Simulation results for the time-averaged size of the bacteria population in the equilibrated system.

<p>The time-averaged size of the bacteria population, , is computed as a function of the clearance rate according to Eq. (68). The proliferation rate and the carrying capacity are kept fixed in the computations for different substitution therapies with frequencies (black line), (red line), (green line), and (blue line). It is clearly observed that the critical value of the clearance rate with depends on the treatment frequency.</p

Qualitative change in the equilibrium probability distribution .

<p>The equilibrium probability distribution Eq. (75) is plotted for different values of the parameter . For (red line) the distribution vanishes, , indicating that the infection is not cleared. For (green line) the distribution diverges, , making the clearance of infection the most likely event. The transition occurs for (black line) where equals the exponential distribution.</p

Schematic representation of the stochastic immune response model under IgG substitution therapy.

<p>A: The serum IgG trough level is maintained by substitution therapy with treatment frequency and IgG dose per treatment (infusion process: →). The natural degradation of IgG occurs with rate (self-inhibiting process: ⊣) and the binding of IgG to bacteria occurs with clearance rate (binding process: →•←), giving rise to IgG-bacterium complexes that are removed from the system. The bacteria population obeys a logistic growth dynamics that is characterized by the proliferation rate (self-activating process: →) and the carrying capacity (self-inhibiting process: ⊣). B: The shape space is an abstract high-dimensional space, where essential features of IgG binding regions with respect to the considered bacteria species are represented as points. For reasons of clarity, a two-dimensional shape space is depicted where the two axis represent independent features of IgG binding, such as charge and size of the binding site. The inhomogeneous IgG shape space distribution under substitution therapy with pooled IgG (bluish area) is represented by the color intensity (arbitrary units). Even if the total amount of administered IgG per treatment is exactly the same, with respect to a specific bacterium, i.e. a specific shape space area (red point), fluctuations occur from treatment to treatment.</p

Simulation results of immune responses to infections for different clearance rates in the presence of fluctuations in the administered IgG dose per treatment with fluctuation strength .

<p>The serum IgG concentration and the bacteria population are plotted as function of time and for substitution therapies with frequencies (black line) and (blue line). A and B: For clearance rate the infection is cleared after 43 days under substitution therapy with frequency , whereas a chronic infection develops under substitution therapy with frequency . C and D: For clearance rate the infection is cleared within 1 and 3 days under substitution therapy with frequency and , respectively. E and F: For the same parameters as in C and D but for a different random realization of fluctuations in the administered IgG dose per treatment. Even though the infection is ultimately cleared, this is only achieved after 44 days under substitution therapy with frequency .</p

Overview of model parameters.

<p>Overview of model parameters.</p

Analytical results for the critical clearance rates in the presence of fluctuations.

<p>A: The ratio [cf Eq. (80)] reveals that the critical clearance rate decreases with increasing fluctuation strength (black line), (red line), (green line), and (blue line). B: For clearance rates above (solid lines) the infection is cleared [cf Eq. (80)], whereas a chronic infection develops for clearance rates below (dashed lines) [cf Eq. (84)]. The dependence of the critical clearance rate on the applied substitution therapy becomes increasingly significant for increasing fluctuation strength (black line), (red line), (green line), and (blue line).</p

Simulation results of the immune response to infection for different clearance rates in the absence of fluctuations in the administered IgG dose per treatment.

<p>The serum IgG concentration and the bacteria population are plotted as a function of time and for substitution therapies with frequencies (black line) and (blue line). A and B: Limit of high clearance rate with . Within one day the infection is cleared by extinction of the growing bacteria population under both substitution therapies. C and D: Limit of low clearance rate with . A chronic infection rapidly develops under both substitution therapies which is accompanied by a reduced serum IgG trough level. E and F: In the intermediate regime of clearance rates with a qualitative difference is observed between substitution therapies with frequencies and . The former succeeds in clearing the infection within 14 days time, whereas the latter can not hold the bacteria population at bay such that a chronic infection develops.</p

Infection clearance time as a function of shape space inhomogeneities for different values of the fluctuation strength.

<p>The clearance time [cf Eq. (87)] as a function of [cf Eq. (86)] for different fluctuation strengths (red line), (green line), and (blue line). For larger than a few percent, the duration of infection rapidly exceeds months and years.</p

Analytical results for the critical clearance rates in the absence of fluctuations.

<p>A: The intersections between (horizontal lines) and (grey lines) indicate the critical clearance rates (crossing points) according to Eq. (73). This is shown for different values of and substitution therapies with frequencies (black line), (red line), (green line), and (blue line). B: Comparison of substitution therapies with different treatment frequencies by the ratio according to Eq. (74).</p

Comparison of serum IgG concentrations for different substitution therapies in the absence of infections.

<p>A: Simulation results based on Eq. (1) with and the source term Eq. (5) for different substitution therapies in the absence of fluctuations (). The frequencies of the substitution therapies are: (black line), (red line), (green line), and (blue line). B: Analytical calculation of the relative difference in the IgG doses per treatment (red line) and time-integrated (green line) for different substitution therapies according to Eqs. (59) and (60), respectively. C: Analytical calculation of the serum IgG peak level (red line) and the time-averaged dose (green line) for different substitution therapies according to Eqs. (61) and (62), respectively. D: The same as in A but in the presence of fluctuations in the administered IgG amount per treatment with fluctuation strength .</p