key: cord-1001296-i6i5xenx authors: Blanco-Rodríguez, Rodolfo; Du, Xin; Hernández-Vargas, Esteban title: Untangling the cell immune response dynamic for severe and critical cases of SARS-CoV-2 infection date: 2021-03-24 journal: bioRxiv DOI: 10.1101/2021.03.23.436686 sha: d8de462264fb03ee424b585db562c67d7251df8d doc_id: 1001296 cord_uid: i6i5xenx COVID-19 is a global pandemic leading high death tolls worldwide day by day. Clinical evidence suggests that COVID-19 patients can be classified as non-severe, severe and critical cases. In particular, studies have highlighted the relationship between the lymphopenia and the severity of the illness, where CD8+ T cells have the lowest levels in critical cases. In this work, we aim to elucidate the key parameters that define the course of the disease deviating from severe to critical case. To this end, several mathematical models are proposed to represent the dynamic of the immune response in patients with SARS-CoV-2 infection. The best model had a good fit to reported experimental data, and in accordance with values found in the literature. Our results suggest that a rapid proliferation of CD8+ T cells is decisive in the severity of the disease. COVID-19 caused by SARS-CoV-2 infection is a global pandemic which has caused more than 40 millions con-22 firmed cases and more than 1 million deaths worldwide. People of all age can be infected where around 20% of the 23 cases are asymptomatic, 60% appear with mild or moderate conditions, and 20% are severe or critical cases [2] . Most 24 of the countries have taken emergency actions, these actions include confinement of their population, travel restric-25 tions, forced use of mask in public spaces, and even a nighttime curfew. In this situations, epidemiological models 26 have been key to mitigating COVID-19 pandemic and many others [15] . 27 There are three coronaviruses (SARS-CoV, MERS-CoV and SARS-CoV-2) that can cause pneumonia, which can 28 be fatal. SARS-CoV-2 is transmitted mainly via respiratory droplets, the median incubation period is around 4 days 29 before symptom onset [13], most of symptomatic patients developing symptoms within 11.5 days [21] . The viral load The data are displayed in Fig. 1 , the median is represented as points and the dashed lines represent the interquartile 95 range (IQR). Data are reproduced from the original paper [48] using plotDigitizer. In [16] has been reported a mathematical model to represent the interaction between SARS-CoV-2 infection and 98 immune response dynamics. The model is given by: where is the virus level, the number of CD8 + T cells, the viral replication rate with maximum carrying capacity and (0) the initial number of them. In figure 2 is shown a schematic representation of this model. Contrary to [16] , in this work we fitted the data of CD8 + T cells to the model (2) to [16] . Also we use the initial viral level (0) = 0.31 copies/ml. The parameter for each case was fixed with the 109 respective initial value (0). Due to lack of data before illness onset, we assumed the initial level of T cells equal to 110 the median of the CD8 + T cells in the day 3 ( (0) = (3)) from the reported data for each case. Infection time was 111 assumed at -3 days after illness onset (daio). In our model, we included CD8 + T cells in the peripheral blood of patients described above, who have a wide range 113 of comorbidities. We only considered several and critical cases, since data from moderate cases showed no marked 114 changes during the disease course. Similar to CD8 + T cells, there is a evidence of activation and/or exhaustion markers at CD4 + T cells [6] . Even it has 116 been suggested that CD4/CD8 ratio is significantly higher in critical patients than non-critical patients [31] . Because 117 of that, we modified the model. We considered that CD4 + helps to proliferation of CD8 + T cells which occurs at rate 118 4 where 4 is CD4 + T cell level and is a free parameter to be estimated. We use piecewise linear fits to generate 119 a time-dependent function 4 ( ) using the experimental data displayed in Fig. 1b . We also explored different ways to 120 integrate CD4 + T cell data to our model, however, we do not obtain good results. innate and adaptive immune responses [42] . It has been correlated the number and function of NK during SARS-CoV-123 2 infection with the severity of the disease [50, 26] . Therefore, we explored the viral clearence due to NK ( ) at rate 124 . Similar to modification above, we use piecewise linear fits to generate ( ) using data in Fig 1c. The ordinary differential equations of the model were solved using a Dormand-Prince fifth-order Runge-Kutta algorithm. The estimation of the free parameters was performed by minimize the Root Mean Square Error (RMSE) using the difference between the experimental measurement ( ) and the predictive output (̄ ) as follows: where is the corresponding sample and is the total number of measurement. In order to compare between different models, we used the Akaike information criterion (AIC) defined by: where is the number of data points and is the number of unknown parameters. A lower AIC values means a better 132 description of the data. However, if the data are insufficient and manifest large variability, the parameter could be practically non-identifiable. This can be visualized as a relatively flat valley in the profile likelihood. A structurally non-identifiable parameter has 142 a profile that maintains a constant RMSE when the parameter is varied. we generated 27 discrete data between lower limit and upper limit of the IQR (including the median value) for each 148 point of the daio (x-axis). We then performed a nonparametric bootstrap approach using Monte Carlo resampling. Data were resampled with replacement to have a sample of equal size to the generated values above. The parameters 150 are estimated from the resampling. We adapted our DE code to perform 100 parameter estimations at the same time 151 using GPU parallelization in order to save computational time. A total of 1000 optimizations (10 runs of our DE code) 152 were performed using different sets of resampled data. We obtained the corresponding parameter distribution from 153 refit our model in each of these repetitions. The total count of cells (accumulative sum) between both cases are in the same order of magnitude although is lower 161 for critical cases (1.3 × 10 7 ) than severe cases (2.2 × 10 7 ). The viral load obtained using the model with the parameter fitted to CD8 + T cells experimental data is displayed 163 in Fig. 3b . The viral load peaks around 40 daio for critical cases and 20 daio for severe cases. There is a delay in the 164 peak of the viral load for critical cases compared to severe cases; critical viral load peak is two orders of magnitude lower than severe viral load peak. This result is consistent with those reported in [39], although in those results the 166 difference is of one order of magnitude between severe and non-severe. The total viral count is higher for severe case 167 (1.2 × 10 8 ) than critical cases (4.5 × 10 6 ). The best fitted parameters are presented in Table 1 for CD8 + T cells and the two cases of illness severity. The 174 viral replication rate for critical cases is a half of that for severe cases. The viral clearance for critical cases 175 is approximately one third of that for severe cases. The CD8 + T cell proliferation rate is higher for critical cases. These results suggest that the rapid proliferation of T cells and the low clearance rate are key in the development of 177 the disease. The role of the immune system during SARS-CoV-2 infection is fragmented. The T cell kinetics seem to be decisive those that model severe cases. The principal difference is in the rate of T cell proliferation , this rate is high in critical 208 cases. This is in accordance with [20, 46], suggesting a hyperactivation and overaggressive CD8 + T cell response. However, it is still unclear whether the T cells in COVID-19 patients are exhausted or just highly activated [6] . 210 On the other hand, fitting results show a viral clearance rate for severe cases is higher than that for critical cases. The viral replication rate for severe cases is also higher than that for critical cases which translates to a higher viral peak. Therefore, although the severe cases have a low production of CD8 + T cells compared with that critical cases, The model that best describe the data is considering CD8 + T cell response. The best fits show a delay viral peak 233 for critical cases, the difference is 20 days; while CD8 + T cells levels peak approximately in the same time for both 234 cases and with almost the same level. Critical cases have the T cell response peak 5 days after their viral load peak. [24] Lopez, L., Rodo, X., 2020. The end of the social confinement in spain and the COVID-19 re-emergence risk. medRxiv . [39] Tan, W., Lu, Y., Zhang, J., Wang, J., Dan, Y., Tan, Z., He, X., Qian, C., Sun, Q., Hu, Q., et al., 2020. Viral kinetics and antibody responses in How will country-based mitigation measures influence the Innate or adaptive 324 immunity? the example of natural killer cells Sparse production but preferential incorporation of recently produced naive T cells in the human 327 peripheral pool Clinical characteristics of 138 329 hospitalized patients with 2019 novel coronavirus-infected pneumonia in Wuhan Modeling the viral dynamics of SARS-CoV-2 infection The dynamics of immune response in COVID-19 patients with different 335 illness severity Clinical characteristics of 82 cases of death from 337 COVID-19 Virulence and pathogenesis of 339 SARS-CoV-2 infection in rhesus macaques: A nonhuman primate model of COVID-19 progression Functional exhaustion of antiviral lymphocytes in COVID-19 341 patients Clinical course and risk factors for 343 mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study