Modelling the control of epidemics by behavioural changes in response to awareness of disease
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Modelling the control of epidemics by behavioural changes in response to awareness of diseaseSavi Maharaj(joint work with Adam Kleczkowski)University of StirlingMotivationIt 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.
50x50 square lattice with every cell occupiedIndividuals may be susceptible, infected, or removed (SIR system).Individuals make contact within radius zi.Susceptibles respond to infection load within an awareness neighbourhood, zaAwareness 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, raSpatial structure of the modelSusceptible Infected
probability of infection per single contact, piprobability of removal, prIndividual dynamics: no controlcontact radius, ziSusceptible contact radius, ziInfected
awareness radius, zaprobability of infection per single contact, piprobability of removal, prrisk attitude, ramodify contact radius, ziIndividual dynamics: stay at homeSusceptible probability of infection per contact, piInfected
awareness radius, zacontact radius, ziprobability of removal, prrisk attitude, ramodify infection probability, piIndividual dynamics: wash handsSusceptible probability of infection per contact, picontact radius, ziInfected
awareness radius, zaprobability of removal, prrisk attitude, raModify contact radius and infection probabilityIndividual dynamics: combined responseRisk 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.ToolsSimulations 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 controlzi = 2pi = 0.1pr = 0.2
Without control, the epidemic invades almost the whole population.
Simulation run: effective suppressionzi = 2pi = 0.1pr = 0.2
stay at home with:
no memoryza = 3ra = 0.2
The effect of control on the final size of the epidemicSometimes stay at home reduces the epidemic most, sometimes wash your hands does. Combining both has the greatest effect.
Maharaj & Kleczkowski, Summer Computer Simulation Conference, 2010The effect of memoryIf 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 awarenesszi = 2pi = 0.1pr = 0.2
Control B with:
no memoryza = 1ra = 0.2
The effect of awareness radiusFor the epidemic to be suppressed, the awareness radius, za, must be at least as big as the infection radius, zi.
Simulation run: too relaxedzi = 2pi = 0.1pr = 0.2
Control B with:
no memoryza = 3ra = 0.3
Effect of risk attitudeThe epidemic is reduced most when individuals are highly risk-averse (very low ra).
Comparison with a (non-spatial) mean field approximationPart 2: considering economic costs and benefitsNetworks are there for a purpose: they serve peoples 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
piControl can reduce the final size of the epidemicReduction in number infected
piBut control can also reduce the number of contactsReduction in number of contacts
PanicRelaxTotal benefit: Switching between strategiesRisk attitudeOptimal attitude in this case: panic!Small awareness radiusLargeawareness radius Impact of risk attitude depends on awareness radius
When awareness radius is large enough, there is a switch between successful and unsuccessful control
Once the threshold is passed, the cost of losing contacts is very severe: better to refrain from actionMaharaj & Kleczkowski, in prep, 2010Total benefit: Relative costOptimal attitude: panic!Optimal attitude: relaxSmall weight on contactsLarge weight on contacts
Increasing pi shifts the transition from panic to relaxTotal benefit: Increasing infectiousnessOptimal attitude: panic!Optimal attitude: relax?Moderately infectious diseaseHighly infectious disease The more infectious the disease, the more vigilant we need to be (smaller ra).
Conclusions and future workSome intriguing results so far - further examination needed!Extend cost-benefit analysis to wash hands controlExamine 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!