Dynamical characterization of antiviral effects in COVID-19

Abstract

Mathematical models describing SARS-CoV-2 dynamics and the corresponding immune responses in patients with COVID-19 can be critical to evaluate possible clinical outcomes of antiviral treatments. In this work, based on the concept of virus spreadability in the host, antiviral effectiveness thresholds are determined to establish whether or not a treatment will be able to clear the infection. In addition, the virus dynamic in the host - including the time-to-peak and the final monotonically decreasing behavior - is characterized as a function of the time to treatment initiation. Simulation results, based on nine patient data, show the potential clinical benefits of a treatment classification according to patient critical parameters. This study is aimed at paving the way for the different antivirals being developed to tackle SARS-CoV-2.

Keywords: Antiviral effectiveness; Dynamic characterization; In-host model; SARS-CoV-2.

Fig. 1

${\eta }_p^c\left(t_{tr}\right)$ vs $t_{tr}$ corresponding to the nine patients simulated in Section 4. . The nine COVID-19 patients (A, B, C, D, E, F, G, H and I) identified from data sets in (Wölfel et al. (2020)) are represented here as ”pat”.

Fig. 2

${\mathcal{H}}^c\left(t_{tr}\right)$ is given by the yellow and the green regions, considering the following system parameters: $\beta =0.5$, $\delta =0.2$, $p=2$ and $c=5$, with $U\left(t_{tr}\right)=3$. The critical boundary of ${\mathcal{H}}^c\left(t_{tr}\right)$, denoted by $\partial {\mathcal{H}}^c\left(t_{tr}\right)$, represents the critical pairs of ${\eta }_{\beta }$ and ${\eta }_p$ such that $\mathcal{R}\left(t_{tr}\right)=1$. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3

$U$, $I$, $V$ time evolution for the untreated case. As stated in Theorem 4.1, in Abuin et al. (2020), ${\check{t}}_V<{\hat{t}}_I<t_c<{\hat{t}}_V$, where ${\check{t}}_V$ and ${\hat{t}}_V$ are the times at which $V\left(t\right)$ reaches a local minimum and a local maximum, respectively, ${\hat{t}}_I$ is the time at which $I\left(t\right)$ reaches a local maximum, and $t_c$ is the time at which $U\left(t\right)$ reaches ${\mathcal{U}}_c$. Furthermore, as stated in Appendix A.2, ${\check{t}}_V\approx 0$ and ${\hat{t}}_I\approx t_c\approx {\hat{t}}_V$. $V_{undet}=100$ (copies/mL) stands for the undetectable level of the virus load.

Fig. 4

Uncertainty analysis on viral load evolution for $p$-parameter 95% confidence interval. Note the uncertainty in viral load peak. Empty dots are data of COVID-19 patients.

Fig. 5

Free virus behavior when treatment is started at viral load detection level ($t_{tr}=t_{DL}$). Values of ${\eta }_p$ smaller (${\eta }_p^1,{\eta }_p^2$), approximately equal (${\eta }_p^3,{\eta }_p^4$) and greater (${\eta }_p^5,{\eta }_p^6$) than ${\eta }_p^c$ are simulated to demonstrate the results in Theorem 3.1. The black line denotes the untreated case (${\eta }_p=0$).

Fig. 6

Infection-related metrics as function of ${\eta }_p$ ($t_{tr}=t_{DL}$), for antiviral effectiveness assessment (all patients). Note that, ${\eta }_p^1=0.5{\eta }_p^c\left(t_{tr}\right)$, ${\eta }_p^2=0.75{\eta }_p^c\left(t_{tr}\right)$, ${\eta }_p^3=0.90{\eta }_p^c\left(t_{tr}\right)$, ${\eta }_p^4={\eta }_p^c\left(t_{tr}\right)+0.1(1-{\eta }_p^c\left(t_{tr}\right))$, ${\eta }_p^5={\eta }_p^c\left(t_{tr}\right)+0.25(1-{\eta }_p^c\left(t_{tr}\right))$, and ${\eta }_p^6={\eta }_p^c\left(t_{tr}\right)+0.5(1-{\eta }_p^c\left(t_{tr}\right))$, being ${\eta }_p^c\left(t_{tr}\right)$ the critical drug efficacy of each patient at time to treatment initiation, $t_{tr}$.

Fig. 7

Viral load time evolution with treatment initial time given by $t_{tr}=0.7t^e$. Values of ${\eta }_p$ smaller, approximately equal and greater than ${\eta }_p^c$ are simulated to demonstrate the results in Theorem 3.1. The black line denotes the untreated case (${\eta }_p=0$).

Fig. 8

Viral load time evolution with treatment initial time given by $t_{tr}=t^e$. Values of ${\eta }_p$ smaller, approximately equal and greater than ${\eta }_p^c$ are simulated to demonstrate the results in Theorem 3.1. The black line denotes the untreated case (${\eta }_p=0$).

Fig. 9

Infection-related metrics as function of ${\eta }_p$ ($t_{tr}=0.7t^e$), for antiviral effectiveness assessment (all patients).

Fig. 10

Infection-related metrics as function of ${\eta }_p$ ($t_{tr}=t^e$), for antiviral effectiveness assessment (all patients).

Fig. 11

Virus time evolution for different treatment times, $t_{tr}=4,6,8,17,20,25$ (dpi). Two fixed values of ${\eta }_p$ were used, smaller and bigger than ${\eta }_p^c\left(t_{tr}\right)$: ${\eta }_p=0.73$ (left) and ${\eta }_p=0.9$ (right), respectively. ${\eta }_p^c\left(t_{tr}\right)\approx 0.81$ for $t_{tr}<t^e$. Patient B. .

Fig. 12

Infection-related metrics for antiviral effectiveness assessment as function of ${\eta }_p$ and ${\eta }_{\beta }$ ($t_{tr}=0.7t^e$). Patient A.. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 13

Qualitative plot of function $f\left(V\right)$ (arbitrary parameters) for different values of $\mathcal{R}$. As it can be seen – independently of the parameter values – if $\mathcal{R}\left(t_{tr}\right)>1$ it there exists an interval of values of $V$, and a corresponding period of time such that $f\left(V\right)>V$.

Fig. 14

Evolution of $\mathcal{R}\left(t\right)$ when an antiviral treatment is initiated at a time of approximately $0.75{\hat{t}}_V$, and different antiviral inhibition effects ${\eta }_p$ are considered, smaller and greater than the critical value ${\eta }_p^c$. The black dashed line represents $\mathcal{R}=1$. As it can be seen, for values of ${\eta }_p<{\eta }_p^c$, $\mathcal{R}\left(t\right)$ crosses $1$ at larger times for larger values of ${\eta }_p$ as it is stated in Lemma 1. This implies that if ${\eta }_p<{\eta }_p^c$, larger values of ${\eta }_p$ delays the virus peak time, as it is stated in Theorem 3.1.iii. Furthermore, the figure confirms that, for real patient date, $t^e$ is close to ${\hat{t}}_V$.

Fig. 15

Phase portrait of system (2.1) with parameters $\beta =0.5$, $\delta =0.2$, $p=2$ and $c=5$, for different initial conditions not necessarily representing realistic cases. Empty circles represent the initial state, while solid circles represent final states. The red hyperplane corresponds to $U\left(t\right)\equiv {\mathcal{U}}_c$ (i.e., the critical value of $U$, when $\mathcal{R}\left(t\right)=1$) while the blue hyperplane corresponds to the fast manifold in which $I\left(t\right)$ and $V\left(t\right)$ are proportional (.i.e, $I\left(t\right)=c∕pV\left(t\right)$). Note that only the initial states with $U_0>{\mathcal{U}}_c=1$ corresponds to scenarios with ${\mathcal{R}}_0>1$.