Respiratory Bacteria Vaccines:
Model Analyses for Vaccine and
Vaccine Trial Design
Jim Koopman MD MPH
Ximin Lin MD MPH
Tom Riggs MD MPH
Dept. of Epidemiology &
Center for Study of Complex Systems
University of Michigan
Questions Addressed
• What role does immunity affecting
pathogenicity vs. transmission play in the
sharp drop with age in NTHi otitis media?
• What vaccine effects should be sought and
measured in trials?
• How should vaccine trials be designed to
insure adequate power to detect important
effects?
General Issues Regarding NTHi
• Causes 20-40% of acute otitis media
• Vaccine market 1 billion $ per year in U.S.
• Infection, immunity, and disease data is
meager, non-specific, & highly variable
• Knowledge of natural history of infection
and immunity is deficient
• Unquestioned assumption that vaccine trials
will be individual based and assess disease
outcomes
Aspects of NTHi (& many other
bacterial) infections
• Partial immunity, rarely sterilizing
– IgA proteases show evolutionary importance of
immunity
• Many variants arise due to transformation
competency
– No permanent strains yet identified
• Immunity to colonization or infection,
disease, & transmission can be distinct
Using NTHi Models for Inference
•
Models with diverse natural Hx of infection and
immunity, age groupings, and contact patterns were
constructed
•
•
•
•
•
Deterministic compartmental (DC) models built first
Gradual acquisition of immunity with each colonization and continuous
loss over time
All models were fit to the full range of data
conformations deemed plausible using least squares
Projections of vaccine effects made for all fits of all
models (about 1000 total)
Individual event history stochastic models
corresponding to the DC models were used for
vaccine trial design
Natural history of NTHi
colonization
Susceptible
Ca
Cb
D
Cc
Susceptible
with*
Susceptibility: θ n (n=0, …, m)
Infectiousness: n (n=0, …, m)
Pathogenecity: n (n=0, …, m)
Colonization
& Disease
Waning of
Immunity
with*
Susceptibility: θ min(n+1, m)
Infectiousness: min(n+1, m)
Pathogenecity: min(n+1, m)
FA model
D
S

Ca




Cb
S’
Modeling partial immunity
Model agent variation and host response as
single process
Assumptions
• equal immunity from each colonization
• multiplicative effects of sequential infections
• immunity limit (m levels)
• immunity waning
Modeling partial immunity:
S1I1S2I2S3I3……Sm-1Im-1SmIm vs. SIR/SIRS/SIS
S1

Ca1 


S2

Ca2

Cb1
Sm

D2

Cc1


Cb2


D1
Cc2
m-1
m-1
m-1





C
am
 
Cbm
Dm m-1
m-1
m-1
  
Ccm
m-1
m-1
Aspects of Immunity Modeled
•
Susceptibility
•
Contagiousness
•
Pathogenicity
•
Duration
Population structure
•
Preschool children (0.5-5 years)
1. Day-care + Non-day-care
2. 9 age groups with 6-month interval
•
School children (5-15 years)
•
Adults
Population structure
Deaths
Births
Births
N1
6-12 Mos
daycare
N10
6-12 Mos
no daycare
Aging
Aging
Deaths
Deaths
...
N9
54-60 Mos
daycare
Aging
N19
5-15
Years
...
N18
54-60 Mos
no daycare
Aging
Deaths
Deaths
Aging
Deaths
N20
>=15
Years
Contact structure
General Mixing
G
Daycare
N1-N9
G
Non-Daycare
N10-N18
D
G
G
Adult
N20
School
N19
S
Daycare
Mixing
School
Mixing
Population parameters
Death rate of individuals less than 1 year
0.00181
Death rate of individuals aged 1-2 years
0.00036
Death rate of individuals aged 3-4 years
0.00036
Death rate of individuals aged 5-15 years
0.00021
Death rate of individuals aged 15 years and over
0.01086
Annual birth rate into 7-12 month age group
0.00938
Rate at which children enter daycare
0.174
Rate at which children leave daycare
0.0358
Day-care attendance at 6 months
0.0785
* The units of all rates are year-1.
Limited & Highly Variable
Epidemiologic data
• NTHi prevalence by age & daycare attendance
(diverse methods)
• AOM incidence < age 5 by daycare (combine incidence
studies & fraction with NTHi studies)
• Antibody levels by age (diverse methods)
• Colonization duration (quite limited)
• Daycare risk ratios for AOM
Colonization prevalence values fitted
Low
Values
High
Values
Colonization prevalence ages 0-5 when in
daycare
Colonization prevalence ages 0-5 when not
in daycare
Colonization prevalence ages 6-15
23%
51%
9.5%
21%
7%
15%
Colonization prevalence in adults
4%
9%
Annual NTHi AOM incidence age* <1
0.08
0.22
Annual NTHi AOM incidence age 1-2
0.13
0.33
Annual NTHi AOM incidence age 2-3
0.08
0.22
Annual NTHi AOM incidence age 3-4
0.06
0.18
Annual NTHi AOM incidence age 4-5
0.05
0.17
AOM Incidence values fitted
Other Data
• Antibody levels peak during elementary
school
• Daycare Risk Ratios from 2 to 3
• Colonization mean of 2 months but many
transient episodes and some long (limited
data)
• Waning “seems” to be relatively fast
Presumptions Before Our
Work
• Very different from Hi Type B
• Colonization is so frequent, even at older
ages, that immunity to transmission cannot
be important
• Trials should assess effects on AOM, not
colonization
General assumptions of our model
• Every colonized individual is infectious
• Acute otitis media (AOM) is the only relevant
disease (Unlike Hi Type B or Strep pneumo)
• Maternal immunity (Children aged 0-6
months totally immune from colonization)
Fitting model to
epidemiologic data
• Berkeley Madonna: “boundary value
ODE…” & optimize functions
• Empirical identifiability checking
• Extensive robustness assessment for both
data conformation and model conformation
rather than estimating variance of estimates
Fitting Results
• Most efficient level # is 4
• Needed immunity profile includes
– Susceptibility
– Contagiousness
– Pathogenicity
• Contagiousness and Duration Effects are
highly co-linear when fitting equilibrium
Parameter values that fit NTHi prevalence & AOM
incidence for models without all immunity effects.
Immune Effects In The Model
(Path effects in all models)
Susc
S&
Infect
S &
Durat
D &I
Goodness of Fit
(Root Mean Square Error)
0.01
0.02
0.03
0.37
Duration of immunity (years) 
84.7
9.8
4.0
5.1
Relative susceptibility
after each colonization 
0.55
0.519
0.535
1
Relative contagiousness
when re-infected 
1
0.76
1
0.301
Relative duration of colonization
when re-infected 
1
1
0.839
0.599
Colonization prevalence and
AOM incidence data fit*
H col
H AOM
H col
L AOM
L col
H AOM
L col
L AOM
Goodness of fit (root mean square error)
0.07
0.05
0.05
0.02
Duration of each level of immunity (years),
3.7
4.7
3.4
9.8
Duration / stage colonization | lowest immunity
0.104
0.107
0.0613
0.0549
P(AOM | colonization at the lowest immunity)
0.343
0.127
0.374
0.136
% decrease in AOM probability per immunity
level (pathogenicity effect),
0.334
0.301
0.294
0.279
% decrease in susceptibility per immunity level,
0.597
0.594
0.732
0.481
% decrease in contagiousness / immunity level,
0.582
0.237
0.116
0.24
Effective contact rate per year at general site,
173
80.1
50.3
94.4
Effective contact rate per year at daycare site,
655
218
359
113
Effective contact rate per year at school site,
301
68
217
61
Data
Conformation
Fitted
Colonization
Prevalence
High
High
Low
Low
AOM
Incidence
High
Low
High
Low
Sensitivity Analysis to 10% Change In
Pathogenicity or Transmission Immunity
AOM Incidence Decrease
Immunity
Type
Decreased
0-1
year
1-2
years
2-3
years
3-4
years
4-5
years
Pathogenicity
1.6%
3.9%
7.9%
10.9%
12.5%
Transmission
12.0%
9.5%
11.8%
17.8%
23.4%
Pathogenicity
1.6%
3.8%
7.6%
10.2%
13.2%
Transmission
23.4%
14.6%
15.3%
23.6%
32.8%
Pathogenicity
1.4%
2.9%
5.1%
6.8%
8.1%
Transmission
15.9%
19.2%
32.6%
48.7%
62.7%
Pathogenicity
1.8%
3.7%
6.7%
9.0%
10.4%
Transmission
59.7%
34.1%
33.5%
53.2%
70.3%
Age
0-1
Age
1-2
Age
2-3
Age
3-4
Age
4-5
Base analysis from previous Table
16.5
5.5
3.7
4.2
4.8
Only susceptibility effects on
transmission
15.6
6.0
3.9
4.3
4.7
Susceptibility and duration effects on
transmission
8.4
2.6
1.4
1.5
1.8
Susceptibility, contagiousness, & duration
effects on transmission
10.2
3.3
2.1
2.5
2.8
Eight levels of immunity
4.6
5.1
2.0
1.5
1.7
Alternate ratios of contact rates by age at
the general mixing site
39.5
11.0
5.9
6.7
7.6
Prevalence and incidence fall more
steeply with age
19.2
4.7
0.6
0.6
1.2
Prevalence and incidence fall less steeply
with age
9.5
3.3
2.0
2.0
2.0
Simpler pattern of compartments for the
natural history of infection and immunity
36.3
6.4
3.2
3.4
3.9
Further Sensitivity Analysis
Vaccination
Immunity acquisition & waning for P
vaccine (Vaccine effects don’t exceed
natural immunity effects)
Vaccination
Immunity acquiring & waning in
vaccinated population: SIP vaccine
Vaccination strategy
All children at age of 6 months vaccinated
% reduction in AOM incidence among all preschool
children as the result of vaccination at birth
P
% Reduction of AOM Incidence
90%
IP
SP
SIP
80%
70%
60%
50%
40%
30%
20%
10%
0%
4_LL
8_LL
4_HH
Models
8_HH
% reduction in AOM incidence among preschool
children due to vaccination at birth.
P_Daycare
% Reduction of AOM Incidence
100%
90%
SIP_Daycare
P_Non-daycare
80%
SIP_Non-daycare
70%
60%
50%
40%
30%
20%
10%
0%
6-12
12-18 18-24 24-30 30-36 36-42 42-48 48-54 54-60
Age (months)
Absolute reduction of AOM incidence by age and
daycare attendance among preschool children due to
vaccination at birth.
AOM Cases per 100 Person-years
30
P_Daycare
25
SIP_Daycare
P_Non-daycare
20
SIP_Non-daycare
15
10
5
0
6-12
12-18
18-24
24-30
30-36
36-42
Age (months)
42-48
48-54
54-60
AOM cases among daycare and non-daycare children
from a population of 1,000,000 before and after
vaccination at birth with SIP vaccines.
700
Before vaccination_daycare
No. of AOM Cases
600
After vaccination_daycare
Before vaccination_non-daycare
500
After vaccination_non-daycare
400
300
200
100
0
6-12
12-18 18-24 24-30 30-36 36-42 42-48 48-54 54-60
Age (months)
Summary of Deterministic
Model Findings
• Wide range of feasible models fit to a wide
range of feasible data
• Over this entire huge range, the intuition
that immune effects on pathogenicity are the
major determinants of AOM incidence
proves to be wrong
• Trials must assess transmission
Model Refinements Desirable
• Model agent strains with different degrees
of cross reacting immunity
• Incorporate evolution of agent into vaccine
effect assessment
• Make maternal immunity and acquisition
time for vaccine immunity more realistic
Additional Practical Need for
Indirect Effects
• Very young age of highest risk means little
time to get all the booster effects needed
Using NTHi Models for Inference
About Vaccine Trial Design
•
•
•
•
Convert deterministic compartmental model
to individual event history model
Add distinct daycare units and families
Construct vaccine trials assessing
colonization in the IEH models with varying
randomization schemes, vaccine effects
exceeding natural immunity, sample
collection periods, serology & typing results
Hundreds of thousands of vaccine trial
simulations performed
Conclusions from Vaccine Trial
Simulations
• Most efficient randomization unit is daycare
– Individual randomized trials run too much risk of
missing important vaccine effects
• Standard power calculation methods for Group
Randomized Trials are far off because they are
based on individual effect
• Role of inside vs. outside transmission in daycare
significantly affects power
• Molecular assessment of transmission worthwhile
Standard variance calculation in
Group Randomized Trials (GRTs)
P
(
1

P
)
• variance:
(1  ( N  1) ICC )
N
• ICC: intraclass correlation
• Assumes objective is measurement of
individual effects
ICC & Vaccine effect
Change of ICC with Prevalence, Daycare Size, and Vaccine Effect
1
0%
5%
10%
15%
20%
DC Size: 25;
Before Vax
DC Size: 25; S: 90%
DC Size: 25; S: 0%
0.1
ICC
I0%
I50%
I60%
I70%
I80%
0.01
I90%
I70% I65%
I85%
I90%
I95%
I60%
…...
DC Size: 200; S: 0%
0.001
25%
DC Size: 200; S: 90%
NTHi prevalence
DC Size: 200;
Before Vax
Change in Variance with Daycare Size &
Sample
Size
Change of
Variance with
Daycare Size
0.1
0%
5%
10%
15%
20%
Variance
0.01
0.001
DC Size: 25; Subsample Size: 25
DC Size: 200; Subsample Size: 25
DC Size: 200; Subsample Size: 200
0.0001
NTHi prevalence
25%
Preliminary results (1): variance &
immunity
Change of Variance with Prevalence, Daycare Size, and Vaccine Effect
0.100
0%
5%
10%
15%
20%
25%
DC Size: 25;
Before Vax
DC Size: 25; S: 0%
Variance
0.010
0.001
I65% I60%
DC Size: 25; S: 90%
I70%
I0%
I75%
I80%
I50%
I85%
I60%
I90%
I70%
I95%
I80%
I65%
I90%
I75% I70%
I90%
I80%
I90%
I85%
I90%
I90%
I90%
I95%
I90%
DC Size: 200; S: 0%
I90%
DC Size: 200; S: 90%
0.000
NTHi prevalence
I60%
DC Size: 200;
Before Vax
Simple Model For Insight
S
I
S*
Equilibrium distribution of states solved theoretically
for daycare with 12 children
Vaccine effect decreases susceptibility by 50%
Unvacc mostly within trans 30%Prev Unvacc mostly outside trans
SIS* cum prob distn (UNVAC)
1
1
0.8
0.8
0.6
S
0.4
I
S*
0.2
Prob. distn
Prob. distn
SIS* cum prob distn (UNVAC)
0.6
S
0.4
I
S*
0.2
0
0
1 2 3 4 5 6 7 8 9 10 11 12 13
1
2
3
4
5
# infected
0.8
0.8
0.6
0.4
S_vac
0.2
I_vac
S*_vac
0
5
6
7
8
# infected
9
10 11 12 13
Prob. distn
Prob. distn
1
4
9 10 11 12 13
SIS* cum prob distn (VAC)
1
3
8
Vacc mostly outside trans
SIS* cum prob distn (VAC)
2
7
# infected
Vacc mostly within trans
1
6
0.6
0.4
S_vac
0.2
I_vac
S*_vac
0
1
2
3
4
5
6
7
8
# infected
9
10 11 12 13
Unvacc mostly within trans 50%Prev Unvacc mostly outside trans
SIS* cum prob distn (UNVAC)
1
1
0.8
0.8
0.6
S
0.4
I
Prob. distn
Prob. distn
SIS* cum prob distn (UNVAC)
S*
0.2
0.6
S
0.4
I
S*
0.2
0
0
1
2
3
4
5
6
7
8
1
9 10 11 12 13
2
3
4
5
7
8
9 10 11 12 13
# infected
# infected
Vacc mostly within trans
Vacc mostly outside trans
SIS* cum prob distn (VAC)
SIS* cum prob distn (VAC)
1
1
0.8
0.8
0.6
0.4
S_vac
0.2
I_vac
Prob. distn
Prob. distn
6
0.6
0.4
S_vac
0.2
I_vac
S*_vac
0
S*_vac
0
1
2
3
4
5
6
7
8
# infected
9
10 11 12 13
1
2
3
4
5
6
7
8
# infected
9
10 11 12 13
Significance of S & S*
Contribution to Power Calculation
• Serological ability to assess cumulative
infection level would contribute
considerably to power
Empirical power calculation
Empirical power & the number of the
pairs of daycare centers
Why standard power calculations
for GRTs are way off
• ICC is determined by transmission dynamics
• Effect is determined by transmission dynamics
• Power is not just determined a single outcome
state but by correlated infection and immunity
states
Thank You