SIR ILM Epidemic Modelling

The data set, A3EpidemicSheep.csv (contained in the Assignment 3 folder) describes an an epi[1]demic of a novel emerging sheep disease which took place in a small region containing 107 predom[1]inantly sheep farms, though some farms contain a mix of sheep and other animals (mostly cattle, though this has not been recorded). Once the disease was discovered on a farm, animals on that farm were destroyed (culled) as quickly as possible to try and contain the spread of the epidemic. Epidemiologists believe the disease is mainly spread via wind plumes, with the potential for quite long range spread.

The data set contains:

(x, y) locations;

the time at which infection was believed to have occurred on the farm (inftime);

the time at which animals on the farm were culled, thus, removing infection from the farm (remtime);

an indicator variable describing the relative size of the farm (large=1 denotes more than 200 sheep, large=0 denotes less than 200 sheep);

an indicator variable describing whether the farm contains sheep alone (sheep=1) or a mix of sheep and other animals (sheep=0);

a measure of herd density on the farm standardized to lie between 0 and 1 (density).

Here, time is measured in weeks, and there are reasons to believe that each of the three covariates, sheep, large or density may be related to farm susceptibility. Throughout, we shall assume that there are no spark infections.

a) Load the data set contained in the Assignment 3 folder into R. Using EpiILM or otherwise produce plots showing the epidemic spread over time and spatially.

b) Fit a power-law spatial SIR model to this data set using MCMC. Put a flat uniform prior U[0, 10000] on the spatial parameter and a gamma prior (1, 1) on the baseline susceptibility parameter. Produce trace plots and posterior summary statistics (means and 95% credible intervals) for the model parameters. Find the deviance information criterion (DIC) for this model.

c) Now consider similar models to decide which covariates (sheep, large and/or density) should be included in the susceptibility function of the ILM to be better able to describe the epidemic. Do not consider interactions or transformations (e.g. quadratic) of these variables. For your final model of choice (the best fitting) use a posterior predictive approach to judge if your model has any obvious deficiencies.

d) Code up a contact network-based ILM which as closely mimics your final chosen spatial ILM from part c). Use MCMC to fit this model to the data. Produce trace plots and posterior summary statistics (means and 95% credible intervals) for the model parameters. Finally, use a posterior predictive approach to show how well this model fits the data, and comment on how the performance of this model compares with that in part c).

e) During the outbreak a pre-emptive mass culling campaign was considered to bring the epidemic under control. In the end this campaign was not carried out, since it was judged that killing animals as part of a pre-emptive culling policy is no better than killing animals that already have the disease. We want to use our model to determine whether it would have made sense to carry out a pre-emptive cull Note: Obviously that reasoning fails if by culling animals on some farms early on, we can save farms that were later to have been infected and underwent culling as a result Use a posterior predictive approach to try to devise a culling policy to have been carried out in week 4 (t=4) of the epidemic. We will assume that we had resources to cull animals on as many farms as we like (107 in theory) in week 4. However, our goal is to minimize the total number of farms on which culling took place, whether that be pre-emptive or as a result of infection, by the time the epidemic ends. We will assume that any farms undergoing culling during week 4 could not have caused any extra infections at any point later in the epidemic. How you select which farms you carry out the culling control policy on is an open question! Produce plots of the posterior predictive distribution of salient statistics as well as numerical summaries of those statistics. What do you conclude about the original judgement not to carry out a pre-empty of cull from your simulations?

	V1	V2	V3	V4	V5	V6	V7
1	93.7	109.1	6	8	1	1	0.932809
2	101.8	103.8	5	7	1	1	0.201983
3	121.6	126.8	6	9	1	1	0.792379
4	116	93.6	7	9	0	1	0.224631
5	103.3	95.4	4	5	0	0	0.030757
6	121.8	124.3	6	8	0	1	0.862034
7	104.9	93.5	6	7	0	0	0.685108
8	107.4	97.9	6	8	1	0	0.942075
9	105.8	96.1	5	6	1	1	0.675854
10	96.9	96.8	5	9	0	1	0.84312
11	115.1	97.2	4	6	0	1	0.361894
12	103.9	104.9	6	9	1	0	0.392366
13	93.8	98.2	4	5	0	1	0.567687
14	77.9	94.9	9	12	0	1	0.095152
15	141.2	123.4	0	0	0	0	0.193784
16	99.6	97.9	5	6	0	0	0.588066
17	99.8	98.2	4	5	0	1	0.751504
18	109.4	99	5	7	0	1	0.867239
19	108.2	107.1	0	0	0	0	0.371796
20	105.9	99.3	5	8	1	0	0.798815
21	109.2	99.6	6	8	0	1	0.058314
22	107.8	93.2	5	7	0	1	0.623436
23	100.7	96.8	5	8	1	1	0.356641
24	80.1	100.6	10	11	0	0	0.587928
25	106.2	94.1	5	10	0	0	0.913785
26	99.4	105.3	5	7	0	0	0.199442
27	98.4	84.8	6	8	0	1	0.369084
28	85.3	103.1	1	2	0	1	0.671408
29	95.2	84.6	6	9	0	0	0.768145
30	104.2	97	5	6	1	0	0.522248
31	113.6	94.7	4	11	0	0	0.828075
32	99	93.5	5	8	0	1	0.527096
33	103.9	99.4	6	8	0	0	0.501755
34	99.5	80.9	6	8	0	1	0.419973
35	116.2	121.8	2	4	0	0	0.362298
36	95.9	83.4	7	10	0	1	0.123429
37	96.1	95.4	4	5	0	1	0.298162
38	99.4	88.8	2	4	1	1	0.276676
39	111	92.5	5	6	1	1	0.770225
40	137.6	130.9	0	0	0	0	0.778181
41	98.4	100.2	4	5	0	1	0.143787
42	97.5	87.1	5	6	0	0	0.515526
43	107	83.6	6	7	0	1	0.59724
44	105.6	104.5	6	10	1	0	0.505843
45	93.1	99.8	6	7	1	0	0.3861
46	92.9	96.8	3	5	0	1	0.426098
47	103.6	90.7	8	10	0	0	0.01176
48	107.7	85.1	9	10	0	0	0.919332
49	98.9	89.2	3	5	1	1	0.07944
50	138.8	120	9	12	0	1	0.507374
51	104	93.8	4	5	0	1	0.820172
52	93.9	86.2	6	9	0	1	0.598395
53	133.4	128.7	7	8	1	1	0.424154
54	88.7	104.3	8	12	0	0	0.55931
55	114.3	97.6	5	8	1	0	0.789094
56	149.8	120.6	8	9	1	1	0.167715
57	96.3	108.9	6	8	0	1	0.970452
58	89.6	93.8	5	6	0	0	0.473503
59	135.7	132.1	8	12	0	0	0.929743
60	98.6	97.4	6	7	0	0	0.900939
61	124	85.8	3	4	1	0	0.750882
62	99.6	98.6	5	6	0	1	0.676569
63	106.9	102.1	6	8	0	1	0.648013
64	130.3	133.1	5	6	1	1	0.073247
65	92.6	101.1	5	9	0	1	0.423558
66	101.9	104.6	5	7	0	0	0.530824
67	82	99.2	8	10	0	0	0.942705
68	114.7	96.7	5	9	0	1	0.712225
69	101.5	99.7	6	9	0	0	0.724491
70	121.7	107.9	6	7	0	0	0.470129
71	134.8	130.8	7	9	1	0	0.120282
72	122.9	120.3	7	8	0	1	0.783098
73	136.1	122.1	7	9	0	1	0.438157
74	90.7	87.7	6	9	0	0	0.431456
75	87.5	109.8	8	11	0	1	0.027498
76	102.9	102.2	6	9	0	0	0.146562
77	95.6	85.3	5	6	0	1	0.422595
78	100	105.2	5	6	1	1	0.767137
79	100.7	98.4	5	6	0	1	0.004766
80	124.1	124.6	5	6	1	0	0.603596
81	94.3	92.3	4	6	1	1	0.905577
82	98.6	95.7	5	7	0	1	0.706662
83	111.8	90.7	4	7	1	1	0.262537
84	84.8	98.2	0	0	0	0	0.851076
85	105.9	104	6	9	0	1	0.333606
86	103.3	92.7	5	7	1	0	0.578284
87	110.6	108.3	6	8	1	1	0.432773
88	97	87.9	6	7	0	1	0.051595
89	103.7	89.5	4	7	0	1	0.729803
90	132.7	124.4	6	10	0	1	0.54817
91	94.6	89.8	6	8	0	0	0.751222
92	112.1	104.1	5	9	0	1	0.050771
93	111.6	96.2	5	6	1	0	0.714966
94	107	104.1	7	8	1	1	0.297694
95	145.9	126.9	0	0	0	0	0.283477
96	135.6	125.9	8	10	0	0	0.829877
97	87.2	96.7	6	8	0	1	0.086395
98	94.3	77.1	4	5	1	1	0.042657
99	117.8	135	8	10	1	0	0.348741
100	95.3	106.7	4	6	1	1	0.542336
101	93.8	105.4	5	6	1	1	0.609461
102	100.4	99.9	5	9	1	1	0.271372
103	90.9	105.1	5	8	0	1	0.205231
104	101.6	98.4	6	7	0	0	0.381632
105	93.5	104.2	6	10	0	0	0.472558
106	117.7	96	6	7	0	0	0.835518
107	107.2	86.3	5	6	0	1	0.121471