Prioritizing Risks and Uncertainties from
Intentional Release of Selected Category A
Pathogens
Tao Hong*1, Patrick L. Gurian2, Yin Huang3, and Charles N. Haas2
1. National Exposure Research Laboratory, EPA, Athens, GA, USA, 2. Department of
Civil, Architectural, and Environmental Engineering, Drexel University, Philadelphia,
PA, USA, 3. Office of Biostatistics and Epidemiology, FDA, Rockville, MD, USA
*[email protected]
SUPPORTING INFORMATION
Supporting Information S1
The simplification of risk assessment model
1
Supporting Information S1
This supporting information describes how the system of equations given by Equation
1 in the main body of the paper can be simplified to allow reduced form solutions to be
developed.
Retrospective scenario
In the retrospective scenario, resuspension makes a negligible contribution to dose,
due to the relatively short exposure period compared the rates of resuspension (in this
case the exposure period is assumed to be is 8 hours, the duration of a working day).
Neglecting re-suspension separates the air compartment from the effects of other
compartments so that air concentration follows a simple first order decay model. The
inhaled dose can be calculated as:
a
doseinh_retro 
Inh
M airo e retrot1 dt1
Vol 0
Inh
1 retrot

 M airo 
e
Vol
retro
t1
0

Inh
1

 M airo 
1  e retro 480
Vol
retro

(Eq. A)
where
retro  ts  tf  utf  w 
Inhen
Q
  air  [1  (1  e) p]
Vol
Vol
(Eq. 31)
The ingested pathogens (doseing_retro) are the organisms which deposit on the touched
surface (Mts) from the initial release in the air (Mair0) (Equation B):
a
M ts   ts M airo e retrot1 dt1
(Eq. B)
0
2
remain alive until being transferred to the hand (Mhand) (Equation C):
a
M hand   rsh f sh M ts e
 pros t2
dt2
(Eq. C)
0
and are ingested (doseing_retro) during surface-hand-mouth contact in the exposure period
(Equation D):
a
doseing_retro   rhm f hm M hand e hand t3 dt3
0
a

 t
  rhm f hm   rsh f sh M ts e pros 2 dt2 e hand t3 dt3
0
0

a
a
a
a
  t 
=  rhm f hm   rsh f sh   ts M airo e retrot1 dt1  e pros 2 dt2 e hand t3 dt3
0
0

 0

(Eq. D)


 1
1
 a  1
 rhm f hm  rsh f sh  M airo ts
(1  eretro a ) 
(1  e pros ) 
(1  e hand a )
retro

 hand

  pros
where
 pros  2   fomite  rsh f sh
(Eq. 32)
hand  rhm f hm + fomite + rhs f hs
(Eq. 33)
To simplify the calculation, three assumptions are made during the derivation of
Equation D: 1) pathogen resuspension and back transfer from hands to the surface are
omitted due to their relatively low rates resulting in small fractions being back transferred,
which is also health conservative; 2) all integration steps are from t=0 to t=a which
provides an upper bound on the amount of pathogen transferred to the next step; 3) the
pathogens will not be transferred to hands until depositing on the touched surface, and the
pathogens will not be ingested until they are transferred to hands.
Prospective scenario
3
In the prospective scenario, the majority of the inhaled dose (doseinh_pros) comes from
two sources (Figure S1). The first source consists of organisms that are inhaled right after
being resuspended (doseinh1_pros) (Equation E).
a

Inh 
 pros t1
dt1  eretrot2 dt2
  2 M tsoe
Vol  0
0

a
doseinh1_pros  
(Eq. E)
The second are those organisms which experienced a certain number of "surfacehand-surface" travels before being resuspended and inhaled doseinh2_pros (Equation F).
a

Inh 
 pros t1
dt1  eretrot2 dt2
  2M tsoe
Vol  0
0

a
doseinh 2_pros  
(Eq. F)
where Θ is the total fraction of resuspended pathogens surviving a number of n "surfacehand-surface" cycles:
a

a

 t
     rhs f hs   rsh f sh M tso e pros a1 dta1  e hand tb1 dtb1 
n 1 

0

0

n
(G)
Θ is composed as a summation of a geometric series with element of Θn, where n indexes
the number of "surface-hand-surface" cycle. In the nth cycle, pathogens survived from the
n-1th cycle (Θn-1Mtso) are first transferred to hands (Equation H), and then back transferred
to the surface (Equation I).
a
n _ sh   rsh f sh n 1M tso e
 pros tan
dtan
(H)
0
a
n  n _ sh n _ hs   rhs f hs n _ sh ehand tbn dtbn
(I)
0
Combine Equation H and I:
4
a
a

n
 t
n   rhs f hs   rsh f sh n1e pros an dtan  ehand tbn dtbn   1 
0
0

(J)
For both sources, pathogen resuspension happens before inhalation. The maximum
inhalation dose is reached when the exposure duration goes to infinity (Equation K):
doseinh _ pros  doseinh1_ pros  doseinh 2 _ pros
a

Inh 
 pros t1
dt1  eretrot2 dt2
  2 (1  ) M tso e
Vol  0
0

a

n
a

a
a
 hand tb1  
Inh    
 retrot2
 pros ta1
 pros t1


1

r
f
r
f
M
e
dt
e
dt
M
e
dt
dt2
  2     hs hs   sh sh tso
a1 
b1  
tso
1e
Vol
0
0

 0  n 1  0

 
(Eq. K)
a 
2 Inh 1  rsh f sh rhs f hs
 M tso

 pros Vol retro n0  pros hand
a
2 Inh 1
 Vol retro
 M tso pros
r f r f
(1  sh sh hs hs )
 pros hand
Similarly, the prospective ingestion dose (doseing_pros) comes from two sources
(Figure S2). The first source is direct ingestion of the pathogens released on the touched
surface (doseing1_pros) (Equation L):
a
a

 t
doseing1_pros   rhm f hm   rsh f sh M ts 0e pros 1 dt1 ehand t2 dt2
0
0

a
where  rsh f sh M ts 0 e
 pros t1
(Eq. L)
dt1 is the mass transferred to victims hand.
0
The second source is those organisms which experience a certain number of "surfacehand-surface" travels before being ingested (doseing2_pros) (Equation M), which contains
the common factor Θ as described above. The maximum ingestion dose is reached when
exposure goes to infinity (Equation N).
5
a
a

 t
doseing 2_pros   rhm f hm   rsh f sh M ts 0e pros 1 dt1 e hand t2 dt2
0
0

(Eq. M)
doseing_pros  doseing1_ pros  doseing 2 _ pros
a
a

 t
  rhm f hm   rsh f sh (1  ) M ts 0e pros 1 dt1 e hand t2 dt2
0
0

n
a

a

 a
 
a




 t
 t
  rhm f hm   rsh f sh 1     rhs f hs   rsh f sh M tso e pros a1 dta1  e hand tb1 dtb1   M ts 0 e pros 1 dt1 e hand t2 dt2
n

1


0
0

 0

0
 

a 
 M tso
rsh f sh rhm f hm
 pros hand

rsh f sh rhs f hs

n 0
(Eq. N)
hand
pros
rsh f sh rhm f hm
 M tso
 pros hand
r f r f
(1  sh sh hs hs )
 pros hand
Table S1 compares the exposure dose approximated by the above-mentioned
equations with the exact results from solving Equation 18. The overall risk is acquired by
inputting inhalation and ingestion doses into Equation 14 separately for the retrospective
(Equation 26) and the prospective scenario (Equation 27).
riskretro  1  (1  riskinh_retro )(1  riskinh_retro )
 1  (1  k  doseinh_retro )(1  k  doseing_retro )

Inh 1
 1  1  k  M airo

1  eretro 480
Vol retro



 1  k  M
 



 1
1
 480  1
r f  rsh f sh  ts
(1  e retro 480 ) 
(1  e pros ) 
(1  ehand 480 ) 

 hand

 retro
  pros
airo hm hm
(Eq. 26)
risk pros  1  (1  riskinh_pros )(1  riskinh_pros )
 1  (1  k  doseinh_pros )(1  k  doseing_pros )
 1  (1  k  M tso
2 Inh 1
 pros Vol retro
(1 
rsh f sh rhs f hs
 pros hand
rsh f sh rhm f hm
)(1  k  M tso
)
 pros hand
(1 
rsh f sh rhs f hs
 pros hand
)
(Eq. 27)
)
6
Figure S1. Pathogen flow for estimating the inhalation dose in the prospective scenario.
7
Figure S2. Pathogen flow for estimating the ingestion dose in the prospective scenario.
8
Table S1. Comparison Exposure Dose between Approximated Analytical Equation and Simulated Results
(1 µm)
Release
Inhalation dose
Ingestion dose
Pathogen
scenario
Approximated
Full numerical
Approximated
Full numerical
analytical equation
simulation
analytical equation
simulation
Retrospective*
1.12×104
1.12×104
2.75×101
2.37×101
Prospective*
2.07×103
2.10×103
7.62×105
7.64×105
Retrospective*
3.32×103
3.42×103
1.98
1.96
Prospective*
8.83×10-1
8.83×10-1
9.84×102
9.83×102
Retrospective*
3.13×103
3.23×103
3.22
3.15
Prospective*
1.56
1.56
1.94×103
1.94×103
Retrospective*
1.08×104
1.08×104
2.57×101
2.23×101
Prospective*
1.65×102
1.65×102
6.33×104
6.32×104
Retrospective*
3.48×103
3.58×103
1.14
1.14
Prospective*
5.34×10-1
5.30×10-1
5.28×102
5.27×102
B. anthracis
Y. pestis
F. tularensis
Variola major
Lassa
*Total release quantity is 1 million spores for both retrospective and prospective scenario. The simulation
period in retrospective scenario is 8 hours, while it is one year in prospective scenario.
9