Download The Mathematical Formulation of the Foot-and-mouth

The LLNL FMD Decision Support System:
Concise Description of Features and Output
Tanya Kostova
T. Bates, C. Melius, S. Smith, A. Robertson, S. Hazlett, P. Hullinger,
Lawrence Livermore National Laboratory
DIMACS Workshop March 2006
“Data Mining and Epidemiological Modeling”
LLNL is developing a decision support system for evaluation
of the economic impact of FMD epidemics
•Effort funded by the Department of Homeland Security
•DHS has numerous S&T investments in research projects for
agriculture security countermeasures and requires tools to help evaluate
future investments
LLNL is developing a decision support system for evaluation
of the economic impact of FMD epidemics
•Effort funded by the Department of Homeland Security
•DHS has numerous S&T investments in research projects for
agriculture security countermeasures and requires tools to help evaluate
future investments
•Numerous FMD epidemiological models exist but…
–They are not national in scale
–Current models target natural or accidental introduction not an
intentional act
–Epidemiological and economic models are not coupled
GENERAL FEATURES OF THE EPIDEMIC MODEL
Agent-based spatially-explicit discrete-time computational model
Time progresses in increments of 1 unit (=1 day)
GENERAL FEATURES
Agent-based spatially-explicit discrete-time computational model
Time progresses in increments of 1 unit (=1 day)
In a time stepping agent based model, at each time increment
some of the agents change some of their attributes depending
on their previous state and on the previous states of some of
the other agents.
GENERAL FEATURES
Agent-based spatially-explicit discrete-time computational model
Time progresses in increments of 1 unit (=1 day)
In a time stepping agent based model, at each time increment
some of the agents change some of their attributes depending
on their previous state and on the previous states of some of
the other agents.
The FMD model agents are the animal facilities.
GENERAL FEATURES
Agent-based spatially-explicit discrete-time computational model
Time progresses in increments of 1 unit (=1 day)
In a time stepping agent based model, at each time increment
some of the agents change some of their attributes depending
on their previous state and on the previous states of some of
the other agents.
The FMD model agents are the animal facilities.
Facilities are groups of animals managed in a specific manner.
Farms, Markets, Feedlots, Slaughter houses …
THE ATTRIBUTES OF THE FACILITY AGENT
Type (incl. species, size and operation)
Static
Spatial coordinates
Average Number of Contacts (to and from),
Method of disease spread – specific network of contacts
Dynamic
Disease states
Change due to interaction
Availability
Seasonal factors
Change externally and
independently of interaction
THE ATTRIBUTES OF THE FACILITY AGENT
Type
The current model version deals with 34 types of animal facilities:
Beef(B), Dairy(S), Dairy(M), Dairy(L), Dairy(B), Grazing(S), Grazing(L), Feedlot(S), Feedlot(L),
Stocker(S), Stocker(L)
Swine(B), SwineFWean(S), SwineFWean(L), SwineFinish(S), SwineFinish(L), SwineNursery(S),
SwineNursery(L), SwineFFeeder(S), SwineFFeeder(L), SwineFarFin(S), SwineFarFin(L),
Sheep(S), Sheep(L), Sheep(B),
Goats, Goats(B),
Market, Market(Cattle), Market(Swine), Market(Other), Market(L), Market(C-L), DCalfHeifer(L)
THE ATTRIBUTES OF THE FACILITY AGENT
The spatial coordinates of each facility are
exact “up to the county level”
The NASS data
supplies the numbers
of different facility types
in each county
Dairy (S)
Beef (S)
Swine (S)
There are 1.2M facilities (according to NASS data) with 160M animals.
These do not include markets which come from another database.
Thus, we model 1.2M+ facilities and their contacts.
THE ATTRIBUTES OF THE FACILITY AGENT
The spatial coordinates of the
facilities are generated using
a random algorithm based on
the county-based data.
Hogs and pigs
Cattle and cows
Sheep
THE ATTRIBUTES OF THE FACILITY AGENT
Type (incl. species, size and operation)
Static
Spatial coordinates
Average Number of Contacts (to and from),
Method of disease spread – specific network of contacts
Dynamic
Disease states
Change due to interaction
Availability
Seasonal factors
Change externally and
independently of interaction
THE ATTRIBUTES OF THE FACILITY AGENT
Depends on the size and type of facility and
determined for each specific facility as random
number drawn from a given probability distribution
obtained from survey data
Average Number of Contacts (to and from),
Method of disease spread – specific network of contacts
THE ATTRIBUTES OF THE FACILITY AGENT
Depends on the size and type of facility and
determined for each specific facility as random
number drawn from a given probability distribution
obtained from survey data
Average Number of Contacts (to and from)
Method of disease spread – specific network of contacts
Direct (regional and inter-state)
Indirect (high risk and low risk)
THE ATTRIBUTES OF THE FACILITY AGENT
Type (incl. species, size and operation)
Static
Spatial coordinates
Average Number of Contacts (to and from),
Method of disease spread – specific network of contacts
Dynamic
Disease states
Change due to interaction
Availability
Seasonal factors
Change externally and
independently of interaction
THE ATTRIBUTES OF THE FACILITY AGENT
Disease states
S - Susceptible (healthy) Culled
L- Latent
U- Subclinically infectious
I- Clinically infectious
W – Vaccinated and susceptible
V- Vaccinated
M- Immune
P- Suspected
Susceptible
F- Confirmed
X - Culled
Confirmed
Waning of immunity
L1 L
L2
Latent
(infected)
Infection
Vaccinated
Suspected
Immune
L3
Subclinically
infectious
Clinically
infectious
?
THE ATTRIBUTES OF THE FACILITY AGENT
The disease state attributes of each facility are
calculated by an “intra-facility model” (IFM)
The intra-facility model is a “time-since infection” Reed-Frost type model
Represents a discrete-time system of difference equations representing
the number of animals on the facility that are in each state
S, L, I, U , V, W, M
THE ATTRIBUTES OF THE FACILITY AGENT
The disease state attributes of each facility are
calculated by an “intra-facility model” (IFM)
The intra-facility model is a “time-since infection” Reed-Frost type model
Represents a discrete-time system of difference equations representing
the number of animals on the facility that are in each state
S, L, I, U , V, W, M
The output of the IFM is used to calculate the probability that an infected
facility will infect other facilities
This is done by using a “spread model “
THE ATTRIBUTES OF THE FACILITY AGENT
Average Number of Contacts (to and from)
Method of disease spread – specific network of contacts
Availability
Seasonal factors
These attributes are used by the Spread Model to calculate the
newly infected facilities
The Spread Model calculates the newly infected facilities
Infected agents can spread the epidemic via various
methods along method-specific networks
Examples of methods
- direct (movement of animals)
- indirect: personnel movements;
- inter-state direct movements
“Truck routes”
network
For each method, the infection can be spread
within a predefined set of facilities specific to
the method.
Thus, an infected facility will spread the
infection to the facilities within the networks to
which it belongs.
“Vet routes”
network
not infected
infected
The epidemic spread is modeled by a random process
S
T
E
P
1
Uses information about the Average Number of
adequate Contacts ANC of the infected facility by each
of the methods
A contact originating from a
facility that can cause infection
is an adequate contact.
The daily number of adequate contacts RANCmi is
obtained from a Poisson process with mean ANC
An adequate contact that
actually infects a target facility
is an effective contact.
For each method of infection m
S
T
E
P
2
For each infected facility i:
- A probability density function Pmi(j)
defined on each of the nodes j of the
Pmi(j) is the probability that
facility j will get a contact with
facility i by method m.
Distance dependent
network Smi of m and i is calculated
- For each node j of Smi the probability Cmj
is calculated
Cmj is the probability that an
adequate contact to facility j
will cause infection.
S
T
E
P
Pmi(j) is used in a roulette algorithm to determine which facilities receive an
adequate contact
3
RANCmi, Pmi(j) and Cmj are used to trace back the cause of infection of j
Cmj is used to determine which of the contacted facilities become infected
The Spread Model involves factors sampled from PDFs
Pmi(j) depends on
- the average number of m-type contacts received by j
- size of the facility j
- seasonal factors
- control measure factors
- distance between i and j
- frequency of contacts between i and j
Cmj depends on
- the fraction of vaccinated animals on the facility
- control measure factors
- probability that a contact of type m would cause infection
Many of these factors are uncertain or involve variability and are
sampled from probability density functions.
The Control Measures Component
“Control measures” include
Vaccination
Culling
Contact restrictions
Isolation
Increased detection
Control measures are applied regionally
Control Policy A
Control Measure A1:
Culling on Circle
Control Policy B
Control Measure B1:
Vaccination on County
Control Measure A2:
Vaccination on Ring
Control Measure A3:
Movement Restrictions on State
Control Measure B2:
Movement Restrictions on State
Events during one increment of time
Spread model
IFM
Control
Measures
AGGREGATION ALGORITHMS
Our model is of US - national scale; however to keep calculations to a
minimum:
- We do not calculate all facilities at all times.
- Only facilities in infected and their neighboring counties are initialized
- Intra-facility model calculated only for infected facilities
- Counties and states that have not been yet infected are considered as
aggregated entities; if a contact happens to in such a county, it gets
disaggregated.
OUTPUTS
A simulation is made of N MC runs
N
 O(102) - O(103)
OUTPUTS
A simulation is made of N MC runs
N
 O(102) - O(103)
A run implements M time steps
M
 O(10 ), usually 200-330 days or
2
until a certain criterion is met (epidemic comes to end)
OUTPUTS
A simulation is made of N MC runs
 O(102) - O(103)
N
A run implements M time steps
M
 O(10 ), usually 200-330 days or
2
until a certain criterion is met (epidemic comes to end)
At each time step we keep track of the number P of facilities that are currently
involved in the epidemic (i.e. the ones that are infected or in the
neighborhoods of infected facilities.
P
 O(10 ) - O(10 )
2
5
???
OUTPUTS
A simulation is made of N MC runs
 O(102) - O(103)
N
A run implements M time steps
M
 O(10 ), usually 200-330 days or
2
until a certain criterion is met (epidemic comes to end)
At each time step we keep track of the number P of facilities that are currently
involved in the epidemic (i.e. the ones that are infected or in the
neighborhoods of infected facilities.
P
 O(10 ) - O(10 )
2
5
???
For each facility the important data (current states, costs, trace-back facilities)
is O(101)
OUTPUTS
Thus, the total output of a simulation could be in the range of O(1010)
or more.
Naturally, we do not keep all this output although what we do not keep
may be important for the analysis
What do we keep currently?
OUTPUTS
Daily Numbers of facilities of the 34 types that are in the 9 disease states
L- Latent
U- Subclinically infectious
I- Clinically infectious
W – Vaccinated and susceptible
V- Vaccinated
M- Immune
P- Suspected
F- Confirmed
X - Culled
Numbers of facilities that have just acquired a new state
Numbers of facilities that have ever been in some disease state
Total numbers of infected, vaccinated, culled facilities
Daily and total numbers of infected, vaccinated, culled animals of different
species
OUTPUTS
Durations:
Lengths of time for which the 34 types of facilities were in some disease state
Duration of total epidemic
Costs associated with epidemic and control measures
OUTPUTS
Currently, output is in Excel spreadsheet format and is used for visualization
As well as to calculate
statistics (means, quantiles,
skewness, kurtosis, etc.) of
MC output.
Cumulative Frequency
Duration of epidemic
Duration
Days after index herd infected
Epidemic model outputs and data mining
Question:
How can modern data mining tools help in the analysis of output data
generated by a large-scale epidemic model?
Epidemic model outputs and data mining
Question:
How can modern data mining tools help in the analysis of output data
generated by a large-scale epidemic model?
Specifically, can data mining help uncover important relations between
- scope of epidemic and spatial distributions of facilities?
- how control measures are applied and the cost of the epidemic?
Epidemic model outputs and data mining
Further, can data-mining tools help …
Identify sources (infected facilities), likely transmission mechanisms?
Classify of outbreaks into "natural" vs. "intentional" to help policy makers
develop correct response strategies?
Epidemic model outputs and data mining
Further, can data-mining tools help …
Identify sources (infected facilities), likely transmission mechanisms?
Classify of outbreaks into "natural" vs. "intentional" to help policy makers
develop correct response strategies?
Identify key facilities/locations for surveillance?
Identify which control mechanisms are having the largest impact?
Epidemic model outputs and data mining
Further, can data-mining tools help …
Identify sources (infected facilities), likely transmission mechanisms?
Classify of outbreaks into "natural" vs. "intentional" to help policy makers
develop correct response strategies?
Identify key facilities/locations for surveillance?
Identify which control mechanisms are having the largest impact?
Evaluate new technologies?
Evaluate vulnerability of different industries and regions of the country?
If the answer is “yes” to at least some of our questions,
which are the recommended data mining tools?
Are they available?