Mathematical Modelling Of M. Tuberculosis Transmission

Mathematical Modelling Of M. Tuberculosis Transmission - With few exceptions, every case of TB results from the airborne transport of MTB from an infectious source to a vulnerable host. Primarily an intracellular pathogen, MTB is adapted to replicate extracellularly in necrotic lung cavities, to endure the rigors of aerosolization in mucus as tiny respiratory droplets, to survive the process of rapid drying into droplet nuclei, and to remain infectious after airborne transport through a variety of harsh environments. Potentially lethal environmental exposures include the extremes of air temperature and humidity, ambient levels of air pollution, and natural ozone and irradiation (Fig. 2.2). 

Tuberculosis Transmission

Dots in air represent droplet nuclei containing M. Tuberculosis

Little is known about these adaptive processes, but the selective pressure is strong since only adapted strains can propagate. Finally, organisms surviving the gauntlet of airborne transmission face innate and adaptive host defences honed, in some human populations, by generations of natural selection. Virulence, the ability of the pathogen to overcome host defences and cause infection, is also selected for by the necessity of transmission. Microbes, of course, adapt much faster than can humans, even to the antimicrobials invented to tilt the contest in our favour. The mathematical modelling that follows attempts to quantify the relationship between the infectious doses generated by the source case(s) and the number of infected humans, but ignores the fact that most tubercle bacilli released from the source case are unlikely to cause infection due to factors cited earlier. In addition, probability plays a role in whether a vulnerable host inhales an infectious dose of MTB.

In an attempt to characterize the transmission process as a probability that a vulnerable host will acquire TB infection, Wells and Riley (Richard’s brother, Edward, an engineer) developed a mathematical model that expands upon and modifies the Soper mass balance equation for epidemiological investigations, incorporating the following operational assumptions:

1. steady-state conditions in which the infectious source is

constant;

2. complete air mixing within a defined volume or space being

studied;

3. equal susceptibility among exposed individuals to infection;

4. Poisson’s law of small chances which employs the natural

logarithm e;

5. uniform virulence of organisms released into the air space; and

6. random distribution of infectious particles within a defined

space.

This model states that for a single generation of infection:

C ¼ Sð1 e

Iqpt=QÞ;

where:

C ¼ number of new cases,

S ¼ number of susceptibles exposed,

e ¼ natural logarithm,

I ¼ number of infectious sources,

q ¼ number of quanta (infectious doses) generated per unit time

in minutes,

p ¼ human ventilation rate in L/minute,

t ¼ exposure duration, and

Q ¼ infection-free ventilation in the room in L/second.

Although this model applies to situations that meet the assumptions above (i.e. a single room or enclosed space with a defined ventilation), it can be applied, although with less validity, to spaces served by a single central ventilation system (i.e.HVACsystem) andwhose air spaces are connected – an essential component for infections which are airborne. The larger the space being considered, the less evenly distributed (mixed) airborne particles may be, thus also potentially reducing the validity of this model for that air space. Despite these limitations, the model has been useful in examining the relative importance of transmission factors in real-life exposure situations. In epidemiological investigations of epidemics in which C, S, I, p, t, and Q were known or estimated, values for q could be calculated as a representation of the infectiousness of the index source case. This was done for a few scenarios of TB transmission: the patients on Riley’s TB ward in the 1950 experiment with guinea pigs, an office outbreak, and a bronchoscopy case on a patient with TB in an intensive care unit (Table 2.2). What follows is a discussion of how the Wells–Riley equation has been used to compare the intensity of exposure to TB and how it can inform approaches to reducing transmission. In the first situation, Riley’s TB patients were receiving TB drug treatment and the infectiousness values represent that of the entire six-bed ward. In the second scenario, a 30-year-old woman returned

towork in a welfare office for an additionalmonth before shewas diagnosed with cavitary, sputumsmear-positive TB, atwhich time contact with fellow employees ended.85 Of 67 co-workers who were initially tuberculin skin test negative, 27 (40%) had conversions to positive upon repeat testing 3 months later. One non-infectious secondary case resulted in a worker who had declined treatment of latent infection.

The office building had been the subject of repeated air quality complaints, and several air quality assessments had been done before and after the TB exposure. A mathematical analysis of the exposure was prompted by the suspicion of several workers that inadequate ventilation was responsible for the large number of infected workers. All values of the Wells–Riley equation were known or estimable except q, the number of infectious quanta generated by the source case.

By calculation, the source generated 13 infectious quanta per hour.

Further calculations showed that outdoor air ventilation at the low end for acceptable air quality (15 cubic feet perminute (cfm) per occupant, based on average roomCO2 values of 1000 ppm) contributed to transmission. However, the model was useful also for indicating that doubling the ventilation would reduce the risk of infection by approximately half (Fig. 2.3). Thirteen workers would still have been infected, according to the model. Moreover, an additional doubling of ventilation, to 60 cfm per occupant, would again reduce the risk by half, leaving approximately six workers unprotected. Both the potential of a moderately infectious patient to infect many contacts over a prolonged period of time and the limitations of building ventilation to prevent transmission were demonstrated, within the assumptions and limitations of the modelFor the third scenario, 

Catanzaro applied the Wells–Riley equation to an episode of transmission in an intensive care unit where an unsuspected patient, initially smear negative for TB, underwent intubation and bronchoscopy. During the 2½ hours of the procedures, 10 of 13 susceptible room occupants became infected. By calculation, the source produced a remarkable 250 infectious quanta per hour. However, the ventilation rate in the intensive care unit was extremely low, and further calculations predicted substantial improvements by increases in ventilation that were