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Rapid loss of immunity is necessary toexplain historical cholera epidemics
Ed Ionides & Aaron King
University of MichiganDepartments of Statistics and Ecology & Evolutionary Biology
![Page 2: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/2.jpg)
Bengal: cholera’s homeland
![Page 3: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/3.jpg)
Bengal: cholera’s homeland
![Page 4: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/4.jpg)
Bengal: cholera’s homeland
![Page 5: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/5.jpg)
Bengal: cholera’s homeland
![Page 6: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/6.jpg)
Bengal: cholera’s homeland
![Page 7: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/7.jpg)
Cholera: unsolved puzzlesI Mode of transmission
I contaminated water: environmental reservoirI food-borne/direct fecal-oralI transient hyperinfectious state (<18 hr)
I Seasonality
I two peaks per yearI regional climate drivers: monsoon rainfall and winter
temperatures?
I Multi-year climate driversI El Nino-Southern Oscillation (ENSO)
I ImmunityI volunteer studies: > 3 yr immunity following severe infectionI community study of reinfections in Matlab, Bangladesh:
I risk of reinfection equal to risk of primary infectionI 1.6 yr average duration between primary infection and
reinfectionI retrospective statistical analyses (Koelle & Pascual 2004):
7–10 yr immunity following severe infection
![Page 8: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/8.jpg)
Cholera: unsolved puzzlesI Mode of transmission
I contaminated water: environmental reservoirI food-borne/direct fecal-oralI transient hyperinfectious state (<18 hr)
I SeasonalityI two peaks per year
I regional climate drivers: monsoon rainfall and wintertemperatures?
I Multi-year climate driversI El Nino-Southern Oscillation (ENSO)
I ImmunityI volunteer studies: > 3 yr immunity following severe infectionI community study of reinfections in Matlab, Bangladesh:
I risk of reinfection equal to risk of primary infectionI 1.6 yr average duration between primary infection and
reinfectionI retrospective statistical analyses (Koelle & Pascual 2004):
7–10 yr immunity following severe infection
![Page 9: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/9.jpg)
Cholera: unsolved puzzles
yr
J F M A M J J A S O N D
010
0030
0050
00D
acca
cho
lera
mor
talit
y
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Cholera: unsolved puzzlesI Mode of transmission
I contaminated water: environmental reservoirI food-borne/direct fecal-oralI transient hyperinfectious state (<18 hr)
I SeasonalityI two peaks per yearI regional climate drivers: monsoon rainfall and winter
temperatures?
I Multi-year climate driversI El Nino-Southern Oscillation (ENSO)
I ImmunityI volunteer studies: > 3 yr immunity following severe infectionI community study of reinfections in Matlab, Bangladesh:
I risk of reinfection equal to risk of primary infectionI 1.6 yr average duration between primary infection and
reinfectionI retrospective statistical analyses (Koelle & Pascual 2004):
7–10 yr immunity following severe infection
![Page 11: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/11.jpg)
Cholera: unsolved puzzlesI Mode of transmission
I contaminated water: environmental reservoirI food-borne/direct fecal-oralI transient hyperinfectious state (<18 hr)
I SeasonalityI two peaks per yearI regional climate drivers: monsoon rainfall and winter
temperatures?I Multi-year climate drivers
I El Nino-Southern Oscillation (ENSO)
I ImmunityI volunteer studies: > 3 yr immunity following severe infectionI community study of reinfections in Matlab, Bangladesh:
I risk of reinfection equal to risk of primary infectionI 1.6 yr average duration between primary infection and
reinfectionI retrospective statistical analyses (Koelle & Pascual 2004):
7–10 yr immunity following severe infection
![Page 12: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/12.jpg)
Cholera: unsolved puzzlesI Mode of transmission
I contaminated water: environmental reservoirI food-borne/direct fecal-oralI transient hyperinfectious state (<18 hr)
I SeasonalityI two peaks per yearI regional climate drivers: monsoon rainfall and winter
temperatures?I Multi-year climate drivers
I El Nino-Southern Oscillation (ENSO)I Immunity
I volunteer studies: > 3 yr immunity following severe infectionI community study of reinfections in Matlab, Bangladesh:
I risk of reinfection equal to risk of primary infectionI 1.6 yr average duration between primary infection and
reinfectionI retrospective statistical analyses (Koelle & Pascual 2004):
7–10 yr immunity following severe infection
![Page 13: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/13.jpg)
Inapparent infections
I most cholera infections are mild or asymptomatic
I reports of the asymptomatic:symptomatic ratio range from3 to 100
I it is easy to underestimate the degree to which oneunderestimates a quantity one cannot observe
I Needed: an approach that allows indirect inference aboutunobserved variables
I historical cholera mortality records are a rich source ofinformation, but have been difficult to fully exploit
![Page 14: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/14.jpg)
Inapparent infections
I most cholera infections are mild or asymptomaticI reports of the asymptomatic:symptomatic ratio range from
3 to 100
I it is easy to underestimate the degree to which oneunderestimates a quantity one cannot observe
I Needed: an approach that allows indirect inference aboutunobserved variables
I historical cholera mortality records are a rich source ofinformation, but have been difficult to fully exploit
![Page 15: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/15.jpg)
Inapparent infections
I most cholera infections are mild or asymptomaticI reports of the asymptomatic:symptomatic ratio range from
3 to 100I it is easy to underestimate the degree to which one
underestimates a quantity one cannot observe
I Needed: an approach that allows indirect inference aboutunobserved variables
I historical cholera mortality records are a rich source ofinformation, but have been difficult to fully exploit
![Page 16: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/16.jpg)
Inapparent infections
I most cholera infections are mild or asymptomaticI reports of the asymptomatic:symptomatic ratio range from
3 to 100I it is easy to underestimate the degree to which one
underestimates a quantity one cannot observeI Needed: an approach that allows indirect inference about
unobserved variables
I historical cholera mortality records are a rich source ofinformation, but have been difficult to fully exploit
![Page 17: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/17.jpg)
Inapparent infections
I most cholera infections are mild or asymptomaticI reports of the asymptomatic:symptomatic ratio range from
3 to 100I it is easy to underestimate the degree to which one
underestimates a quantity one cannot observeI Needed: an approach that allows indirect inference about
unobserved variablesI historical cholera mortality records are a rich source of
information, but have been difficult to fully exploit
![Page 18: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/18.jpg)
Historical cholera mortality
24 Parganas
Bakergang
Bankura
Birbhum
Bogra
Burdwan
Calcutta
Chittagong
Dacca
Dinajpur
Faridpur
Hooghly
Howrath
Jaipaguri
Jessore
Khulna
Malda
Midnapur
Mohrshidabad
Mymensingh
Nadia
Noakhali
Pabna
Rangpur
Rashahi
Tippera
![Page 19: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/19.jpg)
Historical cholera mortality
24 Parganas
Bakergang
Bankura
Birbhum
Bogra
Burdwan
Calcutta
Chittagong
Dacca
Dinajpur
Faridpur
Hooghly
Howrath
Jaipaguri
Jessore
Khulna
Malda
Midnapur
Mohrshidabad
Mymensingh
Nadia
Noakhali
Pabna
Rangpur
Rashahi
Tippera
![Page 20: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/20.jpg)
Historical cholera mortality
x$time
x[[
dis
t]]
1890 1900 1910 1920 1930 1940
01
00
02
00
03
00
04
00
05
00
0
year
ch
ole
ra m
ort
alit
y
Dacca
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Historical cholera mortality
24 Parganas
Bakergang
Bankura
Birbhum
Bogra
Burdwan
Calcutta
Chittagong
Dacca
Dinajpur
Faridpur
Hooghly
Howrath
Jaipaguri
Jessore
Khulna
Malda
Midnapur
Mohrshidabad
Mymensingh
Nadia
Noakhali
Pabna
Rangpur
Rashahi
Tippera
![Page 22: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/22.jpg)
Historical cholera mortality
x$time
x[[
dis
t]]
1890 1900 1910 1920 1930 1940
01
00
02
00
03
00
0
year
ch
ole
ra m
ort
alit
y
Midnapur
![Page 23: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/23.jpg)
Historical cholera mortality
24 Parganas
Bakergang
Bankura
Birbhum
Bogra
Burdwan
Calcutta
Chittagong
Dacca
Dinajpur
Faridpur
Hooghly
Howrath
Jaipaguri
Jessore
Khulna
Malda
Midnapur
Mohrshidabad
Mymensingh
Nadia
Noakhali
Pabna
Rangpur
Rashahi
Tippera
![Page 24: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/24.jpg)
Historical cholera mortality
x$time
x[[
dis
t]]
1890 1900 1910 1920 1930 1940
01
00
02
00
03
00
0
year
ch
ole
ra m
ort
alit
y
Jaipaguri
![Page 25: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/25.jpg)
Historical cholera mortality
24 Parganas
Bakergang
Bankura
Birbhum
Bogra
Burdwan
Calcutta
Chittagong
Dacca
Dinajpur
Faridpur
Hooghly
Howrath
Jaipaguri
Jessore
Khulna
Malda
Midnapur
Mohrshidabad
Mymensingh
Nadia
Noakhali
Pabna
Rangpur
Rashahi
Tippera
![Page 26: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/26.jpg)
Historical cholera mortality
x$time
x[[
dis
t]]
1890 1900 1910 1920 1930 1940
01
00
02
00
03
00
04
00
05
00
0
year
ch
ole
ra m
ort
alit
y
Bakergang
![Page 27: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/27.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
parameter symbolforce of infection λ(t)recovery rate γdisease death rate mmean long-term immune period 1/ε
CV of long-term immune period 1/√
k
![Page 28: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/28.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
parameter symbolforce of infection λ(t)recovery rate γdisease death rate mmean long-term immune period 1/ε
CV of long-term immune period 1/√
k
![Page 29: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/29.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
parameter symbolforce of infection λ(t)recovery rate γdisease death rate mmean long-term immune period 1/ε
CV of long-term immune period 1/√
k
![Page 30: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/30.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
parameter symbolforce of infection λ(t)recovery rate γdisease death rate mmean long-term immune period 1/ε
CV of long-term immune period 1/√
k
![Page 31: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/31.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
parameter symbolforce of infection λ(t)recovery rate γdisease death rate mmean long-term immune period 1/ε
CV of long-term immune period 1/√
k
![Page 32: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/32.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
λ(t) =(
eβtrend t βseas(t) + ξ(t)) I(t)
P(t)+ ω
![Page 33: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/33.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
λ(t) =(
eβtrend t βseas(t) + ξ(t)) I(t)
P(t)+ ω
βtrend = trend in transmission
![Page 34: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/34.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
λ(t) =(
eβtrend t βseas(t) + ξ(t)) I(t)
P(t)+ ω
βseas(t) = seasonality in transmission
semimechanistic approach: use flexible function
![Page 35: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/35.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
λ(t) =(
eβtrend t βseas(t) + ξ(t)) I(t)
P(t)+ ω
ξ(t) = environmental stochasticity
![Page 36: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/36.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
λ(t) =(
eβtrend t βseas(t) + ξ(t)) I(t)
P(t)+ ω
ω = environmental reservoir
![Page 37: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/37.jpg)
SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
λ(t) =(
eβtrend t βseas(t) + ξ(t)) I(t)
P(t)+ ω
P(t) = censused population size
![Page 38: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/38.jpg)
SIRS model
dSdt
=dP(t)
dt+ δ P(t)− (λ(t) + kε Rk + δ) S
dIdt
= λ(t) S − (m + γ + δ) I(t)
dR1
dt= γ I − (k ε + δ) R1
...dRk
dt= k ε Rk−1 − (k ε + δ) Rk
Stochastic force of infection:
λ(t) =(
eβtrend t βseas(t) + ξ(t)) I(t)
P(t)+ ω
![Page 39: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/39.jpg)
Likelihood maximization by iterated filtering
I new frequentist approach(Ionides, Breto, & King, PNAS 2006)
I can accommodate:I continuous-time modelsI nonlinearityI stochasticityI unobserved variablesI measurement errorI nonstationarityI covariates
I based on well-studied sequential Monte Carlo methods(particle filter)
I “plug and play” propertyI Implemented in pomp, an open-source R package
(www.r-project.org)
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Duration of immunity
●
●
●●
●●
●●
●●
●●●●●●
●●●
●●
●●
●●
●●●●
●●●●●●●●●●●●
●●●●●●●
0.01 0.05 0.10 0.50 1.00 5.00
−384
0−3
820
−380
0
logl
ik
immune period (yr)
prof
ile lo
g lik
elih
ood
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SIRS model
P b - S I R1 Rk
M
-λ(t)
-γ
-m
-kε . . . -kε
kε6
parameter symbolforce of infection λ(t)recovery rate γdisease death rate mmean long-term immune period 1/ε
CV of long-term immune period 1/√
k
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SIRS model predictions
I estimated cholera fatality (across districts):0.0039± 0.0021.
I in the historical period, fatality in severe cases was c. 60%I model prediction: lots of silent sheddersI evidence for silent shedders is mixed and weakI Q: is it necessary that all exposed individuals become
infectious?
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SIRS model predictions
I estimated cholera fatality (across districts):0.0039± 0.0021.
I in the historical period, fatality in severe cases was c. 60%
I model prediction: lots of silent sheddersI evidence for silent shedders is mixed and weakI Q: is it necessary that all exposed individuals become
infectious?
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SIRS model predictions
I estimated cholera fatality (across districts):0.0039± 0.0021.
I in the historical period, fatality in severe cases was c. 60%I model prediction: lots of silent shedders
I evidence for silent shedders is mixed and weakI Q: is it necessary that all exposed individuals become
infectious?
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SIRS model predictions
I estimated cholera fatality (across districts):0.0039± 0.0021.
I in the historical period, fatality in severe cases was c. 60%I model prediction: lots of silent sheddersI evidence for silent shedders is mixed and weak
I Q: is it necessary that all exposed individuals becomeinfectious?
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SIRS model predictions
I estimated cholera fatality (across districts):0.0039± 0.0021.
I in the historical period, fatality in severe cases was c. 60%I model prediction: lots of silent sheddersI evidence for silent shedders is mixed and weakI Q: is it necessary that all exposed individuals become
infectious?
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Two-path model
P -b S
I
M
R1 Rk
Y
-m
-γ -kε . . . -kε
� kε
-cλ(t)
-(1− c)λ(t)
ρ
�
parameter symbolforce of infection λ(t)probability of severe infection crecovery rate γdisease death rate mmean long-term immune period 1/ε
CV of long-term immune period 1/√
kmean short-term immune period 1/ρ
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Two-path model
P -b S
I
M
R1 Rk
Y
-m
-γ -kε . . . -kε
� kε
-cλ(t)
-(1− c)λ(t)
ρ
�
parameter symbolforce of infection λ(t)probability of severe infection crecovery rate γdisease death rate mmean long-term immune period 1/ε
CV of long-term immune period 1/√
kmean short-term immune period 1/ρ
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Two-path model
P -b S
I
M
R1 Rk
Y
-m
-γ -kε . . . -kε
� kε
-cλ(t)
-(1− c)λ(t)
ρ
�
parameter symbolforce of infection λ(t)probability of severe infection crecovery rate γdisease death rate mmean long-term immune period 1/ε
CV of long-term immune period 1/√
kmean short-term immune period 1/ρ
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Two-path model
dSdt
= kε Rk + ρ Y +dP(t)
dt+ δ P(t)− (λ(t) + δ) S
dIdt
= c λ(t) S − (m + γ + δ) I(t)
dYdt
= (1− c) λ(t) S − (ρ + δ) Y
dR1
dt= γ I − (k ε + δ) R1
...dRk
dt= k ε Rk−1 − (k ε + δ) Rk
Stochastic force of infection:
λ(t) =(
eβtrend t βseas(t) + ξ(t)) I(t)
P(t)+ ω
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Model comparison
model log likelihood AICtwo-path -3775.8 7591.6SIRS -3794.3 7622.6SARMA((2,2)×(1,1)) -3804.5 7625.0Koelle & Pascual (2004) -3840.1 —seasonal mean -3989.1 8026.1
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Goodness of fit
district r2 district r2
Bakergang 0.856 Jessore 0.754Bankura 0.559 Khulna 0.735Birbhum 0.509 Malda 0.596Bogra 0.570 Midnapur 0.666Burdwan 0.589 Mohrshidabad 0.631Calcutta 0.756 Mymensingh 0.805Chittagong 0.712 Nadia 0.734Dacca 0.848 Noakhali 0.701Dinajpur 0.078 Pabna 0.690Faridpur 0.785 Rangpur 0.594Hooghly 0.569 Rashahi 0.690Howrath 0.769 Tippera 0.767Jaipaguri 0.109 24 Parganas 0.839
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Goodness of fit
r2
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Simulated vs. actual data
c(1891, 1941)
c(0,
550
0)
Dacca actual0
2500
5000
actu
al
c(1891, 1941)
c(0,
550
0)
Dacca simulated
025
0050
00si
mul
ated
1891 1901 1911 1921 1931 1941
c(1891, 1941)
c(0,
550
0)
Faridpur actual
c(1891, 1941)
c(0,
550
0)
Faridpur simulated
1891 1901 1911 1921 1931 1941
SIRS model
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Simulated vs. actual data
c(1891, 1941)
c(0,
550
0)
Dacca actual0
2500
5000
actu
al
c(1891, 1941)
c(0,
550
0)
Dacca simulated
025
0050
00si
mul
ated
1891 1901 1911 1921 1931 1941
c(1891, 1941)
c(0,
550
0)
Faridpur actual
c(1891, 1941)
c(0,
550
0)
Faridpur simulated
1891 1901 1911 1921 1931 1941
two-path model
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Parameter estimatesI cholera fatality: 0.07± 0.05
I probability of severe infection: c = 0.008± 0.005I R0 = 1.10± 0.27I duration of long-term immunity: 2.6± 1.8 yrI duration of short-term immunity: 0.3–8.3 wkI geographical variability in force of infection
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Parameter estimatesI cholera fatality: 0.07± 0.05I probability of severe infection: c = 0.008± 0.005
I R0 = 1.10± 0.27I duration of long-term immunity: 2.6± 1.8 yrI duration of short-term immunity: 0.3–8.3 wkI geographical variability in force of infection
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Parameter estimatesI cholera fatality: 0.07± 0.05I probability of severe infection: c = 0.008± 0.005I R0 = 1.10± 0.27
I duration of long-term immunity: 2.6± 1.8 yrI duration of short-term immunity: 0.3–8.3 wkI geographical variability in force of infection
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Parameter estimates
c(0,
max
(sea
s))
J F M A M J J A S O N D J
0.0
2.5
5.0
7.5
10.0
R0((t
))
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Parameter estimatesI cholera fatality: 0.07± 0.05I probability of severe infection: c = 0.008± 0.005I R0 = 1.10± 0.27
I duration of long-term immunity: 2.6± 1.8 yrI duration of short-term immunity: 0.3–8.3 wkI geographical variability in force of infection
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Parameter estimatesI cholera fatality: 0.07± 0.05I probability of severe infection: c = 0.008± 0.005I R0 = 1.10± 0.27I duration of long-term immunity: 2.6± 1.8 yr
I duration of short-term immunity: 0.3–8.3 wkI geographical variability in force of infection
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Parameter estimatesI cholera fatality: 0.07± 0.05I probability of severe infection: c = 0.008± 0.005I R0 = 1.10± 0.27I duration of long-term immunity: 2.6± 1.8 yrI duration of short-term immunity: 0.3–8.3 wk
I geographical variability in force of infection
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Parameter estimatesI cholera fatality: 0.07± 0.05I probability of severe infection: c = 0.008± 0.005I R0 = 1.10± 0.27I duration of long-term immunity: 2.6± 1.8 yrI duration of short-term immunity: 0.3–8.3 wkI geographical variability in force of infection
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Geographical patterns
environmental reservoir
ωω
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Geographical patterns
transmission, Jan–Feb
b0
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Geographical patterns
transmission, Mar–Apr
b1
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Geographical patterns
transmission, May–Jun
b2
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Geographical patterns
transmission, Jul–Aug
b3
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Geographical patterns
transmission, Sep–Oct
b4
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Geographical patterns
transmission, Nov–Dec
b5
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Contrasting views of endemic cholera dynamics
View of Koelle & Pascual (2004):I long-term immunity (7–10 yr)I modest asymptomatic ratio (c. 70:1)I spatial heterogeneity slows epidemicI seasonal drop in transmission stops epidemicI waning of immunity interacts with climate drivers on
multi-year scale
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Contrasting views of endemic cholera dynamics
New view:I rapidly waning immunityI high asymptomatic ratio (>160:1)I susceptible depletion slows and stops epidemicI loss of immunity replenishes susceptibles rapidlyI waning of immunity interacts with climate drivers on
seasonal scaleI multi-year climate drivers?
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Public health implications
I What determines the switch probability c?
I serological profile data suggest c declines with ageI dose/hyperinfectiousness
I foodborne/direct fecal-oral mode is most importantI lower doses provide short-term immunity
I unintended consequences of neglect of householdtransmission?
I presence of vibriophage in environment?
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Public health implications
I What determines the switch probability c?I serological profile data suggest c declines with age
I dose/hyperinfectiousnessI foodborne/direct fecal-oral mode is most importantI lower doses provide short-term immunity
I unintended consequences of neglect of householdtransmission?
I presence of vibriophage in environment?
![Page 75: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/75.jpg)
Public health implications
McCormack et al. (1969)
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Public health implications
I What determines the switch probability c?I serological profile data suggest c declines with ageI dose/hyperinfectiousness
I foodborne/direct fecal-oral mode is most importantI lower doses provide short-term immunity
I unintended consequences of neglect of householdtransmission?
I presence of vibriophage in environment?
![Page 77: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/77.jpg)
Public health implications
I What determines the switch probability c?I serological profile data suggest c declines with ageI dose/hyperinfectiousness
I foodborne/direct fecal-oral mode is most importantI lower doses provide short-term immunity
I unintended consequences of neglect of householdtransmission?
I presence of vibriophage in environment?
![Page 78: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/78.jpg)
Public health implications
I What determines the switch probability c?I serological profile data suggest c declines with ageI dose/hyperinfectiousness
I foodborne/direct fecal-oral mode is most importantI lower doses provide short-term immunity
I unintended consequences of neglect of householdtransmission?
I presence of vibriophage in environment?
![Page 79: Rapid loss of immunity is necessary to explain …dept.stat.lsa.umich.edu/~ionides/pubs/cholera-talk.pdfRapid loss of immunity is necessary to explain historical cholera epidemics](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa39c191e9573383a05ba4c/html5/thumbnails/79.jpg)
Alternative hypotheses?
I inhomogeneitiesI behavioral effects
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ExtensionsI Recent data (1966–2005, Matlab, Bangladesh)I Discrete population continuous time modelsI Other diseases
I malariaI measlesI influenza
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Seasonality
R. G. Feachem (1982) on the seasonal patterns of choleraepidemics in Bengal:
They are such a dominant feature of choleraepidemiology, and in such contrast to the otherbacterial diarrhoeas which peak during the monsoonin mid-summer, that their explanation probably holdsthe key to fundamental insights into choleratransmission, ecology, and control.
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Seasonality
R. G. Feachem (1982) on the seasonal patterns of choleraepidemics in Bengal:
They are such a dominant feature of choleraepidemiology, and in such contrast to the otherbacterial diarrhoeas which peak during the monsoonin mid-summer, that their explanation probably holdsthe key to fundamental insights into choleratransmission, ecology, and control.
to understand seasonality, we must allow for the interaction ofshort-term immunity with seasonal environmental drivers
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Thanks to . . .I Mercedes PascualI Menno BoumaI Carles BretoI Katia KoelleI Diego Ruiz MorenoI Andy DobsonI the R project (www.r-project.org)I NSF/NIH Ecology of Infectious Diseases
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Thank you!