Download infected - University of Stirling (Computing Science and Mathematics)

Modelling the control of epidemics
by behavioural changes in response
to awareness of disease
Savi Maharaj
(joint work with Adam Kleczkowski)
University of Stirling
Motivation
• It is natural to change ones
behaviour in response to knowing
there is disease present in the
neighbourhood.
– Reducing contact with others (e.g. avoiding
public spaces or non-essential travel)
– Reducing infectiousness of contact (e.g.
wearing face-masks, washing hands).
Questions:
•Can such controls reduce the final size of an epidemic?
•Given a disease with particular characteristics, which control is best at
suppressing the epidemic?
•Control may have an economic cost. For example, if workers stay at
home, the economy suffers. Which response yields the best cost/benefit
tradeoff?
Overview
• Spatial, individual based model of SIR epidemic system.
• Individuals react to awareness of the amount of disease locally. Responses:
• “stay at home” (changing network structure)
• “wash hands” (changing infectiousness of disease)
• Change of behaviour regulated by:
• radius of awareness neighbourhood (local vs global knowledge)
• attitude to risk (panic or relax)
•Part 1 looks at comparing the two responses.
• Result: Sometimes “stay at home” is better at reducing the final size of
the epidemic, sometimes “wash hands” is better. Combining both is best.
•Result: Awareness radius should be at least as big as infection radius.
•Part 2 introduces economic cost and looks at cost/benefit tradeoff for the
“stay at home” response:
•Result: If epidemic can be suppressed, panic! Otherwise, relax.
Spatial structure of the model
•
•
•
•
•
•
50x50 square lattice with every cell occupied
Individuals may be susceptible, infected, or removed (SIR system).
Individuals make contact within radius zi.
Susceptibles respond to infection load within an awareness neighbourhood, za
Awareness is of infected individuals only (no memory) or both infecteds and
removeds (full memory)
Size of response depends on a parameter representing attitude to risk, ra
Individual dynamics:
no control
Susceptible
contact
radius, zi
Infected
probability of infection per single
contact, pi
Removed
probability of
removal, pr
risk attitude,
ra
awareness radius,
za
modify contact
radius, zi
Susceptible
Individual dynamics:
“stay at home”
Infected
Removed
contact radius,
zi
probability of infection per single
contact, pi
probability of
removal, pr
risk attitude,
ra
awareness radius,
za
modify infection
probability, pi
Susceptible
Individual dynamics:
“wash hands”
Infected
Removed
probability of
infection per
contact, pi
contact radius, zi
probability of
removal, pr
risk attitude,
ra
awareness radius,
za
Modify contact
radius and
infection
probability
Susceptible
Individual dynamics:
combined response
Infected
Removed
probability of
infection per
contact, pi
contact radius, zi
probability of
removal, pr
Risk attitude
•Risk attitude, ra, represents
how strongly individuals react
to a given infection pressure.
•Infection pressure: the
fraction of neighbours within
radius za who are infected (no
memory) or either infected or
removed (full memory).
•Smaller values of ra mean
individuals are more panicky,
and will respond more
strongly to a given infection
pressure.
•Larger values of ra mean that
individuals are more relaxed
and have a weaker response.
Tools
• Simulations created with NetLogo
http://ccl.northwestern.edu/netlogo/
• Experiments executed on a network of PC
workstations via Condor
http://www.cs.wisc.edu/condor/
• Data analysed with the R statistical tool
http://www.r-project.org/
Simulation run: no control
zi = 2
pi = 0.1
pr = 0.2
Without control,
the epidemic
invades almost
the whole
population.
Simulation run: effective suppression
zi = 2
pi = 0.1
pr = 0.2
“stay at home”
with:
no memory
za = 3
ra = 0.2
The effect of control on the final size
of the epidemic
• Sometimes “stay at home” reduces the
epidemic most, sometimes “wash your hands”
does. Combining both has the greatest effect.
Maharaj & Kleczkowski, Summer Computer Simulation Conference, 2010
The effect of memory
• If individuals can remember and respond to past cases
of infection, the epidemic is much smaller than if they
only know about current cases of infection.
Simulation run: insufficient awareness
zi = 2
pi = 0.1
pr = 0.2
Control B with:
no memory
za = 1
ra = 0.2
The effect of awareness radius
• For the epidemic to be suppressed, the awareness
radius, za, must be at least as big as the infection
radius, zi.
Simulation run: too relaxed
zi = 2
pi = 0.1
pr = 0.2
Control B with:
no memory
za = 3
ra = 0.3
Effect of risk attitude
• The epidemic is reduced most when individuals
are highly risk-averse (very low ra).
Comparison with a (non-spatial) mean field
approximation
  I    SI
dS
 b  1    
  N   N
dt
  I    SI
dI
 b1    
 gI
  N   N
dt
dR
 gI
dt
Part 2: considering economic costs and benefits
• Networks are there for a purpose: they serve people’s needs and are
not primarily designed to prevent disease spread.
• We can control disease by modifying the networks – but at a cost!
• Gain of healthy individuals: final epidemic size, R compared to the
case with no control, R(no control) − R(with control)
• Loss of contacts: reduction in number of contacts over a designated
accounting period, contacts (no control) − contacts (with control)
• Relative economic importance of each contact, c
• Benefit of control: Gain of healthy individuals − loss of contacts * c
Control can reduce the final size of the epidemic
2500
2000
Reduction in number infected
1500
1000
500
0
0.0
0.2
0.4
0.6
pi
0.8
1.0
But control can also reduce the number of contacts
15000
14000
13000
Reduction in number of contacts
12000
11000
0.0
0.2
0.4
0.6
pi
0.8
1.0
Total benefit: Switching between strategies
Optimal
attitude
in this
case:
panic!
• Impact of risk attitude
depends on awareness
radius
• When awareness radius
is large enough, there is a
switch between successful
and unsuccessful control
Small
awareness
radius
• Once the threshold is
passed, the cost of losing
contacts is very severe:
• better to refrain
from action
Large
awareness
radius
Risk attitude
“Panic”
Maharaj & Kleczkowski, in prep, 2010
“Relax”
Total benefit: Relative cost
Optimal
attitude:
panic!
Small weight on contacts
Optimal
attitude:
relax
Large weight on contacts
Total benefit: Increasing infectiousness
Optimal
attitude:
relax?
Optimal
attitude:
panic!
Moderately infectious disease
Highly infectious disease
Increasing pi shifts the transition from ‘panic’ to ‘relax’
• The more infectious the disease,
the more vigilant we need to be (smaller ra).
Conclusions and future work
• Some intriguing results so far - further examination
needed!
• Extend cost-benefit analysis to “wash hands” control
• Examine different contact networks (small-world,
scale-free,…)
• Validation against data from real epidemics (Can you
help us get such data?)
• Formalization of model in process algebra.
– Are current PAs sufficiently expressive?
• More accurate mean-field approximation (perhaps
using pair-approximation techniques?)
Thank you!