Download R 0

R0 and other reproduction numbers
for households models
... and epidemic models with other social structures
Lorenzo Pellis, Frank Ball, Pieter Trapman
MRC Centre for Outbreak analysis and modelling,
Department of Infectious Disease Epidemiology
Imperial College London
Edinburgh, 14th September 2011
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
R0 in simple models
R0 in other models
INTRODUCTION
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:
 X n( N )  number of infected cases in generation n (pop. size N )
 For every fixed n ,
lim X n( N )  X n
N 
where X n is the n -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 P   k    k , k  0,1,... be the offspring distribution.
 Define
R0  E  k 
 We have “linearised” the early phase of the epidemic
Properties of R0
1
 Threshold parameter:
 If R0  1 , only small epidemics
 If R0  1 , possible large epidemics
 Probability of a large epidemic
0.8
R = 0.5
*
0.6
R* = 1
0.4
R* = 1.5
R =2
*
 Final size:
0.2
1  z  e R0 z
 Critical vaccination coverage:
pC  1 
1
R0
 If X 0  1 , then
R0  E  X1   lim E  X n( N ) 
N 
0
0
0.2
0.4
0.6
0.8
1
Properties of R0
1
 Threshold parameter:
 If R0  1 , only small epidemics
 If R0  1 , possible large epidemics
 Probability of a large epidemic
0.8
R = 0.5
*
0.6
R* = 1
0.4
R* = 1.5
R =2
*
 Final size:
0.2
1  z  e R0 z
 Critical vaccination coverage:
pC  1 
1
R0
 If X 0  1 , then
R0  E  X1   lim E  X n( N ) 
N 
0
0
0.2
0.4
0.6
0.8
1
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):
 k11 k12
k
k22
21

K


 kn1
k1n 




k nn 
where k ij is the average number of type-i cases generated by a
type- j case, throughout the entire infectious period, in a fully
susceptible population
Properties of the NGM:
 Non-negative elements
 We assume positive regularity
Perron-Frobenius theory
 Single dominant eigenvalue  , which is positive and real
 “Dominant” eigenvector V 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 V , with per-generation
multiplicative factor 
 Define R0  
 Interpret “typical” case as a linear combination of cases of each
type given by V
Formal definition of R0
Start the BP with a j -case:
 X n ( j; i )  number of i -cases in generation n

X n ( j )   X n ( j; i )  total number of cases in generation n
i
 Then:
R0 : lim lim n E  X n( N ) ( j ) 
n N 
Compare with single-type model:
R0 : lim E  X1( N ) 
N 
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
 E  X1   1 is not a threshold
 Define:
R0  E  X 2 | X1  1
 Difficult case: short loops, clustering
 Maybe not even possible to use branching
process approximation or define R0
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
 E  X1   1 is not a threshold
 Define:
R0  E  X 2 | X1  1
 Difficult case: short loops, clustering
 Maybe not even possible to use branching
process approximation or define R0
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
 E  X1   1 is not a threshold
 Define:
R0  E  X 2 | X1  1
 Difficult case: short loops, clustering
 Maybe not even possible to use branching
process approximation or define R0
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
Reproduction numbers
Definition of R0
Generalisations
HOUSEHOLDS MODELS
Model description
Example: sSIR households model
 Population of m households with of size nH
 Upon infection, each case i :
 remains infectious for a duration I i I , iid i
 makes infectious contacts with each household member
according to a homogeneous Poisson process with rate  L
 makes contacts with each person in the population according to
a homogeneous Poisson process with rate G N
 Contacted individuals, if susceptible, become infected
 Recovered individuals are immune to further infection
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:
  L  average household final size,
excluding the initial case
 G  average number of global
infections an individual makes
 “Linearise” the epidemic process at the
level of households:
R : G 1   L 
Household reproduction number R
 Consider a within-household epidemic
started by one initial case
 Define:
  L  average household final size,
excluding the initial case
 G  average number of global
infections an individual makes
 “Linearise” the epidemic process at the
level of households:
R : G 1   L 
Individual reproduction number RI
 Attribute all further cases in a household to the primary case
 G
MI  
 L

G 
0 
RI is the dominant eigenvalue of M I :
G 
4 L 
RI 
1  1 

2 
G 
 More weight to the first case than it should be
Individual reproduction number RI
 Attribute all further cases in a household to the primary case
 G
MI  
 L

G 
0 
RI is the dominant eigenvalue of M I :
G 
4 L 
RI 
1  1 

2 
G 
 More weight to the first case than it should be
Further improvement: R2
 Approximate tertiary cases:
 1  average number of cases infected by the primary case
 Assume that each secondary case infects b further cases
 Choose b  1  1  L , such that
1 1  b  b  b  ... 
2
1
3
1 b
 L ,
so that the household epidemic yields the correct final size
 Then:
 G
M2  
 1
G 
b 
and R2 is the dominant eigenvalue of M 2
Opposite approach: RHI
 All household cases contribute equally
L
RHI : G 
1  L
 Less weight on initial cases than what it
should be
Opposite approach: RHI
 All household cases contribute equally
L
RHI : G 
1  L
 Less weight on initial cases than what it
should be
Vaccine-associated
reproduction numbers RV and RVL
Perfect vaccine
Leaky vaccine
 Assume R  1
 Assume R  1
 Define pC as the fraction of
the population that needs to
be vaccinated to reduce R
below 1
 Define EC as the critical
vaccine efficacy (in reducing
susceptibility) required to
reduce R below 1 when
vaccinating the entire
population
 Then
RV : 1 
1
pC
 Then
RVL : 1 
1
EC
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 0  1, 1 , 2 ,..., nH 1 the expected number of cases in
each generation
 Let G be the average number of global infections from each case
 The next generation matrix is:
 G

 1
K 




G
G
G
G 
2 1
n
H 1
n
H 2
0 




0 
More formal approach (I)
 Notation:
 xn,i  average number of cases in generation n and householdgeneration i
nH 1
 xn  i 0 xn,i  average number of cases in generation n and
any household-generation

 System dynamics:
nH 1
xn,0  G  xn1,i
i 0
xn,i  i xni ,0
 Derivation:
1  i  nH  1
xn,0  G xn1
xn,i  i G xni 1
xn 
nH 1
x
i 0
1  i  nH  1
nH 1
n ,i
 G  i xni 1
i 0
More formal approach (II)
 System dynamics:
xn,0  G xn1
xn,i  i G xni 1
xn 
 Define x
( n)

nH 1
x
i 0
nH 1
n ,i
 xn , xn1 ,..., xnnH 1
 System dynamics:
x
where
AnH
( n)
 G 0

 1





1  i  nH  1
 AnH x
 G  i xni 1

i 0
( n1)
G 1 G 2
G  n
0
1
1
0
H 1








More formal approach (III)
 Let
   dominant eigenvalue of AnH
 V  (v0 , v1 ,..., vn 1 )  “dominant” eigenvector
H
 Then, for n   :
x
x
( n)
(n)
( n)
V
( n 1)

x
x
xn xn1  
 Therefore:
  R0
Similarity
 Recall:
 G

 1
K 




AnH
 Define:
G
G
G
G 
0 




0 
2 1
 G 0

 1





n
H 1
G 1 G 2
n
H 2
G  n
0
1
1
0
H 1








 0


S 




1
n
H 2






nH 1 
 Then:
K  SAnH S 1
 So:
  K     An
H
R
0
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  H , W and G of making infectious contacts in each
environment
 No loops in how households and workplaces are connected, i.e.
locally tree-like
Construction of R0
W
W
W
W
H
H
H
H
 Define 0  1, 1 , 2 ,..., nH 1 and 0  1, 1 , 2 ,...,  nW 1
for the households and workplaces generations
 Define nT  nH  nW
 Then
R0 is the dominant eigenvalue of
AnH
where
ck  G
 c0

1






i  j k
0 i  nH 1
0 j  nW 1
c1
1
cnT 2 

0 
,


0 

iH  Wj ,
cnT 3
1
iH  Wj 
i  j k 1
1 i  nH 1
1 j  nW 1
0  k  nT  2
Fundamental inequalities
Insight
COMPARISON BETWEEN
REPRODUCTION NUMBERS
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
R  1  RVL  1  Rr  1  RV  1  RHI  1
Comparison
between reproduction numbers
 Goldstein et al (2009) showed that
R  1  RVL  1  Rr  1  RV  1  RHI  1
 In a growing epidemic:
R
R

RVL


Rr
RV

RHI
Comparison
between reproduction numbers
 Goldstein et al (2009) showed that
R  1  RVL  1  Rr  1  RV  1  RHI  1
 In a growing epidemic:
R

RV

RHI
Comparison
between reproduction numbers
 Goldstein et al (2009) showed that
R  1  RVL  1  Rr  1  RV  1  RHI  1
 To which we added

RI  1 
R0  1 
R2  1
 In a growing epidemic:

RI

RV

R0HI 
R2

Comparison
between reproduction numbers
 Goldstein et al (2009) showed that
R  1  RVL  1  Rr  1  RV  1  RHI  1
 To which we added

RI  1 
R0  1 
R2  1
 In a growing epidemic:
R

RI

RV

R0

R2

RHI

RHI
 To which we added that, in a declining epidemic:
R

RI

RV

R0

R2
Practical implications
1
 RV  R0 , so vaccinating p  1 
is not enough
R0
 Goldstein et al (2009):
R  RV
 RHI
 Now we have sharper bounds for RV :
R  RI
 RV
 R0
 RHI
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 R0 is the dominant eigenvalue of
AnH
 G 0

 1





G 1 G 2
G  n
0
1
1
0
H 1








 From the characteristic polynomial, we find that R0 is the only
positive root of
nH 1
g0 (  )  1  
i 0
G i
 i 1
Discrete Lotka-Euler equation
 Continuous-time Lotka-Euler equation:

 r

(

)
e
d  1
 H
0
 Discrete-generation Lotka-Euler equation:
 H ( )   k k ( )
 0  0
  k  G k 1 for
 k  0
for


  e 
k 0
k  1, 2,..., nH
k  nH
 Therefore, R0  e is the solution of
r
r k
k
nH 1
G i
R
i 0
1
0
i 1
1
Fundamental interpretation
 For each reproduction number RA , define a r.v. X A describing the
generation index of a randomly selected infective in a household
epidemic
iA
 Distribution of X A is P  X A  i 
,
1  L
0  i  
Non-normalised cumulative distribution of XA
2.8
 From Lotka-Euler:
st
XA



XB
2.6
RA

2.4
RB
R
RI
2.2
R0
2
Therefore:
R2
RHI
1.8
R  RI
 R0
 R2
 RHI
1.6
1.4
1.2
1
0
1
2
3
4
5
6
7
8
9
CONCLUSIONS
Why so long to come up with R0?
 Typical infective:
 “Suitable” average across all cases during a household epidemic
 G

 1
K 




G
G
G
G 
2 1
n
H
n
H 1
0 




0 
 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
Conclusions
 After more than 15 years, we finally found R0
 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 RV :
R  RI
 RV
 R0
 RHI
Acknowledgements
 Co-authors:
 Pieter Trapman
 Frank Ball
 Useful discussions:
 Pete Dodd
 Christophe Fraser
 Fundings:
 Medical Research
Council
Thank you all!
SUPPLEMENTARY MATERIAL
Outline
 Introduction
 R0 in simple models
 R0 in other models
 Households models
 Many reproduction numbers
 Definition of R0
 Generalisations
 Comparison between reproduction numbers
 Fundamental inequalities
 Insight
 Conclusions
R0 in simple models
R0 in other models
INTRODUCTION
Galton-Watson branching processes
 Threshold property:
 If R0  1 the BP goes extinct with
probability 1 (small epidemic)
 If R0  1 the BP goes extinct with
probability given by the smallest
solution s  [0,1] of

s   k s k
1
k 0
Assume X 0  1 . Then:
 E  X n    R0 

n
n
E  X n   R0
 E  X n1  E  X n   R0
0.8
R = 0.5
*
0.6
R =1
*
0.4
R = 1.5
*
R* = 0.5
0.2
0
0
0.2
0.4
0.6
0.8
1
Galton-Watson branching processes
 Threshold property:
 If R0  1 the BP goes extinct with
probability 1 (small epidemic)
 If R0  1 the BP goes extinct with
probability given by the smallest
solution s  [0,1] of

s   k s k
1
k 0
Assume X 0  1 . Then:
 E  X n    R0 

n
n
E  X n   R0
 E  X n1  E  X n   R0
0.8
R = 0.5
*
0.6
R =1
*
0.4
R = 1.5
*
R* = 0.5
0.2
0
0
0.2
0.4
0.6
0.8
1
Galton-Watson branching processes
 Threshold property:
 If R0  1 the BP goes extinct with
probability 1 (small epidemic)
 If R0  1 the BP goes extinct with
probability given by the smallest
solution s  [0,1] of

s   k s k
1
k 0
Assume X 0  1 . Then:
 E  X n    R0 

n
n
E  X n   R0
 E  X n1  E  X n   R0
0.8
R = 0.5
*
0.6
R =1
*
0.4
R = 1.5
*
R* = 0.5
0.2
0
0
0.2
0.4
0.6
0.8
1
Galton-Watson branching processes
 Threshold property:
 If R0  1 the BP goes extinct with
probability 1 (small epidemic)
 If R0  1 the BP goes extinct with
probability given by the smallest
solution s  [0,1] of

s   k s k
1
k 0
Assume X 0  1 . Then:
 E  X n    R0 

n
n
E  X n   R0
 E  X n1  E  X n   R0
0.8
R = 0.5
*
0.6
R =1
*
0.4
R = 1.5
*
R* = 0.5
0.2
0
0
0.2
0.4
0.6
0.8
1
Properties of R0
1
 Threshold parameter:
 If R0  1 , only small epidemics
 If R0  1 , possible large epidemics
 Probability of a large epidemic
0.8
R = 0.5
*
0.6
R* = 1
0.4
R* = 1.5
R =2
*
 Final size:
0.2
1  z  e R0 z
 Critical vaccination coverage:
pC  1 
1
R0
 If X 0  1 , then
R0  E  X1   lim E  X n( N ) 
N 
0
0
0.2
0.4
0.6
0.8
1
Properties of R0
1
 Threshold parameter:
 If R0  1 , only small epidemics
 If R0  1 , possible large epidemics
 Probability of a large epidemic
0.8
R = 0.5
*
0.6
R* = 1
0.4
R* = 1.5
R =2
*
 Final size:
0.2
1  z  e R0 z
 Critical vaccination coverage:
pC  1 
1
R0
 If X 0  1 , then
R0  E  X1   lim E  X n( N ) 
N 
0
0
0.2
0.4
0.6
0.8
1
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
 

1  exp 
Ii 
 n 1 
 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
0.8
0.6
1.9