Disease dynamics on community networks Joe Tien GIScience Workshop - - PowerPoint PPT Presentation

Disease dynamics on community networks Joe Tien

GIScience Workshop September 18, 2012

Collaborators

• Zhisheng Shuai
• Pauline van den Driessche
• Marisa Eisenberg

John Snow and the Broad Street Pump

Geographic analysis of cholera spatial spread

Journal of Geographic Analysis 1: 59-75. 1969.

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

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

Community networks

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.

Patch dynamics

S

µS

I

µI

R

µR γI µ

W

ξW bWSW αI W - pathogen concentration in water reservoir

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
• R0 > 1  disease can invade

Defining the basic reproduction number

• Second generation matrix approach (Diekmann, Heesterbeek,

and Metz (1990); van den Driessche and Watmough (2002)).

Network R0

What happens in the limits of fast / slow water movement?

• Interplay of two time scales:

δi -- pathogen decay rate in the water for patch i dW – movement rate of water

R0 and time scales of movement / decay

Fast decay limit: GW approaches the singular matrix L Scaled time: Fast movement / slow decay limit:

Laurent series expansion for R0

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…

R0 and spanning trees

• Weighted average according to the rooted spanning trees ui

(network risk)

• Patch risks qi ri / (mi + γi + di) – “Transmission”
• - “Average pathogen lifetime”

Rooted spanning trees: “rivers”

• Network risk increases by a factor of a/b each step downstream
• Worst place for disease hot spot -- downstream

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:

ui = uj for all i, j

Identical R0 in the limit of fast water movement

R0, bottlenecks, and clustering

Interpreting next term X0: Impact on R0:

R0, bottlenecks, and clustering

Biological interpretation:

• Clustering hot spots together will increase R0
• Worse hot spots  greater impact of clustering on R0
• Bottlenecks to mixing  greater impact of clustering on R0

R0, bottlenecks, and clustering

Accuracy when movement is fast

Insights when movement is slow

Theorem R0 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 R0

• Clustering hot spots together always increases R0

Epidemic centroid movements

Spatial models

• Linking disease dynamics at the

Department level

• Patch models:

– Patch = Department – Dynamics within each patch (e.g. SIWR) – Coupling between patches

Patch coupling via “gravity”

λj -- “Force of infection” on patch j Contribution from patch k to j: – Proportional to Nj Nk – Inversely proportional to distance between patches Tuite et al. 2011.

• Ann. Internal Med.

Gravity model: comparison with data

Tuite et al. 2011. Ann. Internal Med.

Moran’s I

• Clustering statistic: correlation

according to connectivity matrix

• Initial invasion period when spatial

clustering is evident

• Strongest clustering according to

physical adjacency

Opportunities in the geosciences

• Need for data
• Geographic networks (e.g. river networks, habitat)

– Topology – Weights

• Human networks
• Patch characteristics

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 National Science Foundation – EEID Program Mathematical Biosciences Institute

References

Tien JH, Earn DJD. 2010. Bull. Math. Biol. 72(6): 1506-1533. Tuite et al. 2011. Annal. Int. Med. 154(9): 593-601. Eisenberg, Shuai, Tien, van den Driessche. 2012. In prep. Hopital Albert Schweitzer U.N. WASH Cluster International Organization of Migration NASA Tropical Rainfall Measuring Mission CDC USGS Ohio Water Science Center