key: cord-1003547-q28f1kaa
authors: Hernandez-Vargas, Esteban A.; Velasco-Hernandez, Jorge X.
title: In-host Mathematical Modelling of COVID-19 in Humans
date: 2020-09-30
journal: Annu Rev Control
DOI: 10.1016/j.arcontrol.2020.09.006
sha: e8ce36fa3b4c3bc51220634546907d8aa1cb6cfb
doc_id: 1003547
cord_uid: q28f1kaa

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. Considering different starting times of infection, mathematical models are proposed to represent SARS-CoV-2 dynamics in infected patients. Based on the target cell limited model, the within-host reproductive number for SARS-CoV-2 is consistent with the broad values of human influenza infection. The best model to fit the data was including immune cell response, which suggests a slow immune response peaking between 5 to 10 days post-onset of symptoms. The model with the eclipse phase, time in a latent phase before becoming productively infected cells, was not supported. Interestingly, all models predict that SARS-CoV-2 may replicate very slowly in the first days after infection, and viral load could be below detection levels during the first 4 days post infection. A quantitative comprehension of SARS-CoV-2 dynamics and the estimation of standard parameters of viral infections is the key contribution of this pioneering work. These models can serve for future evaluation of control theoretical approaches to tailor new potential drugs against COVID-19.

Epidemics by infectious pathogens are a major threat to humankind. The year 2020 has uncovered one 19 of the biggest pandemics in history, the novel coronavirus SARS-CoV-2 that was first reported in 20 Wuhan, Hubei Province, China in December 2019 [1] . While China did a large effort to shrink the 21 outbreak, COVID-19 has developed into a pandemic in more than 210 countries moving the epicentre 22 from China to Europe and consequently to America [1] . Several countries are planning to relax the 23 strict social distancing regulations. Nevertheless, epidemic rebound risks are latent [2, 3] . 24 Coronaviruses are found in different species of animals (e.g. bats and camels) and can evolve to 25 infect humans by droplets from coughing or sneezing. In February 2003, the Severe Acute Respiratory 26 Syndrome (SARS-CoV) was reported in Asia resulting in 8422 cases with a case-fatality rate of 11% [1] . 27 Later, in 2012, the Middle East respiratory syndrome (MERS-CoV) was identified in Saudi Arabia with 28 about 2506 cases, killing 862 between 2012 and 2020 [1] . Metagenomics studies previous to the 29 COVID-19 outbreak envisaged the possibility of future threats due to the identification of several 30 sequences closely related SARS-like viruses circulating in the Chinese bat populations [4, 5] . 31 So far, no vaccine or antiviral drug is likely to be available soon. Either monoclonal antibody or 32 vaccine approaches have failed to neutralize and protect from coronavirus infections [5] . Therefore, 33 individual behaviour (e.g early self-isolation and social distancing), as well as preventive measures such 34 as hand washing and covering when coughing are critical to control the spread of COVID-19 [6] [7] [8] . 35 Additionally to these measures, several travel restrictions and quarantines have taken place in many 36 countries around the globe. 37 Epidemiological models have highlighted that social distancing interventions to mitigate the 38 epidemic is a key aspect [2, 3, 7, 8] . Nevertheless, there are many epidemiological unknowns with the 39 COVID-19 pandemic [7] . The case fatality rate for COVID-19 is about 0·3-1% [1] . However, adjusted 40 estimation by [9] indicates that the COVID-19 mortality rate could be as high as 20% in Wuhan. In its 41 early stages, the epidemic has doubled in size every 7.4 days [10] . Moreover, the basic reproductive 42 number was estimated to be 2.2 (95% CI, 1.4 to 3.9) [10] . Based on the relatively long incubation 43 period for COVID-19, about 5-6 days [1], Anderson et al. [7] suggested a considerable pre-symptomatic 44 infectiousness.

Parameter identification aims to estimate model parameters based on minimizing the error between 66 model prediction and experimental data. The viral load is measured in Log10 scales, thus, parameter 67 fitting is performed using the following cost function: 68 Definition 1 Root Mean Squared Logarithmic Error (RMSLE). This is the difference on 69 Log10 scales between the model predictive output (ȳ i ), and the experimental measurement (y i ). This 70 writes as follows

where n is the number of measurements. 72 Turn out that the minimization of a cost function implies a nonlinear optimization problem, 73 complex, with several variables, and multiple minima. This complexity can be tackled using 74 evolutionary optimization algorithms such as the Differential evolution (DE) algorithm [35] . Note that 75 3/24 several optimization solvers were considered in previous modeling work [29] , including both 76 deterministic (f mincon Matlab routine) and stochastic (e.g Genetic and Annealing algorithm) methods. 77 Simulation results revealed that the DE global optimization algorithm is robust to initial guesses of 78 parameters than other mentioned methods [36] . 79 Note that multiple models can provide the same fit with observed experimental data. Thus, it 80 becomes necessary to choose between different models. The standard approach to model selection is 81 first estimate all model parameters from the data, then select the model with the best-fit error and 82 some penalties on model complexity. A very used model selection criteria is defined next.

Definition 2 Akaike information criterion (AIC). The corrected (AIC) writes as follows:

where N is the number of data points, M is the number of unknown parameters and RSS is the residual 85 sum of squares obtained from the fitting routine. AIC is used here to compare the goodness-of-fit for 86 models that evaluate different hypotheses [37] . A lower AIC value means that a given model describes 87 the data better than other models with higher AIC values. Small differences in AIC scores (e.g. <2) are 88 not significant [37] .

Redundant parametrization in models provides difficulties to estimate the parameters uniquely.

Ambiguous parameters θ sub ⊂ θ may be varied without changing the output y resulting in constant 91 values for the cost function to minimize e.g residuals. This is particularly important for biological 92 systems, where a large variability is presented from one host to another host, limiting the prediction 93 value of mathematical models and estimated parameters [38] . Next, we introduce some important 94 definitions:

95 Definition 3 Identifiability [38] . A mathematical model is identifiable if θ can be uniquely 96 determined from the measurable output y(t); otherwise, the system is unidentifiable.

Note that a mathematical model is algebraically identifiable may still be practically non-identifiable 98

if the amount and quality of the data are insufficient and the data manifest large deviations. The 99 computational approach by Raue et al. [39] exploits the profile likelihood to determine identifiability 100 and is considered here. This methodology can detect both structurally and practically non-identifiable 101 parameters [31] . 102 

The mathematical model used here to represent SARS-CoV-2 dynamics is based on the target 104 cell-limited model [31, 40, 41] , which writes as follows:

Equation ( organs as a direct consequence of viral involvement [42] .

109 Viral dynamics are represented by (5) . Viral particles (V ) infect susceptible cells with a rate β 110 ((copies/ml) −1 day −1 ). Once cells are productively infected, they release virus at a rate p (copies/ml 111 day −1 cell −1 ) and virus particles are cleared with a rate c (day −1 ). Infected cells are cleared at rate δ 112

(day −1 ) as consequence of cytopathic viral effects and immune responses.

It is still debatable in the literature which compartments SARS-CoV-2 can infect, however, there is a 114 common agreement that the infection mainly takes place in respiratory epithelial cells [43] . Previous that V (0) cannot be measured as it is below detectable levels (about 100 copies/ml ) [33] . Previous 118 modelling work has suggested using half of the detection levels (less than 50 copies/ml) [29, 44] . Here, 119 using a regression model, the initial viral concentration V (0) was estimated to be about 0.31 copies/ml. 120

Remark 1 SARS-CoV-2 kinetics are measured after the onset of symptoms [33] . However, it is 121 unknown when the initial infection took place. Patients infected with MERS-CoV in [45] showed that the 122 virus peaked during the second week of illness, which indicated that the median incubation period was 7 123 days (range, 2 to 14) [45] . The incubation period for SARS-CoV-2 has a median time of 2-5 days from 124 exposure to symptoms onset [46] . Therefore, for simulation purposes, we explore different scenarios of 125 initial infection day (t i ), that is, -7, -3 days before the onset of symptoms.

126 Definition 4 Infectivity. This is the ability of a pathogen to establish an infection [47] . To quantify 127 infectivity, the within-host reproductive number (R 0 ) is employed to compute the expected number of 128 

secondary infections produced by an infected cell [48] . When R 0 < 1, one infected cell would infect less 129 than one cell. Thus, the infection would be cleared from the population. Otherwise, if R 0 > 1, the 130 pathogen would be able to invade the target cell population. For the model (3)- (5), the reproductive 131 number is

Remark 2 Using only the viral titters for parameter fitting procedures in the target cell limited model 133 would result in identifiability problems [49] . That is, parameter values are rescaled and consequently the 134 validity of parameters would be doubtful to extract biological meaning. As the viral clearance is attributed 135 to a process not directly related to the immune system or viral particle per se, the viral clearance 136 parameter (c) is fixed here with previous estimates in humans, e.g approximately 2.4 for influenza and 137

HIV [21, 27] .

Assuming the day of infection closer to the post symptom onset (pso), day 0, numerical results show 139 high reproductive numbers (R 0 ) and higher infection rates (β) as presented in Table 1 . Considering the 140 Remark 1, it is assumed the initial day of infection is -7 or -3 pso, therefore, the rate of infection (β) 141 would be slow but associated with a high replication rate (p). Note that individual parameter values 142 should not be interpreted because of identifiability problems. Fig. 1 Remark 3 An additional state known as the "Eclipse Phase" has been proposed by [50, 51] to represent the time frame of the infection more adequately. Newly infected cells spend time in a latent phase (E) before becoming productively infected cells (I), this can be written as follows:

Cells in the eclipse phase (E) can become productively infected at rate k. For SARS-CoV-2 , we found 147 that the eclipse phase model (AIC≈34) does not improve the fitting respect to the target cell model. This 148

can be attributted to identifiability problems as we only have data for the viral titter [49] . [29, [52] [53] [54] [55] [56] . Here, we adapted a minimalistic model derived by [57, 58] to represent the interaction between influenza and immune response dynamics. The model assumes that the virus (V )

level induces the proliferation of T cells (T ) as follows:

Equation (11) refers to SARS-CoV-2 dynamic. Viral replication is modelled with a logistic function 151 with a maximum carrying capacity K and replication rate p. K is the maximum viral load for each of 152 the patients in [33] . The virus is cleared at a rate c, which is considered as in the Remark 2. The term 153 c T V T represents the rate of killing infected cells by the immune response. 

The summary of fitting procedures at t i = −3 dpso is presented in Table 2 , which highlights that the 162 mean of AICs values of the model (11)- (12) are smaller than those presented in the target cell limited 163 model. Results of parameter fitting are portrayed in Fig.2 . Independently of the starting infection day, 164 the immune response by T cells peaks between 5 to 10 dpso as shown in Fig.3 . Note that this model 165 represents viral clearance only for patients D, H and I. In the other patients viral clearance is not 166 reported in the data [33] . Table 2 . Estimations for model with immune system (11)-(12) using experimental data from [33] assuming m = 2 and infection time -3 dpso. 

Data from [45] showed that MERS-CoV levels peak during the second week with a median value of 176 7.21 (log10 copies/mL)in the severe patient group, and about 5.54 (log10 copies/mL) in the mild group. 177

For SARS, the virus peaked at 5.7 (log10 copies/mL) between 7 to 10 days after onset [60] . For 178 COVID-19, the viral peak was approximately 8.85 (log10 copies/mL) before 5 dpso [33] . Liu et al. [61] 179 found that patients with severe disease reported a mean viral load on admission 60 times higher than 180 mild disease cases. Additionally, high viral levels persisted in severe patients for 12 days after onset [61] . 181

The reproductive number for human influenza ranges from 3.5 -75 [27, 62] , which is consistent with 182 the values reported here for COVID-19. Interestingly, both of our models when fitted to the data set of 183

patient A predict that the virus could replicate below detection levels for the first 4 dpi. This may explain why infected patients with SARS-CoV-2 would take from 2-14 dpi to exhibit symptoms.

The model with immune system (Fig.3) highlights that the T cell response is slowly mounted against 186 SARS-CoV-2 [7] . Thus, the slow T cell response may promote low inflammation levels during the first 187 days post infection [62] , which might be a reason to the observations during COVID-19 pandemic of the 188 detrimental outcome on French patients that used non-steroidal anti-inflammatory drugs (NADs) such 189 as ibuprofen. However, so far, there is not any conclusive clinical evidence on the adverse effects by 190 NADs on SARS-CoV-2 infected patients.

Accelerating therapeutic and prophylactic medication against SARS-CoV-2 is unprecedented. For instance, mathematical terms to represent antiviral effects in the model (3)-(5) writes as follow

where u h and u r would represent the effect of inhibiting the replication cycle and the release of virus, 203 respectively. In this direction, [15] found that in order to reduce the peak viral load by more than 2 logs, 204 drug efficacy needs to be greater than 80% if treatment is administered after symptom onset. In a 205 similar direction, the modelling work by [16] predicted that to lower viral area under the curve therapies 206 need to be given before the SARS-CoV-2 viral load peaks.

Additionally, mathematical models have served to investigate the effect of immune therapies such as in vivo neutralization of pro-inflammatory cytokines during secondary streptococcus pneumoniae 9/24 infection post influenza infection [65] . In the context of COVID-19, our model with immune response (11)-(12) could integrate a term to represent antiviral effects (u r ) as well as immune modulation (u m )

to promote the proliferation of T cells, this would be represented as follows:

In other viral infections such as Ebola [26] , monoclonal antibodies (mAbs) played an important role 208

to inhibit viral replication, e.g the term (1 − u r )p while increasing viral clearance e.g cV . Another way 209

to suppress SARS-CoV-2 replication is when in co-infections with other viruses because of SARS-CoV-2 210 has a slower growth rate than the other viruses examined in [66] . Our parameters fitted with the data 211 sets from [33] support also the idea of a slow replication by SARS-CoV-2.

The humoral response against SARS-CoV-2 is urgently needed to evaluate the protection to infection in a non-human primate model [64] . Furthermore, several benefits were reported for treatment 220 if provided during 12 hours post MERS-CoV infection [64] . Our study here mainly addressed T cell 221 responses, therefore, future modelling attempts should be directed to establish a more detailed model of 222 antibody production and cross-reaction [68] as well as in silico testing of different antivirals [69] .

There are technical limitations in this study that need to be highlighted. The data for SARS-CoV-2 224 kinetics in [33] is at the onset of symptoms. This is a key aspect that can bias parameter estimation as 225 mathematical models initiate on the day of infection. In fact, we could miss viral dynamics at the onset 226 of symptoms as well as the SARS-CoV-2 viral load peak. For example, from throat samples in Rhesus 227 macaques infected with SARS-CoV-2, two peaks were reported on most animals at 1 and 5 dpi [70] .

In a more technical aspect using only viral load on the target cell limited model to estimate 229 parameters may lead to identifiability problems [49, [71] [72] [73] for future mathematical models of COVID-19 to reveal prophylactic and therapeutic interventions at 236 multi-scale levels [74] [75] [76] [77] [78] [79] . Further insights into immunology and pathogenesis of SARS-CoV-2 will help 237 to improve the outcome of this and future pandemics.

Ultimately, previous modelling efforts here and from others [15, 16, 66] to represent SARS-CoV-2 239 could be further extended with control theoretical approaches such as optimal control and model (5) . Blue circles represents the data from [33] . Due to the most complete data sets in [33] were for patients A and B, thus these are the only presented in panel (A) and (B), respectively. Infection time was assumed at -7 and -3 days post symptoms onset. (12) . Blue circles represent the data from [33] . Infection time was assumed at -3 days post symptom onset. (12) . Blue circles represent the data from [33] . Infection time was assumed at -3 days post symptom onset. 

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