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Transcript of R 0 and other reproduction numbers for households models MRC Centre for Outbreak analysis and...
R0 and other reproduction numbers for households models
MRC Centre for Outbreak analysis and modelling, Department of Infectious Disease Epidemiology
Imperial College London
Edinburgh, 14th September 2011
Lorenzo Pellis, Frank Ball, Pieter Trapman
... and epidemic models with other social structures
Outline
Introduction
R0 in simple models
R0 in other models
Households models Reproduction numbers
Definition of R0
Generalisations
Comparison between reproduction numbers Fundamental inequalities Insight
Conclusions
INTRODUCTION
R0 in simple models
R0 in other models
Outline
Introduction
R0 in simple models
R0 in other models
Households models Reproduction numbers
Definition of R0
Generalisations
Comparison between reproduction numbers Fundamental inequalities Insight
Conclusions
SIR full dynamics
Basic reproduction number R0
Naïve definition:
“ Average number of new cases generated by a typical case, throughout the entire infectious period, in a large and otherwise fully susceptible population ”
Requirements:
1) New real infections
2) Typical infector
3) Large population
4) Fully susceptible
Branching process approximation
Follow the epidemic in generations: number of infected cases in generation (pop. size ) For every fixed ,
where is the -th generation of a simple Galton-Watson branching process (BP)
Let be the random number of children of an individual in the BP, and let be the offspring distribution.
Define
We have “linearised” the early phase of the epidemic
( )NnX n N
( )lim Nn n
NX X
n
nX n
0,1,...,kk k P
0 kR E
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
R* = 0.5
R* = 1
R* = 1.5
R* = 2
Properties of R0
Threshold parameter: If , only small epidemics If , possible large epidemics
Probability of a large epidemic
Final size:
Critical vaccination coverage:
If , then
01 e R zz
0
11
Cp R
0 1R
0 1R
0 1X
( )0 1 lim
N
NnR X X
E E
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
R* = 0.5
R* = 1
R* = 1.5
R* = 2
Properties of R0
Threshold parameter: If , only small epidemics If , possible large epidemics
Probability of a large epidemic
Final size:
Critical vaccination coverage:
If , then
01 e R zz
0
11
Cp R
0 1R
0 1R
0 1X
( )0 1 lim
N
NnR X X
E E
Outline
Introduction
R0 in simple models
R0 in other models
Households models Reproduction numbers
Definition of R0
Generalisations
Comparison between reproduction numbers Fundamental inequalities Insight
Conclusions
Multitype epidemic model
Different types of individuals
Define the next generation matrix (NGM):
where is the average number of type- cases generated by a type- case, throughout the entire infectious period, in a fully susceptible population
Properties of the NGM: Non-negative elements We assume positive regularity
22
11 12 1
21
1 n
n
n n
k k k
k kK
k k
ijk ij
Perron-Frobenius theory
Single dominant eigenvalue , which is positive and real
“Dominant” eigenvector has non-negative components
For (almost) every starting condition, after a few generations, the proportions of cases of each type in a generation converge to the components of the dominant eigenvector , with per-generation multiplicative factor
Define
Interpret “typical” case as a linear combination of cases of each type given by
V
V
0R
V
Formal definition of R0
Start the BP with a -case:
number of -cases in generation
total number of cases in generation
Then:
Compare with single-type model:
( ; )nX j i i
j
n
( ) ( ; )ni
nj jX iX n
( )0 : lim lim ( )n
nn N
NR X j
E
( )0 1lim: N
NR X
E
Basic reproduction number R0
Naïve definition:
“ Average number of new cases generated by a typical case, throughout the entire infectious period, in a large and otherwise fully susceptible population ”
Requirements:
1) New real infections
2) Typical infector
3) Large population
4) Fully susceptible
Network models
People connected by a static network of acquaintances
Simple case: no short loops, i.e. locally tree-like Repeated contacts First case is special is not a threshold Define:
Difficult case: short loops, clustering Maybe not even possible to use branching
process approximation or define
0 2 1| 1R X X E
1 1X E
0R
Network models
People connected by a static network of acquaintances
Simple case: no short loops, i.e. locally tree-like Repeated contacts First case is special is not a threshold Define:
Difficult case: short loops, clustering Maybe not even possible to use branching
process approximation or define
0 2 1| 1R X X E
1 1X E
0R
Network models
People connected by a static network of acquaintances
Simple case: no short loops, i.e. locally tree-like Repeated contacts First case is special is not a threshold Define:
Difficult case: short loops, clustering Maybe not even possible to use branching
process approximation or define
0 2 1| 1R X X E
1 1X E
0R
Basic reproduction number R0
Naïve definition:
“ Average number of new cases generated by a typical case, throughout the entire infectious period, in a large and otherwise fully susceptible population ”
Requirements:
1) New real infections
2) Typical infector
3) Large population
4) Fully susceptible
HOUSEHOLDS MODELS
Reproduction numbers
Definition of R0
Generalisations
Model description
Example: sSIR households model
Population of households with of size
Upon infection, each case : remains infectious for a duration , iid makes infectious contacts with each household member
according to a homogeneous Poisson process with rate makes contacts with each person in the population according to
a homogeneous Poisson process with rate
Contacted individuals, if susceptible, become infected
Recovered individuals are immune to further infection
m Hn
L
G N
i
iI I i
Outline
Introduction
R0 in simple models
R0 in other models
Households models Reproduction numbers
Definition of R0
Generalisations
Comparison between reproduction numbers Fundamental inequalities Insight
Conclusions
Household reproduction number R
Consider a within-household epidemic started by one initial case
Define: average household final size,
excluding the initial case average number of global
infections an individual makes
“Linearise” the epidemic process at the level of households:
: 1G LR
L
G
Household reproduction number R
Consider a within-household epidemic started by one initial case
Define: average household final size,
excluding the initial case average number of global
infections an individual makes
“Linearise” the epidemic process at the level of households:
: 1G LR
L
G
Individual reproduction number RI
Attribute all further cases in a household to the primary case
is the dominant eigenvalue of :
More weight to the first case than it should be
0G G
IL
M
41 1
2G L
IG
R
IR IM
Individual reproduction number RI
Attribute all further cases in a household to the primary case
is the dominant eigenvalue of :
More weight to the first case than it should be
0G G
IL
M
IR IM
41 1
2G L
IG
R
Further improvement: R2
Approximate tertiary cases: average number of cases infected by the primary case Assume that each secondary case infects further cases Choose , such that
so that the household epidemic yields the correct final size
Then:
and is the dominant eigenvalue of
1 b
11 Lb
2 3 11 .1 ,
1.. Lb bb
b
21
G GMb
2R 2M
Opposite approach: RHI
All household cases contribute equally
Less weight on initial cases than what it should be
:1
LHI G
L
R
Opposite approach: RHI
All household cases contribute equally
Less weight on initial cases than what it should be
:1
LHI G
L
R
Perfect vaccine
Assume
Define as the fraction of the population that needs to be vaccinated to reduce below 1
Then
Leaky vaccine
Assume
Define as the critical vaccine efficacy (in reducing susceptibility) required to reduce below 1 when vaccinating the entire population
Then
Vaccine-associated reproduction numbers RV and RVL
1R 1R
Cp
RR
CE
11
:VC
Rp
11
:VLC
RE
Outline
Introduction
R0 in simple models
R0 in other models
Households models Reproduction numbers
Definition of R0
Generalisations
Comparison between reproduction numbers Fundamental inequalities Insight
Conclusions
Naïve approach:next generation matrix
Consider a within-household epidemic started by a single initial case. Type = generation they belong to.
Define the expected number of cases in each generation
Let be the average number of global infections from each case
The next generation matrix is:
0 1 2 11, , ,...,Hn
1
2
21
1
0
0H H
G G G G G
n n
K
G
More formal approach (I)
Notation: average number of cases in generation and household-
generation average number of cases in generation and
any household-generation
System dynamics:
Derivation:
,n ix ni
0 ,
1H
n n i
n
ix x
n
1
,0 1,
, ,0
0
11
Hn
in G n i
n i i n i H
x x
x x i n
,0 1
, 1
1 1
, 10 0
11
H H
n G n
n i i G
n n
i i
n i H
n n i G i n i
x
x
x
x
x i
x
n
x
More formal approach (II)
System dynamics:
Define
System dynamics:
where
,0 1
, 1
1 1
, 10 0
11
H H
n G n
n i i G
n n
i i
n i H
n n i G i n i
x
x
x
x
x i
x
n
x
( )
1 1, ,...,H
nn n n nx xx x
( ) ( 1)
H
n nnx A x
0 1 2 1
1 0
1
1 0
H
H
G G G G n
nA
More formal approach (III)
Let dominant eigenvalue of “dominant” eigenvector
Then, for :
Therefore:
0 1 1, ,.. , )( .Hn
v vV v Hn
A
( ) ( )
( ) ( 1)
1
n n
n n
n n
V
x
x x
x
x x
n
0R
Recall: Define:
Then:
So:
Similarity
1
2
21
1
0
0H H
G G G G G
n n
K
0 1 2 1
1 0
1
1 0
H
H
G G G G n
nA
0
1
2
1
H
H
n
n
S
0HnK A R
1
HnK SA S
Outline
Introduction
R0 in simple models
R0 in other models
Households models Reproduction numbers
Definition of R0
Generalisations
Comparison between reproduction numbers Fundamental inequalities Insight
Conclusions
Generalisations
This approach can be extended to:
Variable household size
Household-network model
Model with households and workplaces
... (probably) any structure that allows an embedded branching process in the early phase of the epidemic
... all signals that this is the “right” approach!
Households-workplaces model
Model description
Assumptions:
Each individual belongs to a household and a workplace
Rates and of making infectious contacts in each environment
No loops in how households and workplaces are connected, i.e. locally tree-like
,H W G
Construction of R0
Define and for the households and workplaces generations
Define
Then is the dominant eigenvalue of
where
0 1 2 11, , ,...,H
H H H Hn 0 1 2 11, , ,...,
W
W W W Wn
1 1 10 1 1 1
10
, 0 2
H H
W W
H W H Wk G i j i j
i n i nj n j n
Ti j k i j k
k nc
0 1 3 2
1 0
,1
1 0
T
H
Tn n
n
c c c c
A
T H Wn n n
0R
COMPARISON BETWEEN REPRODUCTION NUMBERS
Fundamental inequalities
Insight
Outline
Introduction
R0 in simple models
R0 in other models
Households models Reproduction numbers
Definition of R0
Generalisations
Comparison between reproduction numbers Fundamental inequalities Insight
Conclusions
Comparison between reproduction numbers
Goldstein et al (2009) showed that
1R 1VLR 1rR 1VR 1HIR
HIR
Comparison between reproduction numbers
Goldstein et al (2009) showed that
In a growing epidemic:
1R 1VLR 1rR 1VR 1HIR
R VLR VR
R rR
HIRR
Comparison between reproduction numbers
Goldstein et al (2009) showed that
In a growing epidemic:
1R 1VLR 1rR 1VR 1HIR
VR
Comparison between reproduction numbers
Goldstein et al (2009) showed that
To which we added
In a growing epidemic:
1R 1VLR 1rR 1VR 1HIR
1IR 0 1R 2 1R
IR VR 0R 2R R HIR
Comparison between reproduction numbers
Goldstein et al (2009) showed that
To which we added
In a growing epidemic:
To which we added that, in a declining epidemic:
1R 1VLR 1rR 1VR 1HIR
1IR 0 1R 2 1R
R IR VR 0R 2R HIR
R IR VR 0R 2R HIR
Practical implications
, so vaccinating is not enough
Goldstein et al (2009):
Now we have sharper bounds for :
0VR R0
11 p
R
0V HIIR R RR R
V HIR RR
VR
Outline
Introduction
R0 in simple models
R0 in other models
Households models Reproduction numbers
Definition of R0
Generalisations
Comparison between reproduction numbers Fundamental inequalities Insight
Conclusions
Insight
Recall that is the dominant eigenvalue of
From the characteristic polynomial, we find that is the only positive root of
0 1 2 1
1 0
1
1 0
H
H
G G G G n
nA
0R
0R
0
1
0 1( ) 1
HnG
i
ii
g
Discrete Lotka-Euler equation
Continuous-time Lotka-Euler equation:
Discrete-generation Lotka-Euler equation:
for for
Therefore, is the solution of
0
( ) 1deHr
1
100
1H
ii
Gn
i
R
0
1( ) ( ) ek
H k kr
kk
0 0
1k G k 0k
1,2,..., Hk n
Hk n
0 erR
Fundamental interpretation
For each reproduction number , define a r.v. describing the generation index of a randomly selected infective in a household epidemic
Distribution of is
From Lotka-Euler:
Therefore:
AR AX
AX , 01
Ai
AL
i iX
P
0 1 2 3 4 5 6 7 8 91
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
Non-normalised cumulative distribution of XA
R
RI
R0
R2
RHI
st
A B A BX RX R
0 2 HIIR R RR R
CONCLUSIONS
Why so long to come up with R0?
Typical infective: “Suitable” average across all cases during a household epidemic
Types are given by the generation index: not defined a priori appear only in real-time
“Fully” susceptible population: the first case is never representative need to wait at least a few full households epidemics
1
2 1
1
0
0H H
G G G G G
n n
K
Conclusions
After more than 15 years, we finally found
General approach clarifies relationship between all previously defined reproduction
numbers for the households model works whenever a branching process can be imbedded in the
early phase of the epidemic, i.e. when we can use Lotka-Euler for a “sub-unit”
Allows sharper bounds for :VR
0V HIIR R RR R
0R
Acknowledgements
Co-authors: Pieter Trapman Frank Ball
Useful discussions: Pete Dodd Christophe Fraser
Fundings: Medical Research
Council
Thank you all!
SUPPLEMENTARY MATERIAL
Outline
Introduction in simple models in other models
Households models Many reproduction numbers Definition of Generalisations
Comparison between reproduction numbers Fundamental inequalities Insight
Conclusions
0R
0R
0R
INTRODUCTION
in simple models
in other models0R
0R
Galton-Watson branching processes
Threshold property: If the BP goes extinct with
probability 1 (small epidemic) If the BP goes extinct with
probability given by the smallest solution of
Assume . Then:
0 1X
[0,1]s
0
k
kks s
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
R* = 0.5
R* = 0.5
R* = 1
R* = 1.5
0n
nX RE
0nn X RE
0 1R
0 1R
1 0 n nX X RE E
Galton-Watson branching processes
Threshold property: If the BP goes extinct with
probability 1 (small epidemic) If the BP goes extinct with
probability given by the smallest solution of
Assume . Then:
0 1X
[0,1]s
0
k
kks s
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
R* = 0.5
R* = 0.5
R* = 1
R* = 1.5
0 1R
0 1R
0n
nX RE
0nn X RE
1 0 n nX X RE E
Galton-Watson branching processes
Threshold property: If the BP goes extinct with
probability 1 (small epidemic) If the BP goes extinct with
probability given by the smallest solution of
Assume . Then:
0 1X
[0,1]s
0
k
kks s
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
R* = 0.5
R* = 0.5
R* = 1
R* = 1.5
0 1R
0 1R
0n
nX RE
0nn X RE
1 0 n nX X RE E
Galton-Watson branching processes
Threshold property: If the BP goes extinct with
probability 1 (small epidemic) If the BP goes extinct with
probability given by the smallest solution of
Assume . Then:
0 1X
[0,1]s
0
k
kks s
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
R* = 0.5
R* = 0.5
R* = 1
R* = 1.5
0 1R
0 1R
0n
nX RE
0nn X RE
1 0 n nX X RE E
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
R* = 0.5
R* = 1
R* = 1.5
R* = 2
Properties of R0
Threshold parameter: If , only small epidemics If , possible large epidemics
Probability of a large epidemic
Final size:
Critical vaccination coverage:
If , then
01 e R zz
0
11
Cp R
0 1R
0 1R
0 1X
( )0 1 lim
N
NnR X X
E E
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
R* = 0.5
R* = 1
R* = 1.5
R* = 2
Properties of R0
Threshold parameter: If , only small epidemics If , possible large epidemics
Probability of a large epidemic
Final size:
Critical vaccination coverage:
If , then
01 e R zz
0
11
Cp R
0 1R
0 1R
0 1X
( )0 1 lim
N
NnR X X
E E
HOUSEHOLDS MODELS
Within-household epidemic
Repeated contacts towards the same individual Only the first one matters
Many contacts “wasted” on immune people Number of immunes changes over time -> nonlinearity
Overlapping generations Time of events can be important
Rank VS true generations
sSIR model: draw an arrow from individual to each other individual with
probability
attach a weight given by the (relative) time of infection
Rank-based generations = minimum path length from initial infective
Real-time generations = minimum sum of weights
1 exp1 iIn
0.8
0.61.9