stochastic.lect

SEC Introduction

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Modelling individual events

	Differential equations model continuous processes

	Disease spreads in the real world through discrete events

	Discrete events are fundamentally stochastic

		Even in theory we don't know when the next event will occur, nor
		even what it will be

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Types of stochasticity

	Demographic stochasticity is caused by the existence of individual
	people and discrete events

	Environmental stochasticity refers to events that affect more than
	one person at a time

		Weather

		Politics

		Economics

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NONOTES Modelling individual events

TEMPFIG trans 0.9

A stochastic potential-transmission event

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NONOTES Deterministic spread

TEMPLAND onedet

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NONOTES Demographic spread

TEMPLAND onesto

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Demographic spread

TEMPLAND onestoch
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SEC Describing a stochastic process 

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States and rates

	We describe our system in terms of the {\em probability rates}
	of events happening

		If the rate of event $E$ is $r_E(t)$, the probability of the
		event occurring in the time interval $(t, t+dt)$ is	$r_E(t) dt$

	If the system is {\em Markovian}, $r_E(t)$ depends only on the state
	of the system at time $t$

		The Markovian assumption is convenient, but can
		have unwanted consequences

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NONOTES States and rates (Deterministic)

SLIP sirs3

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States and rates (Demographic)

SLIP stochsirs2

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Analogy

	The demographic model is an exact analogue of the deterministic one

		Conceptually

		In the limit as $N\to\infty$

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Realizations and ensembles

	How do we think about the behavior of a stochastic process?

		A single example of how the process could go (e.g., from a
		stochastic simulation) is called a {\em realization}

		The universe of possible realizations is called the {\em
		ensemble}.

		The probability distribution that describes what state we expect
		the population to be in at time $t$ is called the {\em ensemble
		distribution}

			Knowing how the ensemble distribution evolves is not the same
			as understanding the whole ensemble

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Some techniques

	Simulate one or many realizations

	Simulate the ensemble distribution

		Requires one state variable for each possible state of the system

	Solve the ensemble distribution dynamics exactly!

		Rarely possible

	Analytic approximations to the ensemble distribution

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Simulating a realization

	Given a state of the system:

		List possible events, and associated rates

		Calculate the total rate $r_T$: this gives the rate at which the
		next event (whatever it is) will happen

			i.e., an exponential waiting time with mean 1/$r_T$

		The probability of event $E$ is $r_E/r_T$

		Randomly select the time and nature of the next event

		Change the system state appropriately

		Repeat forever

			Or until system is extinct

			Or until you are tired

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NONOTES Deterministic model

TEMPLAND comp0

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Demographic model

TEMPLAND comp

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Modelling the ensemble distribution

	We model the ensemble distribution by creating one conceptual `box'
	for each possible state of the system, and asking what is the
	probability that the system is in each box.

		This can be a lot of boxes

	The probabilities change as follows:

	$\displaystyle
		\dot p_S = \sum_{S'}{p_{S'} r_{S'\to S}}
			-  p_S \sum_{S'}	{r_{S\to S'}}
	$

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Modelling the ensemble distribution

	If our system is small enough (particularly, if it has one state
	dimension) we might be able to simulate the ensemble distribution

	We might seek to solve these equations analytically

		Only in special cases

	We might seek analytic approximations to increase our understanding

		Moment approximations

		Diffusion approximations

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Questions

	What kind of questions do we want to ask with a stochastic model?

		How does stochasticity affect disease dynamics?

			Spatial distribution

			Establishment

			Persistence

		How much variance do we expect stochasticity to cause?

		Under what circumstances can we eliminate or eradicate a disease?

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SEC Equilibrium and quasi-equilibrium

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Equilibrium

	Define equilibrium as an ensemble distribution that does not change
	with time

	What are the equilibria of our stochastic SIR system?

		Disease free equilibrium

		Others?

	There is no equilibrium corresponding to the endemic equilibrium of
	the deterministic system!

		As long as any populations not extinct, proportion extinct will
		increase.

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Quasi-equilibrium

	Consider the ensemble distribution confined to the subset of states
	where nothing is extinct

	This system can be described as a {\em fully connected} (you can get
	anywhere from anywhere else, {\em open} (you can leave the set) flow.

	Linear algebra tells us that such a system will converge to a stable
	{\em relative} distribution of probabilities of being in each
	non-extinct box

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Interpreting the quasi-equilibrium

	The quasi-equilibrium is the asymptotic distribution of system states
	given that nothing has gone extinct

	The eigenvalue $\lambda_q$ associated with the quasi-equilibrium
	distribution is the rate at which the probability of non-extinction
	decays (exponentially)

		The distribution of persistence times must be asymptotically
		exponential

		The expected persistence time (looking forward) approaches
		$-1/\lambda_q$ as the system continues to persist

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Modelling the ensemble distribution

	We model the ensemble distribution by creating one conceptual `box'
	for each possible state of the system, and asking what is the
	probability that the system is in each box.

		This can be a lot of boxes

	The probabilities change as follows:

	$\displaystyle
		\dot p_S	= \sum_{S'}	{p_{S'}	r_{S'\to S}}
			-	p_S	\sum_{S'}	{r_{S\to S'}}
	$

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Modelling the quasi-equilibrium

	We can also model the probability of being in a particular state
	given that extinction has not occured

		Computationally convenient

		Can also keep track of cumulative extinction probability

	Define $q_S	= p_S/	(1-p_N)$ where $N$ is a `null' state that we
	cannot escape from.

	Use quotient rule to find dynamic equations for $q_S$.

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The fate of infectious disease

	Fizzle

		Disease fails to ``establish'' 

		We will make this precise later

	Burn-out

		Disease goes extinct after first epidemic

	Fade-out

		Disease goes extinct after system approaches quasi-equilibrium

		Can take a {\em long} time

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SEC Analytic methods

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Linearization

	Two of the most useful tools for understanding deterministic disease
	models are linearizations:

		{\bf Disease-free equilibrium:} what factors control whether the
		disease can invade and persist?

		{\bf Endemic equilibrium:} tendency to cycle, damping or
		persistence of cycles

	Both of these methods have analogues in demographic models

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Linear birth-death process

	We do an invasion analysis by asking how the number of infectives
	behaves in the limit where we assume that virtually the whole
	population is susceptible.

	This corresponds to a demographic model with the state determined by
	the number of infectious individuals $I$ 
	
	This system has only two events:

		Infection at rate $R_0 I$

		Recovery at rate $I$

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Long-term behavior

	Unlike the finite systems discussed before, the probability of
	eventual extinction in this system is not one!

	Why not?

		Probability of extinction given persistence goes to zero, as
		expected number of infectious individuals goes to $\infty$

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Extinction probability

	Chains of infection are independent in this model

	We can use this fact to solve directly for the probability of
	extinction when starting from $I$ infections, $E_I$

		$\displaystyle E_I = R_0^{-I}$, when $R_0>1$

		$1$, otherwise

	We can define this as the `fizzle' probability: the disease would
	have gone extinct even without depleting any susceptibles.

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Moment calculations

	Ask: what is the expected behavior of the mean, variance, \ldots of
	the ensemble?

	Define: $\ds \mu = \sum_I	{I p_i}$

	How does $\mu$ change through time?

		$\ds \dot \mu = \sum_I	{I \dot p_I}$

		$\ds = \sum_I	{(b_I	-	m_I)	p_I}$, where	$b(I) = R_0I$ is the
		`birth' rate, and $m(I) = I$ is the `death' rate

	These equations can be solved in the linear system, or approximated
	(by ``moment closure") for non-linear systems

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Diffusion approximations

	We can approximate the discrete-valued demographic system with a
	real-valued system that reflects the mean {\em and variance} of the
	demographic system

		Thus we can incorporate demographic stochasticity in a continuous
		system

		An excellent approximation except when some values are very small

	In a linear (or linearized) system, we can solve the equilibrium
	distribution of the continuous equations, and thus approximate
	the quasi-equilibrium distribution

		Disease persistence

		Size of demographic fluctuations

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Diffusion approximations

	We linearize about the endemic equilibrium, in exact analogy to Jacobian
	methods for stability in deterministic models

	Diffusion (and thus demographic stochasticity) is relatively
	unimportant when the square of the number infected is large compared
	to the demographic variance

		Number infected at equilibrium: $\ds \frac{(R_0-1)\rho N}{R}$

			$\ds \approx \rho N$

		Demographic variance: $\approx N$

	Diffusion index $\ds \approx \rho^2 N$.  If $\rho$ is small,
	demographic stochasticity can be important even for very large
	populations.

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TSEC Conclusions

	Treating individuals as individuals can have dramatic effects on
	models of disease transmission

		Acquired immunity is an important part of this phenomenon

	Stochastic models are hard, and we usually combine techniques to
	understand them:

		Analytic approximation

		Simulating ensemble distributions

		Simulating realizations

	Remember: demographic stochasticity is real!