rapid loss of immunity is necessary to explain...

Rapid loss of immunity is necessary toexplain historical cholera epidemics

Ed Ionides & Aaron King

University of MichiganDepartments of Statistics and Ecology & Evolutionary Biology

Bengal: cholera’s homeland

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



I SeasonalityI two peaks per year

I regional climate drivers: monsoon rainfall and wintertemperatures?






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



I SeasonalityI two peaks per yearI regional climate drivers: monsoon rainfall and winter

temperatures?









temperatures?I Multi-year climate drivers

I El Nino-Southern Oscillation (ENSO)








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:




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


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





3 to 100I it is easy to underestimate the degree to which one

underestimates a quantity one cannot observe






underestimates a quantity one cannot observeI Needed: an approach that allows indirect inference about

unobserved variables





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

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


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


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


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


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


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


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


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

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

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

SIRS model

P b - S I R1 Rk

M

-λ(t)

-γ

-m

-kε . . . -kε

kε6

λ(t) =(


P(t)+ ω

βtrend = trend in transmission

SIRS model

P b - S I R1 Rk

M

-λ(t)

-γ

-m

-kε . . . -kε

kε6

λ(t) =(


P(t)+ ω

βseas(t) = seasonality in transmission

semimechanistic approach: use flexible function

SIRS model

P b - S I R1 Rk

M

-λ(t)

-γ

-m

-kε . . . -kε

kε6

λ(t) =(


P(t)+ ω

ξ(t) = environmental stochasticity

SIRS model

P b - S I R1 Rk

M

-λ(t)

-γ

-m

-kε . . . -kε

kε6

λ(t) =(


P(t)+ ω

ω = environmental reservoir

SIRS model

P b - S I R1 Rk

M

-λ(t)

-γ

-m

-kε . . . -kε

kε6

λ(t) =(


P(t)+ ω

P(t) = censused population size

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) =(


P(t)+ ω

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)

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

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


k

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?



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?



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?



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?



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?

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


kmean short-term immune period 1/ρ

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) =(


P(t)+ ω

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

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

Goodness of fit

r2

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

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

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

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

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

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

))

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

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

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

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

Geographical patterns

environmental reservoir

ωω


transmission, Jan–Feb

b0


transmission, Mar–Apr

b1


transmission, May–Jun

b2


transmission, Jul–Aug

b3


transmission, Sep–Oct

b4


transmission, Nov–Dec

b5

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

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?

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?


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




McCormack et al. (1969)


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



Alternative hypotheses?

I inhomogeneitiesI behavioral effects

ExtensionsI Recent data (1966–2005, Matlab, Bangladesh)I Discrete population continuous time modelsI Other diseases

I malariaI measlesI influenza

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.

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

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

Thank you!

rapid loss of immunity is necessary to explain...

Documents