Modeling frameworks
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Modeling frameworks. Today, compare:Deterministic Compartmental Models (DCM) Stochastic Pairwise Models (SPM) for (I, SI, SIR, SIS) R est of the week: Focus on Stochastic Network Models. Deterministic Compartmental Modeling. Susceptible. Infected. Recovered.

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Modeling frameworks

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Modeling frameworks

Modeling frameworks

Today, compare:Deterministic Compartmental Models (DCM)

Stochastic Pairwise Models (SPM)

for (I, SI, SIR, SIS)

Rest of the week: Focus on Stochastic Network Models

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Modeling frameworks

Deterministic Compartmental Modeling

Susceptible

Infected

Recovered

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Modeling frameworks

Compartmental Modeling

  • A form of dynamic modeling in which people are divided up into a limited number of “compartments.”

  • Compartments may differ from each other on any variable that is of epidemiological relevance (e.g. susceptible vs. infected, male vs. female).

  • Within each compartment, people are considered to be homogeneous, and considered only in the aggregate.

Compartment 1

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Modeling frameworks

Compartmental Modeling

  • People can move between compartments along “flows”.

  • Flows represent different phenomena depending on the compartments that they connect

  • Flow can also come in from outside the model, or move out of the model

  • Most flows are typically a function of the size of compartments

Susceptible

Infected

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Modeling frameworks

Compartmental Modeling

  • May be discrete time or continuous time: we will focus on discrete

  • The approach is usually deterministic – one will get the exact same results from a model each time one runs it

  • Measures are always of EXPECTED counts – that is, the average you would expect across many different stochastic runs, if you did them

  • This means that compartments do not have to represent whole numbers of people.

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Modeling frameworks

Constant-growth model

Infected population

t= time

i(t)= expected number of infected people at time t

k= average growth (in number of people) per time period

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Modeling frameworks

Constant-growth model

difference equation

(three different notations for the same concept – keep all in mind when reading the literature!)

recurrence equation

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Modeling frameworks

Example: Constant-growth model

i(0) = 0; k = 7

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Modeling frameworks

Proportional growth model

Infected population

t = time

i(t)= expected number of infected people at time t

r= average growth rate per time period

difference equation

recurrence equation

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Modeling frameworks

Example: Proportional-growth model

i(1) = 1; r = 0.3

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Modeling frameworks

SI model

New infections per unit time (incidence)

Susceptible

Infected

t = time

s(t)= expected number of susceptible people at time t

i(t) = expected number of infected people at time t

What is the expected incidence per unit time?

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Modeling frameworks

SI model

t = time

s(t)= expected number of susceptible people at time t

i(t) = expected number of infected people at time t

A new infection requires: a susceptible person to have an act with an infected person and for infection to transmit because of that act

Expected incidence at time t

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Modeling frameworks

SI model

t = time

s(t)= expected number of susceptible people at time t

i(t) = expected number of infected people at time t

A new infection requires: a susceptible person to have an act with an infected person and for infection to transmit because of that act

Expected incidence at time t

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Modeling frameworks

SI model

t = time

s(t)= expected number of susceptible people at time t

i(t) = expected number of infected people at time t

= act rate per unit time

A new infection requires: a susceptible person to have an act with an infected person and for infection to transmit because of that act

Expected incidence at time t

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Modeling frameworks

SI model

t = time

s(t)= expected number of susceptible people at time t

i(t) = expected number of infected people at time t

= act rate per unit time

A new infection requires: a susceptible person to have an act with an infected person and for infection to transmit because of that act

Expected incidence at time t

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Modeling frameworks

SI model

t = time

s(t)= expected number of susceptible people at time t

i(t) = expected number of infected people at time t

= act rate per unit time

= “transmissibility” = prob. of transmission given S-I act

A new infection requires: a susceptible person to have an act with an infected person and for infection to transmit because of that act

Expected incidence at time t

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Modeling frameworks

SI model

t = time

s(t)= expected number of susceptible people at time t

i(t) = expected number of infected people at time t

= act rate per unit time

= “transmissibility” = prob. of transmission given S-I act

n(t)= total population = s(t) + i(t)

A new infection requires: a susceptible person to have an act with an infected person and for infection to transmit because of that act

Expected incidence at time t

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Modeling frameworks

SI model

t = time

s(t)= expected number of susceptible people at time t

i(t) = expected number of infected people at time t

= act rate per unit time

= “transmissibility” = prob. of transmission given S-I act

n(t)= total population = s(t) + i(t) = n

A new infection requires: a susceptible person to have an act with an infected person and for infection to transmit because of that act

Expected incidence at time t

Careful: only because this is a “closed”population

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Modeling frameworks

SI model

Susceptible

Infected

Expected incidence at time t

What does this mean for our system of equations?

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Modeling frameworks

SI model

Susceptible

Infected

Expected incidence at time t

What does this mean for our system of equations?

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Modeling frameworks

SI model - Recurrence equations

Remember:

constant-growth model could be expressed as:

proportional-growth model could be expressed as:

The SI model is very simple, but already too difficult to express as a simple recurrence equation.

Solving iteratively by hand (or rather, by computer) is necessary

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Modeling frameworks

SIR model

What if infected people can recover with immunity?

Susceptible

Infected

Recovered

And let us assume they all do so at the same rate:

t = time

s(t) = expected number of susceptible people at time t

i(t) = expected number of infected people at time t

r(t) = expected number of recovered people at time t

= act rate per unit time

= prob. of transmission given S-I act

r= recovery rate

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Modeling frameworks

Relationship between duration and recovery rate

  • Imagine that a disease has a constant recover rate of 0.2. That is, on the first day of infection, you have a 20% probability of recovering. If you don’t recover the first day, you then have a 20% probability of recovering on Day 2. Etc.

  • Now, imagine 100 people who start out sick on the same day.

  • How many recover after being infected 1 day?

  • How many recover after being infected 2 days?

  • How many recover after being infected 3 days?

  • What does the distribution of time spent infected look like?

  • What is this distribution called?

  • What is the mean (expected) duration spent sick?

100*0.2 = 20

80*0.2 = 16

64*0.2 = 12.8

Right-tailed

Geometric

5 days ( = 1/.2)

D = 1/ r

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Modeling frameworks

SIR model

Expected number of new infections at time t still equals

where nnow equals

Expected number of recoveries at time t equals

So full set of equations equals:

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Modeling frameworks

SIR model

= 0.6, = 0.3, = 0.1

Initial population sizes s(0)=299; i(0)=1; r(0) = 0

susceptible

recovered

infected

What happens on Day 62? Why?

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Modeling frameworks

Qualitative analysis pt 1:

Epidemic potential

Using the SIR model

  • R0= the number of direct infections occurring as a result of a single infection in a susceptible population – that is, one that has not experienced the disease before

  • We saw earlier that R0= . So for the basic SIR model, it also equals /

  • Tells one whether an epidemic is likely to occur or not:

  • If R0 > 1, then a single infected individual in the population will on average infect more than one person before ceasing to be infected. In a deterministic model, the disease will grow

  • If R0 < 1, then a single infected individual in the population will on average infect less than one person before ceasing to be infected. In a deterministic model, the disease will fade away

  • If R0 = 1, we are right on the threshold between an epidemic and not. In a deterministic model, the disease will putter along

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Modeling frameworks

SIR model

= 4, = 0.2, = 0.2

Initial population sizes s(0)=999; i(0)=1; r(0) = 0

Compartment sizes

Flow sizes

Transmissions (incidence)

Recoveries

Susceptible

Infected

Recovered

R0= /= (4)(0.2)/(0.2) = 4

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Modeling frameworks

SIR model

= 4, = 0.2, = 0.8

Initial population sizes s(0)=999; i(0)=1; r(0) = 0

Compartment sizes

Flow sizes

Transmissions (incidence)

Recoveries

Susceptible

Infected

Recovered

R0= /= (4)(0.2)/(0.8) = 1

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Modeling frameworks

SIR model with births and deaths

death

death

death

Susceptible

Infected

Recovered

recov.

birth

trans.

t = time

s(t) = number of susceptible people at time t

i(t) = number of infected people at time t

r(t) = number of recovered people at time t

= act rate per unit time

= prob. of transmission given S-I act

r= recovery rate

f= fertility rate

ms= mortality rate for susceptibles

mi = mortality rate for infecteds

mr = mortality rate for recovereds

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Stochastic pairwise models spm

Stochastic Pairwise models (SPM)

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Basic elements of the stochastic model

Basic elements of the stochastic model

  • System elements

    • Persons/animals, pathogens, vectors

  • States

    • properties of elements

      As before, but

  • Transitions

    • Movement from one state to another: Probabilistic

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Deterministic vs stochastic models

Deterministic vs. stochastic models

Simple example: Proportional growth model

  • States: only I is tracked, population has an infinite number of susceptibles

  • Rate parameters: only 𝛽, the force of infection (b= ta)

Incident infections are determined by the force of infection

Incident infections are drawn from a probability distribution that depends on

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What does this stochastic model mean

What does this stochastic model mean?

Depends on the model you choose for P(●)

P(●) is a probability distribution.

  • Probability of what? … that the count of new infections dI= kat time t

  • So what kind of distributions are appropriate? … discrete distributions

  • Can you think of one?

  • Example: Poisson distribution

  • Used to model the number of events in a set amount of time or space

  • Defined by one parameter: it is the both the mean and the variance

  • Range: 0,1,2,… (the non-negative integers)

  • The pmf is given by:

  • P(X=k) =

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How does the stochastic model capture transmission

How does the stochastic model capture transmission?

The effect of l on a Poisson distribution

  • P(dIt=k) =

Mean:E(dIt)=lt

Variance:Var(dIt)=lt

If we specify:lt = It dt

Then: E(dIt)=  It dt , the deterministic model rate

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What do you get for this added complexity

What do you get for this added complexity?

  • Variation – a distribution of potential outcomes

    • What happens if you all run a deterministic model with the same parameters?

    • Do you think this is realistic?

      • Recall the poker chip exercises

      • Did you all get the same results when you ran the SI model?

      • Why not?

  • Easier representation of all heterogeneity, systematic and stochastic

    • Act rates

    • Transmission rates

    • Recovery rates, etc…

  • When we get to modeling partnerships:

    • Easier representation of repeated acts with the same person

    • Networks of partnerships

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Example a simple stochastic model programmed in r

Example: A simple stochastic model programmed in R

  • First we’ll look at the graphical output of a model

  • … then we’ll take a peek behind the curtain

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Behind the curtain a simple r code for this model

Behind the curtain: a simple R code for this model

# First we set up the components and parameters of the system

steps <- 70 # the number of simulation steps

dt <- 0.01 # step size in time units

total time elapsed is then steps*dt

i <- rep(0,steps) # vector to store the number of infected at time(t)

di <- rep(0,steps) # vector to store the number of new infections at time(t)

i[1] <- 1 # initial prevalence

beta <- 5 # beta = alpha (act rate per unit time) *

# tau (transmission probability given act)

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Behind the curtain a simple r code for this model1

Behind the curtain: a simple R code for this model

# Now the simulation: we simulate each step through time by drawing the

# number of new infections from the Poisson distribution

for(k in 1:(steps-1)){

di[k] <- rpois(n=1, lambda=beta*i[k]*dt)

i[k+1] <- i[k] + di[k]

}

In words:

For t-1 steps (for(k in 1:(steps-1))) Start of instructions( { )

new infections at step t <- randomly draw from Poisson (rpois)

di[k] one observation (n=1)

with this mean (lambda= … )

update infections at step t+1 <- infections at (t) + new infections at (t)

i[k+1] i[k] ni[k]

End of instructions( } )

lt = It dt

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The stochastic deterministic relation

The stochastic-deterministic relation

  • Will the stochastic mean equal the deterministic mean?

    • Yes, but only for the linear model

    • The variance of the empirical stochastic mean depends on the number of replications

  • Can you represent variation in deterministic simulations?

    • In a limited way

      • Sensitivity analysis shows how outcomes depend on parameters

      • Parameter uncertainty can be incorporated via Bayesian methods

      • Aggregate rates can be drawn from a distribution (in Stella and Excel)

    • But micro-level stochastic variation can not be represented.

  • Will stochastic variation always be the same?

    • No, can specify many different distributions with the same mean

      • Poisson

      • Negative binomial

      • Geometric …

    • The variation depends on the probability distribution specified

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To epimodel

To EpiModel…

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Modeling frameworks

Appendix:

Finding equilibria

Using the SIR model without birth and death

also

written

as

or

or

is required by condition 3, and also satisfies conditions 1 and 2

Without new people entering the population, the epidemic will always die out eventually.

Note that s(t) and r(t) can thus take on different values at equilibrium

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