Modeling Viral Dynamics of COVID-19 Coronaviruses

We used a simple target cell limited model to describe SARS-CoV-2, SARS-CoV, and MERS-CoV viral dynamics [20 (link),24 (link),67 (link)]. Target cell limited models have proved very valuable in understanding infection dynamics and therapy for chronic viral infections such as HIV [61 (link),68 (link)], HCV [69 (link)], and HBV [70 (link)] and for acute infections such as influenza [71 (link)], West Nile virus [72 (link)], Zika virus [73 (link)], and SARS-CoV-2 [17 (link),74 (link),75 (link)]. Although the model does not explicitly describe immune responses, the effects of immune responses are implicitly included in model parameters such as the infection rate, which can be influenced by innate responses, and the death rate of infected cells, which can be influenced by adaptive immune responses. Because of the simplicity of the model, these parameters can be estimated and compared among the 3 different coronaviruses. The form of the model that we use was first introduced to model influenza infection [71 (link)] and is given by
$\frac{dT\left(t\right)}{dt}=-\mathrm{\beta }T\left(t\right)V\left(t\right),$
$\frac{dI\left(t\right)}{dt}=\mathrm{\beta }T\left(t\right)V\left(t\right)-\mathrm{\delta }I\left(t\right),$
$\frac{dV\left(t\right)}{dt}=pI\left(t\right)-cV\left(t\right),$
where the variables T(t), I(t), and V(t) are the number of uninfected target cells, the number of infected target cells, and the amount of virus at time t (note: we used time after symptom onset as the timescale), respectively. Symptom onset is defined slightly differently between papers, but it essentially means when any coronavirus-related symptoms (fever, cough, and shortness of breath) appear [76 ]. The parameters β, δ, p, and c represent the rate constant for virus infection, the death rate of infected cells, the per cell viral production rate, and the per capita clearance rate of the virus, respectively. Since the clearance rate of the virus is typically much larger than the death rate of the infected cells in vivo [27 (link),67 (link),77 ], we made a quasi-steady state (QSS) assumption, dV(t)/dt = 0, and replaced Eq 3 with V(t) = pI(t)/c. Because data on the numbers of coronavirus RNA copies, V(t), rather than the number of infected cells, I(t), were available, I(t) = cV(t)/p was substituted into Eq 2 to obtain
$\frac{dV\left(t\right)}{dt}=\frac{p\mathrm{\beta }}{c}T\left(t\right)V\left(t\right)-\mathrm{\delta }V\left(t\right).$
Furthermore, we replaced T(t) by the fraction of target cells remaining at time t, i.e., f(t) = T(t)/T(0), where T(0) is the initial number of uninfected target cells. Note f(0) = 1. Accordingly, we obtained the following simplified mathematical model, which we employed to analyze the viral load data in this study:
$\frac{df\left(t\right)}{dt}=-\mathrm{\beta }f\left(t\right)V\left(t\right),$
$\frac{dV\left(t\right)}{dt}=\mathrm{\gamma }f\left(t\right)V\left(t\right)-\mathrm{\delta }V\left(t\right),$
where γ = pβT(0)/c corresponds to the maximum viral replication rate under the assumption that target cells are continuously depleted during the course of infection. Thus, f(t) is equal to or less than 1 and continuously declines.
In our analyses, the variable V(t) corresponds to the viral load for SARS-CoV-2, MERS-CoV, and SARS-CoV (copies/ml). Because all of them cause acute infection, loss of target cells by physiological turnover can be ignored, considering the long lifespan of the target cells.