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Basic stochastic simulation models

Clinic on the Meaningful Modeling of Epidemiological Data, 2017

African Institute for Mathematical Sciences

Muizenberg, South Africa

Rebecca Borchering, PhD, MS

Postdoctoral Fellow

Biology Department & Emerging Pathogens Institute

University of Florida

Slide Set Citation: DOI: 10.6084/m9.figshare.5044624

The ICI3D Figshare Collection 1

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Goals• Understand motivation for using stochastic models

• Understand two common approaches to stochastic simulation: discrete time vs. event-driven

• Be able to relate these models to the differential equation models we have studied

• Recognize potential benefits and limitations of these stochastic models

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CONTINUOUS TIME

• Gillespie algorithm

DISCRETE TIME

• Chain binomial type models

(eg, Stochastic Reed-Frost models)STO

CH

AST

IC

CONTINUOUS TIME

• Stochastic differential equations

DISCRETE TIME

• Stochastic difference equations

DISCRETE TREATMENT OF INDIVIDUALS

Model taxonomyD

ETER

MIN

ISTI

C

CONTINUOUS TREATMENT OF INDIVIDUALS

(averages, proportions, or population densities)

CONTINUOUS TIME

• Ordinary differential equations

• Partial differential equations

DISCRETE TIME

• Difference equations

(eg, Reed-Frost type models)

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Animal Rabies Cases in Tanzania

Data Credit: Hampson et al. 2009

Serengeti and Ngorongoro Districts

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Why stochastic?

• Small populations, extinction

• Noisy data• imperfect observation

• small samples

• Environmental stochasticity• long term variation in external drivers

• changes in rates, including birth and death rates

• Demographic stochasticity• comes out of having discrete individuals

www.imagepermanenceinstitute.org

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Population size -

Continuous Time Markov Chain (CTMC)- finite population size

- stochastic

Ordinary Differential Equation (ODE)- large (infinite) population size- deterministic 6

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Reed-Frost (Chain Binomial)

• Fixed infectious period duration

• Generations of infectious individuals don’t overlap

• Define

probability of infection

infectious individuals (cases) at time t

susceptible individuals at time t

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The Reed-Frost model

Infectious

Susceptible

Recovered

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• Let

• So the probability of getting infected by any infectious individual is

• For each susceptible individual, at time t:

The Reed-Frost model

prob. not getting infected by a particular infectious individual

prob. not getting infected by any infectious individual

prob. of getting infected by any infectious individual

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• The probability of getting infected by any infectious individual is

• The expected number of cases in the next time unit is

The Reed-Frost model

• Susceptible individuals in the next time unit

• Recovered individuals in the next time unit

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The Reed-Frost model

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• The full set of equations describing the deterministic population update is:

• If is the total population size, the basic reproductive number for this model is

The Reed-Frost model

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• For each susceptible individual, at time t:

• Let

• So the probability of getting infected by any infectious individual is

Building stochastic R-F model

prob. not getting infected by a particular infectious individual

prob. not getting infected by any infectious individual

prob. of getting infected by any infectious individual

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• For each susceptible individual, at time t:

Building stochastic R-F model

prob. not getting infected by a particular infectious individual

prob. of not getting infected by any infectious individual

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The stochastic R-F modelPutting it all together:

prob. of x individuals getting infected by any infectious individual

number of ways to choose x individuals

prob. of individuals not getting infected by any infectious individual

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The Reed-Frost model

Stochastic: Deterministic:

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Chain binomial models• Chain binomial models can also be formulated based on the same

parameters we used in the ODE models and with overlapping generations

• Instantaneous hazard of infection for an individual susceptible individual is

• For a susceptible at time t, the probability of infection by time

is

• Similarly, for an infectious individual at time t, the probability of recovery by time is

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Chain binomial models

The stochastic population update can then be described as

new infectious individuals

new recovered individualsrandom variables

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Chain binomial models• For this model, if D is the average duration of infection, the basic

reproductive number is:

• Non-generation-based chain binomial models can be adapted to include many variations on the natural history of infection.

• Discrete-time simulation of chain binomials is far more computationally efficient than event-driven simulation in continuous time.

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Chain binomial simulationwhile (I > 0 and time < MAXTIME)

Calculate transition probabilities

Determine number of transitions for each type

Update state variables

Update time

end

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Another way to simulate stochastic epidemics…

event-driven simulation

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Stochastic SIR dynamics

Small population

Susceptible

Infectious

Recovered

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Stochastic SIR dynamics

Small population

Susceptible

Infectious

Recovered

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Stochastic SIR dynamics

Small population

Susceptible

Infectious

Recovered

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Stochastic SIR dynamics

Small population

Susceptible

Infectious

Recovered

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Stochastic SIR dynamics

Small population

Susceptible

Infectious

Recovered

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Exponential waiting times

waiting time distribution: distribution of times until an event occurs

time between events

0 2 4 6 8 10

0.0

0.5

1.0

1.5

Days since infection

Pro

ba

bili

ty d

en

sity rate

0.511.5

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Summary: Gillespie algorithm

• finite, countable populations

• well-mixed contacts

• exponential waiting times (memory-less)

Assumptions:

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Summary: Gillespie algorithm

• finite, countable populations

• well-mixed contacts

• exponential waiting times (memory-less)

Assumptions:

• noise (stochasticity) is introduced by the discrete nature of individuals

• event-driven simulation

• computationally slow• especially for large populations

Notes:

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• What happened ?

• When did it happen?

Need to know

Two event types:

Transmission

Recovery

Susceptible to Infectious

Infectious to Recovered

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Need to know• What happened ?

• When did it happen?

Two event types:

Transmission

Recovery

ODE analogue:

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Need to know• What happened ? EventType

• When did it happen? EventTime

Two event types:

Transmission

Recovery

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The Gillespie algorithmTwo event types:

Transmission

Recovery

Time to the next event:

Probability the event is type i: 34

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Simulating the Gillespie modelwhile (I > 0 and time < MAXTIME)

Calculate rates

Determine time to next event

Determine event type

Update state variables

Update time

end

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Break

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Rabies example

?Maintenance population Target population

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Types of transmissionSpillover infections

Within-population

Maintenance population Target population

Target population 38

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R code example

SIR model with spillover

Associated file is linked from the schedule

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R code example

• population size

• spillover rate

• transmission rate

• recovery rate

Try changing:

SIR model with spillover

Associated file is linked from the schedule

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Sample output

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Sub-critical or super-critical?Basic reproduction number for SIR model:

Sub-critical Super-critical

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Jackal rabies application

Rhodes et al. (1998) "Rabies in Zimbabwe: reservoir dogs and the implications for disease control." Philosophical Transactions of the Royal Society B.

(rabies is sub-critical)

Question: how many additional infections are needed in order for rabies to be super-critical in the jackal population?

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Summary• There are many sources of stochasticity including small

population size, imperfect observation of cases, and environmental variability

• The Reed-Frost model is a discrete time model, where the time step is based on the infectious period

• Chain binomial models can be thought of as a generalization of the stochastic Reed-Frost model with an arbitrary time step and infectious period duration

• Event-driven simulations where each event is modeled separately are intuitive, but can be computationally slow especially for large populations

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This presentation is made available through a Creative Commons Attribution license. Details of the license and permitted uses are available at

http://creativecommons.org/licenses/by/3.0/

© 2012 International Clinics on Infectious Disease Dynamics and DataBorchering R, Pulliam JRCP. “Basic Stochastic Simulation Models” Clinic on the Meaningful Modeling of

Epidemiological Data. DOI: 10.6084/m9.figshare.5044624.For further information or modifiable slides please contact figshare@ici3d.org.

See the entire ICI3D Figshare Collection. DOI: 10.6084/m9.figshare.c.3788224.

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