Effects of heterogeneity in
hosts and pathogens on
effectiveness of vaccination
Mirjam Kretzschmar
RIVM, Department of Infectious Diseases
Epidemiology
The Netherlands
Populations are
heterogeneous ...
Why do we have to think about
heterogeneity?
Measles outbreak (almost 3000 cases) despite coverage of 96%
Host heterogeneity
• Disease independent (can be measured also for
non-infected individuals):
– Age, sex, other demographic variables
– Behaviour (e.g. number of contacts, compliance with
vaccination)
• Disease dependent (only for infected individuals):
– Transmission route
– Disease stage; primary versus secondary infection
– Clininal symptoms or asymptomatic
Pathogen heterogeneity
• Heterogeneity between strains:
– Virulence (defined as host mortality or severity of
disease)
– Vulnarability to host immune response
– Competition via cross-immunity
• Within host heterogeneity:
– Immunogenic variability (HIV)
– Different location within host leads to different effects
(invasive infection versus carrier)
Effects of heterogeneity on vaccination
depend on vaccination strategy
• Universal vaccination
– Rationale: create herd immunity such that unvaccinated
individuals are also protected
• pockets of not vaccinated persons (MMR vaccination in the
Netherlands)
• core groups of individuals with very many contacts (STD,
Hepatitis B)
• non-homogeneous contact patterns, i.e. household contacts,
spatial patterns,
• repeated contacts with same individuals (long term
partnerships, networks)
• Targetted vaccination of risk groups
– Rationale: protect those individuals who are at
greatest risk for being infected
• Takes heterogeneity in risk into account, but what
are effects of mixing between risk groups?
• Critical coverage per risk group?
• Effects of vaccination on non-risk groups?
• Ring vaccination
– Rationale: vaccinate direct contacts of infected
individuals to interrupt transmission chain
• Heterogeneity in contact patterns is taken into
account, only persons at risk are included in
vaccination.
• But: how to estimate fraction of contacts that have
been found and vaccinated?
• Modelling: Contact tracing requires a network
modelling approach
Contact patterns: who has
contact with whom?
Contact and transmission route
• Influenza (airborne infection):
– talking with each other at close distance
– coughing at each other
• Gonorrhoea (sexually transmitted dis.):
– sexual intercourse
• Hepatitis C (bloodborne infection):
– sharing contaminated needles
– blood transfusion
Knowledge about contact patterns leads
to insight into transmission routes
• Contact network AIDS cases (Auerbach et
al. 1984)
– Probability that cluster of cases is connected by
contact on the basis of random events
– timing of contacts and onset of disease
• Hypothesis: AIDS is transmitted by
homosexual contact
Cluster of AIDS patients
number: order of diagnosis
0 index case
A.S. Klovdahl. Social networks and the spread of infectious diseases:
The AIDS example. Soc. Sci. Med. 21 (1985): 1203-1216.
Contacts are non-random
• Population heterogeneity
– Age structure, social economic structure,
education
• Social grouping
– Families, working environment, recreation
• Geographical distribution
– Cities, rural areas, mobility between regions
 People are not the same and they choose contacts with certain preferences
 these choices influence the way infectious diseases spread
Influence of contact patterns
on epidemiological outcome
•
•
•
•
Age distribution of cases in STDs for men and women
Biannual measles epidemics in prevaccination era
High prevalence of STDs in high activity core groups
Widespread heterosexual transmission of HIV in subSaharan Africa
• Hepatitis A outbreaks in day care centers
• Increasing incidence of HIV in monogamous married
women in Thailand
• Increasing incidence of malaria in Western Europe
Modelling heterogeneity
• Heterogeneity in number of contacts
– Core groups
– Stratification by activity
– Mixing?
• Local/global contacts
– Households
– Metapopulation models
• Partnership duration: pair formation models, pair
approximation models
• Networks
Vaccination in a population
stratified by households
local contacts
global contacts
Equalizing strategy: Choose individuals for
vaccination sequentially from those households with largest number of
susceptibles.
Minimizes the number of vaccinations needed to reduce R to below 1.
Ball, Mollison & Scalia-Tomba.Ann. Appl. Prob. 7 (1997) 46.
The basic reproduction
number R0
The expected number of secondary cases caused
by one index case during his entire infectious
period in a completely susceptible population.
homogeneous population: R0=cD
heterogeneous population: number of secondary
cases has to be averaged in the right way.
Heterogeneous population
Diekmann, Heesterbeek, Metz. J. Math. Biol. 1990; 28:365-382
Diekmann, Heesterbeek. Mathematical Epidemiology of Infectious
Diseases, Wiley, 2000.
Next generation operator
Number of cases in the (n+1)-th generation of infections
given the distribution of infectious individuals (with
respect to population structure) in the n-th generation.
Basic reproduction number
Dominant eigenvalue of the next-generation operator
Explicit calculation of R0 for separable mixing
Contact funtion c(a,b)=f(a)g(b)
Host heterogeneity: example
Hepatitis B vaccination
• Background:
– Introduction of universal infant vaccination in the
Netherlands?
– Low prevalence, high costs of vaccination
– How many cases of chronic hepatitis B infection can
potentially be prevented?
• Project including case-control study, modelling
and cost-effectiveness analysis
Hepatitis B: many types of
heterogeneity
• Transmission routes:
– Sexual transmission
– Vertical to babies at birth
– Horizontal close contact (household)
• Age:
– Age dependent immune response (clinical symptoms and
development of chronic carrier state)
– Age dependent sexual activity level
• Behaviour:
– High versus low activity within age groups
• Disease states:
– Latent (1-2 months), acute (3-4 months), and chronic stages
Model structure
Williams et al. (1996), Epidemiol & Infect. 116: 71-89
Kretzschmar et al. (2002) Epidemiol & Infect. 128: 229-244.
• Population stratified by age and sexual activity (6
activity classes)
• Two transmission routes (vertical and sexual)
• Different stages of infection (acute, chronic
carrier)
susceptible
latent
acute
vaccinated
carrier
immune
Model
• System of partial differential equations (age
structure)
• Proportionate mixing
• Separate models for hetero/homosexual
populations
• Included immigration and age dependence
in probability to become carrier
• Explicit formula for R0
Calculation of R0
Individuals can be infected via two routes.
R0 is the dominant eigenvalue of next generation matrix
R

R

ss
0
vs
0
s
v
sexual transmission
vertical transmission
sv
0
vv
0
R
R




Calculation of R0ss
6 L
R0ss   
k 1 0
with
k ck (a) La
LK
 c (a   )(  P ( )   P ( , a))dda
k
1 Y
2 C
0
6 L
LK    k ck (a)da
k 1 0
L
k
ck(a)

i
maximum lifetime
fraction in activity class k =1,...6
age dependent contact rate in activity class k
time since infection
transmission probability per partnership
Age-dependence of becoming
carrier
PC(,a) has factor p(a), the probability of becoming carrier
when infected at age a
 0 for a=0

p(a)  
2
exp(  1a ) for a>0
Edmunds et al. 1993:
Point estimate of parameters 1 and 2 from data
from 29 different studies
Probability of becoming carrier after
infection
1
ml estimate
R0 minimal in 95% conf region
R0 maximal in 95% conf region
observations
probability
0.8
0.6
0.4
0.2
0
0
10
20
age (years)
30
Partner change rates from sexual
behaviour surveys (UK and NL)
new partners per year
1,2
heterosexual
homosexual
1
0,8
0,6
0,4
0,2
0
15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59
age
Estimates for R0
(heterosexual population)
constant
age-dependent
R0
1
0
1.11
0.79
estimates UK
0.69
0.53
estimates NL
Estimate R0
• Homosexual men R0>1:
– Hepatitis B virus can persist
– Immigration of infected persons has little
influence
• Heterosexual population R0<1:
– short transmission chains
– immigration of infected persons determines
prevalence
Compare with data
• Case control study:
– heterosexual cases (N=41): 60% of cases
infected by immigrant from high endemic
country
– homosexual cases (N=44): 16% infected by
immigrant from medium or high endemic
country
Conclusions for vaccination
• Vaccinating general population can reduce
incidence of new infections within the
country, but has little influence on
prevalence of carriers.
• Vaccination of risk groups is being
intensified
• Vaccination is offered to children of whom
at least one parent is an immigrant from
country with higher prevalence
Vertical transmission
• In highly endemic countries it is believed that
vertical transmission and horizontal transmission
to children are the most important transmission
routes.
• In low endemic countries the role of horizontal
transmission to children is not known.
• Can we use R0 to analyse importance of those
transmission routes?
• Assume sexual behaviour comparable to UK data
vv
Consider R0
L
R0vv   ( )(b1PY ( )  b2 PC ( ,0)) d
0
with
()
bi
fertility rate at age 
transmission probabilities per offspring
(b1=0.724, b2=0.115)
Fertility distributions
0.14
0.12
UK
NL
live births / woman / year
0.1
0.08
0.06
0.04
0.02
0
0-5
5-10
10-15
15-20
20-25
25-30
30-35
Age (years)
35-40
40-45
45-50
50-55
Horizontal transmission
• Horizontal transmission in households can be
approximately described by increasing bi
• The fertility function can vary in age distribution and total
number of offspring during lifetime.
• Example: mean offspring number 3, b2=0.5
 R0ss
 vs
R
 0
sv
0
vv
0
R
R
  0.75 0.05 
  
 4.17 0.89 

 
R0 = 1.29
 Neither of the transmission modes alone could sustain
endemic prevalence, together they can
Horizontal transmission as main
transmission mode?
0.9
transmission probability b 2
0.8
0.7
Rvv>1
0.6
0.5
R0>1,
0.4
Rvv<1
0.3
R0<1
0.2
0.1
0
0
1
2
3
4
mean offspring number
5
6
7
Conclusions
• Explicit expression for R0 in heterogeneous
populations can help to get insight into influence
of different types of heterogeneity on transmission
dynamics and their interaction
• Drawback: proportionate mixing assumption
• How does R0 depend on underlying model
assumptions?
• How well does R0 reflect heterogeneity?
• Hepatitis B: different vaccination strategies
depending on population heterogeneity?
Heterogeneity in the pathogen
population and vaccination
• When can serotype replacement occur?
• Indirect effects of serotype replacement:
partial immunisation by replacing strains?
• Optimal composition of vaccine (trade-off
between breadth and effectiveness)?
• Evolution to higher virulence?
• Vaccination against disease or against
infection?
Competition of 2 strains
Model McLean:
Assumptions:
– 2 strains, total cross-immunity
– Vaccinated individuals can become infected
with a small probability, vaccine efficacy
differs between strains
– after infection permanent immunity
– Strain 1 outcompetes 2 in absence of
vaccination
A.R. McLean. Proc R Soc Lond B (1995) 261: 389-393.
McLean model
vaccination
birth
S
transmission
V
2I2S
death
1I1S
I1
death
(1-r)2I2V
(1s)1I1V
recovery + death
I2
Effects of vaccination
• Vaccination reduces competitive pressure on weaker strain
2 -> outbreaks
• indirect effect: more herd immunity against strain 1
Dynamicsof 2 competing strains
500
400
300
200
100
0
20
40
60
80
100
Superinfection
Model Lipsitch:
Assumptions:
– no immunity, after recovery susceptible again
– Individual can be infected by 2 strains
simultaneously
– Cross-immunity
– vaccine 100% effectiv for target strain
M. Lipsitch. Emerging Infectious Diseases (1999) 5: 336-345
Model with superinfection
birth
death
transmission
recovery
S
vaccination
V
2(I2+I12)S
1(I1+I1v+I12)S
(1s)1(I1+I1v+I12)V
I2
I1
c22(I2+I12)I1
c11(I1+I1v+I12)I2
aI1I2
I12
I1v
I1v
Effects of vaccination
• Vaccination enables coexistence of strains
• serotype replacement can occur
• If vaccine is also effective for other than the target
strains, higher coverage is needed for eradication
Model Lipsitch
1
prevalence
0.8
0.6
0.4
0.2
15
20
25
HL
30
time years
35
40
45
50
Example pertussis
• Since middle of the 90‘s increase in
incidence of pertussis in NL
• Increase in incidence among vaccinated
children
• Large incidence of subclinical infections in
adults
• Hypothesis: vaccine not as effective against
presently circulating strains
Model with 2 strains
Birth
Birth
I1
S
J1
R
I2
red = transmission
green = recovery
blue = loss of immunity
W
V
J2
Assumptions
• Full cross immunity after natural infection
• vaccine protects fully against strain 1, partly
against strain 2
• vaccine induced immunity lasts shorter than
natural immunity
Equilibrium under vaccination
primary infection more transmissible, R01>R02
0.005
I1
0.004
I2
J1
J2
prevalence
0.003
total
0.002
0.001
0
0
0.2
0.4
0.6
-0.001
vaccination coverage
0.8
1
Dynamics with vaccination
0.003
I1
I2
J1
J2
0.0025
prevalence
0.002
0.0015
0.001
0.0005
0
0
100
200
300
time
400
500
600
Conclusions
•
•
•
•
Strains can coexist for certain range of
vaccination coverage
for high coverages strain 2 is dominant
total prevalence of infection decreases
with increasing coverage
elimination for p larger than critical
vaccination coverage
Equilibrium under vaccination
secondary infection more transmissible
0.005
0.004
I1
prevalence
0.003
I2
J1
0.002
J2
total
0.001
0
0
0.2
0.4
0.6
-0.001
vaccination coverage
0.8
1
Dynamics with vaccination
0.0045
I1
I2
J1
J2
0.004
0.0035
prevalence
0.003
0.0025
0.002
0.0015
0.001
0.0005
0
0
100
200
300
time
400
500
600
Conclusions
• For high coverages coexistence of both strains
• total prevalence of infection increases when 2
strains are present
• infection remains present even with 100%
vaccination coverage
Summary
•
•
•
•
•
Vaccination can lead to coexistence of strains
Contribution of secondary infections determines success
of vaccination
Even very high coverage might not suffice for
elimination
Changes of transmission rate of primary infections may
lead to sudden shifts in prevalence
Need more empirical data about secondary infections