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Mathematical modeling the dynamics of infection
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Mathematical modeling
The dynamics of infection
Niel Hens
www.simid.be
FTNLS
24 April 2015
Mathematical modeling the dynamics of infection
Overview
1
Prologue
Introduction
Discrete time models
The basic reproduction number
2
Who acquires infection from whom?
How it all started . . .
The social contact approach
Research based on social contact data
3
Epilogue
Further research
Discussion
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Mathematical modeling the dynamics of infection
Prologue
Introduction
Mathematical Modelling of Infectious Diseases
Purposes:
prediction: requires the inclusion of known complexities and
population-level heterogeneity
understanding: investigating the factors that drive dynamics
Building a model presents a trade-off:
accuracy: reproduce what is observed and predict future dynamics
transparency: ability to understand how model components influence
the dynamics and interact
flexibility: ease of adapting the model to new situations
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Mathematical modeling the dynamics of infection
Prologue
Introduction
Mathematical Modelling of Infectious Diseases
Limitations:
models present a simplification of reality
chance events of infectious disease transmission hinder perfect
prediction
A good model:
suited to its purpose: simple as possible, but no simpler
balance accuracy, transparency, flexibility
parametrisable from available data
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Mathematical modeling the dynamics of infection
Prologue
Introduction
Mathematical Modelling of Infectious Diseases
Daniel Bernoulli was the first to present a mathematical model for
smallpox in 1760
Since then many people have developed models to describe
infectious disease dynamics, see e.g. Bailey (1975); Anderson and
May (1991); Grenfell and Dobson (1995); Daley and Gani (1999);
Hethcote (2000)
Historical perspectives by Klaus Dietz.
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Mathematical modeling the dynamics of infection
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Prologue
Introduction
Contacts between individuals
Predicting the number of infections at time t + 1 based on the
circumstance at time t
The force of infection λ
the per capita rate at which a susceptible individual contracts
infection
it is assumed proportional to the number of infectious persons at
time t and depending on how the contact structure is assumed to
change with population size N it is given by:
λt = βIt
λt = βIt /Nt
Mathematical modeling the dynamics of infection
Prologue
Introduction
Contacts between individuals
The number of new infections at time t + 1 is given by λt St and
thus:
It+1 = βSt It
It+1 = βSt It /Nt
This is referred to as the mass action principle
density-dependent transmission: It+1 = βSt It :
- as the population size increases, so does the contact rate
- mostly applicable to plant and animal diseases (homogeneity)
frequency-dependent transmission: It+1 = βSt It /Nt :
- the contact rate is assumed constant regardless of a change in
population size
- mostly applicable to human and vectorborne diseases (heterogeneity)
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Mathematical modeling the dynamics of infection
Prologue
Introduction
Contacts between individuals
Heterogeneity
airborne infections: age - example: children at school have more
contacts with children of the same age
sexually transmitted infections: age and sexual behavior
temporal heterogeneity
seasonality,
week vs weekend,
holiday vs non-holiday, . . .
...
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Mathematical modeling the dynamics of infection
Prologue
Introduction
Modeling Frameworks
compartmental models
the population is subdivided into broad subgroups (compartments)
individuals are tracked collectively
roughly either deterministic or stochastic (probabilistic)
deterministic models describe what happens ‘on average’ in a
population
stochastic models allow the number of individuals who move between
compartments to vary through chance
transmission dynamic or static models
microsimulation or agent-based models
network models
metapopulation models
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Mathematical modeling the dynamics of infection
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Prologue
Discrete time models
Discrete time deterministic models
SIR compartmental model
Equations:
St+1
It+1
Rt+1
= St − λt St
= It + λt St − νIt
= Rt + νIt
with Nt+1 = St+1 + It+1 + Rt+1 = St + It + Rt = Nt .
λt = βIt and ν are risks
risks are related to rates as follows:
risk = 1 − e−rate
Mathematical modeling the dynamics of infection
Prologue
Discrete time models
Discrete time stochastic models
The number of newly infected cases arises from a stochastic process
with mean βIt St
The number of newly recovered individuals arises from a stochastic
process with mean νt It
0
50000
100000
150000
200000
250000
Our best option: the binomial distribution
1 9 18 29 40 51 62 73 84 95 107 121 135 149 163
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Mathematical modeling the dynamics of infection
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Prologue
Discrete time models
Discrete time models
When do you expect the number of new infections to decrease?
Focus on the second equation:
It+1 = It + βIt St − νIt
clearly It+1 = It if βSt = ν
The epidemic will
die out if St < ν/β
continue if St > ν/β
At the start of an epidemic in a susceptible population: S0 = N
The epidemic will
die out if N β/ν < 1
take of if N β/ν > 1
Mathematical modeling the dynamics of infection
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Prologue
The basic reproduction number
The basic reproduction number
Consider the total number of new infections in the population
between time t and t + 1:
βIt St
At the start of an epidemic, say t = 0: I0 = 1 and S0 = N and thus
the total number of new infections between t = 0 and t = 1 equals
βN
By the end of the infectious period of duration D = 1/ν time units,
the infectious person would have infected βN D individuals
N βD is called the basic reproduction number R0
Therefore R0 is the number of secondary cases caused by a single
infective introduced into a wholly susceptible population of size N
during the infective’s infectious period.
Mathematical modeling the dynamics of infection
Prologue
The basic reproduction number
The basic reproduction number
R0 constitutes a threshold:
if R0 > 1 then the epidemic can grow
if R0 ≤ 1 then the epidemic will die out
Using R0 , one defines the number of effective contacts by each
person per unit time by
ce = R0 /D.
Therefore β = ce /N is the “per capita number of effective contacts
made by a given individual per unit time”, or equivalently “the per
capita rate at which two specific individuals come into effective
contact per unit time”
For a given pathogen, it is difficult to define an effective contact
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Mathematical modeling the dynamics of infection
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Prologue
The basic reproduction number
The herd immunity threshold
The epidemic will die out if St < ν/beta:
Equivalently st R0 < 1, where st = St /N
Re = st R0 is called the effective reproduction number
Monitoring an epidemic is best done using Re
Vaccination: lowering st → control: Re < 1
Critical vaccination coverage
pc = 1 − 1/R0 .
Examples:
Measles: R0 = 20 → pc = 0.95,
Varicella R0 = 8 → pc = 0.875
But issues of primary and secondary vaccine failure complicate
matters
Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
How it all started . . .
Who Acquires Infection From Whom?
Many infections are
transmitted by contact or air
Influenza, Varicella, Measles,
Parvovirus B19, . . .
The transmission rate β depends on
person-to-person contact c
the probability of transmission given a contact q
but this is not the same for everyone
β ≡ β(a, a0 ) = q(a, a0 ) × c(a, a0 )
Two decades ago math. convenient WAIFW-structures
(Anderson and May, 1991)
Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
How it all started . . .
Who Acquires Infection From Whom?
Heterogeneity: Age
The age-heterogeneous mass action principle:
ND
λ(a) =
L
Z
L
A
0
0
β(a, a )λ(a ) exp −
Z
a0
A
!
λ(s)ds da0
with life expectancy L, population size N and mean infectious period D
and age A the age of maternal antibody loss
The Next Generation Operator
The operator that defines the next generation of infected individuals
g(a, a0 ) =
ND
β(a, a0 )
L
Mathematical modeling the dynamics of infection
Who acquires infection from whom?
How it all started . . .
Who Acquires Infection From Whom?
Basic reproduction number R0
The dominant eigenvalue of the ‘next generation operator’
Initial Spread
Simulate the initial epidemic phase by iterating the next generation
operator
Identical to the right eigenvector of that operator
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Mathematical modeling the dynamics of infection
Who acquires infection from whom?
How it all started . . .
The Traditional ‘WAIFW’ approach
Discretization into several age-categories
Anderson and May (1991): mixing patterns
impose mixing pattern on βij
constrain # distinct elements
based on prior knowledge of social mixing behaviour
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Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
How it all started . . .
The Traditional ‘WAIFW’ approach
Discretization into several age-categories
Anderson and May (1991): mixing patterns
age
age
age
age
class
class
class
class
1→
2→
3→
4→
age
class 1
↓
β1
β4
β4
β4
age
class 2
↓
β4
β2
β4
β4
age
class 3
↓
β4
β4
β3
β4
age
class 4
↓
β4
β4
β4
β4
Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
How it all started . . .
The Traditional ‘WAIFW’ approach
Discretization into several age-categories
Anderson and May (1991): mixing patterns
age
age
age
age
class
class
class
class
1→
2→
3→
4→
age
class 1
↓
β1
β1
β3
β4
age
class 2
↓
β1
β2
β3
β4
age
class 3
↓
β3
β3
β3
β4
age
class 4
↓
β4
β4
β4
β4
Mathematical modeling the dynamics of infection
Who acquires infection from whom?
How it all started . . .
The Traditional ‘WAIFW’ approach
Anderson and May (1991): mixing patterns
→ disadvantages:
low dimensional matrices
non-realistic discontinuities
choice age classes: ad hoc
Farrington and Whitaker (2005): continuous contact surface
→ both methods rely on strong parametric assumptions
Wallinga et al. (2006): use data on social contacts to inform
estimation of age-dependent transmission rates
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Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
The social contact approach
Social Contact hypothesis
Social contact hypothesis (Wallinga et al., 2006)
β(a, a0 )
@
@
R
@
c(a, a0 )
k
k
proportionality constant
contact rate
@
estimation
R
@
serological survey
social contact survey
q
·
Mathematical modeling the dynamics of infection
Who acquires infection from whom?
The social contact approach
Social Contact Approach
Alternative approach:
Using data on social contacts to estimate age-specific transmission
parameters for respiratory-spread infectious agents.
Objectives
Disentangle contact behaviour from transmission process
Get insights in predictiveness of social contact data
Get new insights in the transmission process
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Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
The social contact approach
Social Contact Survey
Wallinga et al. (2006): Utrecht
POLYMOD
pilot study: Beutels et al. (2006)
main study: Mossong et al. (2008)
PLoS MEDICINE
Social Contacts and Mixing Patterns Relevant to
the Spread of Infectious Diseases
Joël Mossong1,2*, Niel Hens3, Mark Jit4, Philippe Beutels5, Kari Auranen6, Rafael Mikolajczyk7, Marco Massari8,
Stefania Salmaso8, Gianpaolo Scalia Tomba9, Jacco Wallinga10, Janneke Heijne10, Malgorzata Sadkowska-Todys11,
Magdalena Rosinska11, W. John Edmunds4
1 Microbiology Unit, Laboratoire National de Santé, Luxembourg, Luxembourg, 2 Centre de Recherche Public Santé, Luxembourg, Luxembourg, 3 Center for Statistics,
Hasselt University, Diepenbeek, Belgium, 4 Modelling and Economics Unit, Health Protection Agency Centre for Infections, London, United Kingdom, 5 Unit Health Economic
and Modeling Infectious Diseases, Center for the Evaluation of Vaccination, Vaccine & Infectious Disease Institute, University of Antwerp, Antwerp, Belgium, 6 Department of
Vaccines, National Public Health Institute KTL, Helsinki, Finland, 7 School of Public Health, University of Bielefeld, Bielefeld, Germany, 8 Istituto Superiore di Sanità, Rome,
Italy, 9 Department of Mathematics, University of Rome Tor Vergata, Rome, Italy, 10 Centre for Infectious Disease Control Netherlands, National Institute for Public Health
and the Environment, Bilthoven, The Netherlands, 11 National Institute of Hygiene, Warsaw, Poland
Funding: This study formed part of
POLYMOD, a European Commission
project funded within the Sixth
Framework Programme, Contract
number: SSP22-CT-2004–502084.
The funders had no role in study
design, data collection and analysis,
decision to publish, or preparation
of the manuscript.
ABSTRACT
(WoK: 400 citations)
Competing Interests: The authors
have declared that no competing
interests exist.
Background
Mathematical modelling of infectious diseases transmitted by the respiratory or close-contact
route (e.g., pandemic influenza) is increasingly being used to determine the impact of possible
interventions. Although mixing patterns are known to be crucial determinants for model
outcome, researchers often rely on a priori contact assumptions with little or no empirical basis.
We conducted a population-based prospective survey of mixing patterns in eight European
countries using a common paper-diary methodology.
Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
The social contact approach
Social Contact Survey
Belgian Contact Survey
Part of POLYMOD project
Period March - May 2006
750 participants, selected through
random digit dialing
Diary-based questionnaire
Two main types of contact:
non-close and close contacts
Hens et al. (2009a,b)
Total of 12775 contacts (≈ 16
contacts per person per day)
Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
The social contact approach
0
e
ag
t
ac
nt
pa
rti
co
ci
pa
e
ag
nt
20
te
ct ra
age contact
ta
con
40
60
Mixing Patterns
0
20
40
age participant
Assortative mixing with clear (grand)parent-(grand)child
components
Divergence for work contacts
60
Mathematical modeling the dynamics of infection
Who acquires infection from whom?
The social contact approach
EU mixing patterns
common structure
note the converging off-diagonals: parents get older
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Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
Research based on social contact data
Serology and social contacts
Modeling Infectious Disease Parameters Based on
Serological and Social Contact Data
A Modern Statistical Perspective
Mathematical epidemiology of infectious diseases usually involves describing the flow
of individuals between mutually exclusive infection states. One of the key parameters
describing the transition from the susceptible to the infected class is the hazard of
infection, often referred to as the force of infection. The force of infection reflects the
degree of contact with potential for transmission between infected and susceptible
individuals. The mathematical relation between the force of infection and eff ective
contact patterns is generally assumed to be subjected to the mass action principle, which
yields the necessary information to estimate the basic reproduction number, another
key parameter in infectious disease epidemiology.
It is within this context that the Center for Statistics (CenStat, I-Biostat, Hasselt University) and the Centre for the Evaluation of Vaccination and the Centre for Health
Economic Research and Modelling Infectious Diseases (CEV, CHERMID, Vaccine and
Infectious Disease Institute, University of Antwerp) have collaborated over the past 15
years. This book demonstrates the past and current research activities of these institutes
and can be considered to be a milestone in this collaboration.
This book is focused on the application of modern statistical methods and models to
estimate infectious disease parameters. We want to provide the readers with software
guidance, such as R packages, and with data, as far as they can be made publicly available.
Varicella
Ogunjimi et al. (2009)
Statistics / Life Sciences,
Medicine, Health Sciences
ISBN 978-1-4614-4071-0
9 781461 440710
SBH
1
Modeling Infectious Disease Parameters
Based on Serological and Social Contact Data
Hens et al. (2012) −→
Statistics for Biology and Health
Niel Hens · Ziv Shkedy · Marc Aerts · Christel Faes · Pierre Van Damme · Philippe Beutels
Hens · Shkedy · Aerts
Faes · Van Damme · Beutels
Using social contact surveys in
statistical and mathematical
models
Statistics for Biology and Health
Niel Hens · Ziv Shkedy
Marc Aerts · Christel Faes
Pierre Van Damme · Philippe Beutels
Modeling
Infectious Disease
Parameters Based
on Serological and
Social Contact Data
A Modern Statistical Perspective
Goeyvaerts et al. (2010)
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4.21
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Mathematical modeling the dynamics of infection
Who acquires infection from whom?
Research based on social contact data
School Closure
most effective social distancing measure
previous analyses were based on assumptions or sentinel data
(see e.g. Cauchemez et al., 2008)
use contact data to quantify the reduction in R0
POLYMOD data
use the holiday period as a proxy for school closure: 17% reduction
use the weekend as a proxy for social distancing: 21% reduction
Hens et al. (2009a)
school closure: huge economic impact
Keogh-Brown et al. (2010a,b)
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Mathematical modeling the dynamics of infection
Who acquires infection from whom?
Research based on social contact data
School Closure
global school closure and its impact on the health care system
Cauchemez et al. (2009)
Could reactive school closure alleviate the burden upon the NHS
critical care capacity during the H1N1 influenza pandemic?
mathematical influenza model
UK contact data (holiday
pattern as a proxy)
timing is crucial!
in the most realistic situation
12% over maximum capacity
House et al. (2011)
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Mathematical modeling the dynamics of infection
Who acquires infection from whom?
Research based on social contact data
Contact patterns during illness
Van Kerckhove et al. (2013): study in the UK
people were asked to record their contacts
when ill (H1N1-diagnosis - lab-confirmed)
when healthy (few weeks later)
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Mathematical modeling the dynamics of infection
Who acquires infection from whom?
Research based on social contact data
Contact patterns during illness
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Mathematical modeling the dynamics of infection
Who acquires infection from whom?
Research based on social contact data
Contact patterns during illness
assuming 1/3-2/3 asymptomatic individuals (Carrat et al., 2008)
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Inc
Inc
Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
20
20
Research based on social contact data
Contact patterns during illness
0
0−3
4−10 11−21 22−45 46−64
Age, years
≥65
0−3
4−10 11−21 22−45 46−64
Age, years
≥65
Figure 3. Age distribution of cases in the early stages of the 2009 A/H1N1pdm influenza epidemic as predicted by asymptomatic contact pat-
terns (left; n = 140) and symptomatic contact patterns (right; n = 140) in the paired data set, assuming full susceptibility in the population, England,
fitted
to age-specific relative ILI incidence during exponential phase
2009–2010. The indicated 95% confidence intervals (T-shaped bars) were obtained using a nonparametric bootstrap. Because of the skewness
of the underlying distributions, the mean values are not in the middle of the intervals.
2009 A/H1N1pandemic in the UK
results:
them continued to engage in normal social activities, there
would have been a large increase in transmission.
work and social distancing for persons with symptoms. It
appears that during the A/H1N1pdm epidemic, symptomatic
persons were, in general, sufficiently unwell that they moderated their social behavior; had a substantial fraction of
symptomatic individuals are 3 to 12R istimes
asmeasure
infectious
a common
of viral fitness (26). Using this
0
measure, we have demonstrated that there is a considerable
symptomatic individuals cause 66% of all infections
80
80
φ =0
φ = 0.5
φ =1
60
Incidence, %
Incidence, %
60
All Contacts
Skin-to-Skin Contacts
Long-Duration Contacts
40
20
40
20
0
0
0
20
40
Age, years
60
80
0
20
40
60
80
Age, years
Figure 4. Theory (left) and fit (right) of a model to the observed age distribution from general practitioners’ consultation data from England and
Wales for the early phase of the 2009 A/H1N1pdm influenza pandemic (black dots), England, 2009–2010. The left-hand panel shows different φ
values (assuming q = 1, for illustration) leading to different age distributions. The right-hand panel shows the fit obtained using the best-fitting
Mathematical modeling the dynamics of infection
Who acquires infection from whom?
Research based on social contact data
Preferential transmission
Account for
pre-clinical subclinical infectious period
subclinical/asymptomatic infections
Investigate the role of carrying over high viral loads
viral load ∼ infectiousness (intranasal dose: Keitel et al., 1990)
viral load ∼ symptoms (Carrat et al., 2008)
transfer?
supported indirectly by challenge studies & empirical evidence for
other infections
Simplistic approach: compartmental models
(a)symptomatic infections (Ejima et al., 2013)
preferential transmission
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asymptomatic
and moves into the ! compartment at rate (1 − !)!! . Finally, asymptomatic
Mathematical modeling the dynamics of infection
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acquires to
infection
whom?
casesWhomove
the from
! compartment
at rate ! ! and symptomatic cases become recovered at rate
Research based on social contact data
! ! . Figure 2 shows a schematic diagram of the preferential transmission model. The
corresponding system of ODEs is available in Appendix.
Preferential transmission
!!
!! #
!!
!!! #
!!!
!!#
!!
(1 − !)!!
S#
R#
(1 − !)!!
!!
!!! #
!! #
!!
!!#
!!
!!!
Figure 2. Schematic diagram of the preferential compartmental transmission model. Superscripts indicate clinical status of
the infectee: symptomatic (s) or asymptomatic (a). Subscripts indicate whether the infector was symptomatic (s) or not (a).
In this paper it is assumed that !! = !! = ! and !! = !! = !. Under these assumptions,
the preferential transmission model simplifies to the non-preferential transmission model if
! = 1 − !. In Section 2.2, a social contact network between different age groups is
introduced which allows using social contact data of ill and healthy people to describe the
Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
Research based on social contact data
Preferential transmission: results
Non-preferential transmission if φ = 1 − φ̃
φ̃ = 0.4209 (95% CI : 0.3258, 0.5222)
1 − φ = 0.1618 (95% CI : 0.1227, 0.2103)
Sensitivity analysis
12 scenarios (estimated - referenced values)
best scenario
(γ, θ, σ a , σ s ) = {(1.5, 0.5, 1, 5.6)days}−1
φ̃ = 0.5706 (95% CI : 0.4784, 0.6583)
1 − φ = 0.2664 (95% CI : 0.2187, 0.3203)
Outcome
rel. trans. of symptomatic individuals is 2.87 (95% CI: 2.20, 3.73)
evidence suggests preferential transmission is possible
Mathematical modeling the dynamics of infection
Who acquires infection from whom?
Research based on social contact data
Preferential transmission: control strategies
Isolation (at home)
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Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
Research based on social contact data
Susceptibility patterns for H1N1 in Vietnam
household-based survey in rural Vietnam
80
80
264 households
3.0
1.2
60
1.0
60
2.5
0.8
0.6
40
contact's age
40
1.5
0.4
20
20
1.0
0.2
0
0.5
0
contact's age
2.0
0
20
40
participant's age
60
80
0
20
40
participant's age
60
80
Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
Research based on social contact data
Susceptibility patterns for H1N1 in Vietnam
0.8
0.8
0.8
1.0
H1N1 incidence
1.0
Seroprevalence post−H1N1
1.0
Seroprevalence pre−H1N1
20
40
●
●
60
80
0.6
0.6
incidence
●
0.4
●●
0
●
●
20
40
60
age
eigenvector (susc)
eigenvector
●
●
80
0
0.20
60
80
ratio of incidence
and eigenvector
→ q(a)
8
q(a)
●
6
●
●
0
●
20
40
age
60
80
0
●
●
2
●
20
40
age
60
80
●
0
●
0.05
●
0
serial
seroprevalence
→ H1N1
incidence estimate
q(a)
●
●
4
incidence
40
0.10
●
●
●
●
0.15
0.20
0.15
●
0.10
incidence
●
20
●
age
10
age
●
●●
0.2
●
0.0
●
0.4
seroprevalence
●
●
0
●
0.0
●●
0.2
0.6
0.4
0.0
0.2
seroprevalence
●
●
20
●
●
40
age
●
60
●
80
Mathematical modeling the dynamics of infection
Who acquires infection from whom?
Research based on social contact data
Comparative analysis of the spread of H1N1 in Europe
relative incidence
example UK:
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Mathematical modeling the dynamics of infection
Who acquires infection from whom?
Research based on social contact data
Comparative analysis of the spread of H1N1 in Europe
based on the European contact data
estimating the relative susceptibility
meta-analysis over countries
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Mathematical modeling the dynamics of infection
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Who acquires infection from whom?
Research based on social contact data
Inferring networks from egocentric data
households of size 4
egocentric data
latent network
64 possible networks
63 parameters
4 × 23 = 32
observations
penalized likelihood
approach
Potter and Hens
(2013)
Mathematical modeling the dynamics of infection
Epilogue
Further research
Further research
Serology-contact papers
Goeyvaerts et al. (2011): immunological processes for B19
Santermans et al. (2014): inferring infectivity from serology
Hens et al. (2009c); Abrams et al. (2014): frailty models
Other contact-related work
Willem et al. (2012): Weather & contacts
Grijalva et al. (2014): Contact Patterns in Peru
Goeyvaerts et al. (in prep): Household contact survey in Flanders
Van Kerckhove et al. (in prep): Spatial networks of social contacts
Luca et al. (in prep): A spatio-temporal model of social contacts
Béraud et al. (in prep): Social contacts in France: temporal effects
...
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Discussion
Infectious disease dynamics
mass action principle
contact data prove to be useful
new epidemiological hypotheses
Bruges 2015: 7-11 September 2015:
2nd network course by Martina Morris and colleagues
6th SIMID course (www.simid.be; Hens et al. (2012))
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Discussion
Acknowledgements
All colleagues @CenStat and Vaxinfectio
All collaborating centres nationally and internationally
John Edmunds, Ken Eames, Stefan Flasche
(London School of Hygiene and Tropical Medicine, UK)
Peter Horby (Oxford University, UK)
Thomas House (University of Warwick, UK)
Gail Potter (University of Washington, US)
Chair in evidence-based vaccinology
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