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Epidemics. 2017 December; 21: 30–38.
PMCID: PMC5729200

A human time dose response model for Q fever

Abstract

The causative agent of Q fever, Coxiella burnetii, has the potential to be developed for use in biological warfare and it is classified as a bioterrorism threat agent by the Centers for Disease Control and Prevention (CDC) and as a category B select agent by the National Institute of Allergy and Infectious Diseases (NIAID). In this paper we focus on the in-host properties that arise when an individual inhales a dose of C. burnetii and establish a human time-dose response model. We also propagate uncertainty throughout the model allowing us to robustly estimate key properties including the infectious dose and incubation period. Using human study data conducted in the 1950's we conclude that the dose required for a 50% probability of infection is about 15 organisms, and that one inhaled organism of C. burnetti can cause infection in 5% of the exposed population. In addition, we derive a low dose incubation period of 17.6 days and an extracellular doubling time of half a day. In conclusion this paper provides a framework for detailing the parameters and approaches that would be required for risk assessments associated with exposures to C. burnetii that might cause human infection.

Keywords: Coxiella burnetii, Mathematical model, Hypergeometric, Bacteria, Macrophage, Deposition

1. Introduction

Coxiella burnetii, a Gram-negative obligate intracellular bacterium, is the causative agent of Q fever and an environmentally-persistent ubiquitous organism. This agent has the potential to be developed for use in biological warfare and is classified as a bioterrorism threat agent by the CDC and as a category B select agent by NIAID. The interest and concern with this organism is due to its physical properties; a very low infectious dose and its stability and persistence in an environment as a small cell variant form for a long period of time on fomites (Marrie, 2003, McCual and Williamns, 1981, Raoult and Drancourt, 1991). The natural routes of infection are inhalation, ingestion or through skin abrasions and the route of infection has an influence on minimum infectious inoculum size, the severity of the disease, and clinical manifestations (Raoult et al., 2005). Disease presentation is extremely variable and may for most be a self-limiting illness (Centers for Disease Control and Prevention). The infection is characterized by high fever, severe headache, asymptomatic seroconversion, flu-like symptoms, pneumonia, chronic infection, myalgia, malaise and hepatitis (Raoult et al., 2005, Schaik et al., 2013). Host factors and strain specificity may also influence presentation (Raoult et al., 2005).

Human infection is usually caused by primary aerosols (particles introduced directly into the atmosphere) from an infected animal, including goats, sheep and cattle, although in more urban situations, pet cats, rabbits and dogs have been highlighted as sources (Raoult and Drancourt, 1991). Alternatively infection can occur via secondary aerosols (re-aerosolization of settled particles), produced by some factor, action or incident temporally or spatially removed from those giving rise to primary aerosols, such as handling or processing of contaminated materials (Welsh et al., 1958). For example, C. burnetii can be transmitted via contact with re-aerosolised contaminated dust particles (Evstigneeva et al., 2005). Many human Q fever cases are sporadic, but outbreaks do occur and these typically affect urban populations (i.e. susceptible persons with no pre-existing immunity) near to affected farms (Hackert et al., 2012, Hawker et al., 1998, Murga et al., 2001, Schimmer et al., 2010, Wallensten et al., 2010). Many Q fever outbreaks have occurred over the years; one notable outbreak is the huge epidemic in the Netherlands (2007–2010), with over 3500 reported human cases (Roest et al., 2011). This was due to high numbers of C. burnetii spores introduced into the environment from ‘abortion storms’ on large dairy goat farms, and to a lesser extent, dairy sheep farms. These organisms can be dispersed as aerosols downwind under dry and dusty conditions, from the source itself (infected animals) or from soil dust from around the source. This outbreak provided a clear demonstration of the advantage to public health if adequate diagnostic, therapeutic and epidemiological tools are developed and available (Dijkstra et al., 2012).

In this paper we focus on the in-host properties that arise when an individual inhales a dose of C. burnetii over a relatively short time frame. In the next section we discuss the immune response and disease pathogenesis. We then adapt and extend the birth-death-survival model created in Wood et al. (2014) to include deposition in order to establish a human time dose response model for C. burnetii and estimate key properties including the infectious dose and incubation period. We also propagate uncertainty throughout the model to ensure robust results.

2. Immunology and pathogenesis

Histopathological analysis on C. burnetii has identified monocytes and macrophages as the primary infection sites, but epithelial and endothelial cell infection has also been evident (Baca and Paretsky, 1983, Russell-Lodrigue et al., 2006). Once an individual inhales the bacteria, it targets alveolar macrophages and passively enters these cells by actin-dependent phagocytosis (Baca et al., 1993), see Fig. 1. Phagocytosis results in the formation of the phagosome, which matures into a phagolysosome following a series of highly ordered and regulated fusion (with lysosomes) and fission events (Flannagan et al., 2012, Schaik et al., 2013). The phagolysosome-like compartment the bacterium resides in is often known as the Coxiella-Containing Vacuole (CCV) (Flannagan et al., 2012). The CCV compartment matures by fusion of multiple smaller CCVs creating large CCVs as well as fusion with autophagic, endocytic and lysosome vesicles (Howe et al., 2003). The maturation of the CCV compartment involves an expansion to a size that almost completely fills the host cell cytoplasm. The host cell and the bacteria together maintain the CCV; the CCV will keep expanding due to the replicating bacteria inside until it ruptures the cell and escapes into the immediate vicinity, when the whole infection process can start again (infect another macrophage) or bacteria die from a depletion of nutrients.

Fig. 1
C. burnetii binds to macrophages triggering phagocytosis through an actin-dependent mechanism. A normal phagosomal development requires the nascent CCV to evolve through fusion and fission events with endosomes and lysosomes soon after internalization. ...

3. Model formulation and solution

The birth-death-survival model created in Wood et al. (2014) extends the standard birth death process to take account of not only host heterogeneity but also a fundamental mechanism undergone by non toxin producing obligate intracellular bacteria, phagocytosis. Representing the in-host dynamics of infection with a biologically derived mechanistic model provides more meaningful results, especially at the crucial low dose level associated with accidental (Meselson et al., 1994) and deliberate (Legrand et al., 2009) aerosolised release. Three events are assumed to occur within the lung of an infected individual following the deposition of organisms; death (a single bacterium is killed) with rate μ per day, survival (a single bacterium is phagocytosed, but survives and multiplies within the phagocyte) with rate α per day, and birth (G bacteria are released from a single bacteria-containing phagocyte) with rate λ per day. If the number of extracellular bacteria and bacteria-containing phagocytes both reach zero then the infection is said to be resolved. Conversely, if the number of extracellular bacteria reach a threshold, M, then illness is said to occur.

In this paper we extend and modify the birth-death-survival model of Wood et al. (2014) for C. burnetii. We define the removal rate of bacteria from the extracellular lung space to be the sum of the death and survival rates mentioned above, so γ = α + μ, and the probability Θ that a bacterium survives phagocytosis is then given by Θ = α/γ. We also explicitly incorporate the probability of deposition ϕ depending on breathing rates and particle sizes at the point of exposure, so that given some inhaled dose D the dose retained in the lung K is defined by a binomial distribution such that K|D = d ~ B(n = D, p = ϕ). The inhaled dose is often observed in experiments but the retained dose is important as it is one of the initial conditions for the two simultaneous ordinary differential equations (ODEs) proposed in Wood et al. (2014) and outlined below. We assume D to be a random variable drawn from a Poisson distribution with mean d such that D ~ Poi(λPoi = d) and the retained dose K is then K ~ Po(ϕd).

In modelling the dose response process we define the number of extracellular bacteria B and the number of bacteria-containing phagocytes P, which are described by the following ODEs,

dBdt=λG PγB,
(1)
dPdt=ΘγBλP,
(2)

with their corresponding initial conditions,

B(0)=K,
(3)
P(0)=0.
(4)

Fig. 2 displays a compartmental diagram of our model Eqs. (1) and (2).

Fig. 2
A visual interpretation of Eqs. (1) and (2).

If the ratio between Θ and G is expected to be small (Θ [double less-than sign] G), dose response can be determined through a binomial distribution,

P(R|K=k)=1(1Θ)k,
(5)

where R is the event of at least one organism surviving phagocytosis and P represents the probability of infection given a retained dose k. Noting that the above k ~ Poi(λ = ϕd), we may develop a mixture distribution such that

P(k|D=d)=1eϕΘd.
(6)

Solving Eqs. (1) and (2) yields,

B=ϕdr+ρ(r+λ)ert+(ρλ)eρt,
(7)
P=ϕΘγdr+ρerteρt,
(8)

given the initial conditions (3) and (4); where,

r=(γ+λ)+(λγ)2+4ΘγλG2,
(9)
ρ=γ+r+λ.
(10)

Because ρ > 0, the second exponential term in Eq. (7) is always decaying. Thus, there can only be bacterial growth when r > 0, i.e. when ΘG > 1. Rearranging Eq. (9) gives,

γ=r(r+λ)ΘλG(r+λ).
(11)

Given a particular threshold for the bacteria, M, response occurs at time, tM, which, if the decaying term in Eq. (7) is sufficiently small, has an asymptotic solution of the form,

tM1rlogMϕ+logr+ρr+λlog(d).
(12)

We introduce the concept of the incubation period, associated with a single inhaled organism, denoted C, here to help simplify the model notation. We may define,

C=1rlogMϕ+log1+(r+γ)[ΘγG(r+γ)]rΘγG,
(13)

and using Eqs. (10) and (11) write Eq. (12) as

tMClog(d)r.
(14)

4. Model parameterization

4.1. Probability of deposition ϕ

The main factors that can influence the deposition are the physics of the aerosol, the anatomy of the respiratory tract and the airflow of the lung (Yeh et al., 1976). An established software package to estimate the deposition was used, named Multiple Path Dosimetry Model (MPPD), freely downloadable at Price et al. (2009). This software package calculates the deposition and clearance of monodisperse and polydisperse aerosols in the respiratory tract of adults and children (deposition only) for particles ranging from ultrafine (0.1 μm) to coarse (20 μm). The model is based upon single path methods for tracking air flow and calculating aerosol deposition in the lung. With respect to the anatomy of the respiratory tract, the MPPD model takes into account the structure and layout of the respiratory tract and diameters, lengths and branching angles of the airway segments.

It is assumed that there is a constant aerosol particle density of 1 g/cm3, that bacteria are spherical in shape and have a monodisperse particle distribution. For spherical particles, the particle diameter completely defines the characteristics of a monodisperse aerosol distribution. Most aerosols exhibit a skewed distribution with a long tail at large sizes. We apply the most common aerosol size distribution, the lognormal distribution (di Giorgio et al., 1996, Hinds, 1999, Huffman et al., 2010, Kowalski et al., 1999, Nicas et al., 2005, Pinnick et al., 2007), to characterise random variability in aerosol particle size.

The particle size of the aerosol was not reported in the study that we used for parametrisation (see Section 4.2), however, the particle size of C. burnetii has been stated as <1 μm in many papers including a paper by some of the same authors Tigertt et al. (1961). From the literature, Oysten and Davies (2011) and Coleman et al. (2004) measure particle size in the range of 0.2–0.5 μm, with a maximum of 1 μm. Dewé and Stärk (2016) and Jones et al. (2006) state the particle size is less than 1 μm, and Omsland et al. (2009) have measured an overall average diameter of 0.29 μm. Based on these data we derive a lognormal distribution, with a median of 0.29 μm and a 97.5% of 0.5 μm, giving logN(μ=1.2373; σ = 0.2776), to give a illustrative distribution.

The breathing patterns of individuals are critical to determine the inhaled dose and can change substantively due to the various types of activity levels individuals are participating in. We assume that the activity of all individuals coming into contact with the aerosolized form of bacteria are preforming a light intensity activity, i.e. sitting at a computer.

Breathing rates were extracted from the Exposure Factors Handbook for a healthy adult with a low activity level, see chapter 6 in the book Moya et al. (2011). From Table 6.2 in Moya et al. (2011), the recommended short-term exposure value for inhalation (males and females combined, over 20 years old) is 0.012 m3/min, with a 95th percentile of 0.016 m3/min. There is slight variation in the reported breathing rate by 10-year adult age groups but not substantively different from those rates associated with other activities (sedentary activity has roughly half the breathing rate and moderate activity about twice the breathing rate) and so activity type is more important than age. These inhalation values are based on a normal distribution assumption, giving N(μ=0.012,σ=0.002).

Now we have models for the breathing rate and the particle size, we can enter simulated values manually into the MPPD software to get the distribution for the deposition. To save time we have limited ourselves to 100 random estimates, treating particle size and breathing rate as independent events. We display the deposition estimates as a histogram in Fig. 3. In the MPPD model we have assumed a 5 lobe lung model in an adult, uniform flow expansion and constant exposure to the aerosol. We neglect deposition in other areas than the pulmonary region of the lung because they are negligibly small and not of interest.

Fig. 3
A histogram of the 100 deposition estimates produced from the MPPD software. The curve represents our estimated probability density function, beta distributed.

To these 100 deposition estimates we fit a Beta distribution, ϕ ~ Beta(α = 42.7, β = 300.6), which gives a median of 0.124 and a 95% confidence interval of 0.092 to 0.161. Plotting the probability density function of the fitted Beta distribution on top of the histogram in Fig. 3 visually demonstrates the similarities between the two distributions. Allowing ϕ ~ B(α, β) in (6) gives

PR(d)=1F11(α;α+β;Θd)=1F11(42.7;343.3Θd).
(15)

where F11 is Kummer's (confluent hypergeometric) function (Abramowitz and Stegun, 1972); sometimes referred to as the Beta-Poisson dose response model (Armstrong, 2005, Armstrong and Haas, 2007, Brooke et al., 2013, Haas, 2002, Teske et al., 2013).

4.2. Probability that a bacterium survives phagocytosis Θ and median infectious dose d50

Tigertt and Benenson (1956) and Tigertt et al. (1961) investigated serological response in humans given a particular inhaled dose of bacteria, where response was calculated as the number of infected over the number of exposed, represented in Fig. 4. Each volunteer inhaled a 10 l sample from an aerosol of whole egg slurry containing C. burnetii. The size dose per group was varied by dilution of the infected material with non-infected egg slurry prior to aerosolization. Grey literature from the Defence Special Weapons Agency by Anno and Deverill (1998), brings together the above military research volunteers clinical records on Q-fever as well as documentary information obtained from numerous journal articles, textbooks, and technical reports.

Fig. 4
The dose response data from Tigertt and Benenson (1956), Tigertt et al. (1961) and Anno and Deverill (1998) are displayed in this spaghetti plot as filled dots and the dashed curve is the median estimate.

We fit Eq. (15) to the data reported in Anno and Deverill (1998) by applying the maximum-likelihood estimation method. We estimate confidence intervals of the fitted parameters in R by estimating the standard errors and display them in the spaghetti plot in Fig. 4, alongside the data.

This method produces a median for Θ of 0.37, with a 95% confidence interval of 0.27–0.51, respectively. We estimate the median infectious dose, d50, by numerically solving Eq. (15) with respect to d and fixing PR = 0.5. This results in a median infectious dose of 15 organisms within a 95% confidence interval of 11 organisms to 21 organisms.

4.3. Low dose incubation period C and extracellular growth rate r

The low dose incubation period and extracellular growth rate parameters cannot be directly deduced from the literature so instead we estimate them by fitting Eq. (14) to human volunteer data for infection with C. burnetii (Tigertt and Benenson, 1956, Tigertt et al., 1961, Anno and Deverill, 1998) (see Fig. 5). To assess the uncertainty in C and r we estimate and plot the confidence intervals, which are also displayed in the spaghetti plot, Fig. 5.

Fig. 5
In this spaghetti plot the points represent the incubation period data from the Tigertt and Benenson (1956) and Tigertt et al. (1961) studies, the dashed curve is the median estimate and the shaded lines represent a sample of regression lines.

The median incubation period for a low dose (d = 1) of C. burnetii is 17.6 days, with a 95% confidence interval of 15.9 days to 20.4 days. The median extracellular growth rate is estimated as 1.30 per day with a 95% confidence interval of 0.87 per day to 1.84 per day.

4.4. Intracellular growth rate ω, burst number of bacteria released G and birth rate λ

The reciprocal of the birth rate (i.e. rate of release of bacteria from phagocyte) is the mean time until bacterial release, which is associated with phagocyte cell death, and the burst number is how many bacteria are released from the cell once it has burst. In the in vitro studies (Dellacasagrande et al., 1999, Dellacasagrande et al., 2002, Zamboni et al., 2001), Human Monocytic THP-1 Cells were infected with C. burnetii (Nine mile strain in phase II) and cell viability and bacterial population size was measured. The authors found that the bacteria grew in an exponential fashion until reaching a peak on around day 6. Other studies investigating the growth of C. burnetii also demonstrated similar population levels and that intracellular bacteria causes macrophages to rupture around day 6 post infection (Omsland et al., 2009, Ormsbee, 1952, Williams et al., 1991).

In order to interpret the results of these studies we have derived a three phase process to explain the in-host properties of infection. These phases are a lag phase, a growth phase and a stationary phase. First, the lag phase, is when the bacteria do not replicate, however cellular metabolism is accelerated; this takes approximately one day after internalization (Coleman et al., 2004, Howe et al., 2010, Kersh et al., 2011, Omsland et al., 2009, Zamboni et al., 2001). Next is the growth phase, where micro-organisms are in a rapidly growing and dividing state, if they survive phagocytosis. Their metabolic activity increases while exploiting the growth medium resulting in a single cell (assuming that each macrophage engulfs a single bacterium) dividing into two, which replicate into four, eight, sixteen and so on. This follows on to the stationary phase, where the nutrients in the growth medium are diminished as the bacterial population continues to grow. The bacteria growth rate will slow down significantly such that the cells undergoing division equals the number of cell deaths, and finally the bacteria will stop dividing completely. The total number of bacteria is not increased and thus the growth rate is zero. If a bacterium is taken from this stationary phase and introduced into a fresh medium, it can easily move back into the growth phase and is able to perform its usual metabolic activities.

In order to estimate the birth rate parameter λ we used the data from Dellacasagrande et al., 1999, Dellacasagrande et al., 2002 that used nine mile strain phase II bacteria to infect macrophages and apoptosis (a process of programmed cell death, biochemical events changes its morphology that results in death) was recorded. We use these data sets because cell death is governed by non-instantaneous processes (Parmelyet al., 2009), defining an actual time of death in such a study would not be possible. In Dellacasagrande et al. (1999), four independent experiments were performed that recorded cell viability in infected macrophages and in Dellacasagrande et al. (2002), the experiment was repeated but with five independent experiments; both set of results were almost identical. We have combined these data sets and displayed them in Fig. 6a. We assume that the time at which 50% of cells are unhealthy corresponds to the reciprocal of the birth rate parameter, λ. However, such information is not expressly provided in Dellacasagrande et al., 1999, Dellacasagrande et al., 2002, and so the value is estimated by fitting a distribution to the combined data, see Fig. 6a. Because the final stages of the intracellular life cycle are still being researched (Oyston, 2008), the choice of distribution isn’t motivated by any biological mechanisms, but purely by goodness of fit. A logistic distribution function is found to provide the best fit when evaluated alongside alternative potential distributions (Gaussian, log-normal, Cauchy and Weibull distributions). A median time of 4.74 days is found (95% confidence interval of 4.21 days to 5.75 days); the reciprocal of which is the birth rate, i.e. λ = 0.211 per day.

Fig. 6
Parametrization, detailing (a) the proportion of unhealthy cells over time, with a logistic distribution function fitted to data in Dellacasagrande et al., 1999, Dellacasagrande et al., 2002, data points are mean ± SD (b) number ...

The total number of intracellular bacteria was recorded at predefined times, τ [set membership] 1, 2, 3, 4, 5, 6, 7 days post infection in the bacterial study by Zamboni et al. (2001). The data point on day 7 of the experiment we assume is after the cell has burst, thus we neglect it. We derive a piecewise logistic function, Eq. (16), to characterise the bacterial growth. Let f(t) be the total bacteria at time t, Gˆ the upper asymptote of the curve, then

equation image

where ω is the intracellular growth rate parameter. We model the stagnant phase with a linear horizontal line at a bacterial CFU of 1, Eq. (16a), since we assume that there is no bacterial growth within the first day. When the bacterium reaches the cytosol bacterial growth begins and we model this with a general logistic function, Eq. (16b). The birth rate and burst number parameters are biologically linked; we use our estimated time until cell death to estimate the burst number, G. With this distribution a median burst number of 130 organisms is found with a 95% confidence interval of 106 organisms to 150 organisms.

4.5. Removal rate γ and threshold M

Using the derived empirical distributions for Θ, r, G and λ in Eq. (11) reveals an estimate for the removal rate of extracellular bacteria, of γ = 0.23 per day with a 95% confidence interval of 0.09 per day to 0.64 per day.

The final step in the parametrization of the model is the determination of the distribution for the number of extracellular bacteria within the lung space, M. Illness is driven by inflammation, i.e. from the immune response, not from the presence of bacteria alone. Inflammation starts when macrophages start rupturing and the body's internal mechanisms recognise that bacteria are ‘present’. These mechanisms are not modelled here, for parsimony, and we use the number of extracellular bacteria as a surrogate of symptom onset (illness).

Rearranging Eq. (13) in terms of M, reveals

M=(r+λ)(λGΘ(r+λ))λGΘ(2r+λ)(r+λ)2ϕeCr.
(17)

And again when inputting the derived distributions for ϕ, Θ, C, r, G and λ now in Eq. (17), reveals an estimate for the threshold of 8.7 log10 organisms with a 95% confidence interval of 6.4 log10 organisms to 11.7 log10 organisms.

We note that in estimating the threshold in Wood et al. (2014) and Egan et al. (2011) where only the deposition, extracellular growth rate and low dose incubation period affect it, has a similar form to our modified birth death survival model where,

Mbd=ϕeCr.
(18)

5. Discussion

Our model parametrization is summarised in Table 1. The framework of the human time dose response model derived here for C. burnetii can be used for other bacteria that act in a similar way. We also applied our modified birth death survival model to Francisella tularensis with the parametrization summarised in Appendix A. This helps to validate that our refined model is appropriate when modelling dose response in humans and thus a plausible model given the current evidence base.

Table 1
A table of the parameter point estimates and uncertainty on them. Noting, the variation in ϕ is variability while the variation in all other quantities is uncertainty. Parameters marked with a † have been fitted to data whilst other values ...

To the best of our knowledge this is the first human time dose response model for C. burnetii that includes deposition. One of the assumptions in estimating a median deposition value of 0.124 is that each individual is undergoing a light activity level. Lippmann et al. (1980) reviewed other human deposition papers (Chan and Lippmann, 1980, Davies et al., 1977, Heyder et al., 1975, Stahlhofen et al., 1980) alongside their work and estimated that a deposition value of around 0.1 agreed well with the literature (assuming a similar aerodynamic particle diameter to that in our model).

The model presented here predicts that the median inhaled infectious dose (d50) is 15 organisms. Hence a retained dose of about 1.86 organisms of C. burnetii is sufficient to cause infection in 50% of the population. Our model also predicts that a single inhaled organism can cause infection in 5% of hosts. The inhaled dose for 50% infection in humans is estimated in Brooke et al. (2013) as 1.18 bacteria, Brooke et al. (2015) as 0.71 bacteria, Tamrakar et al. (2011) as much as 10 bacteria and Jones et al. (2006) as low as 1 organism. In the Brooke papers they actually use the human Tigertt data and fit their model so it correlates well with their animal based model (guinea pigs), while we have taken the Tigertt data with additional data reported in Anno and Deverill (1998) (based on Anno and Deverill's interpretation of the dose required to cause infection in 50% of guinea pigs via intraperitoneal injection, GPIPID50 (Riley et al., 1964)); this is why our d50 values differ by an order of magnitude, though the difference between 1 and 10 organisms is perhaps microbiologically negligible. In other in vitro studies, detailing the number of bacteria to cause infection, Shannon et al. (2005) state below 10 organisms, Schaik et al. (2013) state 1–10 organisms, and Howe et al. (2010) state 1 organism.

The incubation period stated on the Centers for Disease Control and Prevention website (Centers for Disease Control and Prevention) is in the range of 14–21 days. This is very comparable to our median estimate for the low dose incubation period of 17.6 days within the confidence interval of 15.9 days to 20.4 days. Other papers state typical incubation periods of around 20 days (following a low dose assumption). Baca and Paretsky (1983) used a plaque assay technique, which utilizes primary chicken embryo cells for estimating viable C. burnetii; they detected plaques after 16 days of incubation. Porten et al. (2006) reported on a Q fever outbreak involving 299 cases with a median incubation period of 21 days, with an interquartile range of 16–24 days. Marmion et al. (1990) reported an incubation period from a natural attack of Q fever from 4 people of 20 days. Milazzo et al. (2001) claim that the incubation period after a natural exposure is 15–25 days, which is similar to Tissot-Dupont et al. (1999), where they claim an incubation period in the range of 21–28 days. The lowest median incubation period found in the literature is Derrick (1973) of 10 days.

Our estimates of the time to cell death (4.74 days) agree with the literature (Coleman et al., 2004, Hicks et al., 2010, Kersh et al., 2011, Omsland et al., 2009, Zamboni et al., 2001). For example, Zamboni et al. (2001) estimated bacterial growth within cells per day for 7 days. They detected small and large coxiella containing vacuoles on the first day post infection and observed that the bacteria population increased until it reached a peak on day 6. In other papers (Coleman et al., 2004, Hicks et al., 2010, Kersh et al., 2011, Omsland et al., 2009) also experiments were reported with infected cells and found no bacterial growth within the first day followed by exponential growth until day 6. Ormsbee (1952) inoculated embryonic eggs with C. burnetii; these chick embryos were examined daily for cell viability from 2 to 9 days post infection. Using the same method as in this work, that is looking when 50% of cells are unhealthy, gives a time for an infected cell to die of 5.7 days.

The intracellular doubling time is given by, log(2)/ω = 7.8 h with a 95% confidence interval of 7.1–9.0 h. Baca and Paretsky (1983) estimated the intracellular doubling time in the range of 8–12 h, Coleman et al. (2004) estimated it as 10.2 h, Howe et al. (2010) as 12.6 h, Wiegel and Schlegel (1976) as 3–12 h, Dworkin et al. (2006) as 7–8 h and Omsland et al. (2009) as 9.1 h. Using the same calculation we find that the extracellular doubling time, given by log(2)/r = 0.53 days (12.8 h).

Data from Zamboni et al. (2001) which was used to estimate the number of bacteria released, G, gave almost identical results to the data in Zamboni et al. (2002) for the cell's capacity. Roman et al. (1986) stated that a heavily infected cell has at least 100 bacteria inside. In the study by Coleman et al. (2004), the capacity of a vero cell is found to be around 116 bacteria. In the study by Omsland et al. (2009) they estimate the cell's capacity (vero cells) of 200 bacteria. In Howe et al. (2010) they concluded that 85 bacteria was the cells (human monocyte derived macrophages) capacity, while in Shannon et al. (2005), they found that 60 bacteria was the cell's (dendritic cells) capacity.

To help validate our range for the threshold number of bacteria, we extract estimates from the literature where studies have investigated growth rates (normally recorded per day) of C. burnetii in human host cells. Bacterial loads reached a maximum in the range of 7.6 log10 organisms to 8.8 log10 organisms (Coleman et al., 2004), 8 log10 organisms (Howe et al., 2010), 8.5 log10 organisms (Kersh et al., 2011), 9.8 log10 organisms (Shannon et al., 2005) and 6 log10 organisms (Omsland et al., 2009).

The only validation of the removal rate parameter was found in the study by Ghigo et al. (2002) where they used IFN-γ, which triggers the killing of C. burnetii in monocytes, in order to measure bacterial viability. They estimated the bacterial viability over the course of their experiment and it stayed relatively stable at around 82.5%, hence γ =− log(0.825) = 0.19 per day (from using Eq. (1) with P being zero).

6. Conclusion

In this paper we have extended the human time dose response model developed in Wood et al. (2014) by including a more realistic assessment of deposition and accounting for parameter uncertainty. However, uncertainty still exists in how responses may change in groups of people not matching the characteristics of the volunteer data used to parametrize the model. For example, the elderly or immune-compromised may respond to lower infectious doses, given the already low estimate of the ID50 (Centers for Disease Control and Prevention). Furthermore, the activity level, and thus the inhalation rate and deposition to the lung, will vary between (classes of) individuals and, whilst we have used a breathing rate akin to ‘normal’ activity, a real event may have a broader spectrum of individuals at different activity levels. We acknowledge that the difference in aerosol size between the Tigertt and Benenson (1956) experiment and aerosols generated by natural events is a limitation of the model. Other areas of uncertainty include pathogenicity of the organism involved and local environmental conditions. The latter may cause reduction in dose due to organisms settling on surfaces if raining or wider dissemination if windy. The model presented here is only valid for short term exposures, a long term exposure or repeated but infrequent exposures may affect the immune response and so lead to different outcomes in those exposed.

In conclusion this paper provides an initial framework for delineating the parameters and approaches that would be required for risk assessments associated with exposures to C. burnetii that might cause human infection. A key element of this work has been to present preliminary time-dose response modelling for C. burnetii using an extension to a relatively novel mathematical formulation whilst being parameterised by empirical human data.

Acknowledgements

The research was partly funded by the National Institute for Health Research Health Protection Research Unit (NIHR HPRU) in Emergency Preparedness and Response at King's College London and in Modelling Methodology at Imperial, both in partnership with Public Health England (PHE). Ian Hall is also a Member of NIHR Health protection Research Units in Emerging and Zoonotic Infections and Gastrointestinal Infections at Liverpool. The views expressed are those of the author(s) and not necessarily those of the NHS, the NIHR, the Department of Health or Public Health England.

Appendix A. The parametrisation of our Francisella tularensis time dose response model

Here we apply our modified birth death survival model to F. tularensis with the parametrization summarised in Table A.2. We have compared our updated parameter distributions with those of Wood et al. (2014) and Egan et al. (2011), to see what impact our improvements to the model have imposed on them. The two largest differences between the model presented here and Egan et al. (2011) and Wood et al. (2014) is a more realistic estimate of the deposition and Wood et al. (2014) used non-human primate data to parametrise the model whereas here we have restricted the parametrisation to human data alone. We have extracted the parameter values from Egan et al. (2011) and Wood et al. (2014) and added them to Table A.2 in addition to deriving other parameter estimates (where possible) which were not reported in the original studies.

Table A.2

A table of the estimated parameter distributions for F. tularensis. Parameters marked with a † have been fitted to data whilst other values are derived from or assumed within the study.

ParameterDefinition2.5% CIMedian97.5% CIUnitsPrevious values
Wood et al. (2014)Egan et al. (2011)
ϕDeposition0.0650. 1100.170probability11
ΘProbability of surviving phagocytosis0.3430. 6601probability0.030. 07
d50Median Infectious Dose6918organisms2310
CLow dose incubation period137155172hours131155
rExtracellular growth rate0.0940. 1210.169per hour0.220. 121
λBirth rate0.0210. 0230.025per hour0. 0241
ωIntracellular growth rate0.1980. 2120.224per hour0. 212
GBurst number314364419organisms358
log10MThreshold6.06.98.4organisms11. 48.2
log10MbdThreshold birth death model5.97.29.3organisms11. 48.2
γRemoval rate0.00180.00330.0071per hour3.0939

The difference in the deposition between the C. burnetii and F. tularensis models are due to the different particle sizes of the organisms. In the studies (Saslaw et al., 1961, Sawyer et al., 1966) the authors generate an aerosol by a Collison spray, constructed and calibrated it to give an average particle diameter of 0.7 μm and rarely producing diameters less than 0.2 μm. Therefore, we derive a log-normal distribution for particle diameter that produces a median of 0.7, with 0.2 being the 2.5% quantile; this gives logN(μ=0.3567,σ=0.6392). All of the subjects inhaled the aerosol through their nose.

The value Θ was reported in the discussion of Wood et al. (2014) as ps where a value of Θ = ps = 0.03 was calculated. This value is vastly different from our estimated value of Θ = 0.660. The main reason for the significant difference is that we have implemented deposition into the model, though we have also only used human data. It should be noted that the second exponential term in Eq. (7), with central estimate of the parameter ρ = 0.15 per hour (Eq. (10)), decays to negligible amounts on a timescale of a day (within 10 h it contributes to only 5% of the total extracellular bacterial load). On reflection, it is unlikely that such a term could be meaningfully estimated or validated from experimental non-human primate data without regular and frequent observations (at least hourly for first 12 h) which would be an expensive investigation and would require careful experimental control of retention of data in the lung space.

Results from Egan et al. (2011) indicate a median infectious dose of about d50 = 10 organisms, which is similar to our derived value for the median infectious dose of 9 organisms with a 95% confidence interval of 6 organisms to 18 organisms. Wood et al. (2014) reported a larger value for the median infectious dose of 23 organisms.

The median low dose incubation period C = 155 h (6.46 days) and the extracellular growth rate r = 0.121 per hour is estimated in the same way as in an earlier model (Egan et al., 2011). That is by logging the dose and fitting a linear model though the data. In Wood et al. (2014), these parameters are not estimated directly but can be derived from other parameter estimates (relying on non-human primate data). The low-dose incubation period is approximately one day less than estimated here. However, the extracellular growth rate is slower in the Wood et al. paper meaning that at higher inhaled doses, the incubation period is in fact longer than estimated here. This also means that the extracellular bacterial doubling time, given by log(2)/r = 5.7 h, is almost twice that derived in Wood et al. (2014) (3.1 h).

In comparing the birth rate parameter, λ, we estimate a median value of 0.023 with a 95% confidence interval of 0.021–0.025, while Wood et al. (2014) estimates it to be 0.0241. We expected a slightly different value for this parameter due to different extrapolation techniques from Lindemann et al. (2011). The reciprocal of the birth rate λ is estimated through the median time until cell death-found to be 43.4 h. This duration is supported by another similar in vitro study (Mack et al., 1994), in which at 24 h 100% of cells are still healthy (92% in Lindemann et al. (2011)).

We estimated a median value for the burst number, G, of 364 organisms with a 95% confidence interval of 314 organisms to 419 organisms, while Wood et al. (2014) estimated it be 358 organisms. Again, we expect these values to be similar as well, but since there is a difference in the birth rate between the two models there will be a slight difference in the burst number. The value for the burst number given in Wood et al. (2014) does fall within our estimated range for the burst number, as does the equivalent measure of λ. While there is no comparable study to validate the estimate specifically, the validity of the intracellular doubling time can be assessed. For the first 24 h, this is found to be log(2)/ω, with ω in the range of 0.198–0.224 with a median the same as in Wood et al. (2014) of 0.212, which gives a median of 3.3 h doubling time with a 95% confidence interval of 3.0–3.6 h, which is within the range of 3–8 h found in Butchar et al. (2007) as noted in Wood et al. (2014).

The bacterial threshold parameter is 6.9 log10 organisms with a 95% confidence interval of 6.0 log10 organisms to 8.4 log10 organisms, while Wood et al. (2014) produce a median value of 11.4 log10 organisms with a 95% confidence interval of 6.2 log10 organisms to 16.5 log10 organisms. This difference in thresholds is mainly due to how we have obtained parameters C and r, here we fit them to human data and in Wood et al. (2014) they are derived from other fitted parameters from non-human primate data. Using the parametrization presented in Egan et al. (2011) and Eq. (18) gives 8.2 log10 organisms for the threshold (with ϕ = 1) which is slightly higher than our value of 6.9 log10 organisms. We estimate the threshold by evaluating Eq. (17), where we essentially have a scaling factor (≈0.54ϕ = 0.06) multiplied by eCr. Since there are no in vivo studies that investigate the number of extracellular bacteria on illness for humans, complete validation of the resulting distribution is not possible.

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