disease.txt

UNIT 7: Infectious disease

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SEC Introduction

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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?

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Disease agents

	POLL free_text_polls/CCUi8ULHN5OBMlO Can you 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 influenza virus, Ebola 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, various worms

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SSLIDE Influenza virus

FIG images/flu_virus.jpg

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SSLIDE Tuberculosis bacilli

FIG images/tb.jpg

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SSLIDE Malaria sporozoite

FIG images/sporozoite.jpg

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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

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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

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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 images/trans.jpg

SIDEFIG dd/ewmeas.Rout.pdf

EC

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TSEC Rate of spread

	POLL free_text_polls/Bzr6gpzwQ5k1xR://www.polleverywhere.com/free_text_polls/r5O8ujpEfmdBcc0 For many diseases, especially new diseases, we can \emph{observe}
	and \emph{estimate} $r$. What is r?

		ANS the exponential rate of spread

	POLL free_text_polls/NZAPJMsY64WCB5b Want to know what factors contribute to that, and how it relates to
	\R. What is R?

		ANS number of new cases per case

CHANGE CC: 1st poll wasn't working and replace answer of r by per capita growth rate (unit 1/t)
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Basic reproductive number

	People in the disease field love to talk specifically about \Ro

	But they don't always mean the same thing:

		Actual value of \R\ before an epidemic

		Hypothetical value assuming no immunity

		Hypothetical value assuming no control efforts whatsoever

	Often easier to talk simply about \R.

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Example: the West African Ebola epidemic

DOUBLEPDF ebola/liberia.npc.tsplot.Rout

CHANGE CC: Need to increase the size of axis (numbers + labels)
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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

NC

SIDEFIG generationTime.Rout-0.pdf

EC

CHANGE CC: put the SSLIDE Fighting Ebola picture here?
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Generation intervals

DOUBLEFIG generationTime.Rout-0.pdf generationTime.Rout-1.pdf

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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 generationTime.Rout-0.pdf

EC

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SSLIDE Fighting Ebola

FIG images/burial_team.jpg

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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  the
		\emph{smaller} the strength of the epidemic (small reproductive
		number $\R$)

	$\R = \exp(r \hat G)$

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RSLIDE Generation time and risk

FIG steps.Rout-0.pdf

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RSLIDE Generation time and risk

FIG steps.Rout-1.pdf

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Generation time and risk

DOUBLEPDF steps.Rout

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Generation time and risk

	An intuitive view:

		Epidemic speed = Generation speed $\times$ Generation strength

	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 $\R$)

CHANGE CC: add the previous formula R = \exp(r \hat G)
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TSEC Single-epidemic model

WIDEFIG boxes/sir.np.three.pdf

	Susceptible $\to$ Infectious $\to$ Recovered

	We also use $N$ to mean the total population

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Transition rates

WIDEFIG boxes/sir.three.pdf

	What factors govern movement through the boxes?

		People get better independently

		People get infected by infectious people

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Conceptual modeling

BC

	POLL free_text_polls/DXEewMaDpNhiJyY What happens in the long term if we introduce an infectious individual?

		ANS The \emph{may be} an \textbf{epidemic} -- an outbreak of
		disease

		ANS Disease burns out

		ANS Everyone winds up either recovered or susceptible

		ANS Not everyone gets infected!

NC

SIDEFIG boxes/sir.three.pdf

EC

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Interpreting

	Why might there not be an epidemic?

		ANS If the disease can't spread well enough in the population

		ANS Demographic stochasticity: if we only start with one
		individual, we expect an element of chance

	Why doesn't everyone get infected?

		NOANS

		CHANGE How is this one resolved?

CHANGE CC: add environment as sub-reasons (influenza during summer)
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Implementing the model

BC

	The {simplest} way to implement this conceptual
	model is with differential equations:

		$$\frac{dS}{dt}  =  - \beta \frac{SI}{N} $$

		$$\frac{dI}{dt}  = \beta \frac{SI}{N}- \gamma I $$

		$$\frac{dR}{dt}  = \gamma I $$

NC

SIDEFIG boxes/sir.three.pdf

EC


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Quantities

CLASS WIDEFIG boxes/sir.three.pdf

State variables

	$S, I, R, N$: [people] or [people/ha]

CHANGE CC: Stop here on March 30
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CONT Quantities

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])

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RSLIDE Simulating the model

DOUBLEPDF sims/burnout.plots.Rout

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Simulating the model

DOUBLEFIG sims/burnouts.plots.Rout-0.pdf sims/burnouts.plots.Rout-4.pdf

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Basic reproductive number

	POLL free_text_polls/Dx8yk5UQrFPOJq0 What \emph{unitless} parameter can you make from the model above?
	What unitless parameter can you make?

		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

CHANGE CC: good idea to write the def/unit of the variables on the board
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Basic reproductive number implications

	POLL free_text_polls/Gj5tDb3y6grYJzG 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 cannot grow (disease cannot
		invade)

CHANGE CC: Perhaps consider to also have the 2nd one as a poll too (only 1 person was willing to participate)
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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)

CHANGE CC: perhaps for next year consider to have the boxes plots (or have it small on slides where you are referring to it) (you drew it later for TSS Epidemic size but probably usefull to also have it earlier)
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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

CHANGE CC: probably add the number of sick people at the beginning as the 3rd factor
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Overshoot 

BC

	Why does more susceptibles at the beginning mean fewer susceptibles at the
	end?

		ANS Bigger epidemic $\implies$

		ANS More infections at peak (same number of susceptibles) $\implies$

		ANS More infections after peak \ldots

CHANGE CC: For answer: need more steps / more details

NC

SIDEFIG sims/burnouts.plots.Rout-4.pdf

EC

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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 ebola/liberia.npc.tsplot.Rout

EC

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What limits epidemics?

	POLL free_text_polls/Ww28eOlRUK8f5I2 What limits epidemics in our simple models?

		ANS Depletion of susceptibles

	POLL free_text_polls/GBZdsyZ88grHUbJ What else limits epidemics in real life?

		ANS Interventions

		ANS Behaviour change

		ANS Heterogeneity (differences between hosts, locations, etc.)

CHANGE CC: Stop here on March 31st
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TSEC Recurrent epidemic models

BC

	POLL free_text_polls/lkgsFIpMHpl8V7F
	If epidemics tend to burn out, why do we often see repeated
	epidemics?

		ANS People might lose immunity

		ANS Births and deaths

NC

SIDEFIG dd/ewmeas.Rout.pdf

EC

CHANGE CC: add more details for the ans
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Recurrent epidemics

FIG dd/ewmeas.Rout.pdf

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Closing the circle

WIDEFIG boxes/sirs.four.pdf

	ANS Loss of immunity

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Closing the circle

WIDEFIG boxes/sirbd.four.pdf

	ANS Births and deaths

		ANS Effect on dynamics is similar to loss of immunity

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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

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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

			$S/N = 1/\Ro$
		
		ANS Reciprocal control!

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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

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Reciprocal control

	What happens if we protect susceptibles (move them to $R$ class)?

		ANS Equation for $dI/dt$ does not change

		ANS Number of susceptibles does not change

		ANS Fewer susceptibles need to be removed by infection (some are removed
		by us)
		
		ANS Number of infectious individuals goes down 

	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

CHANGE CC: add that it is at the equilibrium?
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Reciprocal control

	POLL free_text_polls/UaSJBaSOoy0rOKi 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

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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 too many 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 free_text_polls/qvMZLezBsM69b21 If oscillations tend to be damped in simple models, why do they
	persist in real life?

		ANS Weather

		ANS School terms

		ANS Demographic stochasticity

		ANS Changes in Behaviour
		
			ANS People are more careful when disease levels are high

NC

SIDEFIG dd/ewmeas.Rout.pdf

EC

CHANGE CC: ANS seasonality (add/change compared to weather?)?
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TSEC Reproductive numbers and risk

	At equilibrium, the proportion of people who are susceptible to
	disease should be approximately $S/N = 1/\Ro$

	Proportion ``affected'' (infectious or immune) should be
	approximately $V/N = 1-1/\Ro$

	If you have a single, fast epidemic, the size is also predicted by
	\Ro.

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Reproductive numbers and risk

DOUBLEFIG sims/fs.Rout-1.pdf sims/fs.Rout-0.pdf

CHANGE CC: Go slower when you draw the plots on the board: V/N vs R0 and I vs S (for the last one also explain more what are the loops)

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Examples

	Ronald Ross predicted 100 years ago that reducing mosquito densities
	by a factor of 5 or so would \emph{eliminate} malaria

	Gradual disappearance of polio, typhoid, etc., without risk factors
	going to zero

	Eradication of smallpox!

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Threshold for elimination

	What proportion of the population should be vaccinated to eliminate
	a disease?

		ANS Transmission should be reduced by a factor of $\R$, so a
		fraction $1-1/\R$ should be vaccinated

CHANGE CC: add "at least" (+ can also be killing mosquito or other stuff to decrease the repro of virus…)
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Examples: 

	Polio has an $\Ro$ of about 5.

	POLL free_text_polls/rpnF77L9UWWOEki What proportion of the population
	should be vaccinated to eliminate polio?

		ANS At least 1-1/5 = 80%

	Measles has an $\Ro$ of about 20. What proportion of the population
	should be vaccinated to eliminate measles?

		ANS At least 1-1/20 = 95%

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Persistence of infectious disease

	Why have infectious diseases persisted?

		The pathogens \emph{evolve}

		Human populations are \textbf{heterogeneous}

			People differ in: nutrition, exposure, access to care

		Information and misinformation

			Vaccine scares, trust in health care in general

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Heterogeneity and persistence

	Heterogeneity \emph{increases} \Ro

		When disease is rare, it is concentrated in the most vulnerable
		populations 

			Cases per case is high	

			Elimination is harder

	Marginal populations

		Heterogeneity could make it easier to concentrate on the most
		vulnerable populations and eliminate disease

		Humans rarely do this, however: the populations that need the
		most support typically have the least access