Global Dynamics of SARS-CoV-2 Infection with Antibody Response and the Impact of Impulsive Drug Therapy

Abstract

Mathematical modeling is crucial to investigating tthe ongoing coronavirus disease 2019 (COVID-19) pandemic. The primary target area of the SARS-CoV-2 virus is epithelial cells in the human lower respiratory tract. During this viral infection, infected cells can activate innate and adaptive immune responses to viral infection. Immune response in COVID-19 infection can lead to longer recovery time and more severe secondary complications. We formulate a micro-level mathematical model by incorporating a saturation term for SARS-CoV-2-infected epithelial cell loss reliant on infected cell levels. Forward and backward bifurcation between disease-free and endemic equilibrium points have been analyzed. Global stability of both disease-free and endemic equilibrium is provided. We have seen that the disease-free equilibrium is globally stable for R0<1, and endemic equilibrium exists and is globally stable for R0>1. Impulsive application of drug dosing has been applied for the treatment of COVID-19 patients. Additionally, the dynamics of the impulsive system are discussed when a patient takes drug holidays. Numerical simulations support the analytical findings and the dynamical regimes in the systems.

Keywords: antibody response; basic reproduction number; drug holidays; epithelial cell; impulsive control; transcritical bifurcation.

Figure 1

Conceptual diagram of Model (2). It shows the flowchart of antibody responses in SARS-CoV-2 infection within a host.

Figure 2

(a,b) Forward bifurcation diagram of Model (2) using Theorem 6. Red curves represent stable disease-free equilibria (DFE) and the black-dashed line denotes stable endemic equilibria (EE).

Figure 3

Region of stability of disease-free equilibrium (DFE) and endemic equilibrium (EE) shown in (a) $\beta -p$, (b) $\beta -{\mu }_3$ parameter planes. Color represents the value of $R_0$. DFE is stable for $R_0<1$ and unstable for $R_0>1$. EE exists and is stable for $R_0>1$.

Figure 4

(a–d) Numerical solution of Model (2) for the set of parameters in Table 1.

Figure 5

(a–c) Steady-state values of the populations plotted as function of $\beta$. Parameter values are same as in Figure 4.

Figure 6

Phase portraits plotted in $E_S-E_I-V$ phase-space with different initial conditions and $R_0>1$.

Figure 7

(a,b) Numerical solutions of impulsive Model 4 with treatment ($\omega =60,\tau =7$) and without treatment.

Figure 8

(a,b) Solutions of impulsive Model (4) shown for impulse interval $\tau =7$ days with dosing rate $\omega =100$ mg (blue line), and impulse interval of $\tau =14$ days with dosing rate $\omega =200$ mg (green line).

Figure 9

Dynamics of the drug with and without drug holidays, taking an impulse interval of ($\tau =7$ days) and a fixed drug dosing of $\omega =100$ mg with two consecutive drug holidays, $h_1=2$.