Malaria, river-blindness and plague: three case
studies in spatial modelling of tropical diseases
Emanuele Giorgi and Peter J. Diggle
Lancaster Medical School, Lancaster University, Lancaster, UK
LSHTM, 24 April 2015
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
1 / 21
Three issues of disease mapping in low-resource settings
1
Combining data from multiple surveys of different quality:
e.g. randomised and non-randomised surveys.
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
1 / 21
Three issues of disease mapping in low-resource settings
1
Combining data from multiple surveys of different quality:
e.g. randomised and non-randomised surveys.
2
Spatially structured zero-inflation: chance finding or disease-free
community?
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
1 / 21
Three issues of disease mapping in low-resource settings
1
Combining data from multiple surveys of different quality:
e.g. randomised and non-randomised surveys.
2
Spatially structured zero-inflation: chance finding or disease-free
community?
3
Analysis of spatio-temporal data at multiple spatial-scales: how to
integrate district-level disease case reports with high resolution
environmental data?
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
1 / 21
Overview
1
Malaria: combining data from a community and a (biased)
school-based surveys in Nyanza Province, Kenya.
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
2 / 21
Overview
1
Malaria: combining data from a community and a (biased)
school-based surveys in Nyanza Province, Kenya.
2
River-blindness: geostatistical modelling of spatially-structured zero
inflation in prevalence data from Mozambique, Malawi and
Tanzania.
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
2 / 21
Overview
1
Malaria: combining data from a community and a (biased)
school-based surveys in Nyanza Province, Kenya.
2
River-blindness: geostatistical modelling of spatially-structured zero
inflation in prevalence data from Mozambique, Malawi and
Tanzania.
3
Plague: spatio-temporal analysis of monthly plague incidence from
2000 to 2007 in Madagascar.
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
2 / 21
-0.45
-0.50
-0.60
-0.55
Latitude
-0.40
Malaria mapping in Nyanza Province
34.70
34.75
34.80
34.85
34.90
34.95
Longitude
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
3 / 21
35.00
-0.45
-0.50
-0.60
-0.55
Latitude
-0.40
Malaria mapping in Nyanza Province
34.70
34.75
34.80
34.85
34.90
34.95
Longitude
Statistical issues
• Different unmeasured risk factors for malaria.
• Joint model or gold-standard data only?
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
3 / 21
35.00
Accounting for residual spatial bias
• rd (x) = ‘‘odds for the disease’’
• rs (x) = ‘‘odds within the survey’’
• rc (x) = ‘‘relative odds between participants and non-participants
to the survey’’
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
4 / 21
Accounting for residual spatial bias
• rd (x) = ‘‘odds for the disease’’
• rs (x) = ‘‘odds within the survey’’
• rc (x) = ‘‘relative odds between participants and non-participants
to the survey’’
rs (x) = rd (x) × rc (x)
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
4 / 21
Accounting for residual spatial bias
• rd (x) = ‘‘odds for the disease’’
• rs (x) = ‘‘odds within the survey’’
• rc (x) = ‘‘relative odds between participants and non-participants
to the survey’’
rs (x) = rd (x) × rc (x) ⇐⇒ log{rs (x)} = S(x) + B(x)
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
4 / 21
(1)
Accounting for residual spatial bias
• rd (x) = ‘‘odds for the disease’’
• rs (x) = ‘‘odds within the survey’’
• rc (x) = ‘‘relative odds between participants and non-participants
to the survey’’
rs (x) = rd (x) × rc (x) ⇐⇒ log{rs (x)} = S(x) + B(x)
(1)
• Yijk |S(xij ), B(xij ) ∼ Bernoulli(pijk ) (i=‘‘survey", j =‘‘household",
k =‘‘’individual’’).
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
4 / 21
Accounting for residual spatial bias
• rd (x) = ‘‘odds for the disease’’
• rs (x) = ‘‘odds within the survey’’
• rc (x) = ‘‘relative odds between participants and non-participants
to the survey’’
rs (x) = rd (x) × rc (x) ⇐⇒ log{rs (x)} = S(x) + B(x)
(1)
• Yijk |S(xij ), B(xij ) ∼ Bernoulli(pijk ) (i=‘‘survey", j =‘‘household",
k =‘‘’individual’’).
• log{pijk /(1 + pijk )} = d>
ijk β + S(xij ) + B(xij );
2
B(x) = 0, ∀x ∈ R if i is ‘‘community’’.
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
4 / 21
Results (1)
Term
Intercept
Age
Rachuonyo District
Socio-Economic-Status
School survey
Age (bias)
Emanuele Giorgi and Peter J. Diggle
Estimate
-1.412
-0.141
2.006
-0.121
-0.761
0.094
95% CI
(-2.303, -0.521)
(-0.174, -0.109)
(1.228, 2.785)
(-0.169, -0.072)
(-1.354, -0.167)
(0.046, 0.142)
Spatial modelling of tropical diseases
5 / 21
Results (2)
Mapping of (a) rc (x) and (b) 1 − P (0.9 < rc (x) < 1.1|y).
-0.45
0.8
0.7
0.6
0.5
0.4
0.3
0.2
-0.55
Latitude
(a)
34.70
34.80
34.90
35.00
Longitude
-0.45
1.00
0.98
0.96
0.94
0.92
0.90
0.88
-0.55
Latitude
(b)
34.70
34.80
34.90
35.00
Longitude
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
6 / 21
Results (3)
Gain/loss in the accuracy for the estimates of S(x).
(b)
Latitude
-0.55
-0.50
-0.45
-0.40
1.0
0.8
0.6
-0.65
-0.60
0.4
Std. errors (community + school)
-0.35
1.2
(a)
0.4
0.6
0.8
Std. errors (community)
Emanuele Giorgi and Peter J. Diggle
1.0
34.70 34.75 34.80 34.85 34.90 34.95 35.00
Longitude
Spatial modelling of tropical diseases
7 / 21
River blindness
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
8 / 21
Rapid epidemiological mapping of onchocerciasis
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
9 / 21
Prevalence estimates in the 20 APOC countries
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
10 / 21
Prevalence estimates in the 20 APOC countries
Statistical issue
1
3, 320 villages with no case and 11, 154 villages with at least one case.
2
Accounting for zero-inflation: how to identify disease-free areas?
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
10 / 21
Extending the standard geostatistical model
• Standard model: Y |S(x) ∼ Bin(·; n, p(x)) where
log{p(x)/(1 − p(x))} = µ + S(x).
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
11 / 21
Extending the standard geostatistical model
• Standard model: Y |S(x) ∼ Bin(·; n, p(x)) where
log{p(x)/(1 − p(x))} = µ + S(x).
• Allowing for zero-inflation:
(
π(x) + (1 − π(x))Bin(0; n, p(x)) if y = 0
P (Y = y|S1 (x), S2 (x), D(x)) =
(1 − π(x))Bin(y; n, p(x))
if y > 0
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
11 / 21
Extending the standard geostatistical model
• Standard model: Y |S(x) ∼ Bin(·; n, p(x)) where
log{p(x)/(1 − p(x))} = µ + S(x).
• Allowing for zero-inflation:
(
π(x) + (1 − π(x))Bin(0; n, p(x)) if y = 0
P (Y = y|S1 (x), S2 (x), D(x)) =
(1 − π(x))Bin(y; n, p(x))
if y > 0
where
logit(π(x))
= µ1 + S1 (x) + D(x),
logit(p(x))
= µ2 + S2 (x) + D(x).
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
11 / 21
Extending the standard geostatistical model
• Standard model: Y |S(x) ∼ Bin(·; n, p(x)) where
log{p(x)/(1 − p(x))} = µ + S(x).
• Allowing for zero-inflation:
(
π(x) + (1 − π(x))Bin(0; n, p(x)) if y = 0
P (Y = y|S1 (x), S2 (x), D(x)) =
(1 − π(x))Bin(y; n, p(x))
if y > 0
where
logit(π(x))
= µ1 + S1 (x) + D(x),
logit(p(x))
= µ2 + S2 (x) + D(x).
• In the next example D(x) = 0, ∀x.
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
11 / 21
Mapping in Mozambique - Malawi - Tanzania (1)
-5
Difference
-5
Standard GEO
0.8
1.0
-10
-10
1.0
0.8
0.6
0.2
-10
-5
Geo-ZIB
0.1
0.6
0.4
30
35
40
Emanuele Giorgi and Peter J. Diggle
-0.1
-0.2
-25
0.0
-20
0.2
-25
-25
0.0
-20
-20
0.2
-15
-15
-15
0.0
0.4
30
35
40
Spatial modelling of tropical diseases
30
35
12 / 21
40
Mapping in Mozambique - Malawi - Tanzania (2)
-5
Mapping of π(x).
-10
1.0
0.8
-15
0.6
0.4
-20
0.2
-25
0.0
25
Emanuele Giorgi and Peter J. Diggle
30
35
40
45
Spatial modelling of tropical diseases
13 / 21
Plague
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
14 / 21
0.7
0.5
0.3
0 0.1
Incidence per 100,000
0.9
Plague in Madagascar
1960
1963
1966
1969
1972
1975
1978
1981
Emanuele Giorgi and Peter J. Diggle
1984
1987
1990
1993
1996
1999
2002
2005
2008
Spatial modelling of tropical diseases
15 / 21
0.7
0.5
0.3
0 0.1
Incidence per 100,000
0.9
Plague in Madagascar
1960
1963
1966
1969
1972
1975
1978
1981
1984
1987
1990
1993
1996
1999
2002
2005
2008
Statistical issue
1
Ecological bias.
2
How to integrate spatially continuous data with district-level case
reports?
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
15 / 21
Modelling framework
• Ni,t = ‘‘number of cases in the i-th district at time t’’
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
16 / 21
Modelling framework
• Ni,t = ‘‘number of cases in the i-th district at time t’’
• λ(x, t) = ‘‘intensity of an inhomogeneous Poisson process at
location x and time t’’
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
16 / 21
Modelling framework
• Ni,t = ‘‘number of cases in the i-th district at time t’’
• λ(x, t) = ‘‘intensity of an inhomogeneous Poisson process at
location x and time t’’
• Ni,t ∼ Poisson
R
Emanuele Giorgi and Peter J. Diggle
t+1
t
R
Ai
λ(u, v) du dv .
Spatial modelling of tropical diseases
16 / 21
Modelling framework
• Ni,t = ‘‘number of cases in the i-th district at time t’’
• λ(x, t) = ‘‘intensity of an inhomogeneous Poisson process at
location x and time t’’
• Ni,t ∼ Poisson
R
t+1
t
R
λ(u, v) du dv .
R t+1 R
Pni,t
• t
λ(u, v) du dv ≈ ∆ j=1
λ(xj , t).
Ai
Emanuele Giorgi and Peter J. Diggle
Ai
Spatial modelling of tropical diseases
16 / 21
Modelling framework
• Ni,t = ‘‘number of cases in the i-th district at time t’’
• λ(x, t) = ‘‘intensity of an inhomogeneous Poisson process at
location x and time t’’
• Ni,t ∼ Poisson
R
t+1
t
R
λ(u, v) du dv .
R t+1 R
Pni,t
• t
λ(u, v) du dv ≈ ∆ j=1
λ(xj , t).
Ai
Ai
• Xi,t = {Xj,t ∈ Ai }|Ni,t = ni,t has multinomial distribution with
probabilities
λ(xk , t)
pk = Pni,t
, for k = 1, . . . , ni,t .
j=1 λ(xj , t)
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
16 / 21
Intensity of the point process
• m(x, t) = ‘‘number of susceptibles at location x and time t’’
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
17 / 21
Intensity of the point process
• m(x, t) = ‘‘number of susceptibles at location x and time t’’
• λ(x, t) = m(x, t) exp{d(x, t)> β + S(x) + V (t)},
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
17 / 21
Intensity of the point process
• m(x, t) = ‘‘number of susceptibles at location x and time t’’
• λ(x, t) = m(x, t) exp{d(x, t)> β + S(x) + V (t)},
d(x, t)> β = β1 + β2 temp(x, t) + β3 rain(x, t) +
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
17 / 21
Intensity of the point process
• m(x, t) = ‘‘number of susceptibles at location x and time t’’
• λ(x, t) = m(x, t) exp{d(x, t)> β + S(x) + V (t)},
d(x, t)> β = β1 + β2 temp(x, t) + β3 rain(x, t) +
β4 I(elev(x) > 800m) + β5 elev(x) +
β6 elev(x) × I(elev(x) > 800m) +
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
17 / 21
Intensity of the point process
• m(x, t) = ‘‘number of susceptibles at location x and time t’’
• λ(x, t) = m(x, t) exp{d(x, t)> β + S(x) + V (t)},
d(x, t)> β = β1 + β2 temp(x, t) + β3 rain(x, t) +
β4 I(elev(x) > 800m) + β5 elev(x) +
β6 elev(x) × I(elev(x) > 800m) +
β7 log{pop(x)} +
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
17 / 21
Intensity of the point process
• m(x, t) = ‘‘number of susceptibles at location x and time t’’
• λ(x, t) = m(x, t) exp{d(x, t)> β + S(x) + V (t)},
d(x, t)> β = β1 + β2 temp(x, t) + β3 rain(x, t) +
β4 I(elev(x) > 800m) + β5 elev(x) +
β6 elev(x) × I(elev(x) > 800m) +
β7 log{pop(x)} +
[β8 + β9 temp(x, t) + β10 rain(x, t)] sin(2πt/12) +
[β11 + β12 temp(x, t) + β13 rain(x, t)] cos(2πt/12)
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
17 / 21
Intensity of the point process
• m(x, t) = ‘‘number of susceptibles at location x and time t’’
• λ(x, t) = m(x, t) exp{d(x, t)> β + S(x) + V (t)},
d(x, t)> β = β1 + β2 temp(x, t) + β3 rain(x, t) +
β4 I(elev(x) > 800m) + β5 elev(x) +
β6 elev(x) × I(elev(x) > 800m) +
β7 log{pop(x)} +
[β8 + β9 temp(x, t) + β10 rain(x, t)] sin(2πt/12) +
[β11 + β12 temp(x, t) + β13 rain(x, t)] cos(2πt/12)
S(x) = Si , if x ∈ districti
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
17 / 21
Intensity of the point process
• m(x, t) = ‘‘number of susceptibles at location x and time t’’
• λ(x, t) = m(x, t) exp{d(x, t)> β + S(x) + V (t)},
d(x, t)> β = β1 + β2 temp(x, t) + β3 rain(x, t) +
β4 I(elev(x) > 800m) + β5 elev(x) +
β6 elev(x) × I(elev(x) > 800m) +
β7 log{pop(x)} +
[β8 + β9 temp(x, t) + β10 rain(x, t)] sin(2πt/12) +
[β11 + β12 temp(x, t) + β13 rain(x, t)] cos(2πt/12)
S(x) = Si , if x ∈ districti
V (t) | V (t − 1), . . . , V (0) = V (t) | V (t − 1)
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
17 / 21
Results (1)
Term
β1
β2
β3
β4
β5
β6
β7
β8
β9
β10
β11
β12
β13
Emanuele Giorgi and Peter J. Diggle
Estimate
-41.301
0.055
0.028
22.241
0.027
-0.024
0.069
0.735
-0.293
0.025
5.017
0.040
-0.026
95% CI
(-54.246, -30.104)
(-0.008, 0.125)
(-0.009, 0.061)
(10.566, 35.322)
(0.011, 0.043)
(-0.041, -0.008)
(0.014, 0.140)
(-0.103, 1.444)
(-0.314, -0.269)
(-0.003, 0.053)
(4.513, 5.498)
(0.007, 0.078)
(-0.063, 0.013)
Spatial modelling of tropical diseases
18 / 21
Results (2)
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
19 / 21
Results (3)
Spatial residuals
under -1
-1 - -0.05
-0.05 - 0.05
0.05 - 1
1-2
over 2
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
20 / 21
Discussion
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
21 / 21
Discussion
• How to account for spatio-temporally structured bias?
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
21 / 21
Discussion
• How to account for spatio-temporally structured bias?
• Alternative modelling approaches to zero-inflation.
1
2
Admitting discontinuities in prevalence.
Constraining prevalence to smoothly approach zero.
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
21 / 21
Discussion
• How to account for spatio-temporally structured bias?
• Alternative modelling approaches to zero-inflation.
1
2
Admitting discontinuities in prevalence.
Constraining prevalence to smoothly approach zero.
• Comparison of the mechanistic approach with empirical models.
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
21 / 21
Discussion
• How to account for spatio-temporally structured bias?
• Alternative modelling approaches to zero-inflation.
1
2
Admitting discontinuities in prevalence.
Constraining prevalence to smoothly approach zero.
• Comparison of the mechanistic approach with empirical models.
• Acknowledgements: Gillian Stresman (LSHTM), Jennifer Stevenson
(LSHTM), Hans Remme (Ornex), Cyril Caminade (Liverpool),
Matthew Baylis (Liverpool).
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
21 / 21
Discussion
• How to account for spatio-temporally structured bias?
• Alternative modelling approaches to zero-inflation.
1
2
Admitting discontinuities in prevalence.
Constraining prevalence to smoothly approach zero.
• Comparison of the mechanistic approach with empirical models.
• Acknowledgements: Gillian Stresman (LSHTM), Jennifer Stevenson
(LSHTM), Hans Remme (Ornex), Cyril Caminade (Liverpool),
Matthew Baylis (Liverpool).
Thanks for your attention.
Emanuele Giorgi and Peter J. Diggle
Spatial modelling of tropical diseases
21 / 21