Probability of a Major Outbreak for Heterogeneous Populations

Math. Biol. Group Meeting

26 April 2005

Joanne Turner and Yanni Xiao

Previously for 1-Group Model (Homogeneous Case)

• Roger showed that 4 different threshold conditions are equivalent

i.e.

where

– R0 is basic reproduction ratio (number of secondary cases per

primary in an unexposed population)

– z is probability of ultimate extinction (probability pathogen will

eventually go extinct)

– r is exponential growth rate of incidence i(t)

– s() is proportion of the original population remaining

susceptible.

1)( 0 1 10 srzR

1-Group Model: Theory of Probability of Major Outbreak

• When there are a infecteds at time t = 0,

prob. of ultimate extinction =

prob. of major outbreak =

• As Roger showed, q is the unique solution in [0,1) of

• If G = number of new infections caused by 1 infected individual

during its infectious period.

and pG = prob that 1 infected produces G new infections,

then

aq1

aa qz

equivalent to z = g(z) in Roger’s slides

)(qfq

G

G

GG qEqpqf

0

)(

generating function

1-Group Model: Calculation of Probability of Ultimate Extinction

• number of new infections created by 1 infectious individual

= direct transmission parameter

– X* = disease-free equilibrium value for the number of susceptibles

– T = infectious period

• Therefore

– where = X*(1-q) (i.e. is a function of q)

TXG *Poisson ~

TqTXTqE G exp)1(exp *

Poisson distribution dictates this form

TEqEqf G exp)( taking the expectation removes the condition on T

average number of new infs

1-Group Model: Calculation of Prob. of Ultimate Extinction (cont.)

• Infectious period

= rate of loss of infected individuals (i.e. death rate + recovery rate)

• p.d.f. is

• Now need to solve

teth )(

dteeeEqf ttt )(0

1 lExponentia~T

)1(* qXq

average infectious period

1-Group Model (Homogeneous Case)

• We find that

• probability of a major outbreak (when R0 > 1)

where a = initial number of infectious individuals

1 if 1

1 if 1

0

00

Rzq

RR

zq

aa

Rq

0

111

This is NOT true for multigroup models

4-Group Model: Prevalence Plots

• Herd size affects persistence of infection and, hence, probability of a major outbreak.

• Same is true for 1-group models (previous results only true for large N).• When we start with 1 infected (i.e. invasion scenario), average prevalence

for stochastic model does not tend to deterministic equilibrium.

stoch, N = 1120

deter, N = 112

stoch, N = 11200

stoch, N = 1120 300 600 900 1200 1500

time

0.00

0.02

0.04

0.06

0.08he

rd p

reva

lenc

e4-group dairy model

4-Group Model: Estimate of Probability of Major Outbreak

• Prob. of major outbreak

• Stochastic prevalence level depends on proportion of minor outbreaks (long-term zeros drag down the average).

In previous example:

stochastic level

deterministic equilibrium

Further increases in N indicate that the prob. major outbreak tends to a limit of approx 0.14.

prop. sims with prev > 0

stoch prev(t = 1500)

prob major outbreak (est)

N = 112

N = 1120

N = 11200

0 / 100

11 / 100

14 / 100

0

0.0065

0.0105 0.138

0.086

0

results for t = 1500

4-Group Model: Theory of Probability of Major Outbreak

According to Damian Clancy,

• prob. of major outbreak =

– (aU, aW, aD, aL) are numbers of infecteds in each group at time t = 0.

– is the unique solution in [0,1)4 of

• generating function is

– are numbers of new infections in each

group caused by an infected individual that was initially in group i.

– are variables of generating function f.

LDWU aL

aD

aW

aU qqqq1

),,,( LDWU qqqqq )(qfq

)(

)(iG

i sEsf

),,,( )()()()()( iL

iD

iW

iU

i GGGGG

4)1,0[s

need q and a for each group

4-Group Model: Theory of Probability of Major Outbreak

Direct transmission:

• Number of new infecteds in group j created by an infected initially in group i is

j = direct transmission parameter for group j

– Xj* = disease-free equilibrium value for group j

– Tj(i) = time spent in group j by an infected initially in group i

• Therefore

Repeat for indirect transmission (much more complicated) and pseudovertical transmission [see Yanni’s paper for full details].

)(*),( Poisson ~ ijjj

dirij TXG

jj

ijjj

iGL

GD

GW

GU sTXTssssE

diriL

diriD

diriW

diriU )1(exp )(*)(),(),(),(),(

4-Group Model: Theoretical Result

• Theory is only true for large N. Therefore, it gives the upper limit for the probability of a major outbreak.

•

• For previous example:– upper limit for prob major outbreak = q = 0.145. – upper limit for prevalence = 0.011.

prop. sims with prev > 0

prev(t = 1500)

prob major outbreak (est)

N = 112

N = 1120

N = 11200

0 / 100

11 / 100

14 / 100

0

0.0065

0.0105 0.138

0.086

0

upper limit prevalence

upper limit prob major outbreak

deterministic equilibrium prevalence

= x

results for t = 1500

4-Group Model: 1 – qW versus 1 – 1/R0

• 1-group model with a = 1: 1 – q = 1 – 1/R0

• 4-group model with aW = 1 and aU = aD = aL = 0: 1 – qW 1 – 1/R0

e.g. from Yanni’s paper

Conclusions

• Herd size affects persistence of infection and, hence, probability of a major outbreak.

• Theory is only true for large N. Therefore, it gives the upper limit for the probability of a major outbreak.

• 1-group model with a = 1: 1 – q = 1 – 1/R0

• 4-group model with aW = 1

and aU = aD = aL = 0: 1 – qW 1 – 1/R0