Fig 1.
Schematic network d(See Table 1 for notation).
Macrophage M0, from blood monocytes, polarize into either M1 or M2 macrophages under stimulation by different cytokines. Dendritic cells attract T cells from the lymph nodes into the granuloma. When an M2 macrophge phagocytoses extracellular bacteria Be, it becomes an infected macrophage M2i. M1, M2, M2i, D, Th1 and Th2 cells produce a variety of proinflammatory and anti-inflammatory cytokines, which upregulate or downregulate these cells. The growth of bacteria Bi inside macrophages M2i is inhibited by IL-1β, TNF-α and IFN-γ (modulated by IL-6). When an infected macrophage M2i bursts, its Bi are released to become extracellular bacteria.
Table 1.
The variables of the model; concentration and densities are in units of g/cm3.
Table 2.
Parameters’ description and value.
Table 3.
Parameters’ description and value.
Fig 2.
Simulation results over 100 days with α = 1 and initial inhalation of 102 Be and 103 Bi; and all of the units are in g/cm3.
The symbols for cells and cytokines are listed in Table 1, and all the parameters are listed in Tables 2 and 3.
Fig 3.
“Switching time” of macrophages; all the parameters and initial conditions are as in Fig 2.
M1 macrophages become dominant over infected M2 macrophages at day 47.
Fig 4.
“Switching time” v.s. α; all the parameters (except α) and initial conditions are as in Fig 2.
Since switching time cannot be shorter than at least two weeks [44], the biologically range of α is 0 < α < 50.
Fig 5.
The total bacteria load (in g/cm3) v.s. α; all the parameters (except α) and initial conditions are the same as in Fig 2.
The bacterial load decreases as the immune strength parameter α increases.
Fig 6.
The radius of the granuloma at day 100 v.s. the immune strength parameter α; all the parameters (except α) and initial conditions are as in Fig 2.
The radius at day 10 is monotonically increasing with respect to the parameter α.
Fig 7.
The total bacteria under IL-10 Ab treatment for α = 1; the treatment begins at different weeks and goes on for 30 days.
The IL-10 Ab is assumed to reduce the production of IL-10 by 90%, i.e., the parameters λI10 M2i, λI10 D in Eq (14) and λI10 M2 are reduced by factor 1/10 during the 30 days of the treatment. All the other parameter and initial values are the same as in Fig 2.
Fig 8.
The percentage of total bacterial load reduction with anti IL-10 when the Ab is administered at different initial times (for 30 days) to patients with different immune strength α.
The week when treatment begins is noted on the vertical axis. The color column indicates the percentage of bacterial reduction after 100 days from the initial time when the granuloma was formed. The parameter λI10 M2i, λI10 D and λI10 M2 in Eq (14) are reduced by factor of 1/10, and all the other parameters (except α) and initial values are the same as in Fig 2.
Fig 9.
The percentage of total bacteria load reduction with anti IL-13 when the Ab is administered at different initial times (for 30 days) to patients with different ‘immune strength’ α.
The parameter λI13 T2 and λI13 M2 in Eq (16) are reduced by factor of 1/10, and all the other parameters (except α) and initial values are the same as in Fig 2.
Fig 10.
The sensitivity analysis for the cytokine production rates.