Understanding the contribution of space on the spread of Influenza using an Individual-based model approach

Transcription

1 Understanding the contribution of space on the spread of Influenza using an Individual-based model approach Shrupa Shah Joint PhD Candidate School of Mathematics and Statistics School of Population and Global Health Supervisors: Dr. Nicholas Geard Prof. Peter Taylor A/Prof. James McCaw A/Prof. Jodie McVernon The University of Melbourne April 2015 Shrupa Shah - The University of Melbourne ANZAPW 2015 April

2 Overview Influenza Individual based model (IBM) approach Space as a proxy Homogeneous mixing model Spatially-explicit mixing model Measures of disease dynamics Height of peak prevalence Time of peak prevalence Shrupa Shah - The University of Melbourne ANZAPW 2015 April

3 Seasonal Influenza Morbidity Mortality Endemic Epidemic Height of peak prevalence Time of peak prevalence Modelling Figure: Laboratory confirmed cases - Australia Shrupa Shah - The University of Melbourne ANZAPW 2015 April

4 Baseline model - the Classic S-I-R model S I R Assumptions: 1 Homogeneity in individual attributes 2 Homogeneity in individual interactions Reality: 1 Individuals differ in biology, physiology, immunity and behaviour 2 Subset of interactions Shrupa Shah - The University of Melbourne ANZAPW 2015 April

5 Models: 1 Relaxation of basic assumptions Homogeneous individuals Homogeneous mixing 2 Space as a proxy Spatially heterogeneous interventions Spatial models Individual based models Shrupa Shah - The University of Melbourne ANZAPW 2015 April

6 Motivation for an Individual based model (IBM) approach Individual heterogeneity Social interaction and behavioural patterns No issue with unobserved cases Scenario analyses Data - parameterization Chain of transmission Shrupa Shah - The University of Melbourne ANZAPW 2015 April

7 Classic S-I-R model β I N S γi S I R ds dt = β I N S di dt = β I N S γi dr dt = γi N = S + I + R Assume random mixing β - rate of transmission γ - rate of recovery Force of infection, λ = β I N Shrupa Shah - The University of Melbourne ANZAPW 2015 April

8 Equivalent discrete time Individual based model Probability of infection from individual i, p si = c si β β - probability of transmission Homogeneous mixing c si = 1 N 1 Spatially-explicit mixing c si = 1 2πκ exp( 1 2κ x s x i 2 ) Probability of infection from any infectious individuals 1 I i=1 (1 p si) Shrupa Shah - The University of Melbourne ANZAPW 2015 April

9 Homogeneous mixing, single index case 100 uniformly distributed individuals Assume homogeneous mixing Equal probability of contact Equal risk of infection Shrupa Shah - The University of Melbourne ANZAPW 2015 April

10 Spatially explicit mixing, single index case h(p s, p i ) = 1 2πκ exp( 1 2κ p s p i 2 ) 100 uniformly distributed individuals Assume spatiallyexplicit mixing Risk of infection is spatially dependent Shrupa Shah - The University of Melbourne ANZAPW 2015 April

11 Finding comparable parameters for the two mixing models Final size 100 realistions β (probability of transmission) = 0.3 Homogeneous mixing: Average final size Standard deviation of final size Spatially-explicit mixing: Final Size Spatial tolerence - κ Average Standard deviation Shrupa Shah - The University of Melbourne ANZAPW 2015 April

12 Homogeneous mixing, single Index case, at time t 10 β = 0.3 Shrupa Shah - The University of Melbourne ANZAPW 2015 April

13 Spatially explicit mixing, single Index case, at time t 10 β = 0.3 κ = 0.5 Shrupa Shah - The University of Melbourne ANZAPW 2015 April

14 A single realisation, β = 0.3, κ = 0.5 Shrupa Shah - The University of Melbourne ANZAPW 2015 April

15 100 realisations, β = 0.3, κ = 0.5 Shrupa Shah - The University of Melbourne ANZAPW 2015 April

16 Another approach for comparing the two mixing models Let the contact event, in the homogeneous mixing model, between individual i and j be an independent and identical bernoulli trial C ij 0 1 P(C ij = c ij ) 1 1 N 1 1 N 1 The expected number of total contacts of individual i, is: N c i = E N = E[C ij ] = 1 (1) j i C ij j i Shrupa Shah - The University of Melbourne ANZAPW 2015 April

17 (continued...) Similarly, the contact event, in the spatially-explicit mixing model, between individual i and individual j be an independent, but not identical, bernoulli trial C ij [ ( 0 1 P(C ij = c ij ) 1 1 2πκ exp 1 2κ p i p j 2)] ( 1 2πκ exp 1 2κ p i p j 2) The expected number of total contacts of individual i is: c i = N P(C ij = 1) (2) i j Shrupa Shah - The University of Melbourne ANZAPW 2015 April

18 κ = 0.38 Shrupa Shah - The University of Melbourne ANZAPW 2015 April

19 Shrupa Shah - The University of Melbourne ANZAPW 2015 April

20 A single realisation, β = 0.7, κ = 0.38 Shrupa Shah - The University of Melbourne ANZAPW 2015 April

21 Chain of transmission Shrupa Shah - The University of Melbourne ANZAPW 2015 April

22 Concluding remarks Risk of infection Homogeneous mixing - equal Spatially-explicit - decreases with increasing distance Height and Time of peak prevalence Homogeneous mixing - lower and later Spatially-explicit - higher and earlier Calibrate parameters to data Homogeneous mixing - β Spatially-explicit mixing - β and κ Individual-based model approach Spatially referenced individual locations Distance driven contact networks - Social interaction patterns Individual heterogeneity and behavioural patterns Shrupa Shah - The University of Melbourne ANZAPW 2015 April

23 Acknowledgements Supervisors The University of Melbourne - School of Mathematics and Statistics Audience Shrupa Shah - The University of Melbourne ANZAPW 2015 April