Fig 1.
Within-host model of Francisella tularensis pathogenesis.
A single bacterium is taken up by a macrophage, inside a phagosome (top left). Without activating the macrophage, the bacterium escapes and multiplies inside the cytosol (top line) eventually causing the macrophage to rupture and release many bacteria. Free bacteria (center) may infect other macrophages, die or migrate to a different organ (bottom line). Macrophages exist in resting (green), anti-inflammatory (blue) and pro-inflammatory (red) states.
Fig 2.
A birth-and-death process with catastrophe representing division, death and rupture.
The state n represents a macrophage with n cytosolic bacteria. There are three types of events: transition to state n + 1 (division of a bacterium, rate βn), transition to state n-1 (death of a bacterium, rate μn), and transition to state H (rupture of the macrophage with release of n bacteria, rate δn). In this work we assume that βn = βn, μn = μn and δn = δn. The states 0 and H are absorbing. The infected macrophage survives for as long as it does not reach the state H.
Fig 3.
Realisations of the birth-death process with catastrophe.
On the left, the initial number of bacteria, k = 1. On the right, the initial number of bacteria, k = 2. On the right, the two families follow, independently, the same stochastic process as in the case k = 1. However, the catastrophe affects both families at the same instant. The parameter values are β = 0.15 h−1, μ = 0.01 h−1 and δ = 0.01 h−1 [23].
Fig 4.
Polynomial governing the survival function.
F(S) = βS2 − (β + μ + δ)S + μ is the RHS of (4) with parameter values β = 0.15 h−1, μ = 0.01 h−1 and δ = 0.01 h−1. The constants a and b satisfy and
. The value of a is equal to the probability that the infected macrophage eliminates the infection, rather than ruptures.
Fig 5.
Survival function, probability density function and bacterial release on rupture.
Left: the macrophage survival probability function, S(t). Centre: the bacterial load is proportional to . The vertical line is (6). Right: the function
that gives the mean number of bacteria released per macrophage. The parameter values, taken from Ref. [23], are β = 0.15 h−1, μ = 0.01 h−1 and δ = 0.001 h−1. The dashed lines show the corresponding functions when μ = 0.
Fig 6.
Agent-based realisation compared to predicted means: First cohort of macrophages.
In a numerical realisation, N = 30 macrophages are infected, by one bacterium each, at t = 0. The red line shows the number of those macrophages surviving up to time t. The dotted red curve is NS(t), using the survival function (5). Each blue dot in the lower panel coincides with a downward step in the red line, corresponding to a macrophage rupture event. The dotted blue curve is , given by (11). Parameter values have been taken from Ref. [23]: β = 0.15 h−1, μ = 0 and δ = 0.001 h−1. Agent-based realisations are simulated making use of tau-leaping time-stepping with Δt = 0.01 h−1, M = 104, ρ = 0.01 h−1, ϕ = 2 h−1, μE = 0.01 h−1 and γ = 1 h−1.
Fig 7.
Cohorts of bacteria in the lung.
The formulæ for the number of bacteria in phagosomes (left) and cytosols (right), obtained from (16a) and (16b) are shown as dashed lines. Averages over 102 realisations of the agent-based computational model are shown as solid lines. The parameter values, taken from Ref. [23], are β = 0.15 h−1, μ = 0, δ = 0.001 h−1, M = 104, ρ = 0.01 h−1, ϕ = 2 h−1, γ = 0.1 h−1, μE = 0.01 h−1, N = 102 and Δt = 0.01h−1.
Fig 8.
An indication of the most important parameters that describe the dynamics of total bacterial counts in the lung (left) and MLNs(right) during the first 48 hours of infection. The ranges over which each parameter is varied are: (Mρ) ∈ [10−2 h−1, 105 h−1], ϕ ∈ [0.5h−1, 5h−1], β ∈ [10−2 h−1, 1h−1], δ ∈ [10−5 h−1, 10−1 h−1], μE ∈ [10−4 h−1, 10−1 h−1] and γ ∈ [1h−1, 103 h−1].
Fig 9.
Bacterial counts in the lung, MLN, liver, kidney and spleen.
Mice are exposed to either 160.33 CFUs (high), 13.7 CFUs (medium) or 2 CFUs (low) of Francisella tularensis SCHU S4 bacteria. The observed data are denoted by shaded points, whilst the geometric mean and standard deviation are represented by solid points and bars, respectively. Zero counts have been replaced by one in order to calculate the geometric mean and standard deviation.
Table 1.
A description and value of the parameters and prior distributions used to determine the total bacterial load in each organ with approximate Bayesian inference. Migration probabilities are calculated using the proportion of bacteria in each organ after 48 hours, starred parameters are inferred using ABC from the observed bacterial counts in each organ.
Table 2.
Table with experimental data sets. Bacterial counts in the lung, MLN, liver, kidney and spleen used for the parameter inference.
Mice are exposed to either 160.33 CFU (top) or 13.7 CFU (bottom) of Francisella tularensis SCHU S4 bacteria. Geometric means and standard deviations (SD) are also given.
Fig 10.
A comparison between model predictions of total bacterial counts and observed bacteria counts for medium (left) and high (right) initial doses. Solid curves and shaded regions, respectively, denote pointwise median predictions and 95% credible regions. These have been constructed using all parameter sets from the posterior sample.
Fig 11.
Posterior histograms for β and δ.
From the posterior sample, the histogram for β (left) shows the significant learning that has been achieved making use of the experimental data. When comparing to bacterial counts, only an upper bound for δ can be identified (centre). However, this distribution can be refined when also considering model predictions for the mean time until rupture (right).
Fig 12.
Posterior histogram for Mρ and γ.
Bivariate histogram (left) depicting the strong correlation between Mρ and γ and a histogram of the migration probability (right), constructed using posterior samples of Mρ, γ and μE = 0.01 h−1.
Table 3.
Summary statistics of the posterior sample for each parameter included in the approximate Bayesian inference.
Posterior samples contain 5 × 103 values.
Fig 13.
Low infectious dose predictions.
Predictions of total bacterial counts in each organ following infection at a low initial dose. Posterior distributions inferred by performing ABC with the medium and high doses have been used to create pointwise median predictions and 95% credible regions.