Spatial considerations for the allocation of pre-pandemic
influenza vaccination in the United States
Joseph T Wu*†, Steven Riley*, Gabriel M Leung*
*
Department of Community Medicine and School of Public Health, Li Ka Shing Faculty
of Medicine, The University of Hong Kong, Hong Kong SAR, China
†
To whom correspondence should be addressed: [email protected]
Supplementary Information: Text and Figures
Reduction in IAR is a convex function of coverage
We first use the simple homogeneous mixing model to illustrate why the relation between
reduction in IAR and coverage is convex as shown in Figure 1B. For simplicity, we
assume that the vaccine provides 100% reduction in susceptibility. The attack rate a (c )
for a given coverage c can be obtained by solving the following equation (Diekmann and
Heesterbeeck, 2002) :
a(c)  1  c  1  e  R0a ( c ) 
The second derivative of a (c ) can be easily obtained as follows:


 R0 a ( c )
e  R0 a ( c ) 
1   R0 1  c  
d 2 a (c) R0 1  e

1



2
 R0 a ( c )
2

dc
1  c  
1  R0 e  R0 a ( c ) 1  c   1  R0 e

Since

1
1
d 2 a (c )
d 2 a(c)
c

1

and
is
the
only
root
of
in 0  c  1 
, a (c ) is

0
2
2
R0
R0
dc
dc c 0
a concave function of c . Since the number of infections averted is  a(0)  a(c)  N ,
where N is the population size, it is a convex function of coverage c . Computational
results suggest that this convex relation also holds in the vaccine response model used in
this study (see below): We randomly generate 1,000,000 sets of parameters from all
possible parameter values with up to 10 susceptible states ( nS  10 ) and convexity holds
in all these scenarios.
Mathematical details for the example in the Results section: Why is prorata the least efficient policy?
Here, we provide detailed mathematical argument as to why pro-rata is the least efficient
policy for the example in the introduction. Consider K populations of sizes
N1 , , N K that are isolated from each other. Let r (c ) be the reduction in IAR achieved by
vaccine coverage level c . All populations have the same r (c ) , which is convex and
becomes flat only after the critical coverage level c * has been achieved (Figure 1B).
Given V doses of vaccines and an allocation vector α  (1 , ,  K ) , where i is the
proportion of the overall stockpile allocated to population i , the total number of
infections averted is T (α)  N1r (1V N1 )   N K r ( KV N K ) . Note that we must have
K 1
 K  1    i . Substituting this relation into T , we have
i 1
T
T
(α )  V  r ( iV N1 )  r ( KV N K )  . Since
(p)  V  r (V N )  r (V N )   0 , p is
 i
 i
a stationary point of T . Further, since r (c)  0 ( r (c ) is convex when c  c* ), we have
V2
 N r ( KV N K )  0 if i  j ,
 2T
 K
(α )  
 i  j
V 2  1 r ( V N )  1 r ( V N )   0 otherwise.
i
i
K
K 
  N i
NK


Therefore, T is a convex function of α . Taken together, T is minimized at p , which is
the pro-rata allocation. That is, pro-rata allocation gives the least number of infections
averted and is thus the least efficient policy.
Mathematical details for the example in Figure 4: Why is it never
optimal to split the stockpile unless at least one population has achieved
control?
Consider the example in Figure 4 where Population 1 and 2 are non-interacting. For
i 1, 2 , let wi (v) be the number of infections averted in Population i if v doses of
vaccines are allocated to it, and vi* be the number of doses needed to reach critical
coverage for Population i. Note that wi (v)  Ni r (v Ni ) and vi*  c* Ni (see Figure 1B)
where r (c ) is the same reduction in attack rate for coverage c defined above. Given V
doses of vaccines, let WV ( y)  w1 ( yV )  w2 ((1  y)V ) be the total number of infections
averted if a proportion y of the overall stockpile is allocated to Population 1. The goal is
to minimize the total number of infections. When V is not large enough to achieve
control simultaneously in both populations, which is necessary for the allocation problem
to be worthwhile, we only need to consider allocations y  ymin , ymax  where




ymin  max 0,1  v2* V and ymax  min 1, v1* V . In this range, w1 and w2 are strictly
convex (i.e. wi(v)  0 ), so WV( y)  V 2  w1( yV )  w2((1  y)V )  0 . This means that all
stationary points of TV ( y ) are minima and TV ( y ) is maximized at either y  ymin
or y  ymax . If y  ymin , either all vaccines are allocated to Population 2 ( ymin  0 ) or
Population 2 has achieved control ( ymin  1  v2* V ). Similarly, if y  ymax , either all
vaccines are allocated to Population 1 ( ymax  1) or Population 1 has achieved control
( ymax  v1* V ). Taken together, these observations imply that it is never optimal to split
the stockpile unless at least one population has achieved control. These observations also
explain concentration of resources and choppiness (discontinuity in allocation arises
when the optimal allocation switches between y  ymin and y  ymax ).
The Meta-population Model Without Vaccination
Let Si , I i , and Ri be the number of susceptible, infectious, and removed individuals in
population i where i  1,
, K . Given a mixing matrix M  mij  , where mij is the average
proportion of time that a resident of population i spends in population j (see next section),
the model without vaccination is defined by
Total number of infectives weighted
by time spent in Population j
Transmissibility
in Population j
K
dSi
  Si 
dt
j 1
mij
Proportion of time
that a Population i
individual spends
in Population j


j
K
l 1
K
mlj I l
,
m N
l 1 lj l
Total number of individuals weighted
by time spent in Population j
dI i
dS
I
 i  i ,
dt
dt DI
dRi
I
 i ,
dt
DI
for i, j  1, , K , where DI is the mean infectious duration. Since we are only concerned
about the final attack rates, omitting the exposed class, which corresponds to the latent
period, has no effects on our results.
Inter-population Mixing: Construction of the Mixing Matrix
The M matrix. We constructed a K  K mixing matrix M  mij  where mij is the average
proportion of time that a resident of population i spends in population j over one year. To
this end, we first construct the following matrices:
 A  aij  where aii  0 and aij , i  j , is the number of air person-trips from
population i to population j over one year.

W  wij  where wii  0 and wij , i  j , is the number of individuals who reside
in population i and commute to work in population j on a daily basis.

L  lij  where lii  0 and lij , i  j , is the total duration (in days) of non-air

person-trips from population i to population j over one year with travel
distance at least 100 miles.
D  dij  where dii  0 and d ij , i  j , is the average duration of an air person-
trip from population i to population j .
Construction of these matrices is documented below. We estimated the matrix
M  mij  by
Total person
Number of days
 Number of i  j

days of i  j
at work per year
person-trips via
long distance

 air

travel
non-air travel


1
8 5 

aij
dij  lij
 wij  365     for i  j ,


24 7  

 365 N i 
Duration
Number
mij  
per
i

j
of
i

j


Total number
air trip
commute
 of person-days 

i
 inin Population
one year

1   mik otherwise.
 ik
The A matrix: Air Travel Volume. We estimate the US domestic air travel volume
using the 2005 T-100 Domestic Market and the 2005 Airline Origin and Destination
Survey (DB1B) from the Bureau of Transportation Statistics
(www.transtats.bts.gov/Tables.asp?DB_ID=125&DB_Name=Airline%20Origin%20and
%20Destination%20Survey%20(DB1B)). DB1B is 10% sample of airline tickets from
reporting carriers collected by the Office of Airline Information of the Bureau of
Transportation Statistics. Data include origin, destination, and other itinerary details of
passengers transported for each quarter. On average, about 70% of these itineraries
(weighted by the number of passengers in each itinerary) are roundtrips. A unique
destination can be identified for more than 99.5% of these roundtrips (the others have
multiple stops before returning to the origin). For these trips, let bijk be the number of
passengers traveled from population i to population j and ck be the total number of
coupons during quarter k . Let pk be the total number of passengers during quarter k (as
reported in T-100). We estimated the matrix A by
4
p
aij   bijk k for i  j. .
ck
k 1
The W matrix: Workflow Volume. We estimate workflows using the United States
Census 2000 County-To-County Worker Flow Files
(www.census.gov/population/www/cen2000/commuting.html). We choose from the
database only residence-workplace pairs for which the distance between the two counties
is less than 200 km; these account for 99% of all residence-workplace pairs and 83% of
those for which residence and workplace are in different states. We compute the distances
using the latitudes and longitudes provided in the Census 2000 US Gazetteer Files
(www.census.gov/geo/www/gazetteer/places2k.html).
The L matrix: Long-Distance Non-Air Travel Volume. We estimate the long-distance
non-air travel volume using the 1995 American Travel Survey
(www.bts.gov/publications/1995_american_travel_survey/index.html). We set lij  0 if
the number of reported (i, j ) non-air person-trips is positive but less than 15; these origindestination pairs account for only 1.2% of non-air person-trips in the database.
The D matrix: Air-Travel Trip Average Duration. We estimate the interstate air-travel
trip average durations using the 1995 American Travel Survey
(www.bts.gov/publications/1995_american_travel_survey/index.html). Let S be the set of
(i, j ) pairs for which the number of reported air person-trips is positive but less than 15;
these account for 10.8% of the air person-trips in the database. Let d oj and d dj be the
average duration of air person-trips with state j as the origin and destination, respectively,
in the database with S excluded. Let d be the average duration of all air person-trips other
than those in S . We estimate d ij , (i, j )  S , by
 d dj  d oj
if d dj  0 and d oj  0,

2

d
o
dij  d j if d j  0,
d o if d d  0,
j
 j
d otherwise.
Vaccine Efficacy Model
After an individual is vaccinated, his susceptibility and infectiousness (if he becomes
infectious) are reduced by a factor of Z S and Z I , respectively. We assume that
Z S and Z I are independent random variables. We denote the different states of
Z S and Z I in ascending order by z kS , k  1,..., nS and zlI , l  1,..., nI , respectively. We say
that an individual is of class  k , l  , k  1,..., nS and l  1,..., nI , if his relative susceptibility
and relative infectiousness are 1  zkS and 1  zlI , respectively. Let PZ S , Z I  k , l  be the
probability of an individual becoming class  k , l  upon vaccination.
The Meta-population Model With Individual Vaccine Response
Let Sikl be the number of class  k , l  susceptible in population i , where i  1,
k  1,
, nS , l  1,
,K ,
, nI . Define I ikl and Rikl similarly. The full meta-population model is
Total umber of infectives in Population d
weighted by their relative infectiousness
 d 1 mdj
K

dSikl
  Sikl 1  zkS
dt

K
 mij  j
j 1
relative
susceptibility
dI ikl
dS
I
  ikl  ikl ,
dt
dt
DI
dRikl I ikl

,
dt
DI
where i  1, , K , k  1, , nS , l  1,
, nI .
nS
nI
 1  z  I
k '1 l '1
K
 d 1 mdj N d
I
l'
dk ' l '
,
Algorithm for Calculating Attack Rates
We develop an algorithm to calculate the attack rates. This algorithm allows us to obtain
the attack rates by solving a system of K  K nonlinear equations regardless of the
number of immune states introduced in the vaccine response model. In the following, we
first illustrate the algorithm for a single population and then generalize to the metapopulation model.
Attack Rate in a Single Population. Consider a single population of size N . Let
X kl and U kl be the initial number of class  k , l  individuals and final number of infections
nS
in class  k , l  , respectively. The attack rate
nI
U
k 1 l 1
N
kl
can be obtained by solving for U kl ’s
in






nS nI



S R0
I
U kl  X kl 1  exp   1  zk   1  zl ' U k 'l '   , 1  k  nS , 1  l  nI
N k '1 l '1

 Relative

force of infection over the 
 susceptibility Total

entire course of the epidemic

 
(1)
Probability of getting infection during the epidemic
Y
, the initial number of class  k , l  individuals are
N

 N  Y  YPZ S , Z I  k , l  if k  1, l  1,
X kl  
otherwise.

YPZ S , Z I  k , l 
The numbers of infections U kl ’s can be computed using the following algorithm:
Given a vaccine coverage of
1. Express U kl in terms of U11 as follows:
  U 1 zk 

 U11 
11
  ln 1 
  U kl  X kl 1  1 
  . (2)
X
X
 

11 
11 




d ln S kl
d ln S11
 1  zkS 
This relation is obtained using the relation
which is true
dt
dt
for all 1  k  nS and 1  l  nI .
2. Solve for U11 in the Equation (1)
 U
(1  zks ) ln 1  kl
 X kl
S

 R nS nI

U11  X 11 1  exp   0  1  zlI U kl  
 N k 1 l 1


where the double summation can be expressed as a function of U11 only using
Equation (2) as follows:
nS
nI
 1  z U
k 1 l 1
I
l
kl
N

 U 
 U 
U11

 Y  E  Z I   1  11   Gˆ Z S 1  11  1  E  Z I 
X 11
 X 11 

 X 11 



 

where Gˆ Z S  x   E  x1 Z  .


3. Using the solution of U11 obtained in Step 2, the total number of infections (again,
using Equation (2)) is
nS nI
 U 
 U 
U
U kl  N 11  Y 1  11   Gˆ Z S 1  11   .

X 11
k 1 l 1
 X 11  
 X 11 
S
From this algorithm, we see that U kl ’s, and therefore the attack rate, depend on Z I via
only P  Z S  0, Z I  0  and E  Z I  .
Attack Rates in the Meta-Population. Let X ikl and U ikl be the initial number of
class  k , l  individuals and the final number of infections in class  k , l  in population i ,
respectively. As in the case for a single population, the attack rate in a meta-population
K
nS
nI
 U
i 1 k 1 l 1
K
N
i 1
ikl
can be obtained by solving for U ikl ’s in the following system of equations:
i




K


S


U ikl  X ikl 1  exp  1  zk  
mij
DI  j


j 1
Proportion of time
relative

 susceptibility
that a Population i
individual spends


in Population j



K
nS
nI
I
m
1

z
U


 d 1 dj  k '1  l '1 l ' dk 'l ' 
K

 d 1 mdj N d

Total force of infection in Population j

over the course of the epidemic

Probability of a Population i individual getting infected during the epidemic
for 1  i  K , 1  k  nS , 1  l  nI
The numbers of infections U ikl ’s can be computed using the algorithm for a single
population but with the following modification: In Step 2, solve for U i11 ’s in the
following system of equations:
K
n
n

 K
mdj  kS'1  l 'I1 1  zlI' U dk 'l '  

d

1
 , 1  i  K .
U i11  X i11 1  exp   mij DI  j
K

 j 1

m
N
 d 1 dj d



The Next Generation Matrix and Reproductive Number
The next generation matrix is used to compute the reproductive number. For a completely
susceptible meta-population, the next generation matrix is a K  K matrix  Rij  where Rij is
the expected number of secondary infections in the fully susceptible population i resulting
from a randomly selected infectious individual in population j (Heesterbeek, 2002). For
our meta-population model,
K
m N
.
Rij  DI  m ja b K ia i
m
N
a 1
 b1 ba b
The basic reproductive number R0 is equal to the largest eigenvalue of the next
generation matrix. With vaccination, the next generation matrix becomes a
KnS nI  KnS nI matrix Rij( a ,b )( k ,l )  where Rij( a ,b )( k ,l ) is the expected number of secondary
infections among class  a, b  individuals in population i from a randomly
selected  k , l  infectious individual in population j , where
Rij( a ,b )( k ,l )  Rij
X iab
1  zaS 1  zlI  :
Ni
 R11(1,1)(1,1)
 (1,2)(1,1)
 R11

 (1,1)(1,1)
R
R   21(1,2)(1,1)
R
 21

 R ( nS ,nI 1)(1,1)
 K1
( nS , nI )(1,1)
 RK 1
.


R



(1,1)( nS , nI )
R2 K

(1,2)( nS , nI )

R2 K


( nS , nI 1)( nS , nI ) 
RKK

( nS , nI )( nS , nI )

RKK
R11(1,1)(1,2)
R12(1,1)(1,1)
R12(1,1)(1,2)
nS , nI 1)
R1(1,1)(
K
nS , nI )
R1(1,1)(
K
(1,2)(1,2)
11
(1,2)(1,1)
12
(1,2)(1,2)
12
(1,2)( nS , nI 1)
1K
(1,2)( nS , nI )
1K
R
R
R
R
R21(1,1)(1,2)
R22(1,1)(1,1)
R22(1,1)(1,2)
nS , nI 1)
R2(1,1)(
K
R21(1,2)(1,2)
R22(1,2)(1,1)
R22(1,2)(1,2)
nS , nI 1)
R2(1,2)(
K
RK( n1S ,nI 1)(1,2)
RK( n2S , nI 1)(1,1)
RK( n2S , nI 1)(1,2)
( nS , nI 1)( nS , nI 1)
RKK
RK( n1S , nI )(1,2)
RK( n2S , nI )(1,1)
RK( n2S , nI )(1,2)
( nS , nI )( nS , nI 1)
RKK
Minimizaton of Post-vaccination Reproductive Number
Pro-rata is Near-Optimal. We first use a special case of our meta-population model to
illustrate why pro-rata almost minimizes the post-vaccination reproductive number.
Suppose there are only two populations with sizes N1 and N 2 . The two populations have
the same local transmissibility  with a mixing matrix
m12  1  
 
m
M   11


1   
 m21 m22   
for an arbitrary  between 0 and 1. Suppose the vaccine provides 100% reduction in
susceptibility and let 1 be the proportion of a vaccine stockpile of size V allocated to
Population 1. The next generation matrix after vaccination is
m11 (1  c1 ) N1
m12 (1  c1 ) N1

 m11 m N  m N  m12 m N  m N
11 1
21 2
12 1
22 2
 DI 
m21 (1  c2 ) N 2
m22 (1  c2 ) N 2

 m11 m N  m N  m12 m N  m N

11 1
21 2
12 1
22 2
m11 (1  c1 ) N1
m (1  c1 ) N1 
 m22 12
m11 N1  m21 N 2
m12 N1  m22 N 2 

m21 (1  c2 ) N 2
m22 (1  c2 ) N 2 
m21
 m22
m11 N1  m21 N 2
m12 N1  m22 N 2 
m21
where ci is the coverage in Population i . The reproductive number is the maximum
eigenvalue of this next generation matrix. Therefore, the reproductive number is
minimized by finding a 1 that minimizes this eigenvalue. Straightforward calculations
N1
show that the reproductive number is minimized when 1 
, which is the proN1  N 2
rata allocation. Supplementary Figure 5A shows that in the 10-region US model, the
reproductive number under pro-rata is larger than the optimal by no more than 15%.
Supplementary Figure 5B shows that allocations that minimize the attack rate do not
minimize the reproductive number.
Meta-population with the Same Reproductive Number can have Different Attack
Rates. To show this, consider a simple 2-population epidemic system where the two
populations have the same size:
dS1
  11S1 I1  12 S1 I 2
dt
dS 2
   21S 2 I1   22 S 2 I 2
dt
dI1
dS
I
 1 1
dt
dt DI
dI 2
dS
I
 2  2
dt
dt DI
We randomly generate 10,000 transmission matrix   ij  as follows: ij  rxij where xij ’s
are i.i.d. U(0,1) random variables and r is a scaling factor such that the resulting matrix
 ij  gives a reproductive number of 1.8. Supplementary Figure 6 shows that these
simple systems have different attack rates even though they have the same reproductive
number.
Supplementary Figure 1. IAR(100) as a function of basic reproductive number R0 (yaxis) and coverage c (x-axis) in the 4-region and 49-state model. The black contour
marks the boundary below which IAR(100)  6%. A. The 4-region model. B. The 49state model.
Supplementary Figure 2. Reduction in attack rate IAR(100) for the 100%
discretionary policy in the 10-region model when there is no inter-regional mixing (i.e.
regions are isolated from each other). The black contour marks the boundary below
which IAR(100)  6%.
Supplementary Figure 3. Univariate (A-C) and multivariate (D) sensitivity analyses for
the 50% discretionary policy in the 10-region model. (A-C) Coverage is fixed at 20%.
The black contours in each panel mark the boundary at which IAR(50)  0 and the black
dashed vertical line indicates the base case parameter values. A. IAR (50) drops as interregional mixing (  ) increases. B. IAR (50) drops as antigenic match (  ) deviates from
the base case value. IAR (50) can become negative when antigenic match is near perfect.
C. IAR (50) is mostly unaffected when transmissibility is overestimated (   0 ) but
drops to significantly negative values when transmissibility is overestimated (   0 ). D.
Multivariate sensitivity analysis. When base case assumptions (used to construct the
discretionary policies) are not satisfied, the actual IAR (50) is typically lower than the
IAR (50) expected in the base case (see caption of Figure 5B on how the multivariate
analysis is performed).
Supplementary Figure 4. The relation between reduction in IAR and coverage when
there are two high risk groups and vaccination priority is given to the group with higher
infectiousness (see Discussion in main text). c R is the proportion of high-risk group in
the population and c * is the critical coverage.
Supplementary Figure 5. A. The ratio between reproductive number under pro-rata and
the optimal reproductive number. B. The ratio between reproductive number under the
100%-discretionary policy and the optimal reproductive number.
Supplementary Figure 6. Meta-populations with the same reproductive number can
have different attack rates. 10,000 different transmission matrices are randomly generated.
The quantity q   12   21   i , j ij plotted on the x-axis is a proxy of the level of interpopulation mixing. This figure suggests that the higher the level of inter-population
mixing, the narrower the distribution of attack rates for a given reproductive number.
References
Diekmann, O. & Heesterbeeck, J. A. P. (2002) Mathematical Epidemiology of Infectious
Diseases: Model Building, Analysis and Interpretation, New York, Wiley.
Heesterbeek, J. A. P. (2002) A brief history of R-0 and a recipe for its calculation. Acta
Biotheoretica, 50, 189-204.