key: cord-0260331-jdnp609q
authors: Hillen, T.
title: A Standard Virus Load Function
date: 2020-06-22
journal: nan
DOI: 10.1101/2020.06.19.20135814
sha: 903996d6401f294f8dd107cd7170a8dada9150cf
doc_id: 260331
cord_uid: jdnp609q

The idea is to design a simple function that can describe typical virus-load curves without solving a full virus load dynamic model. We present such a standard virus load function and validate it on data from influenza A as well as SARS-CoV-2 infection data for Macaque monkeys and humans. Further, we compare the virus load function to an established {it target model} of virus dynamics as presented by A. Smith in cite{Amber2018}. This virus-load function can be used as input to higher level models for the physiological effects of a virus infection, for models for tissue damage, and to develop treatment strategies.

Virus load curves, as reported in [1, 5, 4] , have a very typical infection progression (see Figure  1 ). According to A. Smith [1] the virus infection presents itself in five phases. In the first phase (Phase Ia) the virus quickly infects cells without being detectable. This phase is followed by exponential growth (Phase Ib) until growth shows signs of saturation and a maximum is reached (Phase Ic). A period of slow exponential decline ensues, which we call Phase II. And finally, we often observe a fast decline that leads to virus clearance (Phase III). Depending on the virus and the response of the patient, these phases can be shorter or longer, or are not shown before the patient parishes.

To develop our virus load function (1) we divide the virus load progression into three phases. We consider the Phase I of simgoid increase between time points a 1 and a 2 (see Figure 1 ). The Phase I includes the three initial phases (Phases Ia, Ib, Ic) of Smith [1] mentioned above. At time a 2 a slow decline of the virus is observed as the immune response kicks in (Phase II between a 2 and b 1 in Figure 1 ) and finally (Phase III) shows a rather sharp decline once the virus is controlled (between b 1 and b 2 ). We are guided by the default virus load curve from [8] Figure 2 , page 100 and write the virus load curve as a product of three functions, representing the three main phases:

where v 1 describes the initial growth phase between a 1 and a 2 , v 2 the intermediate slow decay phase between a 2 and b 1 , and v 3 the final decay phase between b 1 and b 2 . These are given as 1 Figure 1 : Typical virus load curve. The virus load ("titer") is usually reported as a dilution value that is needed to infect 50% of a given cell culture, the TCID50 value. Left: absolute scale, right: TCID50-scale (log-scale).

sigmoid and exponential functions, respectively

where H(t) denotes the Heavyside function. The specific form of sigmoid curves for v 1 and v 3 was developed previously by Olobatuyi in [6] in a cancer model. It allows us to define these functions based on intuitive transition threshold values. The value a 1 describes the onset of growth and a 2 a value when saturation is reached, similarly, b 1 denotes the time where decay switches from slow to fast and b 2 is the time when the virus is effectivley eliminated. The parameter α describes the intermediate exponential decay rate. In Table 1 we list the values used in Figure 1 and Figure 2 and their meaning. The hyperbolic tangent function is a sigmoid step function that connects −1 to 1. In (1) it has been shifted and scaled such that transitions occur between a 1 and a 2 upwards and between b 1 and b 2 downwards, where the maximum is max and the minimum min. The ominous value "6" in the argument of the hyperbolic tangent can easily be explained by looking at a 1 = −1 and a 2 = 1. Then tanh( 6 a 2 −a 1 ) = tanh 3 = 0.995. That means that at a 2 a 99.5% of the saturation level is reached. Similar, at a 1 the function is 0.5% above its minimum.

Virus load functions are in high demand in the virus modelling community. For example in [9] , a large community of researchers develops an individulal based SARS-CoV-2 physiological model that includes virus infection, virus transmission, immune response, and potential damage to the tissue. The immune response and the tissue complications are directly related to the virus load of the tissue. Our standard virus load function will be a welcome modelling tool to describe 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.

The copyright holder for this preprint this version posted June 22, 2020. .

Influenza (Fig 2) In the next sections we compare this new virus load function (1) to a very controlled data set of infection of influenza A of mice, and to SARS-CoV-2 data for humans and Macaque monkeys. Table 1 . Right: Human SARS-CoV-2 data from oropharynx saliva samples [3] , parameter values are in Table 2 . and the virus load in the saliva was measured. From patients who were intubated a endotrachial aspirate sample was taken. As the ciliary activity of the lung epithelium transports mucus to the posterior oropharyngeal area, these samples give a good indication of the viral activity in the lungs. The data were collected on a daily basis and recorded as mean values and standard deviations. In Figure 2 on the right we show the orophalynx measurements as mean values with error bars. We fit our virus load function (1) to these data as shown on the right of Figure 2 as solid line, and the corresponding model parameters are listed in Table 2 . We observe that the initial virus growth phase is rather quick and the virus reaches its carrying capacity within 2.5 days (a 2 − a 1 = 2.5). The virus load reaches a saturation level, which is likely to be related to the innate immune response and it starts a phase of slow decay with a half-life time of T 1 2 = 1.6 days. After 15 days the virus load drops more quickly, possibly due to the adaptive immune response and at day 29 the virus is cleared.

Again we adapted the curve by hand. The error bars are so wide that a optimization of a mean squared error seems to be futile in this case. We see that the virus load function (1) can conveniently describe all three phases of these dynamics, and, as we will see, the dynamics for macaque monkeys is rather similar.

In [2] nine rhesus macaques monkeys were infected with SARS-CoV-2. Three monkeys (group 1) obtained a high initial virus dose, three monkeys (group 2) a medium initial dose, and three (group 3) a small initial dose. The virus load was measured daily or every other day through a bronchoalveolar lavage probe. The recorded data for each individual group and for all three groups together are shown as dots in Figure 3 . All nine monkeys showed only mild disease symptoms and they all fully recovered. Hence the infection cycle here is more indicative of a mild infection, in contrast to the human data considered above.

. 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 June 22, 2020. . In Figure 3 we adapt our virus load function (1) to the rhesus monkey data. We can see in Table 2 that the chosen parameter values show a great level of consistency between the three groups, such that a good fit of all three groups combined is also possible. The characteristic values for the initial virus growth a 1 = 1 and a 2 = 2.5 are identical between the groups, indicating that the amount of initial viral dose is not so important. 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 June 22, 2020. .

Macaque data Group 1 Group 2 Group 3 Groups 1,2,3 max 700000 6 · 10 6 6 · 10 6 10 6 6 · 10 6 min 10 −6 10 −6 10 −6 10 −6 10 −6 a 1 1. Table 2 : Parameters adapted to the human data (column 2), to the macaque data (columns 3, 4, 5) and to the target model (column 6).

In [1] a virus load model has been fit to the influenza data mentioned above. The so called target model comprises of four ordinary differential equations (ODEs) for the target cells T (t), the infected cells, also called the eclipse phase, I 1 (t), the infectious cells I 2 (t) and the virus load

Here β is the virus infection rate, k the infection maturation rate, p the virus production rate, c the virus decay rate, δ d the base decay rate of infectious cells, and K d the half saturation constant for the decay term of the infectious cells. The parameter values which we use here are listed in Table 3 . The parameter values that are published in [1] and their meanings are listed as "Target model 1". These do not give the best fit to the influenza data from Figure 2 and I am fortunate to have parameter values from a direct communication with the authors of [1] , which fit better. We call those alternative choices "Target model 2".

The basic reproduction number R 0 for this model is given as (see [1] )

For the two parameter sets that we consider here, we compute R 0 in Table 3 . It should be noted that the value of R 0 = 8.8, listed in [1] is not quite accurate. In Figure 4 we show the virus load curve from the target model as a thick red line and an overlay of the virus load function (1) as a black thin line. Target model 1 is shown on the left and Target model 2 on the right. We see that the virus load function (1) can reproduce both curves with a high level of accuracy. The parameter values are listed in Table 3 .

The explicit form of this new standard virus load function (1) is simple and convenient. The corresponding parameter values all have easily understandable biological meaning, allowing this 6 . 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 June 22, 2020. . 

83.07 9.268 basic reproduction number Table 3 : Parameters for the target model ODE (2) . Target model 1 refers to the parameter values that are listed in [1] . Target model 2 are those parameter values that produce the best fit to the influenza data from above, and were directly communicated by one of the authors of [1] . 7 . 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 June 22, 2020. (1) will allow researchers to focus their analysis on higher level effects such as immune responses, tissue damage, ARDS, spread to other organs in the body and for treatment designs.

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Potential effects of coronaviruses on the cardiovascular system

Dynamically linking influenza virus infection with lung injury to predicy disease severity. bioRxiv

A reaction-diffusion model for radiation-induced bystander effects

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Host-pathogen kinetics during influenza infection and coinfection: insights from predictive modeling

Acknowledgements: I am grateful to J. Newby and A. Smith for helpful comments to an early version of this idea. I am also most grateful to A. Smith to freely sharing the influenza virus load data, which her group collected in painstakingly tedious work.