ro$
Why doesn't everyone get infected?
ANS When susceptibles are low enough $\Reff<1$
ANS When $\Reff<1$, the disease dies out on its own (less than one case per case)
----------------------------------------------------------------------
TSS Epidemic size
In this model, the epidemic always burns out
No source of new susceptibles
Epidemic size is determined by:
ANS \Ro: larger \Ro\ leads to a bigger epidemic
ANS The number of susceptibles at the beginning of the epidemic
ANS More susceptibles leads to a bigger epidemic
ANS \ldots and \emph{fewer} susceptibles at the end
ON ANS The number of infected individuals at the beginning of the epidemic
ON ANS Usually relatively small (and a relatively small effect)
----------------------------------------------------------------------
Overshoot
BC
Why does more susceptibles at the beginning mean fewer susceptibles at the end?
ANS More susceptibles $\implies$
ANS Faster initial growth $\implies$
ANS Bigger epidemic $\implies$
ANS More infections at peak (same number of susceptibles) $\implies$
ANS More generations needed for disease to fade out $\implies$
ANS More infections after peak \ldots
NC
SIDEFIG sims/burnouts.plots.Rout-4.pdf
EC
----------------------------------------------------------------------
Ebola example
BC
In September, the US CDC predicted ``as many as'' 1.5 million Ebola
cases in Liberia by January
In fact, their model predicted many \emph{more} cases than that by April
What happened?
NC
SIDEFIG WA_Ebola_Outbreak/liberia.npc.tsplot.Rout.pdf
EC
----------------------------------------------------------------------
What limits epidemics?
POLL What limits epidemics in our simple models?
ANS Depletion of susceptibles
POLL What else limits epidemics in real life?
ANS Interventions; changes in government policy, medicine, vaccines
ANS Behaviour change; people stay home, wear masks, avoid sick people
ANS Heterogeneity (differences between hosts, locations, etc.)
----------------------------------------------------------------------
TSEC Recurrent epidemic models
BC
POLL
If epidemics tend to burn out, why do we often see repeated
epidemics?
ANS People might lose immunity
ANS Births and deaths; newborns are susceptible
ANS Pathogen evolution
NC
SIDEFIG dd/ewmeas.Rout.pdf
EC
----------------------------------------------------------------------
Recurrent epidemics
FIG dd/ewmeas.Rout.pdf
----------------------------------------------------------------------
Closing the circle
WIDEFIG boxes/sirs.four.pdf
ANS Loss of immunity
----------------------------------------------------------------------
Closing the circle
WIDEFIG boxes/sirbd.four.pdf
ANS Births and deaths
ANS Effect on dynamics is similar to loss of immunity
----------------------------------------------------------------------
ONSLIDE Births and deaths
BC
$$\frac{dS}{dt} = b N - \beta \frac{SI}{N} - d S$$
$$\frac{dI}{dt} = \beta \frac{SI}{N}- \gamma I -d I $$
$$\frac{dR}{dt} = \gamma I - d R$$
We often assume $b=d$
$\implies$ population is constant
NC
SIDEFIG boxes/sirbd.four.pdf
EC
----------------------------------------------------------------------
SS Dynamics
----------------------------------------------------------------------
Equilibrium
At equilibrium, we know that $\Reff=1$
One case per case
Number of susceptibles at equilibrium determined by the number required
to keep infection in balance
$S/N = 1/\Ro$
What does this remind you of?
ANS Reciprocal control!
----------------------------------------------------------------------
CONT Equilibrium
Number of infectious individuals determined by number required to keep susceptibles in balance.
As susceptibles go up, what happens?
Per capita replenishment goes down
Infections required goes down
----------------------------------------------------------------------
Reciprocal control
What happens to \emph{equilibrium} if we protect susceptibles (move them to $R$ class)?
ANS Equation for $dI/dt$ does not change
ANS Number of susceptibles at equilibrium does not change
ANS Fewer susceptibles removed by infection (some are removed
by us)
ANS Less disease
What else could happen?
ANS If we remove susceptibles fast enough, infection could go extinct
ANS If we keep increasing the rate \ldots
ANS Number of susceptibles goes down
----------------------------------------------------------------------
Reciprocal control
POLL What happens if we remove infectious individuals at a constant rate (find them and cure them or isolate them)?
What happens if we remove infectious individuals at a constant rate?
ANS We need more susceptibles to balance $dI/dt$
ANS If we have more susceptibles, then per capita replenishment goes down
ANS So the number of infectious individuals required for balance goes
down
ANS If we remove infectious individuals fast enough, the infection could
go extinct
ANS $\R_c S/N \leq 1$
----------------------------------------------------------------------
REPSLIDE Single-epidemic model
FIG sims/burnouts.plots.Rout-4.pdf
----------------------------------------------------------------------
Tendency to oscillate
DOUBLEPDF sims/recurrent.plots.Rout
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Tendency to oscillate
``Closed-loop'' SIR models (ie., with births or loss of immunity):
Tend to oscillate
Oscillations tend to be damped
System reaches an \textbf{endemic} equilibrium -- disease
persists
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Source of oscillations
Similar to predator-prey systems
What happens if we start with a lot of susceptibles?
ANS There will be a big epidemic
ANS \ldots then a very low number of susceptibles
ANS \ldots then a very low level of disease
ANS \ldots then an increase in the number of susceptibles
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Persistent oscillations
BC
POLL If oscillations tend to be damped in simple models, why do they
persist in real life?
ANS Weather
ANS Seasonality
ANS Environmental stochasticity
ANS School terms
ANS Demographic stochasticity
ANS Changes in Behaviour
ANS People are more careful when disease levels are high
ANS Pathogen mutations
NC
SIDEFIG dd/ewmeas.Rout.pdf
EC
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