[Emerging Infectious Diseases * Volume 3 * Number 3 * July - September 1997]

Synopses

Using a Mathematical Model to Evaluate the Efficacy of TB Control Measures

Laura Gammaitoni and Maria Clara Nucci
University of Perugia, Perugia, Italy
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      We evaluated the efficacy of recommended tuberculosis (TB)
      infection control measures by using a deterministic
      mathematical model for airborne contagion. We examined the
      percentage of purified protein derivative conversions under
      various exposure conditions, environmental control strategies,
      and respiratory protective devices. We conclude that
      environmental control cannot eliminate the risk for TB
      transmission during high-risk procedures; respiratory
      protective devices, and particularly high-efficiency
      particulate air masks, may provide nearly complete protection
      if used with air filtration or ultraviolet irradiation.
      Nevertheless, the efficiency of these control measures
      decreases as the infectivity of the source case increases.
      Therefore, administrative control measures (e.g., indentifying
      and isolating patients with infectious TB) are the most
      effective because they substantially reduce the rate of
      infection.

After decades of steady decline and a subsequent relaxation of infection
control practices in health-care facilities, the number of tuberculosis
(TB) cases has been increasing dramatically in the United States (1) and
Western Europe (2) since 1985; the increase is due to human
immunodeficiency virus (HIV) infection, immigration, poverty, homelessness,
and drug addiction.

Epidemiologic studies have shown that primary pulmonary TB is caused by
inhaling the tubercle bacillus in a droplet nucleus form (3). Airborne
contagion is crucial in the indoor transmission of all respiratory
infections from person to person (4,5). Infective persons may contaminate
the air by coughing, sneezing, and spitting (4,5), which generate a large
number of small respiratory droplets that evaporate almost instantly into
small droplet nuclei, disperse into the environment, and implant themselves
in the lung when inhaled (4). Droplet nuclei have a leading role in
airborne contagion (3-5).

Outbreaks of TB have been reported from prisons, nursing homes, residential
centers for HIV-infected persons, urban homeless shelters, aircraft,
schools, and bars. All outbreaks occurred under crowded living conditions
with prolonged close exposure to an infectious person. Nosocomial TB
transmission is also associated with cough-generating procedures (6),
bronchoscopy (7), endotracheal intubation and suctioning (8), open abscess
irrigation (9), and autopsy (10). Workers involved in such procedures are
at high and increasing risk for TB (11) because of the resurgence of the
disease, the emergence of multidrug-resistant (MDR) strains causing
outbreaks in hospitals among patients and health-care workers, and
compromised TB control due to decreased funding of health-care agencies
responsible for TB control.

After studying hospital outbreaks that resulted in TB transmission to
health-care workers, various authorities have recommended measures to
prevent nosocomial TB transmission (12-15). The implementation of a TB
infection control program requires risk assessment and development of a TB
infection control plan including early identification, treatment, and
isolation of infectious TB patients; effective engineering controls
(environmental controls such as general ventilation, high-efficiency
particulate air [HEPA] filters, or ultraviolet germicidal irradiation
[UVGI]); the adoption of appropriate respiratory protection (surgical masks
and particulate respirators such as HEPA masks); health-care worker TB
training, education, counseling, and screening; and evaluation of the
program's effectiveness (13). Several reviews of environmental control
measures and respiratory protective devices are available (16-20).
Implementing recommended measures would require massive expenditures in all
hospitals that admit TB patients. At several hospitals where MDR-TB
outbreaks have occurred, measures similar to those recommended in TB
prevention guidelines have been implemented. Such implementation halted and
prevented MDR-TB transmission to health-care workers and patients (21-23),
but which of the many implemented control measures played a key role in
reducing risk is not clear. However, data suggest that implementing
administrative control measures, in particular identifying and isolating
patients with infectious TB, substantially reduces the risk for
transmission (22).

In this article, we review the most important TB infection control measures
and evaluate their efficacy in various settings by a deterministic
mathematical model for airborne contagion (24). The model describes how a
person can become infected by staying in a room where a source of airborne
infection and an air disinfection device are present.

Two existing mathematical models describe the propagation of airborne
contagion indoors. The first model was developed in 1955 by William F.
Wells (4), who introduced the so-called quantum theory. Wells defined a
quantum to be the number of infectious droplet nuclei required to infect 1
- 1/e of susceptible persons. His ninth postulate states, "The response to
inhaled droplet nuclei contagium is quantal; the Poisson equation expresses
reasonably well the relation between dosage and initial response, a quantum
infecting 63.2% of homogeneously exposed hosts by definition." Wells
explains, "When on the average one animal breathes one quantum, (omissis)
36.8% of the animals will survive, since this is the fraction whose
negative natural logarithm is 1. Thus 1 quantum of contagium has been
breathed per animal when 63.2% of the animals become infected (4, p. 124)."
In 1978, Edward R. Riley, G. Murphy, and Richard L. Riley elaborated
another model (25), which deals with the probability of a susceptible
person becoming infected by inhaling a quantum of infection. They used the
Reed Frost modification of the Soper equation for airborne transmission and
Poisson's law of small chances. This model contains Wells' model as a
particular case.

Our model is based on Wells' experiments and postulates and includes the
Riley/Murphy/Riley model as a particular case (24). In this article, we
describe some of the most important TB infection control measures, such as
environmental controls and respiratory protective devices, and review
recommendations made by various agencies and organizations. Then, we
consider four recent TB outbreaks during which health-care workers became
infected while performing bronchoscopy, jet irrigation of a thigh abscess,
autopsy, and intubation. Using our deterministic mathematical model, we
simulate the adoption of different environmental control strategies and
respiratory protective devices, and we give the number of purified protein
derivative (PPD) conversions predicted to occur under various exposure
conditions. Moreover, we use our mathematical model to evaluate the
efficacy of administrative control measures by analyzing data related to
PPD conversions found on an HIV ward before and after implementing such
control measures.

Environmental Control Strategies and Respiratory Protective Devices

A TB infection control program should be based on control measures that
address the most important factors involved in TB transmission: the
concentration of droplet nuclei in the air and the rate at which droplet
nuclei are inhaled by susceptible persons. The concentration of infectious
droplet nuclei in the air can be reduced by environmental controls. These
controls include diluting and removing contaminated air by general
ventilation and air cleaning by air filtration (e.g., through HEPA filters)
or UVGI.

In general ventilation systems, uncontaminated supply air mixes with
contaminated room air (dilution), which is subsequently removed from the
room by the exhaust system (removal). Two types of general ventilation
systems can be used for dilution and removal of contaminated air:
single-pass systems and recirculating systems. In single-pass systems the
supply air is uncontaminated fresh outside air, and after it passes through
the ventilated area, 100% of that air is exhausted to the outside. In a
recirculating system, a small portion of the exhaust air is discharged to
the outside and is replaced with fresh outside air, which mixes with the
portion of exhaust air that was not discharged to the outside. The
resulting mixture, which can contain a large proportion of contaminated
air, is then recirculated. The rate at which airborne particles are removed
from an enclosed space (e.g., a room) by ventilation is usually expressed
as the number of air changes per hour (ACH), which is the ratio between the
volume of air entering the room per hour and the room volume. A minimum of
6 ACH is recommended for TB isolation rooms and treatment rooms (26-28).
Where feasible, this airflow rate should be increased to 12 ACH or more,
and in areas where the nature of work is exceptionally hazardous, such as
autopsy rooms, airflow rates of 15-25 ACH have been recommended (29,30).

HEPA filtration units or UVGI can be used as a supplement to ventilation
control measures in settings where adequate airflow cannot be provided with
the general ventilation system alone (13). HEPA filters have a demonstrated
and documented minimum removal efficiency of 99.97% of particles whose
diameter is greater than or equal to 0.3 m (droplet nuclei are an estimated 1 to 5 m in size
[13], but they can have an aerodynamic diameter of less than 1 m [31]).
They can be used in fixed or portable HEPA recirculation systems that can
achieve 12 ACH or more (13). UVGI can kill or inactivate tubercle bacilli
under experimental conditions (32), especially where air mixing is
accomplished mainly by convection (13). The effect of UVGI in a room
without supplemental ventilation (33) and in another with a ventilation
rate of 6 ACH (34) is estimated at 10 and 39 ACH, respectively. Greater
rates of ventilation may decrease the killing of bacteria (34), but the
optimal relationship between ventilation and UVGI is not known. However,
general ventilation plus UVGI has a disinfection rate of 45 ACH if properly
installed (34).

Personal protective devices are recommended when engineering controls are
not likely to protect against inhaling infectious airborne droplet nuclei
(e.g., in TB isolation rooms, treatment rooms in which cough-inducing or
aerosol-generating procedures are performed, and ambulances during the
transport of infectious TB patients) (13). The respiratory protective
devices used in these settings should have a filter efficiency greater than 
or equal to 95% with an obtainable face-seal leakage not greater than 
10% (13). The efficiency of standard cup-shaped surgical masks
in preventing the inhalation of droplet nuclei with a diameter of 1 to
5 m is less than 50% (11) with a face-seal leakage of 0% to 20% (35).
Since 1990, the use of particulate respirators
including HEPA masks tested to filter 99.97% of 0.3 mm particles with a
face-seal leakage less than 2% has been recommended (13).

Because of the very high costs of HEPA masks, the breathing difficulties
they may cause, and the odd appearance of people wearing them, which may
have a psychological impact on patients, the recommendation for personal
respiratory devices is the most controversial aspect of existing TB
guidelines. Reported cases of TB and skin-test conversions among
health-care workers have led many to conclude that HEPA respirators are
justified. On the other hand, many believe that less stringent personal
protective devices, such as dust/mist respirators, may provide sufficient
protection, with less discomfort and lower cost (31). New certification
rules (42 CFR 84) have been recently approved that allow the use of
dust/mist respirators, in addition to HEPA respirators, in TB control
settings (36).

The Model

In the case of an airborne infection outbreak in a single room (e.g., a
hospital room) of volume V, where a certain number I(subscript 0)
greater than or equal to 1 of infective persons and S(subscript 0)
of susceptible persons are present at initial time t = 0,
the number of susceptible persons can be expressed as a function of the
time interval from the beginning of the exposure (the time of exposure t,
measured in minutes), the rate at which quanta are produced by the
infective persons (the infection rate q), and the number of effective or
equivalent air changes (AC) in the unit time (the disinfection rate C)
(24).
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Table 1. Nosocomial TB outbreaks involving health-care workers who
performed high-risk procedures
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     Pulmonary                  Room          No. of          No. of      Exposure   Infection  Disinfection
       ventilation                 volume         susceptible  uninfected   time           rate          rate
   p(m(superscript3)   V(m(superscript3) persons     persons     t (min)       q (qpm)   C (ACH)
             /min)                                     S(subscript 0)    S
                 
Bron-  10(superscript -2)    200               13               3                 150               6        1.26
choscopy       

Abscess  6 x             200(superscript b)   5                1        150(superscript b)  38        6
    10(superscript -3)                                                                                               (super-
    (superscrip -a)                                                                                                   script b)

Autopsy   10              200(superscript b)   4       0(superscript c)     150              94      11
         (superscript -2)                                                                 (superscript b)

Intubation   6 x                  3000               3                 1                   66             514      2.4   
           10(superscript  -3)                                                                                      (super-
             (superscript -a)                                                                                         script d)
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(superscript a)Surgical masks were adopted.
(superscript b)Estimated value; actual data are not available
(superscript c)We assumed S=0.001 almost equal to 0.
(superscript d)In the emergency department where this outbreak developed, 60% of the air was recirculated without filtration; we assumed an airflow rate of 6 ACH, obtaining an effective disinfection rate equal to 40% of 6 ACH.
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The following assumptions (4,24,25) are taken into consideration:
differences in susceptibility are neglected; the rate at which quanta of
infection are added to the air by infectious persons is considered
constant; the latent period of the disease is longer than the time scale of
the model, and the number of infective persons in the room is constant;
droplet nuclei are instantaneously and evenly distributed in the room;
droplet nuclei are assumed to be removed by fresh air ventilation of the
room and ultraviolet irradiation at a constant rate; fresh air is at the
same temperature and pressure as the air already present in the room; and
the number of infected persons is proportional to the number of encounters between susceptible persons and quanta of infection, (and the encounter
rate is proportional to both the number of susceptible persons and quanta
of infection in the room [law of mass action]).

If the c
onstant of proportionality between the number of infected persons
and the number of susceptible persons times the number of quanta of
infection Q is the ratio between the pulmonary ventilation p (we assume p =
0.01 m(superscript 3)/min) and the volume V of the room, the following equations can be
derived:

[Equations not available in ASCII.]

1.

[Image]
with the initial conditions

2.

[Image]
The solution of equation 1 - equation 2 is

3.

[Image]

4.

[Image]

The percentage of PPD conversion among health-care workers is the
following:

5.

[Image]
The rate of production of quanta of infection during each outbreak is not
known a priori. Therefore, we do not know the real value of q during a
specific epidemic. However, it can be derived a posteriori when the number
S(bar) of susceptible persons who were not infected is known. In fact, solving
equation S(t,q,C) = S(bar) with respect to q, with S given by formula 3 yields

6.

[Image]
By substituting the given values of p, V, S(subscript 0), S(bar), t, and C in the righthand
side of formula 6, we obtain the value for q during each epidemic.

In this article, we simulate the adoption of various environmental control
strategies (e.g., HEPA filtration devices or UVGI combined with
ventilation) by giving suitable values to the parameter C. Also, we
simulate the adoption of personal protective devices (e.g., surgical masks,
dust/mist respirators, and HEPA masks) by scaling the rate at which quanta
of infection are breathed (pulmonary ventilation p), which depends both on
the mask's filter efficiency and face-seal leakage. If the masks adopted
have a filter efficiency of X% and a face-seal leakage of Y%, then we can
describe the effective filter efficiency of the mask as Z%, with Z =
(X-XY/100). The scaling factor for p, therefore, is (100-Z)%.

For example, dust/mist respirators should have a filter efficiency X% equal
to 95% and a face-seal leakage Y% not greater than 10%; then, their
effective filter efficiency is Z = X-XY/100 = 85.5%, and the scaling factor
for p is (100-Z)% = 14.5%. We derive a scaling factor of 60% and 3% for
surgical and HEPA masks, respectively (13). Moreover, we assume C = 6 ACH
(0.1 AC/min) for general ventilation alone, and C = 18 ACH (0.3 AC/min) or
C = 45 ACH (0.75 AC/min) when HEPA filtration or UVGI is adopted in
combination with ventilation (34).

Model Application

We consider two types of nosocomial exposure to TB infection: exposure
during high-risk procedures and exposure during normal working conditions
on a ward where a source of TB infection is present. Using data from
published TB outbreaks among health-care workers and assuming the use of
different environmental control strategies (general ventilation, HEPA
filtration, UVGI) and respiratory protective devices (surgical masks,
dust/mist respirators, HEPA masks), we derive the percentage of PPD
conversions that can occur.

High-Risk Procedures

We use data from four published TB outbreaks during which health-care
workers were infected while performing bronchoscopy (7), jet irrigation of
a thigh abscess (9), autopsy (10), and intubation (8) (Table 1). The values
for q vary from six (7) to 514 (8) quanta per minute (qpm), indicating a
high level of infectiveness of the index case in each outbreak. The patient
with the highest rate of quanta production was considered a dangerous
disseminator of TB (8).

Table 2 shows the percentage of PPD conversions that can occur during four
procedures (bronchoscopy, abscess irrigation, autopsy, intubation) if
different environmental control strategies were adopted, with and without
the adoption of surgical masks, dust/mist respirators, or HEPA masks.
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Table 2. Percentage of purified protein derivative conversions predicted to
occur during various procedures, by using various environmental control
strategies and respiratory protective devices
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                                                          Abscess
                      Bronchoscopy               irrigation                   Autopsy                    Intubation
Control                q=6,                            q=38,                      q=90,                         q=514,
measures     t=150 min(superscript b)   t=150 min                 t=150 min                   t=66 min
(superscript a)

GV                      34.3                                93                           99.9                          61.7
GV+HF               13.6                                 60.5                        90                             30.1
GV+UVGI              5.8                                31.4                        60.5                          13.7
GV+SM               22.3                                79.7                        98.1                           43.8
GV+HF+SM           8.4                                42.7                        74.8                           19.3
GV+UVGI+SM       3.5                                20.2                         42.8                           8.5
GV+DMR              5.9                                32                            61.5                           13
GV+HF+DMR        2.1                               12.6                          28.3                            5.1
GV+UVGI+DMR     0.9                                5.3                         12.6                             2.1
GV+HM                 1.3                                7.7                         17.9                             2.8
GV+HF+HM           0.4                                2.7                           6.7                             1.1
GV+UVGI+HM        0.2                               1.1                           2.8                              0.4
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(superscript a)GV=general ventilation; HF=HEPA filtration; SM=surgical masks;
DMR=dust/mist respirators; HM=HEPA masks.
(superscript b)Q=quanta; t=time.
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The percentage of PPD conversions that can occur as a function of the time
of exposure t during the least risky procedure -- bronchoscopy -- if
different environmental control strategies were adopted, no personal
protective devices were used, and the time of exposure varied from 50 to
400 minutes is shown in Figure 1.

 Figure 1. [Figures not available in ASCII]  Percentage of PPD 
 conversions predicted to occur as
 a function of time of exposure t during bronchoscopy, if
 different environmental control strategies were adopted, no
 personal protective devices were used, and the time of
 exposure varied from 50 to 400 minutes.

Normal Working Conditions

We consider an MDR-TB outbreak that occurred on an HIV ward (21). During
the initial period, the percentage of PPD conversions among health-care
workers was 28%. Inadequate TB control programs or facilities (delays in TB
diagnosis and in determining drug susceptibility and inadequate patient
isolation precautions) facilitated the transmission. Administrative
measures similar to those subsequently recommended (37) were implemented on
the HIV ward, and early follow-up period showed a decrease in PPD
conversions among healthcare workers (18%), who were required to wear
surgical masks.

The actual time of exposure during the initial period is not available.
Assuming that the ratio between the time of exposure t(subscript in) and the time span
T(subscript in)=150 days of the initial period (January-May 1990) is equal to the ratio
between the time of exposure t(subscript ef)=135 days (average between 139 and 129
days) and the time span T(subscript ef)=270 days of the early follow-up period (June
1990-February 1991), we assume that t (subscript in)=T(subscript in) x  t (subscript ef)/T(subscript ef) approximately equal to 75 days.

The remaining data are summarized in Table 3.

We calculate the rate of infection in the initial period to be q = 0.006;
and after administrative control measures were implemented, it decreased to
q = 0.003 in the early follow-up period. Administrative control measures
decreased the rate of infection by 50%, thus confirming their key role in
preventing TB transmission.

Figure 2 shows the percentage of PPD conversions predicted to occur during
the initial period with the use of different environmental control
strategies and respiratory devices.

Figure 2. [Figures not available in ASCII.] Percentage of PPD conversions
predicted to occur on the HIV ward during the initial period, if different 
environmental control  strategies were adopted and with and without the
adoption of surgical  masks (SM), dust/mist respirators (DMR), or HEPA masks (HM).

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Table 3. Nosocomial TB outbreaks involving health-care workers under normal
working conditions
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              Pulmonary          Room            No. of         No. of    Exposure  nfection Disinfection
               ventilation          volume         susceptible  uninfected   time          rate           rate
   p(m(superscript3)   V(m(superscript3)) persons     persons     t (days)   q (qpm)     C (ACH)
             /min)                                   S(subscript 0)     S
                 
 Initial 10(superscript -2) 200(superscript a)   25          18             75       0.006           6
 period
                       
 Early
 follow-up  6 x                  200(superscript a)  17          14            5         0.003            6
 period    10(superscript 3)
                 (superscript b) 
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(superscript a)Estimated value; actual data are not available.
(superscript b)Surgical masks were adopted.
---------------------------------------------------------------------------

Conclusions

Our mathematical model indicates that a certain number of persons exposed
to a source of TB infection will be infected in spite of all precautions.
Nevertheless, if the infection rate is known, this number can be kept low
by increasing the disinfection rate and decreasing the time of exposure.

We have evaluated the probability of acquiring TB infection under various
exposure conditions and found that the efficacy of environmental control
strategies depends on the duration of exposure and the infection rate. In
particular, if the recommended levels of airflow are the only means of
reducing the concentration of droplet nuclei in a room, they can neither
eliminate TB contagion nor provide tolerable value for the risk of
contagion. If HEPA filtration or UVGI is combined with general ventilation,
protection against TB transmission is ensured at low infection rates,
especially for brief exposures. As the infection rate increases, higher
disinfection rates appear less and less effective in reducing the risk for
transmission, and in situations of very intensive exposure, not even HEPA
filtration or UVGI combined with general ventilation can keep the risk for
transmission low enough. In these circumstances, respiratory protective
devices may be used to protect health-care workers from inhaling droplet
nuclei. The adoption of HEPA masks would provide nearly complete
protection, even for long exposures, if used together with HEPA filtration
or UVGI. Nevertheless, the efficacy of these control measures decreases as
the infectivity of the source case increases; therefore, the only control
measures that significantly reduce the infection rate are administrative.

About the Author

Dr. Gammaitoni is a postgraduate fellow at the University of Perugia and a
computer specialist for ALENIA-Italy. Her research interests include
mathematical models applied to environmental health problems.

Dr. Nucci is a researcher at the University of Perugia. Her recent
interests include application of mathematical modeling techniques to
epidemiology. Her work is supported in part by Minister of Universit
Ricerca Scientifica Tecnologica: Fondi (60% and 40%).

Address for correspondence: Maria Clara Nucci, Dipartimento di Matematica,
Universit di Perugia, 06123 Perugia, Italy; fax: 011-39-75-585-5024;
e-mail: nucci@unipg.it.

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Emerging Infectious Diseases
National Center for Infectious Diseases
Centers for Disease Control and Peevention
Atlanta, GA

URL: ftp://ftp.cdc.gov/pub/EID/vol3no3/ascii/nucci.txt