4.5. Model Integration, Analysis and Parameter Estimation

The systems of ordinary differential equations were integrated using the LSODA method [55,56] in COPASI software (http://www.copasi.org, accessed on 1 January 2020).

Estimations of model parameters were undertaken separately for model Ab and model Cv based on experimental data or previously published models, as described below and in the Supporting Information (Texts S1 and S2). Additional parameter adjustment for the complete model was based on experimental data [1,35,57].

For model Ab, values for naive B-cell proliferation, antibody secretion by plasma cells and antibody death rates were taken from published mathematical models and the literature [38,51,58], as indicated in Tables S1–S4. Parameters for the antigen–antibody complex formation were obtained from the average values for binding with monoclonal antibodies [59]. The ranges for time lags t_memory and t_naive were calculated from known data on lymphocyte physiology [60,61] and experimental results [34] (see Text S1 for the calculation details).

For model Cv, parameters were either fitted for personal patient data [1] or taken from [36] (Table S4).

For the complete model, the parameters π_V* and δ_p were estimated from the data on SARS-CoV-2 replication in human airway epithelial (HAE) cells [35]. The set of equations for HAE cell infection is represented in Text S3 in the Supplementary Materials. The parameters for the macrophage activity were calculated based on experimental data and published models [1,38,57], as described in Text S2 in the Supplementary Materials.

The analysis of the models consisted of time-course simulations, analysis of steady states performed using Newton’s method [55] in COPASI and sensitivity analysis [55,62,63]. For the latter, we calculated sensitivities using the “one-at-a-time” approach and the following formula:

where Ri indicates the model response (Va, S_a, A_b, B, Lp, Bm at time 30 days for model Ab) and p_j-s are the models parameters. The detailed results are represented in Table S5.