Uses of Mathematical Models for Understanding Drug Resistance: Considerations (with a brief detour on animal abx use) Marc Lipsitch Harvard School of Public

Uses of Mathematical Models for Understanding Drug

Resistance: Considerations(with a brief detour on animal

abx use)

Marc Lipsitch

Harvard School of Public Health

Outline

• Why model?• How mathematical transmission models work• Choices in modeling drug resistance• Example 1: Agricultural Abx use.

– How should we weigh evidence from modeling?

• Example 2: Models for parameter estimation

Outline




Goals of modeling

• Understand observed phenomena– Why is it possible to clear a resistant strain from a hospital

quickly, but hard to clear a resistant strain from a community?

– How can treatment with one drug be a risk factor for infection with resistance to an unrelated drug?

• Predict the future– How much drug resistance will there be in 10 years?

• Predict the absolute or relative merits of interventions– Can cycling drugs reduce resistance?

• Evaluating interventions using data / parameter estimation– Hidden Markov models

Outline




Classical compartment model

birthSSusceptible

infection

IInfected/ infectious

vaccination

RRecovered (immune)

recovery

bSSusceptible

cSI


vS

RRecovered (immune)

kI

dI

dR

dS

Deterministic implementation

dRvSkIdt

dR

IdkcSIdt

dI

dScSIvSbdt

dS

)(

bSSusceptible

cSI


RRecovered (immune)

kI

dI

dR

dS vS

Stochastic implementation

Event Rate S I R Birth b +1 0 0 Death of S dS -1 0 0 Death of I dI 0 -1 0 Death of R dR 0 0 -1 Infection cSI -1 +1 0 Recovery kI 0 -1 +1 Vaccination vS -1 0 +1

TOTAL b+dS+dI+dR+csI+kI+vS

bSSusceptible

cSI


RRecovered (immune)

kI

dI

dR

dS vS

Realizations of a stochastic model: R0 ~ 2

5 10 15 20 25

20

40

60

80

100

5 10 15

20

40

60

80

100

0.5 1 1.5 2

20

40

60

80

100

2.5 5 7.5 10 12.5 15 17.5

20

40

60

80

100

R

S

I

R

S

I

RS

I

S

RI

Outline




What to include?

• Patients: also HCW vectors?• Drug resistant strain• Drug sensitive strain?• Treatment?

– Event– State– Risk factor

• Disease vs. colonization• Co-colonization: possible?• How do people enter the

model? Colonized or not?

• Considerations:– Biology: what are the

interactions?– Question: need HCW if you

want to assess an intervention on them!

– Duration of treatment

Copyright ©2000 by the National Academy of Sciences

Lipsitch, Marc et al. (2000) Proc. Natl. Acad. Sci. USA 97, 1938-1943


Bergstrom, Carl T. et al. (2004) Proc. Natl. Acad. Sci. USA 101, 13285-13290

Fig. 1. Schematic diagram of the model and the corresponding differential equations


Austin, Daren J. et al. (1999) Proc. Natl. Acad. Sci. USA 96, 6908-6913

No Caption Found

Outline




Antibiotics and Agriculture:Why we have skirted the issue

Quantities: a lot by any measure

“Hogging It”, Union of Concerned Scientists, 2001http://www.ucsusa.org/food_and_environment/antibiotics_and_food/hogging-it-estimates-of-antimicrobial-abuse-in-livestock.html

Abx makers’ estimates

UCS estimates

Poundage may be misleading

• (Higher doses may kill resistant strains)• Resistant bacteria are not a problem unless

they– Make people sick– Donate resistance to other strains

• Most bacteria in animals don’t get into people – from the human “bacteriome” perspective abx in animals may contribute far less– Contribution of ag use ~ (relative poundage in ag

use) x (relative contribution of ag bacteria to human bacteria)

Resistanthuman disease

Animal Use and Human Resistance

Bacteria in Animals

Resistancein Bacteria in Animals

AbxAbx


Resistant bacteriain food


opportunistic


Campy,Salmonella

Resistant human

commensals

Resistance xfer

Transmission

Resistant human

commensals

E coli,Enterococcus

Food-disease linkquantifiable

Food-disease linkidentifiable

Food-disease linkdifficult to document

in any given case

Almost a smoking

gun: Salmonella DT104 with reduced FQ susceptibility in Denmark

Molbak 1999 NEJM

Evidence

• S. typhimurium DT104 food poisoning outbreak with unusual resistance pattern (5 drugs + nalidixic acid) and other genetic markers

• Same strain in samples of pork from one slaughterhouse

• No other samples of this strain identified in extensive surveillance

• Strain not responsive to ciprofloxacin: nal-R = gyrA• Epidemiologic links for several patients to

contaminated pork, possible links for others• No quinolone use on the farms per vet’s written

statement

As AHI would be quick to point out

• No quinolone-resistant organisms cultured from after treatment

• Circumstantial (though compelling) evidence that quinolone resistance led to treatment failure

• No evidence that the precursor to FQ resistance (Nal-R) was acquired via ag use (later UK outbreak did show such an association – Walker 2000 Vet Record)

Yet

• Clear evidence of a transfer of resistant bacteria from food animals to humans, causing disease

• FQ were being used in agriculture in DK

• Difficult to think of another likely source of resistance in animal isolates

Risk factor studies: show that poultry is a source of resistant infection

Kassenborg HA 2004 Clin Infec Dis Suppl

Resistant infections are worse

Nelson JM 2004 et al. J Infec Dis

Resistant infections are worse

Varma JK 2005 et al. J Infec Dis

As I see it….

• Compelling but not smoking gun evidence for every step of the “food poisoning” (food-associated gastroenteritis) process

• Food poisoning is usually self-limiting• Not good but not a major public health

problem• Much of it addressable by hygiene


Animal Use and Human Resistance

Bacteria in Animals


AbxAbx


Resistant bacteriain food


opportunistic


Campy,Salmonella

Resistant human

commensals

Resistance xfer

Transmission

Resistant human

commensals

E coli,Enterococcus

Food-disease linkquantifiable

Food-disease linkidentifiable

Food-disease linkdifficult to document

in any given case

What about the aspects that are harder to document?

• Transfer of resistant E. coli, Enterococcus, etc to humans which later transmit and cause disease (often in others)– VRE example: Europe has much human

carriage of VRE from animals, but little disease!


Smith, David L. et al. (2002) Proc. Natl. Acad. Sci. USA 99, 6434-6439

A modeling approach

Comparison of medical vs. agricultural use


Conclusions


• Impact on human health is likely, though often hard to document; most likely and best-documented for organisms causing food-associated gastroenteritis (Campylobacter, Salmonella)

• Overall magnitude is hard to estimate but likely fairly low

• For commensals/opportunists:– Should try to prevent emergence of novel resistance

determinants via food– Ag use of common human antibiotics for which resistance is

already a clinical problem likely has minimal impact on human health

Outline




Interrupted Time Series: Classic Study Design in Hospital Epidemiology

Ignaz Philipp Semmelweis, 1818-65

1846 1847 1848

Mat

erna

l mor

tali

ty (

%)

5

10

15

Hand disinfection started

[adapted from Rotter, Hyg Med, 1997]

Onesko KM, Infection Control 1987

Nosocomial MRSA (infections?). ICU & 4 South ward combined.

00.5

11.5

22.5

33.5

Aug Oct Dec Feb Apr Jun Aug Oct Dec Feb Apr Jun

% o

f pat

ien

ts

1983 1984 1985

Non-medicated soap Low-iodine soap

Interrupted time series studies

Usual statistics

Pre-intervention Post-intervention

Uninfected 20 5

Infected 180 195

Communicable Diseases Have Outbreaks

P. de Man et al., Lancet 2000

R2 = 0.3292

0

5

10

15

20

0 5 10 15 20

AmCefA col -1

Am

Cef

A c

ol

R2 = 0.5375

0

1

2

3

4

5

6

7

0 5 10

AmCefB-1

Am

Cef

B

=52=20

=32=3

Outbreaks show:

• Serial autocorrelation: this week is like last week

• Overdispersion: Big departures from the mean, due to spread of bugs, or lack of spread

• CONSEQUENCE: Usual statistics, which assume modest deviations from the mean and independent observations, don’t work– 2x2 tables– Poisson regression– Etc.

How wrong will the results be?

How wrong will the results be? To address this question we need data where we know what the correct answer is.

How wrong will the results be? To address this question we need data where we know what the correct answer is.

We can create such data by simulating from a plausible stochastic transmission model:

Cooper, Medley, Scott. J Hosp Infect 1999

1000 such data-sets were generated from an epidemicmodel simulating transmission of an organism such as MRSAor VRE on a single ward.

Outcomes were arbitrarily divided into two phases.

This simulates 1000 interrupted time series studies where the null hypothesis (no intervention effect) is true.

We then applied the standard statistical test (which assume outcomes are independent) to see if there wasevidence to reject the null hypothesis.

We then applied the standard statistical test (which assume outcomes are independent) to see if there wasevidence to reject the null hypothesis.

Using models to assess the interventions appropriately

Susceptiblepatients

Colonized +- infection

Transmission in a hospital ward

transmissionadmissiondischarge

Susceptiblepatients

Colonized or infected

More typically colonization data is incomplete or only those patients who develop infections are observed :

Susceptiblepatients


Simple transmission model for a hospital ward

Assumptions: • Susceptible patients become colonized faster as the number of colonized patients on the ward increases (colonization pressure).•Colonization assumed to last a long time compared to length of stay • The Markov assumption (what happens next depends only on the current state of the system, not on history).

Parameters• Proportion of patients colonized on admission to ward•Transmission rate•Length of stay

Proposed by Pelupessy et al. (PNAS 2002) for analysing hospital infection data. Limitation is that it requires complete colonization data.

Markov assumption violated. Structured Hidden Markov Models are required

Susceptiblepatients


More typically colonization data is incomplete or only those patients who develop infections are observed :

Very simple model

Susceptiblepatients

Colonized +- infection

transmissionadmissiondischarge

Event Ratechange # colonized

transmission b C (N-C) 1

admit C; discharge NC v u (N-C) 1

admit NC; discharge C (1-v) u C -1

transmission rate constant

v admission prevalence

1/ length of stay

probability of case per week per colonized pt.

Pr[ 1] ( ) ( ) ( )

Pr[ 1] (1 ) ( )

t h t t t t

t h t t

C C C N C h v N C h o hN

C C v C h o h

transmissionmodel (hidden)

observationmodel

exp( )( )Pr[ | ]

!

yt t

t t

C CY y C

y

• Test of null hypothesis that there is no transmission:

– For the MRSA and VRE data we reject this hypothesis (p=.0008 and p=0.000002).

– For the R-GNR data there is no evidence to reject this hypothesis (p=0.25) and data are consistent with all cases being colonized with R-GNR on admission.

•To assess the performance of the different models in evaluating interventions we simulated 500 time-series under each 3 scenarios:

1. Intervention had no effect2. Intervention reduced transmission rate by 25%3. Intervention reduced transmission rate by 50%

•Data were simulated using the transmission model and parameters estimated from the VRE time series.

•We used the 3 best statistical methods to test the null hypothesis that there was no change in transmission, and compared these with the current approach (Poisson).

Evaluating interventions

Evaluating interventions: results

Structured hidden Markov (transmission model )

Poisson(assumes independence)

Neg.

binomial

autoreg.

Unstructuredhidden Markov

1. No change

0.06 0.20 0.14 0.08

2. 25% reduction in transmission rate

0.49 0.75 0.71 0.31

3. 50% reduction in transmission rate

0.95 1.00 1.00 0.48

Frequency of obtaining p< 0.05

Summary/Conclusions: Statistical analysis based on transmission models

• Hospital infection data: time series of low-numbered counts, serial autocorrelation, overdispersion

• Standard statistical models, almost universally used, don’t fit– Infectious disease modelled without contagion– Fail to capture autocorrelation, overdispersion– Don’t resemble the data– Inflated type 1 error rate

Summary/Conclusions II: Statistical analysis based on transmission models

• Mechanistic (transmission models) provide a basis for estimation with– Better fit– Inference about parameters of interest– Better hypothesis testing characteristics

Documents

Uses of Mathematical Models for Understanding Drug Resistance: Considerations (with a brief detour on animal abx use) Marc Lipsitch Harvard School of Public