Fig 1.
Schematic of the mathematical model of within-host SARS-CoV-2 infection.
The key quantities and their interactions contained in our model (Eqs 1–4) are illustrated. Arrows and blunt-head arrows depict positive and negative regulation, respectively. The parameters and rate expressions shown next to the arrows are described in the text.
Table 1.
Model parameters and their description.
Fig 2.
Model predictions are consistent with in vivo dynamics.
(A) Best-fits of our model (lines) to patient data (symbols) of normalized sputum viral load as a function of time from viral exposure [38]. The corresponding (B) innate immune responses and (C) CD8 T-cell responses predicted by our model. The asterisks represent rescaled variables (see Methods). Patient IDs, as provided in Bӧhmer et al. [37], are in the top-left corner of each subplot in (A).
Table 2.
Estimated population parameters of the model fit to the sputum viral load dataset [38] (Fig 2).
Table 3.
Estimated individual parameters of the model fit to the sputum viral load dataset [38] (Fig 2).
Units are the same as in Table 2.
Fig 3.
Predator-prey-like oscillations may underlie multiple viral load peaks.
(A) The dynamics of infected cells (I) and innate immune response (X) in the absence of CD8 T-cells (E). The asterisk represents rescaled variables (see Methods). (B) Corresponding trajectory on a phase plane plot of infected cells and innate response. (C) and (D) Predictions as in (A) and (B) but in the presence of E. Inset shows the dynamics of E over time (in days). The individual parameters for patient id 10 (Table 3) were used. Parameter values used are as the following: k1 = 4.41/day, k3 = 0.73/day, /day,
τ = 0.88 day, k6 = 0.2/day, k4 = 1.5/day,
α = 1.0×104, β = 2.0×104/day, γ = 0.5/day. In (A) and (B), we set
whereas in (C) and (D), we let
= 6.42×10−3/day.
Fig 4.
Variations in innate and CD8 T-cell responses capture disease heterogeneity.
(A) Effect of variation of parameters determining the strengths of innate and CD8 T cell responses on the trajectory of the infection. The black annotated triangles at the top and right indicate the nature and the direction of the variation of the indicated parameters. Individual subplots show the dynamics of infected cells, cytokine mediated innate immune response, and effector CD8 T-cell response. In each subplot, the left Y-axis shows the normalized infected cell dynamics and the right Y-axis shows the other two species. The rectangular, colored patch at the top of each subplot represents the extent of immunopathology. The range of immunopathology is given by the color scale at the bottom. On the left-side of the color scale, a separate legend denotes the texture used for depicting unbounded immunopathology. Unity on the colorscale indicates the immunopathology quantified in the central subplot (subplot with an arrowhead), calculated using the population parameters estimated from Fig 2. (B) The tissue damage (D) associated with each subplot in (A) is shown. The area shaded light orange in each panel is used to calculate immunopathology (see S1 Fig), and is also depicted by the colored patches in the subplots of (A). (C) The effect of varying the sensitivity of CD8 T-cell response to antigen, kp. The presentation is similar to (A). The scale for immunopathology is in (A). The population estimates (fixed effects) of the parameters estimated in Table 2 were used. Other parameter values used are: k6 = 0.2/day, k4 = 1.5/day, α = 104, β = 2.0×104/day, γ = 0.5/day. Variations in k3 are obtained as the following fold-changes to the above value: 0.35, 0.75, 1, 2, 3. The fold-changes for variation in
are: 0.5, 0.75, 1, 2, 5. Values of
used in (C) are: 1.0×10−5, 1.0×10−4, 2.497×10−4, 5.0×10−3, 5.0×10−2.
Fig 5.
Fits of the model to viral load data from patients with mild and severe symptoms.
Best-fits of our model (lines) to data (symbols) of nasopharyngeal viral load from patients with (A) mild and (B) severe symptoms. Cross marks represent data points below the limit of detection. Entries in the boxes show patient IDs as provided in [38] in (A) and the location (row and column) of the subplot in the original figure in [60] from which the data was extracted in (B).
Table 4.
Population parameters estimated by fitting the model to NP swab viral load data from mildly infected patients [38] (Fig 5A).
Table 5.
Population parameters estimated by fitting the model to NP swab viral load data set from severely infected patients [60] (Fig 5B).
Fig 6.
Comparison of parameter estimates between mildly and severely infected patients.
Best-fit estimate of (A) T cell proliferation rate (k3), (B) strength of innate response (), (C) initial population of virus-specific T cells (
), (D) antigen threshold for T cell proliferation (
) between mildly (M) and severely (S) infected patients (see Fig 5). ns: not significant, ***: p < 0.0001. (E) The distributions of model-predicted immunopathology of the two patient cohorts using different values of β, reflecting the relative contribution of cytokines to immunopathology. The solid lines in the violin plots are medians.