Online supplementary material
The Markov chain representation of the compartmental model implies a matrix T,
called the transition matrix, for which the entries are the rates of flow from one
compartment to another. The interpretation is that flux occurs among the
compartments within the model according to the various transition probabilities.
These correspond to the rates of mortality, infection, reinfection and progression to
disease. By assuming constant population size, extended to constant compartmental
size, construction of the Markov transition matrix becomes possible. For this matrix
the element at the intersection of a row and a column is the probability of a transition
from the source (row) compartment to the destination (column) compartment (Table
1).
The transition period is chosen to be equal approximately to the delay to diagnosis
and initiation of therapy, and hence all members of P, the compartment of infectious
cases, either die or start therapy during a transition. Thus the transition from P to Lp,
the compartment of recovered cases i.e. latent cases, has the probability 1 - µp
where µp denotes the mortality rate expressed as a probability per unit time step. The
probability of the transition from the mortality compartment to the live births
compartment is shown as 1.0. This is a technical detail to ensure that recruitment to
the live births compartment exactly equals the losses from all of the compartments
due to mortality. The patients who commence therapy are regarded, for simplicity, as
being part of the latent compartment, Ls. A separate compartment could have been
allocated specifically for this group but would merely serve to hold these cases for
1
one Markov time step. So far as the process is concerned this would be an idle
compartment and is therefore not included as such.
Using standard matrix notation, if at some time the distribution of the population is
given by the row vector u1 then one time step later, the distribution vector is given by
u2 = u1T. After many consecutive time steps, it may be the case that a steady state,
v, is achieved, where the relative proportions of the population in the various
compartments no longer undergo further changes. This state v is then given by v =
lim u1Tn, and generally a matrix, T∞, called the long term transition matrix, can be
calculated so that v = u1T∞.
The long-term transition matrix, T∞, can indeed be calculated for the compartmental
model under consideration (because T is regular) and has the property that v = uT∞
where v is the unique steady state distribution and u is an arbitrarily selected initial
distribution. This v corresponds exactly to the determination of the constant
compartment sizes found in the long-term steady state and has the form
b
m
s
im
i
p
lp
ls
en
i*
ex
The values of the elements of v are the proportions of the population in each of the
compartments and correspond respectively to the total number of live births, deaths,
susceptibles, immune, first-time infected cases, infectious, first-time latent, secondtime latent, endogenous, repeat infected cases and exogenous cases. Thus v is an
appropriate representation of the epidemiological conditions prevalent in a region
where TB has been endemic for an extended period and such that the compartment
2
sizes have achieved approximate stability. We refer to such conditions as stationary
epidemiological conditions.
It will be noted that stationary epidemiological conditions, that is constant
compartment sizes, are equivalent to zero rates of changes of compartmental sizes.
Therefore, instead of having to solve a system of differential equations to describe
the epidemic, it is only necessary to solve a system of linear algebraic equations.
The Markov process paradigm as described above is just another, and more
convenient, way of setting out this system of linear equations and obtaining a
solution.
To calibrate the model to achieve the best fit to the data the following is taken into
consideration: Three parameters are used in the construction of the
reinfection/incidence graphs namely the reinfection factor (ρ), the rate of progression
to disease (p), and the mortality rates (μ, as appropriate). Figure 3 (text) shows that
ρ is the parameter that determines the shape of the graph. The slope of the graph is
governed by p (figure 1) while the vertical position of the graph is fixed by the
mortality rate (figure 2). These three parameters thus describe independent
attributes of the graph and the values that produce the best fit to the empirical data
are therefore uniquely determined.
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Table 1: The Markov transition matrix T.
Destination compartment
Source
compartment
B
M
S
Im
I
P
Lp
Ls
En
I*
Ex
B
0
1
0
0
0
0
0
0
0
0
0
M
0
0
µ
µ
µ
µp
µ
µ
µp
µi
µp
S
1-n
0
1-µ-λ
0
0
0
0
0
0
0
0
Im
n
0
0
1-µ
0
0
0
0
0
0
0
I
0
0
λ
0
1-p-k-µ
0
0
0
0
0
0
P
0
0
0
0
p
1-µp - rp
0
0
0
0
0
Lp
0
0
0
0
k
0
1-µ-a
0
0
0
0
Ls
0
0
0
0
0
rp
0
1-µ-ρλ-c
re
0
re*
En
0
0
0
0
0
0
a
c
1-µp-re
0
0
I*
0
0
0
0
0
0
0
ρλ
0
1-µi-gp
0
Ex
0
0
0
0
0
0
0
0
0
gp
1-µp-re*
Notes: The symbols p and λ denote the rate of progress to disease and the rate of infection
respectively. µ, with appropriate subscripts denote the various mortality rates. A sensitivity
analysis shows that without losing significant accuracy one global value for µ can be used for
all compartments. The reinfection factor is denoted by ρ. g is a factor that modifies the rate of
progress to disease after a reinfection. Since the model is designed in a way that in one time
step all people in infectious compartments either die or recover, the rate for the transition
P → P: 1 - µp - rp is equal to 0. This means rp = 1 - µp. By the same reasoning re = 1 - µp
and re* = 1 - µp.
Table 2: Default parameter values
Parameter
n
µ
Value
0.025
0.005
µp
0.05
µi
0.005
λ
0.0001
p
k
k*
a, c
ρ
g
0.02
0.01
0.01
0.001
4
1
Note: The default values are estimates (Vynnycky & Fine 1997;Vynnycky & Fine 2000)
Method of implementing the model in Excel
For a given set of values of the parameters, the matrix T is set out in Excel. Using
the matrix multiplication feature of Excel, T can be multiplied by itself to produce T 2.
Subsequently T2n-1 for n = 3, 4 ,5… is obtained by reiteration using: T2n-1= Tn Tn-1
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where the seed is T3 = T2 T1. As n increases, the point is reached eventually where
Tn Tn-1 is constant for further values of n. T∞ denotes this matrix Tn Tn-1. The rows of
this matrix are all identical and the steady state distribution vector is equal to such a
row:
This means that for the example in Tables 3 and 4 the rate of reinfection as a
proportion of all infections is: i*/(i+i*) = 0.00034 / (0.00034 + 0.00378) = 8.3 %. The
incidence is (0.000037 + 0.000025 + 0.000003) * 100 000 per 100 000, i.e. 6.5 per
100 000 (per annum).
Table 3: An example of a typical T∞
B
M
S
Im
I
P
Lp
Ls
En
I*
Ex
B
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
M
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
S
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
Im
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
I
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
P
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
Lp
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
Ls
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
En
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
I*
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
Ex
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
Note: The values in each row show the proportion of the population in each compartment.
Table 4: The steady state distribution vector corresponding to T∞
B
B
M
S
Im
I
P
Lp
Ls
En
I*
Ex
0.004954
0.004954
0.947111
0.024771
0.002706
0.000054
0.001082
0.014059
0.000022
0.000281
0.000006
Notes: The values in each row show the proportion of the population in each compartment.
The proportion of disease due to reinfection is
Ex/(Ex+En +P) = 0.000006/(0.000006+0.000022+0.000054) = 7.3% which matches closely the
proportion seen in Holland (de Boer et al. 2003), and corresponds to an ARI of 0.0001 .
The simulations lead to robust results that are relatively insensitive to variations
in the model parameters (Table 5). In particular, simulations were performed with
5
a range of values (0.005 to 0.05) for the mortality rate of infectious cases together
with a fixed mortality rate (0.005) for non-infectious cases. These simulations all
produced the same steady state. This means that the simplification of a global
mortality rate in place of different mortality rates for the various compartments is
an acceptable simplification.
Table 5: Sensitivity of the slope of the regression line for proportion of
reinfected to log(incidence) with respect to model parameters.
Standard
Modified
Change in
value
value
slope (%)
0.005
0.0051
-2.2
μ
0.005
0.0051
-1.26
μi
0.05
0.051
-0.004
μp
0.05
0.051
0.65
p
λ
0.0001
0.000102
0.05
k
0.01
0.0102
1.03
a, c
0.001
0.00102
-1.1
Notes: μ, μ i, μp are the mortality rates for susceptible and latent compartment members,
the mortality rate specifically for the compartments of infected and reinfected cases and
the mortality rate for infectious cases respectively. A 2% increase in each parameter in
turn results in at most a 2.2 % change in the slope. This is negligible compared to data
uncertainty.
Parameter
The infection, re-infection and relapse rates
In figure 1 the following transitions, together with the corresponding infection or
reinfection rates are shown: S → I, (λ) ; Ls → I*, (ρ.λ) and Lp → I, (b - rate
not shown as it is incorporated into the counter-flow k). Similarly there are
transitions due to relapse: Ls → En, (c) ;
Lp → En, (a) .
The model is investigated by varying the input parameter λ and is fitted to data by
using a suitable choice of ρ. The remaining infection/relapse rates a and c can be
independently chosen. The impact of these rates is best investigated by a
sensitivity analysis on the parameters a and c (Table 5).
6
Reinfection and p-values
Reinfection cases as a
fraction of all cases (%)
70
60
50
p = 0.01
40
p = 0.02
30
p = 0.04
20
10
0
1
10
100
1000
10000
Incidence (per annum per 100 000)
Figure 1: The reinfection rate as a function of incidence for three different p
values and reinfection-factor = 7.
Reinfection rates and mortality rates
Reinfection cases as a
fraction of all cases (%)
90
80
70
60
m = 0.005
50
m = 0.01
40
m = 0.015
30
m = 0.02
20
10
0
1
10
100
1000
10000
Incidence (per annum per 100 000)
Figure 2: The reinfection rate as a function of incidence for four different
mortality rates and reinfection-factor, ρ = 7.
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References
de Boer, A. S., Borgdorff, M. W., Vynnycky, E., Sebek, M. M., & van, S. D. 2003,
"Exogenous re-infection as a cause of recurrent tuberculosis in a low-incidence area", Int.J
Tuberc.Lung Dis., vol. 7, no. 2, pp. 145-152.
Vynnycky, E. & Fine, P. E. 1997, "The natural history of tuberculosis: the implications of
age-dependent risks of disease and the role of reinfection", Epidemiol.Infect, vol. 119, no. 2,
pp. 183-201.
Vynnycky, E. & Fine, P. E. 2000, "Lifetime risks, incubation period, and serial interval of
tuberculosis", Am J Epidemiol., vol. 152, no. 3, pp. 247-263.
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