Spatial corridors of spread

Transcription

1 Measles Epidemics Iceland, November 1946 June 1947 major epidemic of measles. Monthly cases in each medical district. Reykjaík to other regions. 1

2 Several Epidemics Eight major measles epidemics in Iceland from 1945 to 1970 Figure 2. t ij = month of first case in district i in epidemic j. Average time to infection for district i t i = j t ij / n, where n = 8 number of measles epidemics. 2

3 Four phases 1. Originating phase (< 3 months) measles confined to the capitol, Reykjaík. 2. Localized spread phase (3-5 months) local spread to neighboring regions and long range diffusion to regional centers. (Akureyri, 5.8) 3. Generalized spread phase (6-8 months) all parts of the island expect for remote. 4. Remote spread phase (>= 9 months) delayed penetration into inaccessible parts of the island, northwest fjords. (Flateyjar, 17) 3

4 Trends Spread down the hierarchy of urban size larger districts to smaller. Contagious spread from initial centers of introduction. Strong inverse relationship between the population size and distance from the capitol hard to separate size and distance effects. Similar findings for influenza: 8 months horizon vs. 17 months for measles. Serial interval = time from symptoms in one case to symptoms in directly infected case 3 days for flu and 14 days for measles. 4

5 Maps as graphs: Southwest England Link distance General Register Office (GRO) areas Bristol Plymouth Penzance Total 1,090 1,141 2,576 Mean Maximum % within 3 links Six counties are divided into 179 areas, ignoring Scilly Isles get 178 areas in contiguous network. areas = nodes in graph; common boundary = link between a pair. 5

6 Maps as graphs: Southwest England Shortest number of links to connect any pair of nodes = spatial lag. Average degree is 5.22 (for internal 151 nodes). The largest center Bristol is most connected, table lists three major centers for the spread of measles. 6

7 Maps as graphs: Southwest England Weekly number of cases for each area from 1966 to 1970 (222 weeks). Figure 3 shows a time-space dynamics: correlation of reported cases that are 1, 2,..., 15 spatial lags apart for every one of 222 weeks. Lower graph is smoothed, diagonally shaded histogram is number of cases each week. 7

8 Maps as graphs: Southwest England Recall two components of spread: center to center (city to city, town to town avoiding the intervening countryside) and localized spread into neighboring countryside. Vertically shaded slices generally show exponential drop off support the localized spread. However, two local maxima at lag 5 and lag 12. 8

9 Maps as graphs: Southwest England Lag 5 average separation between major centers: Gloucester-Cheltenham (200), Poole (120), Swindon (120), Torbay (100). Lag 12 spatial lag between Bristol and Plymouth, which comprise one fifth of the population (1,250). The spread appears to show both diffusion components! 9

10 Testing diffusion hypothesis Adjacency graph vs. graphs that supports both hierarchical and contagious diffusion. Simple test: Moran BW test statistic BW = 1 2 i j w ij (x i x j ) 2, where w ij = 1 if link between vertices i and j, x i = 1 if outbreak in region i. Distribution of BW is approximately normal with known mean and variance. General observation: hierarchical spread was more important at the onset of epidemics and the localized spread dominates late stages. 10

11 Disease centroids Define a centroid to show a mean center of disease incidence, e.g. Ū = / I k u k I k k k V = / I k v k I k k k where u and v are horizontal and vertical coordinates respectively, I k number of cases in area k. Then monitor time changes in centroid to see direction and velocity of the spread. Figure 4 trajectories of centroids for 8 measles epidemics in Iceland between 1945 and

12 Disease centroids First one Reykjavík to Dalvíkur. Almost all quickly reach Reykjavík. A general noth-easterly drift after seventh month. A rotationally symmetric trend would not be captured! 12