Characterization of SARS-CoV-2 dynamics in the host

Abstract

While many epidemiological models were proposed to understand and handle COVID-19 pandemic, too little has been invested to understand human viral replication and the potential use of novel antivirals to tackle the infection. In this work, using a control theoretical approach, validated mathematical models of SARS-CoV-2 in humans are characterized. A complete analysis of the main dynamic characteristic is developed based on the reproduction number. The equilibrium regions of the system are fully characterized, and the stability of such regions is formally established. Mathematical analysis highlights critical conditions to decrease monotonically SARS-CoV-2 in the host, as such conditions are relevant to tailor future antiviral treatments. Simulation results show the aforementioned system characterization.

Keywords: Equilibrium sets characterization; In-host model; SARS-CoV-2 infection; Stability analysis.

Fig. 1

Lambert function. W(z) has two branches, denoted as W_p (in blue) and W_m (in red). Both branches are defined for $z\in [-1/e,0]$; however ${\lim }_{z\to 0^{-}}{W}_p=0$ while ${\lim }_{z\to 0^{-}}{W}_m=-\infty ,$ which means that only the branch W_p will be used in our analysis, as it is shown in the proof of Theorem 3.1.

Fig. 2

Function $z({\mathcal{R}}_0,{\mathcal{K}}_0),$ for ${\mathcal{R}}_0\ge 0$ and ${\mathcal{K}}_0\le 0$. Note that $z({\mathcal{R}}_0,{\mathcal{K}}_0)>-1/e=-0.3679$ for all values of ${\mathcal{R}}_0\ge 0$ and ${\mathcal{K}}_0\le 0$.

Fig. 3

Every point in ${\mathcal{X}}_s^1$ is $ϵ-\delta$ stable but not attractive. Initial states x₀ starting arbitrarily close to x_s remain (for all t ≥ 0) arbitrarily close to x_s, but do not converge to x_s. As a consequence, set ${\mathcal{X}}_s^1$ is AS but the points inside it are not.

Fig. 4

According to Eq. (3.13), U_∞(U₀) is plotted for different values of V₀. All parameters are equal to 1 for simplicity, which means that ${\mathcal{U}}_c=1$.

Fig. 5

Phase portrait of system (2.1), with unitary parameters. Empty circles represent the initial states, while solid circles represent final states. Note that only the initial states with $U_0>{\mathcal{U}}_c=1$ corresponds to scenarios with ${\mathcal{R}}_0>1$.

Fig. 6

Time evolution of U, I and V, with unitary parameters β, δ, p, c, for initial conditions $U_0=3,I_0=0,V_0=0.2$ (upper plot) and $U_0=1.2,I_0=0,V_0=0.12$ (lower plot).

Fig. 7

SARS-CoV-2 Dynamics. The continuous blue line is the simulation with parameter values presented in Hernandez-Vargas & Velasco-Hernandez, 2020. The patient labeling is as presented in Wölfel et al. (2020). V_clear denotes a value of 50 [copies/ml] under which the virus is not detectable.

Fig. 8

Susceptible cells dynamics. The continuous blue line is the simulation with parameter values presented in Hernandez-Vargas & Velasco-Hernandez, 2020. The patient labeling is as presented in Wölfel et al. (2020). Simulation for the patient C shows a very low value of U_∞ (practically zero), which suggests that the selected value of $U_0=1.0e^7$ may be large.