key: cord-277237-tjsw205c
authors: Hernandez Vargas, Esteban Abelardo; Velasco-Hernandez, Jorge X.
title: In-host Modelling of COVID-19 Kinetics in Humans
date: 2020-03-30
journal: nan
DOI: 10.1101/2020.03.26.20044487
sha: 
doc_id: 277237
cord_uid: tjsw205c

COVID-19 pandemic has underlined the impact of emergent pathogens as a major threat for human health. The development of quantitative approaches to advance comprehension of the current outbreak is urgently needed to tackle this severe disease. In this work, several mathematical models are proposed to represent COVID-19 dynamics in infected patients. Considering different starting times of infection, parameters sets that represent infectivity of COVID-19 are computed and compared with other viral infections that can also cause pandemics. Based on the target cell model, COVID-19 infecting time between susceptible cells (mean of 30 days approximately) is much slower than those reported for Ebola (about 3 times slower) and influenza (60 times slower). The within-host reproductive number for COVID-19 is consistent to the values of influenza infection (1.7-5.35). The best model to fit the data was including immune responses, which suggest a slow cell response peaking between 5 to 10 days post onset of symptoms. The model with eclipse phase, time in a latent phase before becoming productively infected cells, was not supported. Interestingly, both, the target cell model and the model with immune responses, predict that virus may replicate very slowly in the first days after infection, and it could be below detection levels during the first 4 days post infection. A quantitative comprehension of COVID-19 dynamics and the estimation of standard parameters of viral infections is the key contribution of this pioneering work.

is none so far at within-host level to understand COVID-19 replication cycle ( Fig.1 ) and its interactions 51 with the immune system. Among several approaches, the target cell model has served to represent 52 several diseases such as HIV [7] [8] [9] [10] , Hepatitis virus [11, 12] , Ebola [13, 14] , influenza [15] [16] [17] [18] , among 53 many others. A detailed reference for viral modelling can be found in [19] . Very recent data from 54 infected patients with COVID-19 has enlighten the within-host viral dynamics. Zou et al. [20] presented 55 the viral load in nasal and throat swabs of 17 symptomatic patients. Interestingly, COVID-19 56 replication cycles may last longer than flu, about 10 days or more after the incubation period [4, 20] . 57 Here, we contribute to the mathematical study of COVID-19 dynamics at within-host level based on 58 data presented by Wolfel et al. [21] . 59 

Using ordinary differential equations (ODEs), different mathematical models are presented to adjust the 61 viral kinetics reported by Woelfel et al. [21] in infected patients with COVID-19. Viral load [21] was cost function (14) is minimized to adjust the model parameters based on the Differential Evolution (DE) 67 algorithm [22] . 68 

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03. 26.20044487 doi: medRxiv preprint viral dynamic is divided into two parts, exponential growth (V g ) and decay (V d ) modelled by equations 70

(1) and (2), respectively.

Viral growth is assumed to start at the onset of symptoms, with initial viral concentration V g (0).

The parameter ρ is the growth rate of the virus. The parameter η quantifies the decay rate of the virus, 73

while V d (0) the initial value of the virus in decay phase. Note that the growth phase of the virus was 74 measured only in two patients (A, and B) [21] . Exponential growth and decay model for COVID-19. Continuous line are simulation based on (1) for viral exponential growth (V g ) or on (2) for viral decay (V d ). blue circles represents the data from [21] . Viral growth rate (ρ) was only computed for patients A (till day 6) and B (till day 4) while the rest of patients have missing these measurements. For all patients viral decay rate η in (2) is computed.

Simulation are shown in Fig.2 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03. 26.20044487 doi: medRxiv preprint Table 1 . Estimations for the model (1)-(2) using experimental data from [21] . For the exponential growth phase there were measurements only for patient A and B, for the rest of patients were more in the logarithmic decay phase. This is the reason why patient A and B are the only ones that report estimations of viral growth. 

Host cells can be in one of following states: susceptible (U ) and infected (I). Viral particles (V ) infect 85 susceptible cells with a rate β ((Copies/mL) −1 day −1 ). Once cells are productively infected, they release 86 virus at a rate p (Copies/mL day −1 cell −1 ) and virus particles are cleared with rate c (day −1 ). Infected 87 cells are cleared at rate δ (day −1 ) as consequence of cytopathic viral effects and immune responses.

Coronaviruses infect mainly in differentiated respiratory epithelial cells [25] . Previous mathematical 89 model for influenza [17] have considered about 10 7 initial target cells (U (0)). Initial values for infected 90 cells (I(0)) are taken as zero. V (0) is determined from estimations in Table 1 . Note that V (0) cannot 91 be measured as it is below detectable levels (about 100 Copies/m) [21] .

Viral kinetics are measured after the on-set of symptoms [21] , however, it is unknown when the 93 initial infection took place. Patients infected with MERS-CoV in [26] showed that the virus peaked 94 during the second week of illness, which indicated that the median incubation period was 7 days (range, 95 2 to 14) [26] . For parameter fitting purposes, we explore three different scenarios of initial infection day 96

(t i ), that is, -14, -7, -3 days before the onset of symptoms for patients A and B, see Fig. 3 .

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The copyright holder for this preprint this version posted March 30, 2020. . Blue circles represents the data from [21] . Due to the most complete data sets in [21] were from patient A and B, then these are the only presented in panel (a) and (b), respectively. Infection time was assumed at -14, -7 and 0 days post symptom onset.

Infectivity can be defined as the ability of a pathogen to establish an infection [27] . To quantify 98 infectivity, the within-host reproductive number (R 0 ) was computed. R 0 is defined as the expected 99 number of secondary infections produced by an infected cell [28] . When R 0 < 1, one infected individual 100

can infect less than one individual. Thus, the infection would be cleared from the population.

Otherwise, if R 0 > 1, the pathogen is able to invade the target cell population. This epidemiological 102 concept has been applied to the target cell model (3)- (5), with

Previous studies [13, 29, 30] provided estimates of the infecting time (t inf ), that represents the time 104 required for a single infectious cell to infect one more cell. Viruses with a shorter infecting time have a 105 higher infectivity [29, 30] . From equations (3)-(5), t inf can be explicitly computed as:

Assuming day of infection at day 0 post symptom onset (pso) would result in very high reproductive 107 numbers (R 0 ) and a high infection rate (β) for patients A and B as presented in Table 2 . Alternatively, 108

assuming the initial day of infection is either day -14 or -7 pso, then the rate of infection of susceptible 109 cells (β) would be slow but associated with a high replication rate (p). . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 30, 2020. before becoming productively infected cells (I) [29, 31] . This can be written as follows:

Cells in the eclipse phase (E) can become productively infected at rate k. Holder et al. [29] improve the fitting respect to the target cell model (Table 2 ) even when very long eclipse phase periods 121 are assumed (e.g 100 days), implying that this mechanism could be negligible on COVID-19 infection.

Mathematical Model with Immune Response. Previous studies have acknowledged the relevance of the immune T-cell response to clear influenza [17, [32] [33] [34] [35] [36] . Due to identifiability limitations for the estimation of the parameters of the target cell model using viral load data, a minimalistic model was derived in [37, 38] to represent the interaction between the viral and immune response dynamics. The model assumes that the virus (V ) level induces the proliferation of T cells (T ) as follows:

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. gives a better fitting than previous models (Fig.2-4) . Furthermore, AICs values for patient A and B 135 highlight that t i = −15 dpso give the best fitting. For presentation purposes, numerical results for 136 patient A and B are the only portrayed in Fig.4 . The summary of fitting procedures at t i = −15 dpso is 137 presented in Table 3 . Independently of the starting infection day, the immune response by T cells peaks 138 between 5 to 10 dpso. Interestingly, the longer the period between infection time to the onset of 139 symptoms, the higher the immune response. 140 

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The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03. 26.20044487 doi: medRxiv preprint (12)- (13) . Blue circles represents the data from [21] . Due to the most complete data sets in [21] were from patient A and B, then these are the only presented in panel (a) and (b), respectively. Infection time was assumed at -14, -7 and 0 days post symptom onset. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03. 26.20044487 doi: medRxiv preprint Data from [26] showed that MERS-CoV levels peak during the second week with a median value of 150 7.21 (log10 copies/mL)in the severe patient group, and about 5.54 (log10 copies/mL) in the mild group. 151

For SARS, the virus peaked at 5.7 (log10 copies/mL) between 7 to 10 days after onset [40] . For 152 COVID-19, the viral peak was approximately 8.85 (log10 copies/mL) before 5 dpso [21] . Liu et al. [41] 153 found that patients with severe disease reported a mean viral load on admission 60 times higher than 154 that of the mean of mild disease cases, implying that higher viral loads relate clinical outcomes.

Additionally, higher viral load persisted for 12 days after onset [41] . 156 Using the target cell model, Nguyen et al. [13] computed for Ebola infection an average infecting 157 time of 9.49 hours, while Holder et al. [29] reported that infecting time for the wild-type (WT) 158 pandemic H1N1 influenza virus was approximately 0.5 hours [29] . Here, based on the results of the 159 target cell model in Table 2 , we found that COVID-19 infecting time between cells (mean of 30 days 160 approximately) would be slower than those reported for Ebola (about 3 times slower) and influenza (60 161 times slower). The reproductive number for influenza in mice ranges from 1.7 to 5.35 [42] , which is 162 consistent with the values reported for COVID-19.

Interestingly, both of our models (the target cell model (3)-(5) and the model with immune response 164 (12)-(13)) when fitted to the patient A data, predict that the virus can replicate below detection levels 165 for the first 4 dpi. This could be an explanation of why infected patients with COVID-19 would take 166 from 2-14 dpi to exhibit symptoms.

The model with immune system (Fig.4(b and d) ) highlights that the T cell response is slowly 168 mounted against COVID-19 [4] . Thus, the slow T cell response may promote a limit inflammation 169 levels [42] , which might be a reason to the observations during COVID-19 pandemic of the detrimental 170 outcome on French patients that used non-steroidal anti-inflammatory drugs (NADs) such as ibuprofen. 171

However, so far, there is not any conclusive clinical evidence on the adverse effects by NADs on 172 COVID-19 infected patients.

The humoral response against COVID-19 is urgently needed to evaluate the protection to . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03. 26.20044487 doi: medRxiv preprint infection in a non-human primate model [44] . Furthermore, benefits has been reported for therapeutic 181 treatment if provided during 12 hours MERS-CoV infection [44] . Our study here mainly addressed T cell 182 responses, therefore, future modelling attempts should be directed to establish a more detailed model of 183 antibody production and cross-reaction [45] as well as in silico testing of different antivirals [46] . 184 There are technical limitations in this study that need to be highlighted. The data for COVID-19 185 kinetics in [21] is at the onset of symptoms. This is a key aspect that can render biased parameter 186 estimation as the target cell regularly is assumed to initiate at the day of the infection. In fact, we could 187 miss viral dynamics at the onset of symptoms. For example, from throat samples in Rhesus macaques 188 infected with COVID-19, two peaks were reported on most animals at 1 and 5 dpi [47] .

In a more technical aspect using only viral load on the target cell model to estimate parameters may 190 lead to identifiability problems [48] [49] [50] [51] . Thus, our parameter values should be taken with caution when 191 parameters quantifications are interpreted to address within-host mechanisms. For the model with multi-scale levels [52] [53] [54] [55] [56] [57] . Further insights into immunology and pathogenesis of COVID-19 will help to 198 improve the outcome of this and future pandemics. 199 

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) Mathematical models based on Ordinary Differential Equations (ODEs) are solved using the MATLAB 202 library ode45, which is considered for solving non-stiff differential equations [58] . The clinical data of 9 individuals is from [21] . Due to close contact with index cases and initial 205 diagnostic test before admission, patients were hospitalized in Munich [21] . Viral load kinetics were 206 reported in copies/ml per whole swab for 9 individual cases. All samples were taken about 2 to 4 days 207 post symptoms. Further details can be found in [21] . 

where n is the number of measurements. The minimization of RMS is performed using the Differential 213

Evolution (DE) algorithm [22] . Note that several optimization solvers were considered, including both 214 deterministic (f mincon Matlab routine) and stochastic (e.g Genetic and Annealing algorithm) methods. 215 Simulation results revealed that the DE global optimization algorithm is robust to initial guesses of 216 parameters than other mentioned methods.

Model Selection by AIC. The Akaike information criterion (AIC) is used here to compare the 218 goodness-of-fit for models that evaluate different hypotheses [59] . A lower AIC value means that a given 219 model describes the data better than other models with higher AIC values. Small differences in AIC 220 scores (e.g. <2) are not significant [59] . When a small number of data points, the corrected (AICc) 221 writes as follows:

222

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The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03. 26.20044487 doi: medRxiv preprint where N is the number of data points, M is the number of unknown parameters and RSS is the 223 residual sum of squares obtained from the fitting routine.

The authors declare that the research was conducted in the absence of any commercial or financial 226 relationships that could be construed as a potential conflict of interest. 

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The copyright holder for this preprint this version posted March 30, 2020. . https://doi.org/10.1101/2020.03. 26.20044487 doi: medRxiv preprint 

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