Disease dynamics on community networks. Joe Tien. GIScience Workshop September 18, 2012

Transcription

1 Disease dynamics on community networks Joe Tien GIScience Workshop September 18, 2012

2 Collaborators Zhisheng Shuai Pauline van den Driessche Marisa Eisenberg

3 John Snow and the Broad Street Pump

4 Geographic analysis of cholera spatial spread Journal of Geographic Analysis 1:

5 Present day cholera: Haiti El Tor Ogawa (same strain as found in SE Asia) Start of the outbreak Artibonite Valley Official stats (Haitian Ministry of Health) as of April 10, 2012: 534,647 cases 7,091 deaths Source: MSPP Cholera ward, Hopital Albert Schweitzer

6 Outline Theoretical results Ability of disease to invade a community network Network risk and patch risk Clustering of disease hot spots Haiti cholera Opportunities to merge theory and data

7 Community networks

8 Model Assumptions Patch dynamics correspond to an SIWR system. The network is strongly connected by the movement of water. Infected people are too sick to move. All disease transmission is waterborne.

9 Patch dynamics µs µi µr µ S I γi R b W SW W αi ξw W - pathogen concentration in water reservoir

10 Invasibility and the basic reproduction number Number of secondary infections created by single infected individual in otherwise susceptible population Rate of new infections length of infectiousness R 0 > 1 disease can invade

11 Defining the basic reproduction number Second generation matrix approach (Diekmann, Heesterbeek, and Metz (1990); van den Driessche and Watmough (2002)).

12 Network R 0 Interplay of two time scales: δ i -- pathogen decay rate in the water for patch i d W movement rate of water What happens in the limits of fast / slow water movement?

13 R 0 and time scales of movement / decay Fast decay limit: Fast movement / slow decay limit: Scaled time: G W approaches the singular matrix L

14 Laurent series expansion for R 0 Langenhop (1971): Laurent series for perturbed singular matrices For a given network, these terms can be computed explicitly X -1 involves the rooted spanning trees of the network X _0 involves a fundamental matrix of an associated Markov process These terms have natural biological interpretations

15 R 0 and spanning trees Weighted average according to the rooted spanning trees u i (network risk) Patch risks q i r i / (m i + γ i + d i ) Transmission -- Average pathogen lifetime

16 Rooted spanning trees: rivers Network risk increases by a factor of a/b each step downstream Worst place for disease hot spot -- downstream

17 Rooted spanning trees: balanced graphs Balanced graph the net outflow equals net inflow for every vertex - Need not be symmetric Identical network risk for every vertex: u i = u j for all i, j Identical R 0 in the limit of fast water movement

18 R 0, bottlenecks, and clustering Interpreting next term X 0 : Impact on R 0 :

19 R 0, bottlenecks, and clustering Biological interpretation: Clustering hot spots together will increase R 0 Worse hot spots greater impact of clustering on R 0 Bottlenecks to mixing greater impact of clustering on R 0

20 R 0, bottlenecks, and clustering

21 Accuracy when movement is fast

22 Insights when movement is slow Theorem R 0 for the domain is monotone increasing with ε. Biological significance Moving a hot spot to a node with a larger number of spanning trees always increases R 0 Clustering hot spots together always increases R 0

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24 Epidemic centroid movements

25 Spatial models Linking disease dynamics at the Department level Patch models: Patch = Department Dynamics within each patch (e.g. SIWR) Coupling between patches

26 Patch coupling via gravity λ j -- Force of infection on patch j Contribution from patch k to j: Proportional to N j N k Inversely proportional to distance between patches Tuite et al Ann. Internal Med.

27 Gravity model: comparison with data Tuite et al Ann. Internal Med.

28 Moran s I Clustering statistic: correlation according to connectivity matrix Initial invasion period when spatial clustering is evident Strongest clustering according to physical adjacency

29 Opportunities in the geosciences Need for data Geographic networks (e.g. river networks, habitat) Topology Weights Human networks Patch characteristics

30 Acknowledgements Marisa Eisenberg Mark Guseman Pauline van den Driessche Zhisheng Shuai David Fisman Ashleigh Tuite Greg Kujbida David Earn Junling Ma Ian Rawson Dawn Johnson Renold Estimie Carrie Weinrobe Julio Urruela Patrick Duigan Hopital Albert Schweitzer U.N. WASH Cluster International Organization of Migration NASA Tropical Rainfall Measuring Mission CDC USGS Ohio Water Science Center National Science Foundation EEID Program Mathematical Biosciences Institute References Tien JH, Earn DJD Bull. Math. Biol. 72(6): Tuite et al Annal. Int. Med. 154(9): Eisenberg, Shuai, Tien, van den Driessche In prep.