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Modeling frameworks

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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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Deterministic Compartmental Modeling

Susceptible

Infected

Recovered

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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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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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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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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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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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Example: Constant-growth model

i(0) = 0; k = 7

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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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Example: Proportional-growth model

i(1) = 1; r = 0.3

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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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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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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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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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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

Expected incidence at time t

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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

Expected incidence at time t

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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)

Expected incidence at time t

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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

Expected incidence at time t

Careful: only because this is a “closed”population

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SI model

Susceptible

Infected

Expected incidence at time t

What does this mean for our system of equations?

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SI model

Susceptible

Infected

Expected incidence at time t

What does this mean for our system of equations?

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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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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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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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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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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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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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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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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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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)

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- System elements
- Persons/animals, pathogens, vectors

- States
- properties of elements
As before, but

- properties of elements
- Transitions
- Movement from one state to another: Probabilistic

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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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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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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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- 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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- First we’ll look at the graphical output of a model
- … then we’ll take a peek behind the curtain

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# 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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# 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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- 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.

- In a limited way
- 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

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

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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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