r 0 and other reproduction numbers for households models

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 modelsR0 in other models

Outline



Generalisations


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 nN

X 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* = 1.5R* = 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

1 1Cp R

0 1R

0 1R

0 1X

( )0 1 lim

N

NnR X X

E E

Outline



Generalisations


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 kk k

K

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


Naïve definition:


Requirements:


2) Typical infector

3) Large population


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


Naïve definition:


Requirements:


2) Typical infector

3) Large population


HOUSEHOLDS MODELS

Reproduction numbersDefinition 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

iiI I i

Outline



Generalisations


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

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 12G 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 12G 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

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

1 1:VC

Rp

1 1:VLC

RE

Outline



Generalisations


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 in

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

11H 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

11H 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 01

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 01

1 0

H

H

G G G G n

nA

0

1

2

1

H

H

n

n

S

0HnK A R

1Hn

K SA S

Outline



Generalisations


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 HW 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 inequalitiesInsight

Outline



Generalisations


Conclusions

Comparison between reproduction numbers

Goldstein et al (2009) showed that

1R 1VLR 1rR 1VR 1HIR

HIR



In a growing epidemic:


R VLR VR

R rR

HIRR





VR



To which we added



1IR 0 1R 2 1R

IR VR 0R 2R R HIR



To which we added


To which we added that, in a declining epidemic:


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 pR

0V HIIR R RR R

V HIR RR

VR

Outline



Generalisations


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 01

1 0

H

H

G G G G n

nA

0R

0R

0

1

0 1( ) 1Hn

G

i

iig

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


Conclusions

0R

0R

0R

INTRODUCTION

in simple models in other models0R0R

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

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* = 1.5R* = 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

1 1Cp 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

r 0 and other reproduction numbers for households models

Documents