Modelling periodic ACF Pete Dodd Factors inﬂuencing the … · B T o n y t i l i b a b ro p...

Modelling periodicACF

Pete Dodd

IntroductionScope

Overview

The modelIncidence from prevalence

Prevalence from incidence

Example behaviours

ResultsGeneral observations

Cases found

Cases averted

ConclusionsLessons

Next?

1

Factors influencing the performanceof active case-finding for TBTB screening/ACF meeting, WHO, Geneva.

1 June, 2011

Pete DoddCentre for the Mathematical Modelling of Infectious Diseases

Department of Infectious Disease EpidemiologyLondon School of Hygiene and Tropical Medicine


Pete Dodd

IntroductionScope

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ConclusionsLessons

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2

This talk

What this talk is not. . .

• A fully-costed evaluation of the economics of case-finding.• A detailed modelling exercise parametrized for a specific

setting.• A comparison between multiple strategies.


Pete Dodd

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3

What this talk is . . .

Aims

• Develop a simple model that depends on as fewparameters as possible.(Preferring to keep things in terms of measurablequantities.)

• Investigate periodic rounds of TB case-finding, in thecontext of HIV.

• Obtain insight into community and interventioncharacteristics that most determine ACF outcomes.


Pete Dodd

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4

Overview

Structure

• Description of model structure and inputs.• Features implied by model or typical in results.• Interpretation of results.


Pete Dodd

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Example behaviours


Cases found

Cases averted

ConclusionsLessons

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5

Model Specification


Pete Dodd

IntroductionScope

Overview



Example behaviours


Cases found

Cases averted

ConclusionsLessons

Next?

6

Specifying a model

incidence


Pete Dodd

IntroductionScope

Overview



Example behaviours


Cases found

Cases averted

ConclusionsLessons

Next?

7

Specifying a model

incidence

prevalence


Pete Dodd

IntroductionScope

Overview



Example behaviours


Cases found

Cases averted

ConclusionsLessons

Next?

8

Specifying a model

incidence

prevalence


Pete Dodd

IntroductionScope

Overview



Example behaviours


Cases found

Cases averted

ConclusionsLessons

Next?

9

Combined model for TB after infection

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

time (years)

prob

abili

ty n

o TB

Figure: The probability of avoidingTB after infection.

0 2 4 6 8 100.90

0.92

0.94

0.96

0.98

1.00

time (years)

prob

abili

ty n

o TB

Figure: As left, but differenty-scale.

Natural mixture model for survival function, with two timescales.


Pete Dodd

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10

A simple model: incidence from prevalence.

Aim: very simple, ‘open box’ model with a minimum ofparameters.

Assume HIV, LTBI prevalence, and ‘slow’-incidence ≈ const.

Incidence:

I0 = (1− PR0).I0 + PR0.I0 (1)

I(t) = (1− PR0).I0 + PR0.I0G(t)G0

(2)

where G(t) is a recently weighted average of the FOI, F (t):

G(t) =

∫P(fast delay = x)× F (t − x).dx (3)

F (t) ∝ D(t)N

(4)


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11

Different characteristics of HIV+/- TB

HIV-

HIV+

A: asymptomatic period

S: symptomatic,motivated to seek care

delay

A S delay

time

The outcomes and characteristics of disease are also verydifferent.


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12


0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

2.5

time (years)

Figure: Probability of duration asprevalent case: HIV+/-

Incidence

��Prevalent

((��vvDetection Death Selfcure

Possible outcomes.


Pete Dodd

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13

Prevalence from incidence: The Picture (with ACF).

●

●

●

●

ω

α

τ

t

ti + T

ti + 2T

ti

Calendar time

Time as prevalent case


Pete Dodd

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14

Model assumptions

Recap:

• LTBI prevalence ≈ const.• HIV prevalence ≈ const.• IRR | HIV prevalence ≈ const.• ‘Slow’ constribution to TB incidence ≈ const.• (Proportional hazards model for detection vs

death/self-cure.)• ART, homogeneity,. . .

Interested in 10 year time-frame.


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15

Model parametrization

Parameter Meaning Valuer0 rate for recent activation 0.9/yf relative infectiousness

HIV+ TB0.5

k Weibull shape 2.5L− Weibull timescale (HIV-) 1y †

L+ Weibull timescale (HIV+) 0.25y †

I+0 initial TB incidence(HIV+)

450.10−5/y †

I−0 initial TB incidence(HIV-)

150.10−5/y †

PR0 initial proportion TB ofincident ‘recent’

72% †

(CDR+/−0 )∗ initial TB case-detection

rate (HIV+/-)50% †

† Default example: changes in parameter investigated.∗ Only affects conclusions about numbers of cases found.


Pete Dodd

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16

Example behaviour


Pete Dodd

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17

Prevalence & incidence through time

0 5 10 15 20

020

4060

8010

012

0

time

prev

alen

ce/1

e+5

Figure: Prevalence

0 5 10 15 20

010

020

030

040

0

time

inci

denc

e/1e

+5p

y

Figure: Incidence


Pete Dodd

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18

Cumulative cases found

5 10 15 20

050

010

0015

0020

0025

00

time

cum

ulat

ive

tota

l cas

es fo

und/

1e+

5

Figure: Fewer HIV+ cases found;more HIV- cases found.

5 10 15 20

010

0020

0030

0040

00

time

cum

ulat

ive

tota

l cas

es fo

und/

1e+

5

Figure: Fewer HIV+ cases found;fewer HIV- cases found.


Pete Dodd

IntroductionScope

Overview



Example behaviours


Cases found

Cases averted

ConclusionsLessons

Next?

19

Results


Pete Dodd

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Cases found

Cases averted

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20

General observations

Implied by model formulation:

• cases found/averted ∝ I0

• cases averted ∝ PR0 × I0• HIV+ cases averted = (I+0 /I

−0 )× HIV- cases averted

• given survival as active, cases averted independent ofCDR0


Pete Dodd

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• cases found/averted ∝ I0• cases averted ∝ PR0 × I0

• HIV+ cases averted = (I+0 /I−0 )× HIV- cases averted



Pete Dodd

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20



• cases found/averted ∝ I0• cases averted ∝ PR0 × I0• HIV+ cases averted = (I+0 /I




Pete Dodd

IntroductionScope

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Example behaviours


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Cases averted

ConclusionsLessons

Next?

21

Cases averted and found @ different ε, T and CDR0.

0 10 20 30 40

050

015

0025

00

efficiency x no. rounds

HIV

+ c

ases

ave

rted

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0 10 20 30 40

−60

0−

200

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600


HIV

− e

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The outcomes and characteristics of disease are also verydifferent.


Pete Dodd

IntroductionScope

Overview



Example behaviours


Cases found

Cases averted

ConclusionsLessons

Next?

22


Further generalizations

• cases found/averted ∝ I0• cases averted ∝ PR0 × I0• HIV+ cases averted = (I+0 /I



• cases found/averted ∝ (ε× N), for realistic parameters• whether you find more or fewer cases after 10 years

depends largely on the CDR0


Pete Dodd

IntroductionScope

Overview



Example behaviours


Cases found

Cases averted

ConclusionsLessons

Next?

23

Cases found


Pete Dodd

IntroductionScope

Overview



Example behaviours


Cases found

Cases averted

ConclusionsLessons

Next?

24

When are fewer cases found in total over 10 years?

initial HIV− CDR (%)

initi

al H

IV+

CD

R (

%)

0

0

0

10 20 30 40 50 60 70 80 90

1020

3040

5060

7080

90


Pete Dodd

IntroductionScope

Overview



Example behaviours


Cases found

Cases averted

ConclusionsLessons

Next?

25

What is this sensitive to?

0 20 40 60 80 100

020

4060

8010

0

initial HIV− CDR (%)

initi

al H

IV+

CD

R (

%)

●

●

●

PR0=70%PR0=50%PR0=40%


Pete Dodd

IntroductionScope

Overview



Example behaviours


Cases found

Cases averted

ConclusionsLessons

Next?

26

Cases averted


Pete Dodd

IntroductionScope

Overview



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Proportion of cases averted by 10 yearly rounds

0 20 40 60 80

010

2030

4050

round efficiency (%)

prop

ortio

n of

cas

es a

vert

ed (

%)

●

●

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●

●

PR0=80%PR0=50%PR0=40%


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Cases averted and efficiency by HIV-status

HIV− cases averted

round efficiency HIV− (%)

roun

d ef

ficie

ncy

HIV

+ (

%)

50

100

150

200

250

300

350

400

450

0 20 40 60 80 100

020

4060

8010

0

HIV+ cases averted

round efficiency HIV− (%)

roun

d ef

ficie

ncy

HIV

+ (

%)

100

200

300

400

500

600

700 800

900

1000

1100

1200

1300

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0


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Conclusions


Pete Dodd

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Lessons

Interpretations

• Cases averted ∝ I0 × PR0

=⇒ if the burden is high enough, ACF cost-effective

• If routine case-detection good enough, may need to treatfewer cases (esp. HIV+) over period

=⇒ favourable to cost-effectiveness; hint of cost-saving

• Cases averted ∝ ε× N

=⇒ for fixed N, prefer strategy with cheaper efficiency(expensive/good seldom vs. cheap/bad often)

• Comparison of cases averted by HIV-specific roundefficiency

=⇒ cannot hope to avert many cases by concentratingon HIV+ cases alone.


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Lessons

Interpretations

• Cases averted ∝ I0 × PR0=⇒ if the burden is high enough, ACF cost-effective

• If routine case-detection good enough, may need to treatfewer cases (esp. HIV+) over period

=⇒ favourable to cost-effectiveness; hint of cost-saving






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Lessons

Interpretations


• If routine case-detection good enough, may need to treatfewer cases (esp. HIV+) over period=⇒ favourable to cost-effectiveness; hint of cost-saving






Pete Dodd

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Lessons

Interpretations



• Cases averted ∝ ε× N=⇒ for fixed N, prefer strategy with cheaper efficiency(expensive/good seldom vs. cheap/bad often)




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Lessons

Interpretations



• Cases averted ∝ ε× N=⇒ for fixed N, prefer strategy with cheaper efficiency(expensive/good seldom vs. cheap/bad often)

• Comparison of cases averted by HIV-specific roundefficiency=⇒ cannot hope to avert many cases by concentratingon HIV+ cases alone.


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Future avenues; extra information needed

Important gaps

• Above suggests PR0 a key determinant for usefulness ofACF - not independent from I0. (Nor CDR0 from I0, etc.)

=⇒ need better understanding of how importantquantities (CDR, I0,. . . ) correlate in reality.

• Above has been agnostic about costs and efficiencies ofreal screening methods.

=⇒ need for good information of cost and performance ofoptions, and CE analysis including HIV+/- outcomes

• Above has relatively simple assumptions timing of careseeking.

=⇒ need to understand how this fits in withinfectiousness, detectability, heterogeneity, Rx outcomes.


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Important gaps

• Above suggests PR0 a key determinant for usefulness ofACF - not independent from I0. (Nor CDR0 from I0, etc.)=⇒ need better understanding of how importantquantities (CDR, I0,. . . ) correlate in reality.

• Above has been agnostic about costs and efficiencies ofreal screening methods.

=⇒ need for good information of cost and performance ofoptions, and CE analysis including HIV+/- outcomes




Pete Dodd

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Important gaps


• Above has been agnostic about costs and efficiencies ofreal screening methods.=⇒ need for good information of cost and performance ofoptions, and CE analysis including HIV+/- outcomes




Pete Dodd

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Important gaps


• Above has been agnostic about costs and efficiencies ofreal screening methods.=⇒ need for good information of cost and performance ofoptions, and CE analysis including HIV+/- outcomes

• Above has relatively simple assumptions timing of careseeking.=⇒ need to understand how this fits in withinfectiousness, detectability, heterogeneity, Rx outcomes.


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(extra slides)


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Delay to disease following (re-)infection.

187Natural history of TB

0 10 20Age at infection (years)

(a)

d p(a

,0)

1·000

0·410

0·1300·0860·028

0 1 2 3 4 5Years since ‘conversion’

“Rel

ativ

e ris

k”

1·0

0·8

0·6

0·4

0·2

00 10 20 30 40 50 60 70 80 90

Age (years)

Prop

ortio

n sp

utum

-pos

itive

(d+(

a))

1951–53

Value used in model1960–62

1954–561963–66

1957–591967–69

(b)

(c)

Fig. 2. (a) Relationship between the risk of developing the first primary disease episode and the age at infection assumed inthe model. The relationship (i) between the risk of developing exogenous disease and the age at reinfection, and (ii) betweenthe risk of developing endogenous disease and the current age of an individual are assumed to follow this basic pattern. Notethat the rates of disease onset for 10–20 year olds can be expressed in terms of those for individuals aged 0–10 years, andthose aged over 20 years. (b) Observed and assumed relationship between the rate at which individuals experience their firstprimary episode�exogenous disease in each year following infection�reinfection relative to that during the first year afterinfection�reinfection. These were estimated from the distribution of the time interval between ‘tuberculin conversion’ anddisease onset of those who were tuberculin-negative at the start of the UK MRC BCG trial [34]. The ‘relative risk’ for a givenyear after ‘conversion’ is taken to be the ratio between: (i) the proportion of the total disease incidence among initiallytuberculin-negative individuals which occurred in that year following ‘conversion’, and (ii) the corresponding proportionwhich occurred during the first year after ‘conversion’. (c) Observed and assumed proportion of total respiratory diseaseincidence among cases of age a attributable to sputum-positive forms, d

+(a). All lines (excluding the heavy solid line) show

the relative contribution of sputum-positive disease to age-specific notifications of pulmonary tuberculosis in males inNorway (1951–69). Source: Dr K. Styblo (TSRU) and Dr K. Bjartveit (Norwegian National Health Screening Service).

Figure: Source: Vynnycky & Fine, 1997.

Relative risk of disease in 5 years following (re-)infection. As in[Vynnycky & Fine, 1997]: based on data from 1950s UK MRCBCG trial.


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With survival function, σ(t):

Book-keeping: no intervention

D(t) =

∫ t

0I(t − x)σ(x).dx +

∫ ∞0

I0σ(t + x).dx (5)

Under a PACF intervention with a round efficiency ε:

Book-keeping: intervention

σ(x)→ (1− ε)n(t,x) × σ(x) (6)

where n(t , x) counts the number of rounds experienced by timet for a case who developed disease a time x ago.


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A simple model: incidence from prevalence.

Aim: very simple, ‘open box’ model with a minimum ofparameters.

Assume HIV, LTBI prevalence, and ‘slow’-incidence ≈ const.

Incidence:

I0 = (1− PR0).I0 + PR0.I0 (7)

I(t) = (1− PR0).I0 + PR0.I0G(t)G0

(8)

where G(t) is a recently weighted average of the FOI, F (t):

G(t) =

∫P(fast delay = x)× F (t − x).dx (9)

F (t) ∝ D(t)N

(10)


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Other changes under intervention

0 5 10 15 20

020

4060

8010

0

time

prop

ortio

n re

cent

(%

)

Figure: Proportion recent.

0 5 10 15 20

020

4060

8010

0

time

rout

ine

CD

R (

%)

Figure: Case detection rates.


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Frenquency

0 1 2 3 4 5

0.0

0.2

0.4

0.6

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1.0

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rela

tive

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ber

of H

IV−

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Frenquency

1 2 3 4 5

4.2

4.4

4.6

4.8

round efficiency of 80%

period

HIV

− a

vers

ions

/cos

t

1 2 3 4 5

12.5

13.5

14.5


period

HIV

+ a

vers

ions

/cos

t

1 2 3 4 5

3.70

3.75

3.80


period

HIV

− a

vers

ions

/cos

t

1 2 3 4 5

11.0

11.2

11.4


period

HIV

+ a

vers

ions

/cos

t


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Frenquency

1 2 3 4 5

1214

1618


period

HIV

− a

vers

ions

/cos

t

1 2 3 4 5

3540

4550

55


period

HIV

+ a

vers

ions

/cos

t

1 2 3 4 5

89

1011

12


period

HIV

− a

vers

ions

/cos

t

1 2 3 4 5

2426

2830

3234

36


period

HIV

+ a

vers

ions

/cos

t

Modelling periodic ACF Pete Dodd Factors inﬂuencing the … · B T o n y t i l i b a b ro p...

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