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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 [email protected].
See the entire ICI3D Figshare Collection. DOI: 10.6084/m9.figshare.c.3788224.
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