Supplementary Material A: Matrix-analytic approach for the birthdeath model
Whilst explicit results for the probability of response (illness) and the distribution of
time until response are provided in (1) and (2), these are asymptotic approximations
(which assume a very large threshold compared to initial dose) and also assume that
birth rate is greater than death rate. An alternative approach that does not rely on such
assumptions is detailed as follows (this will provide the foundation for the solution to
the birth-death-survival model, detailed in Supplementary Material B). Time until
absorption (response or resolution) is considered by representing the Markov chain as
a continuous phase-type distribution (Figure A1) with probability density and
distribution function given by
r (t ; S ,  )   exp  tS  S M
eqn. (A.1)
R  t ; S ,    1   exp  tS  1
S
where Q  
0
SM 
and   1 , 2 ,...., M 1  are the transition rate matrix and initial
0 
probability vector respectively. The  M  1 dimension square matrix S is tridiagonal and contains the transition rates between the transient states with non-zero
elements given by
si ,i  i      i  1, M  1
si 1,i   i  1 
si ,i 1  i
i  1, M  2
i  1, M  2
The  M  1 length vector S M  S 1    0
  contains the transition rates
T
from the transient states to the absorbing state, M . Given an initial dose of
k 1, M 1 then k  1 and i  0 for i 1, M 1 , i  k . The mean time until
absorption and the associated variance are standard results given by
 S 1 1
2 S 2 1   S 1 1 
2
eqn. (A.2)
Thus, the vector of length M  1 containing mean absorption times with fixed initial
dose k 1, M 1 is S 1 1 .
< Figure A1 >
However, the results given by eqns. (A.1) and (A.2) are relevant only to absorption
and do not distinguish between response and resolution. Such a distinction is made in
(3). Here, the  M  1 dimension transition rate matrix is defined as
0
Q   S0
 0
0 
S S M  where S is as before, S0    0
0 0 
0
S M  S M  S0 , S   S 0  S  1   0
0  and
T
0   . The distribution function of time until
T
response is written
R  t ; S0 , S   T1 R*  t ; S0 , S 


eqn. (A.3)
 
where R* t; S0 , S  S 1  I  exp tS  S M ,   R*  t     S 1SM and I is the
 M 1 dimension identity matrix. This result (eqn. (A.3)) is a vector of length
 M 1 that gives the cumulative probability density up to time t
given an initial
dose of k 1, M 1 . The range,  k50' , k50'  1 , which contains the median infectious
dose is calculated by finding k50'  1, M  1 such that  k '  0.5 and  k '
50
50 1
 0.5 . In
extending the approach of (3) to consider an initial probability vector the following
results are obtained:
R  t; SM , S ,     S 1  I  exp  tS  SM
eqn. (A.4)
Letting t   gives the ultimate probability of response,
P  S M , S ,     S 1S M
eqn. (A.5)
Thus the dose-response relation is given by
P  S M , S    S 1S M
eqn. (A.6)
Supplementary Material B: Matrix-analytic approach for the birthdeath –survival model
The matrix-analytic approach detailed in Supplementary Material A is now modified.
The formulae for the density functions for time until response and resolution in this
model are equivalent to those of eqn. (A.4). However, adjustments are required to the
transition rate matrix, S , the vectors S 0 and S M and the initial probability vector,  .
Derivation of S
The matrix S is a block matrix of size  M 1   M  1 . Each block s i , j contains the
transition rates for movements from states in which T  i  1, M  1 to states in
which T  j  1, M  1 (see Figure 1B), where T is the total number of extracellular
bacteria, B, and bacteria-containing phagocytes, P. Thus,
 s1,1
 s
2,1
S


 sM 1,1
s1, M 1 
s2, M 1 
where each si , j is of dimension  i  1   j  1 and


sM 1, M 1 
s1,2
s2,2
sM 1,2
the constituent elements si , j
i' , j '
represent the transition rate from state X  i, i '  1
to state X   j , j '  1 with i ' 1, i  1 , j ' 1, j  1 . Thus the state space of S ,
   X | T  1, M  1 , is lexicographically ordered with cardinality
M 1
 M  1
 1
 2 
    i  1  
i 1
eqn. (B.1)
That is, there are  number of transient states. The population of the non-null blocks
are described next.
Death (i.e. killing of bacteria) reduces the number of extracellular bacteria in the lungspace, B , and hence T , by one and thus concerns the sub-diagonal blocks si 1,i for
i 1, M  2 . Non-zero elements form the diagonal of these blocks and are populated
by the formula si 1,i
i ' ,i '
  i  i '  2   for i ' 1, i  1 . For example, i  1 yields
 2 0 
s2,1  0
  . Survival (i.e. phagocytosis not resulting in death) too reduces the
0
0 
number of extracellular bacteria but has no effect on the value of T , and thus
concerns the diagonal blocks si ,i for i 1, M 1 . Non-zero elements form the superdiagonal of these blocks, which are populated by the formula si ,i
i' ,i' 1
  i  i '  1 
for i ' 1, i  . Birth (i.e. release of bacteria from phagocyte) increases B by G  2 and
decreases P by one, hence increasing T by G 1 . If G  2 then the matrix S is
block tri-diagonal. Moreover, birth relates to the blocks si ,i G 1 for i 1, M  G
(since i  G 1  M 1 ). Non-zero elements form the sub-diagonal of these blocks and
are populated by the formula si ,i G 1
elements of S (i.e. each si ,i
i ' ,i '
i' 1,i'
 i ' for i '  1, i  . Finally, the diagonal
for i 1, M 1 , i ' 1, i  1 ) must be populated.
Rather than specifying a formula for this, it is noted that each row of the transition
  M  1 
rate matrix Q must sum to zero, and so, qi ,i   qi , j for i, j  1, 
  1 .
i j
  2  
Since S0   q2,1 q3,1
qM ,1  and SM   q2, M 1 q3, M 1
T
qM , M 1  then such
T
calculation is dependent on these vectors.
Derivation of S 0
Entry into the lower absorbing state can only be achieved through bacterial death
from state 1,0 (see Figure 1B) which occurs with rate  . Therefore,
S0   
0
 M  1
T
0  where S 0 is of length 
  1.
 2 
Derivation of S M
As can be seen from Figure 1B, the state M can be reached from a number of
transient states through birth (i.e. release of bacteria from phagocyte). This represents
scenarios in which the number of bacteria released results in the threshold being met
or exceeded. The threshold is met from states  X | T  M  G  1; P  0 and exceeded
from states  X | M  G  1  T  M  1; P  0 . Defining S M   s1
s2
sM 1 
T
where si are vectors of length i  1 for i 1, M 1 , the non-zero elements of the
non-null vectors are populated by the formula
si
i' 1
 i ' for M  G 1  i  M 1
and i ' 1, i  .
Derivation of 
Let   1  2
 M 1  where i are vectors of length i  1 for i 1, M 1 .
Clearly at the time of challenge there has been no interaction between the inhaled
bacteria and the cells of the immune system; and thus there has been no phagocytosis.
Therefore, the only states that are initially accessible are  X | P  0 , and thus the
(only) non-zero element of  is populated by k
1
 1.
References
1. Saaty TL. Some stochastic processes with absorbing barriers. Journal of the Royal
Statistical Society Series B (Methodological). 1961;319–34.
2. Shortley G. A stochastic model for distributions of biological response times.
Biometrics. 1965;21(3):562–82.
3. Tan WY. On the absorption probabilities and absorption times of finite
homogeneous birth-death processes. Biometrics. 1976;745–52.
Figure captions
Figure A1
Phase type representation of the standard birth-death process. The probability of the
process starting in phase i  1, 2,..., M  1 is i . The birth and death rates are   0
and   0 respectively. Absorption of the process occurs through either resolution (of
the infection) or response (illness).