Epidemics with two levels of mixing

Epidemics with two levels of mixing Frank Ball [email protected]

University of Nottingham Modelling Complex Systems, University of Manchester 21st June 2010 Joint work with Peter Neal (University of Manchester) and Owen Lyne (University of Kent) Research supported by EPSRC and The Leverhulme Trust

Epidemics with two levels of mixing – p.1

Outline of talk Background on homogeneously mixing SIR (susceptible → infective → removed) epidemic models General two-level-mixing SIR epidemic model Special cases — — — —

households model overlapping groups model great circle model network model with casual contacts

Threshold behaviour and final outcome of global epidemic — general results — applications to special cases — numerical illustrations

Vaccination in Households SIR model Concluding comments Epidemics with two levels of mixing – p.2

General deterministic epidemic (x(0), y(0), z(0)) = (N, a, 0) x(t)

y(t)

z(t)

β = infection rate γ = removal rate

susceptibles

infectives

removed

SIR (susceptible → infective → removed) dx dt

dz = −βxy, dy = βxy − γy, dt dt = γy

dy dt

> 0 ⇐⇒ βxy − γy > 0 ⇐⇒ x > γ/β

Epidemic takes off ⇐⇒ N > γ/β ⇐⇒ R0 = N β/γ > 1 (Kermack and McKendrick (1927)) Epidemics with two levels of mixing – p.3

General deterministic epidemic 60 number of infectives y(t)

R0=5.0 R0=2.0 R =1.5 0 R0=1.0 R =0.5

50 40

0

30

N=100, a=3, γ=1

20 10 0 0

5

10 time t

15

20 Epidemics with two levels of mixing – p.4

General stochastic epidemic X(0) = N X(t)

Y(t)

Z(t)

Y (0) = a Z(0) = 0

susceptibles

infectives

removed

Transition t + ∆t n cccc11 (x − 1, y + 1) infecctio c cccc c c c c c (x, y) [[[[[[[[[ [[[[[[[[ -removal (x, y − 1)

Probability

t

βxy∆t + o(∆t) γy∆t + o(∆t) (Bartlett (1949)) Epidemics with two levels of mixing – p.5

General stochastic epidemic X(0) = N X(t)

Y(t)

Z(t)

Y (0) = a Z(0) = 0

susceptibles

infectives

removed

Final size T = N − X(∞) — number of susceptibles ultimately infected Let Pj = P(T = j) (j = 0, 1, . . . , N ). Then a+j i X N −j i N Pj = 1 + 1 − R0 N −i N i j=0

(i = 0, 1, . . . , N ) (Whittle (1955)) Epidemics with two levels of mixing – p.6

Final size 0.5

0.5

0.5

0.4

0.4

R0 = 0.5

0.3

0.2

0.1 0

10

20

30

final size

40

0

50

0.1 0

10

20

30

final size

40

0

50

0.5

0.5

0.5

0.4

R0 = 1.5

0.3 0.2

0

10

20

30

final size

40

50

0

30

40

50

30

40

50

0.2

0.1 0

20

final size

R0 = 5.0

0.3

0.2

0.1

10

0.4

R0 = 2.0

0.3

0

probability

0.6

probability

0.6

probability

0.6

0.4

R0 = 1.25

0.3

0.2

0.1 0

0.4

R0 = 1.0

0.3

0.2

probability

0.6

probability

0.6

probability

0.6

0.1 0

10

20

30

final size

40

50

0

0

10

20

final size

Final size distribution for general stochastic epidemic with 1 initial infective and 50 initial susceptibles Epidemics with two levels of mixing – p.7

Threshold behaviour Consider epidemic with initially few infectives and many susceptibles If R0 ≤ 1, then Only minor outbreaks occur Size of outbreak distributed according to total progeny of approximating branching process, Z say, in which all contacts lead to an infection If R0 > 1, P(major outbreak) = 1 − P(Z goes extinct) > 0 Size of minor outbreak distributed according to total progeny of Z , conditional upon extinction Size of major outbreak satisfies a central limit theorem with mean given by deterministic model (von Bahr and Martin-Löf (1980)) Epidemics with two levels of mixing – p.8

Basic reproduction number R0 R0 = "the expected number of secondary cases produced by a typical infected individual during its entire infectious period, in a population consisting of susceptibles only" (Heesterbeek and Dietz (1996))

Major outbreak can occur ⇐⇒ R0 > 1 If proportion c of susceptibles is vaccinated with a perfect vaccine, R0 is reduced to Rv = (1 − c)R0 , so, if R0 > 1, Rv ≤ 1 ⇐⇒ c ≥ 1 − R0−1 — critical vaccination coverage (Smith (1964)) Epidemics with two levels of mixing – p.9

Multitype general deterministic epidemic m types of individual, labelled 1, 2, . . . , m, reflecting e.g. age, vaccine status, geographical location. m X dxi = −xi βji yj , dt j=1 m X dyi = xi βji yj − γi yi , dt j=1

dzi = γi yi dt

(i = 1, 2, . . . , m),

with (xi (0), yi (0), zi (0)) = (Ni , ai , 0)

Analogues of all previous results hold BUT threshold behaviour requires Ni → ∞ (i = 1, 2, . . . , m), i.e. that the population is LOCALLY LARGE Epidemics with two levels of mixing – p.10

Non-locally-large models Spatial models percolation structure too rigid for human populations Network models Multi-level mixing models metapopulation/households models Complex simulation models more realistic but can be computationally expensive and difficult to interpret


General two-level-mixing epidemic model 3

2

Population

1

N = {1, 2, . . . , N } N

SIR (susceptible → infective → removed) Infectious periods I1 , I2 , . . . , IN iid ∼ I (arbitrary but specified) Infection rates (individual → individual) local global

λL ij λG /N

Latent period (Ball and Neal (2002)) Epidemics with two levels of mixing – p.12

Households model

m households, each of size n N = mn

  λ L λLij =  0

if i and j belong to same household otherwise

Unequal-sized households. ´ (Bartoszynski (1972), Becker and Dietz (1995), Ball, Mollison and Scalia-Tomba (1997))


Overlapping groups model

Workplace

Household

mα households, each of size nα , mβ workplaces, each of size nβ , so N = mα nα = mβ nβ  L  λ if i and j belong to same household   α λL λL if i and j belong to same workplace ij = β    0 otherwise

(Ball and Neal (2002), cf. Andersson (1997)) Epidemics with two levels of mixing – p.14

Great circle model N −1

N

1

2 3

  λL L λij =  0

if i and j are neighbours otherwise

‘Small-world’ networks More general contact distribution

(Ball, Mollison and Scalia-Tomba (1997), Ball and Neal (2002, 2003))


Networks with casual contacts

‘independent’ random graph of possible local contacts with specified degree distribution pk = P(D = k) (k = 0, 1, . . . )   λ if i and j are neighbours L L λij =  0 otherwise (Diekmann et al. (1998), Ball and Neal (2002), (2008), Kiss et al. (2006); Newman (2002)) Epidemics with two levels of mixing – p.16

Directed graph of potential local contacts

i → j if and only if i, if infected, contacts j locally. Given infectious periods I1 , I2 , . . . , IN , P(i → j) = 1 − e−λL Ii independently for distinct (i, j). Epidemics with two levels of mixing – p.17

Local infectious clump CiN

i

CiN = {j ∈ N : i j}, where i j if and only if there exists a chain of directed arcs from i to j, and CiN = |CiN |. Epidemics with two levels of mixing – p.18

Local infectious clumps

i

j CiN = {j ∈ N : i j}, where i j if and only if there exists a chain of directed arcs from i to j, and CiN = |CiN |. Epidemics with two levels of mixing – p.19

Threshold parameter R∗ As N → ∞, process of infected clumps tends to a branching process having offspring random varaible P R ∼ Poisson(λG A), where A = j∈C Ij Global epidemic occurs if and only if this branching process does not go extinct hP i R∗ = E[R] = λG E[A] = λG E j∈N Ij 1{j∈C} = PN λG j=1 E[Ij ]P(j ∈ C) = λG µI E[C] P(global epidemic) > 0 ⇐⇒ R∗ > 1


Local susceptibility set SiN

i

SiN = {j ∈ N : j

i} and SiN = |SiN |.


Final outcome of global epidemic Suppose N is large and there are few initial infectives. Let z be the expected proportion of the population who are infected by the epidemic. Then λG π = P(typical susceptible avoids global infection) = exp − N zµI = exp(−λG zµI ) N and 1 − z = P(typical susceptible avoids infection by epidemic) = P(typical local susceptibility set avoids global infection) =

∞ X

P(S = k)π k = fS (π) = fS (e−λG zµI )

(1)

k=1

R∗ = λG µI E[C] = λG µI E[S] R∗ ≤ 1

z = 0 is the only solution of (1) in [0, 1]

R∗ > 1

unique second solution zˆ ∈ (0, 1), giving mean ‘size’ of global epidemic

Fully rigorous proof and central limit theorem for final size of global epidemic is available using Scalia-Tomba (1985) embedding technique


Great circle model

Si = {i} ∪ SL ∪ SR pL = P(i infects i + 1 locally) = 1 − E[e−λL I ] SR

i

SL

P(SL = k) = P(SR = k) = (1 − pL )pkL (k = 0, 1, . . . ) SL and SL are independent, so P(S = k) = (1 − pL )2 pk−1 (k = 1, 2, . . . ) E[S] = 2p−1 L −1


Households model Consider household of n individuals, labelled 1, 2, . . . , n, and let S be the local susceptibility set of individual 1.

(n)

Let Pj

= P(S = j)

(j = 1, 2, . . . , n)

j−1

n−j

qk = E[e−kλL I ] be the probability that a given set of k susceptibles avoids local infection from a given infective (n) n−1 (j) n−j Pj = j−1 Pj qj (j = 1, 2, . . . , n)

1

k X

(k) Pj

=1

=⇒

j=1

k X k − 1

j=1

=⇒

j−1

(j)

Pj qjk−j = 1

n−k (n) j=1 n−j Pj qjn−k

Pk

=

Triangular system of linear equations for P(S = j)

n − 1 k−1

(k = 1, 2, . . . , n)

(j = 1, 2, . . . , n) Epidemics with two levels of mixing – p.24

Overlapping groups model i

Workplace

Household

Construct local susceptibility set S of typical individual i via a two-type branching process in which individuals beget only the opposite type and the offspring of a type α (β) individual are the individuals in its workplace (household) susceptibility set. If µα (µβ ) is the mean size of a household (workplace) susceptiblity set, then  µα µβ  if (µα − 1)(µβ − 1) < 1 µα +µβ −µα µβ E[S] =  ∞ otherwise Epidemics with two levels of mixing – p.25

NETWORK — Configuration model Population N = {1, 2, . . . , N } D= degree of typical individual pk = P(D = k) (k = 0, 1, . . . )

µD = E[D]

specified

D1 , D2 , . . . , DN iid copies of D, conditioned on SN = D1 + D2 + · · · + DN being even Attach Di half-edges to individual i (i = 1, 2, . . . , N ) Pair up the SN half-edges uniformly at random to form the network 2 IMPERFECTIONS — sparse if σD = var(D) < ∞

(Bollobás (2001))


Networks with casual contacts ˜ = degree of typical neighbour of typical individual in the Let D ˜ Then network and µD˜ = E[D]. 2 var(D)+µ D ˜ = k) = kp /µ P(D (k = 1, 2, . . . ) and µ ˜ = . k

D

D

µD

a.s.

Size of typical local susceptibility set S N −→ S as N → ∞, where S is the total size of a branching process having offspring law Bin(D, pL ) for the initial individual and ˜ − 1, p ) for all subsequent individuals Bin(D L   1+ E[S] =  ∞

µD pL 1−(µD ˜ −1)pL

if (µD˜ − 1)pL < 1 otherwise


‘Deterministic’ households model m households of size n, labelled 1, 2, . . . , m. Let xi (t) and yi (t) be the number of susceptibles and infectives in household i at time t.

m X dxi −1 = −(λL yi + N λG yj )xi , dt j=1 m X dyi = (λL yi + N −1 λG yj )xi − γyi dt j=1

(i = 1, 2, . . . , m),

Basic Reproduction number R0 = (λG + nλL )/γ Proportion of susceptibles ultimately infected, zˆdet given by largest root in [0, 1] of 1 − z = exp(−R0 z) Epidemics with two levels of mixing – p.28

Households and great circle models Households, z, λG=0.75.

Households, R*=1. 1

1

0.8

0.8 n=2

0.6

λG 0.4 0.2 0 0 1

z

n=3 n=4

2n λ 3 L

4

n=4

0.6

n=3

n=2

0.2

0

1

n=5

0.4

n=5

R =1

det

0 0

5

Great circle, R*=1.

1

1

2 nλ 3 L

4

5

Great circle, z, λG=0.5 det

0.8 λ

0.8

0.6

z

G

0.4 0.2 0 0

0.6

sto

0.4 0.2

R0 =1 5 λ

L

10

0 0

0.5

1 λ

1.5

2

L

Critical values of (λL , λG ) so that R∗ = 1 and final outcome zˆ when I ∼ Exp(1). Epidemics with two levels of mixing – p.29

Overlapping groups model, varying R*=1, nα=3, nβ =2

1

z, λG =0.5, nα=3, nβ=2

0.9 λβ=0.0 λβ=0.2 λβ=0.4 λβ=0.6 λβ=1.0 λβ=100.0

0.9 0.8 0.7 0.6

0.8 0.7 0.6

L λβ

λβ=0.2 λ =0.4 β λβ=0.6 λβ=1.0 λ =100.0 β

z 0.5

λG0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0 0

0.5

1

L

λα

1.5

2

0 0

0.5

1

nα λLα

1.5

2

Critical values of (λLα , λG ) so that R∗ = 1 and final outcome zˆ when I ∼ Exp(1) Epidemics with two levels of mixing – p.30

Overlapping groups model, varying nβ R =1, n =3, λL =0.25 *

0.7

α

L λG =0.5, nα=3, λα=0.25

1

α

0.9

λG

0.6

0.8

0.5

0.7

0.3

nβ=3

0.4 0.3

n =2

nβ =4

β

0.2

R =1

0.1

0

0 0

nβ=4

z0.5

nβ =3

0.2 0.1

nβ=5

0.6

nβ=2

0.4

det

n =5 β

0.5

1 L

n λ β

β

1.5

2

0 0

0.5

1 L

nβ λβ

1.5

2

Critical values of (λLβ , λG ) so that R∗ = 1 and final outcome zˆ when I ∼ Exp(1) Epidemics with two levels of mixing – p.31

Networks with casual contacts 7000

3500

6000

3000

5000 4000

N = 250, d = 8

2500

p = 0.1, λ = 0.0

2000

L

G

3000

2500

N = 250, d = 8

2000

p = 0.2, λ = 0.0 L

p = 0.2, λ = 0.5

G

0

1000

1000

500

10

20

30

40

50

60

0 0

1000

50

100

150

200

250

0 0

0

50

100

4500

7000

3500

4000

N = 1000, d = 8

3000

p = 0.1, λ = 0.0

2500

G

2000

150

200

250

800

1000

Total size

4000

L

R = 1.9512

Total size

8000

4000

G

500

Total size

5000

L

1500

R = 1.4

0

2000

6000

N = 250, d = 8

1500

R = 0.7

0 0

3000

3500

N = 1000, d = 8

3000

pL = 0.2, λG = 0.0

2500

N = 1000, d = 8 pL = 0.2, λG = 0.5

2000

3000

R = 0.7

1500

0

2000

1000

1000

500

0 0

20

40

60

Total size

80

100

0 0

R = 1.4 0

1500

R = 1.9512 0

1000 500 200

400

600

Total size

800

1000

0 0

200

400

600

Total size

Histograms of size of 10,000 simulated epidemics per parameter combination, for a constant-degree network with D ≡ d, I ≡ 1 and other parameters as shown. Epidemics with two levels of mixing – p.32

Illustration of CLT 1200 1000 800 600 400 200 0 5600

5800

6000

6200

6400

6600

6800

Total size

Histogram of size of 10,000 simulated global epidemics in a population of size N = 10, 000 when D ≡ 8, λG = 0 and pL = 0.2 (I ≡ 1 and λL = − log 0.8), with asymptotic normal approximation superimposed. Epidemics with two levels of mixing – p.33

Networks with casual contacts 1

d=3 d=4 d=5 d=8 d=∞

0.9 0.8

Critical λG

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

dλL

1.5

2

Critical values of (λL , λG ) so that R∗ = 1 when I ≡ 1. [Expected number of potentially infectious contacts made by an infective is λG + dλL .] Epidemics with two levels of mixing – p.34

Networks with casual contacts 1 0.9 0.8 0.7 0.6

z

0.5 0.4 Scale−free, λG = 0.5 Scale−free, λG = 0.0 D ≡ 8, λG = 0.5 D ≡ 8, λ = 0.0

0.3 0.2 0.1 0 0

G

0.05

0.1

0.15

0.2

pL

0.25

0.3

0.35

0.4

Asymptotic proportion of population infected by global epidemic, zˆ, for constant-degree and scale-free (P(D = k) ∝ k −2.466956 (k = 3, 4, . . .)) networks with µD = 8 when I ≡ 1. Epidemics with two levels of mixing – p.35

Households SIR epidemic model mn households of size n (n = 1, 2, . . . ) P total no. of households m = ∞ n=1 mn P∞ total no. of individuals N = n=1 nmn < ∞

SIR (susceptible → infective → removed) Infectious period ∼ TI , having an arbitrary but specified distribution Infection rates (individual → individual) (i) local (within-household)

λL

(ii) global (between-household)

λG /N

Latent period ´ (Bartoszynski (1972), Becker and Dietz (1995), Ball, Mollison and Scalia-Tomba (1997)) Epidemics with two levels of mixing – p.36

Threshold parameter R∗

GLOBAL INFECTION

R∗ = mean number of global contacts emanating from a typical single-household epidemic ∞ X R∗ = α ˜ n µn (λL )λG E[TI ], n=1

where nmn N

=

P(randomly chosen person lives in a household of size n)

µn (λL )

=

mean size of single (size-n) household epidemic with 1 initial infective

α ñ =

P(global epidemic) > 0 ⇐⇒ R∗ > 1 (Ball, Mollison and Scalia-Tomba (1997), Becker and Dietz (1995)) Epidemics with two levels of mixing – p.37

Vaccination For n = 1, 2, . . . and v = 0, 1, . . . , n, let xnv

=

proportion of size-n households that have v members vaccinated

µnv

=

mean number of global contacts emanating from a single-household epidemic in a household in state (n, v), initiated by an individual chosen uniformly at randomly being contacted globally

Post-vaccination Rv =

∞ X

n=1

α ñ

n X

xnv µnv

v=0

Vaccination coverage n X v c= α ñ xnv n n=1 v=0 ∞ X

Determination of optimal vaccination scheme (e.g. to reduce Rv to 1 with minimum vaccination coverage) is a linear programming problem, whose solution can be constructed explicitly. (Becker and Starczak (1997), Ball and Lyne (2002, 2006)) Epidemics with two levels of mixing – p.38

Calculation of µnv xnv

=

proportion of size-n households that have v members vaccinated

µnv

=

mean number of global contacts emanating from a single-household epidemic in a household in state (n, v), initiated by an individual chosen uniformly at randomly being contacted globally

µnv depends on model for vaccine action. For an all-or-nothing model, in which vaccinees are rendered immune independently with probability ǫ, otherwise the vaccine has no effect µnv

v X v k n−k = ǫ (1 − ǫ)v−k µn−k (λL ) λG E[TI ] n k k=0 | {z } | {z } | {z } (1)

(2)

(1)

P(k vaccinations are successful)

(2)

P(globally contacted individual is susceptible)

(3)

Mean size of single-household epidemic

(3)


Variola Minor, Sao Paulo, 1956 Data comprise final numbers infected in each of 338 households. Household size varied from 1 to 12 (mean = 4.56) Each individual labelled vaccinated or unvaccinated 773 unvaccinated — 425 infected (58%) 809 vaccinated — 85 infected (11%) Fit households SIR model with non-random vaccine response, assuming infectious period TI ≡ 1, using pseudolikelihood method of Ball and Lyne (2010) to obtain the estimates ˆ L = 0.3821, λ ˆ G = 1.4159, aˆ = 0.1182, ˆb = 0.8712 λ


Comparison of vaccination strategies 7 Random individuals Random households Optimal Worst R =1

6

v

5

4 Rv 3

2

1

0

0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Proportion of individuals vaccinated c

0.8

0.9

1


Concluding comments General framework for determining threshold behaviour and final outcome of stochastic SIR epidemics with two levels of mixing. Local (e.g. household) structure matters! Significant impact on threshold and final outcome. Consequent impact on performance of vaccination schemes. Explicit calculation is possible only in a few special cases — need to find other local structures which are both practically relevant and mathematically tractable. Can relax symmetries and/or consider multitype epidemics. Non-SIR models SIS households and great circle SIR households with demography? FADE OUT Epidemics with two levels of mixing – p.42

References Andersson, H. (1999) Epidemic models and social networks. Math. Scientist. 24, 128–147. Ball, F. G. and Lyne, O. D. (2002) Optimal vaccination policies for stochastic epidemics among a population of households. Math. Biosci. 177-178,333–354. Ball, F. G. and Lyne, O. D. (2006) Optimal vaccination schemes for epidemics among a population of households, with application to variola minor in Brazil. Stat. Method Med. Res. 15, 481–497. Ball, F. G. and Lyne, O. D. (2010) Statistical inference for epidemics among a population of households. J. R. Statist. Soc.B, under revision. Ball, F. G., Mollison, D. and Scalia-Tomba, G. (1997) Epidemics with two levels of mixing. Ann. Appl. Probab. 7,46–89. Ball, F. G. and Neal, P. J. (2002) A general model for stochastic SIR epidemics with two levels of mixing. Math. Biosci. 180, 73–102. Ball, F. G. and Neal, P. J. (2003) The great circle epidemic model. Stoch. Proc. Appl. 107, 233–268. Ball, F. G. and Neal, P. J. (2008) Network epidemic models with two levels of mixing. Math. Biosci. 212, 69–87. Epidemics with two levels of mixing – p.43

References ´ Bartoszynski, R. (1972) On a certain model of an epidemic. Applic. Math. 13, 139–151. Becker, N. G. and Dietz,K. (1995) The effect of household distribution on transmission and control of highly infectious diseases. Math. Biosci. 127,207–219. Becker, N. G. and Starczak. (1997) Optimal vaccination strategies for a community of households. Math. Biosci. 139,117–132. Becker, N. G. and Starczak. (1998) The effect of random vaccine response on the vaccination coverage required to prevent epidemics. Math. Biosci. 154,117–135. Bollobás, B. (2001) Random graphs. Cambridge University Press. Diekmann, O., De Jong, M. C. M. and Metz, J. A, J. (1998) A deterministic epidemic model taking account of repeated contacts between the same individuals. J. Appl. Prob. 35, 448–462. Kiss, I., Green, D. and Kao, R. (2006) The effect of contact heterogeniety and multiple routes of transmission on final epidemic size. Math. Biosci. 203, 124–136. Newman, M. (2002) Spread of epidemic disease on networks. Phys. Rev. E 66, 016128. Scalia-Tomba, G. (1985) Asymptotic final size distribution for some chain-binomial processes. Adv. Appl. Prob. 17, 477–495.


Epidemics with two levels of mixing

Epidemics with two levels of mixing

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