Modeling the Within-Host Dynamics of SARS-CoV-2 Infection Based on Antiviral Treatment

Abstract

The COVID-19 pandemic has highlighted the profound impact of the SARS-CoV-2 virus as a significant threat to human health. There is an urgent need to develop a comprehensive understanding of the current outbreak by studying the dynamics of the virus within the human body. In this research, we present a mathematical model that explores the progression of SARS-CoV-2 infection, taking into account both the innate and adaptive immune responses. We calculated the basic reproduction number and analyzed the stability of the equilibria. Additionally, we demonstrated the existence of a periodic solution through numerical simulations. By conducting a global sensitivity analysis, we determined the significance of the model parameters and investigated the influence of key parameters on viral load. The results emphasized the crucial roles of cytokines and antibodies in shaping the dynamics of SARS-CoV-2. Furthermore, we evaluated the effectiveness of antiviral treatment in controlling the dynamics of SARS-CoV-2 infection. Our findings revealed a direct relationship between the basic reproduction number and the impact of antiviral treatment. To evaluate the effect of antiviral treatment on viral load, we conducted numerical simulations.

1. Introduction

COVID-19, a highly contagious epidemic disease caused by the newly discovered coronavirus known as SARS-CoV-2, has been officially declared a pandemic by the World Health Organization (WHO) [1]. The initial outbreak of COVID-19 occurred in early December 2019, and it has continued to spread worldwide up until the present day. According to aggregated worldwide data from the international Worldometer.info website, the number of confirmed COVID-19 cases has surpassed 760 million, with approximately 7 million reported deaths as of 12 June 2023. In response to this ongoing crisis, researchers from around the globe are working tirelessly to find effective solutions and develop strategies to control the pandemic, with the aim of minimizing its impact on both human health and the global economy [2].

SARS-CoV-2, the virus responsible for COVID-19, can be transmitted through various means, including direct, indirect, or close contact with infected individuals through secretions such as saliva and respiratory droplets, and these droplets can be released during activities such as coughing, sneezing, speaking, singing, or even exhaling [3]. The clinical presentation of COVID-19 can vary widely, ranging from asymptomatic cases to mild or moderate respiratory symptoms, as well as severe and potentially life-threatening cardiovascular and pulmonary complications [4]. Fever is often an initial and prominent symptom of COVID-19, accompanied by other manifestations such as dry cough, shortness of breath, muscle aches, dizziness, headache, sore throat, runny nose, chest pain, diarrhea, nausea, and vomiting [3]. It is worth noting that the severity of the disease is influenced by various factors, including underlying conditions such as age, diabetes, cardiovascular disorders, lung and kidney disorders, and an immunocompromised state; genetic factors in the host; and viral variants [5,6].

Research has demonstrated that SARS-CoV-2 primarily infects alveolar epithelial cells, particularly type 2 alveolar epithelial cells (AEC2), by binding to the angiotensin-converting enzyme receptor 2 (ACE2) [7,8]. This interaction leads to the destruction of the epithelial cells and an increase in cell permeability, facilitating the release of the virus into the surrounding environment. In the battle against SARS-CoV-2, the host immune system plays a crucial role in controlling and resolving the infection [9]. Understanding the intricate interplay between the virus and the immune system is of utmost importance for comprehending the pathogenesis and disease progression of COVID-19, devising effective treatment strategies, and creating vaccines to combat the disease.

The immune responses against viral infections such as SARS-CoV-2 involve both innate and adaptive immunity. Innate immunity serves as the initial non-specific defense, responding rapidly to viral invasion and limiting viral entry, replication, and assembly [3]. It also identifies and eliminates infected cells and facilitates the development of adaptive immunity [10]. Innate immune cells release molecular regulators like cytokines, and these cytokines operate through cellular signaling pathways, effectively interfering with viral genome transcription within the nucleus and ultimately leading to the suppression of viral protein synthesis [11,12]. The adaptive immune response is activated after the innate immune system and includes humoral and cellular immunity [13]. These responses enable targeted protection against dangerous agents like viruses and bacteria. While there is some overlap between innate and adaptive immunity, they have significant differences in terms of response time, target antigens, and mechanisms of viral antigen recognition.

Humoral and cellular immunity are essential in the adaptive immune response against SARS-CoV-2. Humoral immunity involves B cells in germinal centers responding to viral antigens, differentiating into plasma cells that secrete specific antibodies to control viral replication [14]. Helper T cells support B-cell activation and antibody production. Antibodies generated during the humoral response can be found in the blood and can also be produced upon re-exposure to viral antigens by memory B cells and plasma cells [15]. On the other hand, cellular immunity involves the activation of phagocytes, antigen-sensitized cytotoxic T cells, and the release of cytokines and chemokines in response to antigens. It is mediated by T lymphocytes and effectively combats intracellular pathogens and cancerous cells [16,17,18]. SARS-CoV-2-specific CD4+ T cells primarily produce cytokines with antiviral functions, supporting B cell antibody production and assisting other immune cell populations [19,20]. Activated T cells differentiate into cytotoxic T cells that directly kill virus-infected cells through the MHC–antigen complex [21]. Overall, both humoral and cellular immunity play critical roles in the immune response to SARS-CoV-2.

Mathematical studies are invaluable in understanding and managing human diseases and viral infections. Studying the transmission dynamics of SARS-CoV-2 at the population level is beneficial for comprehensively understanding the patterns and characteristics of virus transmission, formulating effective prevention and control strategies, and predicting the trends of the epidemic. Currently, a large number of scholars are paying attention to research in this area [22,23,24,25]. For example, Diagne et al. formulated a deterministic model of COVID-19 transmission with an imperfect vaccine, analyzing stability, deriving the critical vaccination threshold, and performing sensitivity analysis [26]. Meanwhile, Watson et al. used a mathematical model to assess the global impact of COVID-19 vaccination and the potential benefits of a more equitable global vaccination campaign [27]. Studying the dynamics of SARS-CoV-2 within the host at the individual level is crucial for gaining a comprehensive understanding of the nature of viral infection, developing effective treatment and prevention measures, and preparing for potential new viral threats in the future. However, research on the within-host dynamics of SARS-CoV-2 is relatively limited [28,29]. Almocera et al. developed an in-host model specifically focusing on the effector T-cell response to SARS-CoV-2, revealing the the rapid replication of the virus to overcome T cells [30]. Additionally, Tavares et al. explored COVID-19 immunotherapy using a mathematical model, emphasizing the roles of natural killer (NK) cells [31]. These two studies mainly focused on investigating the impact of adaptive immunity. Reis et al. further investigated the hypothesis that SARS-CoV-2 infects immune cells through a system of fifteen ordinary differential equations that modeled the immune response to the virus [32]. This study comprehensively considered the mechanisms of viral infection within the host, but the model was overly complex, making it challenging to conduct theoretical analysis.

In this paper, we aimed to gain a better understanding of the mechanisms of viral infection within the host and explore the intricate interactions between the virus and the immune system. This research may provide valuable insights for developing effective strategies to combat COVID-19. To achieve this, a mathematical model was used, examining the interactions between the virus and the host immune system, encompassing both innate immunity and adaptive immunity, while also considering the role of cytokines. The model was subject to both theoretical and numerical analyses to calculate equilibria, assess the stability of the equilibria, and conduct sensitivity analysis on its parameters. Moreover, the study investigated the effectiveness of treatments and vaccines by incorporating therapeutic terms into the model and conducting relevant numerical analyses.

2. Methods

A comprehensive within-host mathematical model was formulated to study the intricate interaction between the virus and the host. The stability theory of dynamical systems was employed to analyze the asymptotic behavior of the model. The study then expanded the model to incorporate antiviral treatment interventions. Numerical simulations were conducted to analyze various solution curves, including periodic solutions. Sensitivity analysis was performed to identify the most influential parameters concerning viral load. Lastly, the impact of antiviral drugs or vaccines was numerically investigated to better understand their effectiveness.

3. Model Formulation

To study the within-host dynamics of SARS-CoV-2 infection, we formulated a mathematical model. Mathematically modeling the actual phenomena of immune cell interactions is very difficult. We present here a general mathematical model to discuss the interaction between SARS-CoV-2 infection and the immune cells of the host. Upon infection by SARS-CoV-2, the human body activates both innate and adaptive immune responses, working in tandem to neutralize and counteract the threat posed by the virus [33,34]. T cells play a pivotal role in the activation of B cells, resulting in the production of antibodies, specifically IgM and IgG, which are essential for neutralizing SARS-CoV-2 viral particles. T cells and macrophages contribute significantly to the synthesis of cytokines, thereby facilitating the activation of acquired immunity [35,36]. Additionally, cytokines are essential for suppressing viral replication and regulating the subsequent effects of the immune response [37]. Hence, we considered macrophage cells, T cells, B cells, antibodies, and cytokines as the immune compartments. Additionally, in order to simplify the system, we did not consider the other role of T cells and B cells, which is related to memory. The model mainly relied on the following assumptions:

A1

The model considered the interactions between eight compartments: susceptible cells, $S\left(t\right)$; infected cells, $I\left(t\right)$; SARS-CoV-2 viral particles, $V\left(t\right)$; T cells, $T\left(t\right)$; macrophage cells, $M\left(t\right)$; B cells, $B\left(t\right)$; antibodies, $A\left(t\right)$; and cytokines, $C\left(t\right)$.

A2

The healthy cells are regenerated and die at rates $\mathrm{\Lambda }$ and $d_s$, respectively. The healthy cells become infected by SARS-CoV-2 viral particles at a rates of $\beta$.

A3

Infected cells $I\left(t\right)$, SARS-CoV-2 viral particles $V\left(t\right)$, T cells $T\left(t\right)$, macrophage cells $M\left(t\right)$, B cells $B\left(t\right)$, antibodies $A\left(t\right)$ and cytokines $C\left(t\right)$ die at rates $d_i$, $d_v$, $d_t$, $d_m$, $d_b$, $d_a$, and $d_c$, respectively.

A4

Infected cells are diminished by T cells and macrophage cells at rates ${\xi }_1$ and ${\xi }_2$, respectively.

A5

Viral particles burst from infected cells at rate p, and they are neutralized by cytokines and antibodies at rates ${\gamma }_1$ and ${\gamma }_2$, respectively.

A6

T cells are activated by cytokines at rate ${\theta }_1$, and the proliferation rate of T cells is ${\theta }_2$.

A7

Macrophage cells are activated by cytokines at rate ${\theta }_3$.

A8

The proliferation rate of B cells is ${\omega }_1$. B cells are activated by viral particles at rate ${\omega }_2$.

A9

B cells are stimulated by T cells at rate ${\eta }_1$, and the virus kills or neutralizes antibodies at rate $\xi$.

A10

The rates at which T cells and macrophage cells stimulate cytokine production are ${\alpha }_1$ and ${\alpha }_2$, respectively.

Based on assumptions A1–A10, we formulated the mathematical model. A schematic flow diagram of the model is depicted in Figure 1.

The ODEs (ordinary differential equations) for the within-host model are as follows:

$$\begin{array}{cc}\hfill S^{\prime }\left(t\right)& =\mathrm{\Lambda }-\beta SV-d_{s}S,\hfill \\ \hfill I^{\prime }\left(t\right)& =\beta SV-d_{i}I-{\xi }_{1}TI-{\xi }_{2}MI,\hfill \\ \hfill V^{\prime }\left(t\right)& =p{d}_{i}I-d_{v}V-{\gamma }_{1}CV-{\gamma }_{2}AV,\hfill \\ \hfill T^{\prime }\left(t\right)& ={\theta }_{1}MC-d_{t}T+{\theta }_{2}MT,\hfill \\ \hfill M^{\prime }\left(t\right)& ={\theta }_{3}C-d_{m}M,\hfill \\ \hfill B^{\prime }\left(t\right)& ={\omega }_{1}BT+{\omega }_{2}BV-d_{b}B,\hfill \\ \hfill A^{\prime }\left(t\right)& ={\eta }_{1}B-d_{a}A-\xi AV,\hfill \\ \hfill C^{\prime }\left(t\right)& ={\alpha }_{1}T+{\alpha }_{2}M-d_{c}C.\hfill \end{array}$$

(1)

The description of the parameters of the model (1) is provided in

Table 1. Parameters of the model.

Parameter	Definition	Estimated Value	Ref.
$\mathrm{\Lambda }$	production rate of healthy cells	4000 cells/mL/day	[37]
$\beta$	rate at which healthy cells are converted to infected cells	$(5-561)\times {10}^{-9}$ mL/cell/day	[38]
$d_s$	natural death rate of healthy cells	0.14/day	[37]
$d_i$	natural death rate of infected cells	0.14/day	[39]
${\xi }_1$	rate at which infected cells are diminished by T cells	$1.89\times {10}^{-6}$ mL/cell/day	[40]
${\xi }_2$	rate at which infected cells are diminished by macrophage cells	$1.21\times {10}^{-7}$ mL/cell/day	[41]
p	burst rate of viral particles from dead infected cells	10,000/day	[42]
$d_v$	natural death rate of viral particles	$1.75$/day	[43]
${\gamma }_1$	rate at which viral particles are neutralized by cytokines	$2\times {10}^{-4}$ mL/cell/day	[44]
${\gamma }_2$	rate at which viral particles are neutralized by antibodies	$2\times {10}^{-4}$ mL/molecule/day	[44]
${\theta }_1$	activation rate of T cells	$1.43\times {10}^{-5}$ mL/cell/day	[32]
$d_t$	natural death rate of T cells	0.3/day	[32]
${\theta }_2$	proliferation rate of T cells	$1\times {10}^{-8}$/day	[32]
${\theta }_3$	activation rate of macrophage cells	$1.1\times {10}^{-6}$ mL/cell/day	[41]
$d_m$	natural death rate of macrophage cells	0.04/day	[32]
${\omega }_1$	proliferation rate of B cells	$1.27\times {10}^{-8}$ mL/cell/day	[32]
${\omega }_2$	activation rate of B cells	$8.98\times {10}^{-5}$ mL/cell/day	[32]
$d_b$	natural death rate of B cells	0.2/day	[45]
${\eta }_1$	rate at which B cells are stimulated by T cells	0.01/day	[37]
$d_a$	natural death rate of antibodies	0.07/day	[46]
$\xi$	rate at which the virus kills or neutralizes antibodies	$3\times {10}^{-7}$/day	[32]
${\alpha }_1$	production rate of cytokines stimulated by T cells	0.0178/day	[32]
${\alpha }_2$	production rate of cytokines stimulated by macrophage cells	328/day	[32]
$d_c$	natural death rate of cytokines	0.07/day	[32]

.

4. Model Analysis

4.1. Basic Reproduction Number and the Existence of Equilibria

Following references [47,48], we obtained the basic reproduction number of our system, which was

$$R_0=\rho \left(F{V}^{-1}\right)=\frac{\mathrm{\Lambda }p\beta }{d_s{d}_v}.$$

The equilibria of the system (1) were determined by the following equations:

$$\left\{\begin{array}{c}\mathrm{\Lambda }-\beta SV-d_{s}S=0,\hfill \\ \beta SV-d_{i}I-{\xi }_{1}TI-{\xi }_{2}MI=0,\hfill \\ p{d}_{I}I-d_{v}V-{\gamma }_{1}CV-{\gamma }_{2}AV=0,\hfill \\ {\theta }_{1}MC-d_{t}T+{\theta }_{2}MT=0,\hfill \\ {\theta }_{3}C-d_{m}M=0,\hfill \\ {\omega }_{1}BT+{\omega }_{2}BV-d_{b}B=0,\hfill \\ {\eta }_{1}B-d_{a}A-\xi AV=0,\hfill \\ {\alpha }_{1}T+{\alpha }_{2}M-d_{c}C=0.\hfill \end{array}\right.$$

(2)

There were five type of equilibria in the system (1), namely:

• The disease-free equilibrium is given by $E_0=(\frac{\mathrm{\Lambda }}{d_s},0,0,0,0,0,0,0)$.
• The virus persistence in the absence of immune response equilibrium is given by $E_1=(S_1,I_1,V_1,0,0,0,0,0)$, where $S_1=\frac{\mathrm{\Lambda }}{d_s+\beta V_1}$, $I_1=\frac{d_s{d}_v}{p\beta d_i}(R_0-1)$, $V_1=\frac{d_s}{\beta }(R_0-1)$. This equilibrium exists only when $R_0>1$.
• The virus persistence in the absence of antibody response equilibrium is given by $E_2=(S_2,I_2,V_2,T_2,M_2,0,0,0)$, where $S_2=\frac{(d_i+{\theta }_9{M}_2)(d_v+{\theta }_{10}{M}_2)}{\beta p{d}_2}$, $I_2=\frac{\mathrm{\Lambda }-d_s{S}_2}{d_i+{\theta }_9{M}_2}$, $V_2=\frac{p{d}_i{I}_2}{d_v+{\theta }_{10}{M}_2}$, $M_2=\frac{d_t(d_m{d}_c-{\alpha }_2{\theta }_3)}{d_m{\theta }_1{\alpha }_1+{\theta }_2(d_m{d}_c-{\alpha }_2{\theta }_3)}$, $T_2=\frac{(d_m{d}_c-{\alpha }_2{\theta }_3)M_2}{{\theta }_3{\alpha }_1}$, ${\theta }_9=\frac{d_m{\xi }_1{d}_c}{{\theta }_3{\alpha }_1}-\frac{{\xi }_1{\alpha }_2}{{\alpha }_1}+{\xi }_2$, ${\theta }_{10}=\frac{{\gamma }_1{d}_m}{{\theta }_3}$. This equilibrium exists only when the inequalities of $d_m{d}_c-{\alpha }_2{\theta }_3>0$ and $\mathrm{\Lambda }-d_s{S}_2>0$ are satisfied simultaneously.
• The virus persistence in the presence of antibody response equilibrium is given by $E_3=(S_3,I_3,V_3,0,0,B_3,A_3,0)$, where $S_3=\frac{\mathrm{\Lambda }}{\beta V_3+d_s}$, $I_3=\frac{\beta S_3{V}_3}{d_i}$, $V_3=\frac{d_b}{{\omega }_2}$, $B_4=\frac{1}{{\eta }_1}(d_a{A}_3+\xi A_3{V}_3)$, $A_3=\frac{p{d}_i{I}_3-d_v{V}_3}{{\gamma }_2{V}_3}$. This equilibrium exists only when $R_0>1+\frac{\beta d_b}{d_s{\omega }_2}$.
• The coexistence of all cells equilibrium is given by $E_4=(S_4,I_4,V_4,T_4,M_4,B_4,A_4,C_4)$, where $S_4=\frac{\mathrm{\Lambda }}{\beta V_4+d_s}$, $I_4=\frac{V_4}{p{d}_i}({\theta }_{22}+{\gamma }_2{A}_4)$, $V_4=\frac{1}{{\omega }_2}(d_b-{\omega }_1{T}_4)$, $T_4=\frac{(d_m{d}_c-{\theta }_3{\alpha }_2)M_4}{{\theta }_3{\alpha }_1}$, $M_4=\frac{d_m{\theta }_1{\alpha }_1+{\theta }_2(d_m{d}_c-{\alpha }_2{\theta }_3)}{d_t(d_m{d}_c-{\alpha }_2{\theta }_3)}$, $B_4=\frac{1}{{\eta }_1}(d_a+\xi V_4)A_4$, $A_4=\frac{p{d}_i}{{\gamma }_2}(\frac{\beta S_4}{d_i+{\theta }_9{M}_4}-\frac{{\theta }_{22}}{p{d}_i})$, $C_4=\frac{1}{d_c}({\alpha }_1{T}_4+{\alpha }_2{M}_4)$, ${\theta }_{22}=d_v+\frac{1}{d_c}({\gamma }_1{\alpha }_1{T}_4+{\gamma }_2{\alpha }_2{M}_4)$. This equilibrium exists only when the inequalities of $d_m{d}_c-{\alpha }_2{\theta }_3>0$ and $d_b-{\omega }_1{T}_4>0$ and $p{d}_i\beta S_4-{\theta }_{22}(d_i+{\theta }_9{M}_4)>0$ are satisfied simultaneously.

Stability of Equilibria

Studying the stability of equilibrium in this model was essential for understanding the developmental dynamics of SARS-CoV-2 within the host. By analyzing the stability of the equilibrium, it was possible to determine whether SARS-CoV-2 will persist or eventually be eradicated within the host. A stable equilibrium would indicate the controlled development of SARS-CoV-2, while an unstable equilibrium may lead to outbreaks or decline. These analyses are valuable for formulating effective interventions and providing theoretical guidance for clinical treatment.

To study the stability of the different equilibria, we linearized the system about the arbitrary equilibria $E=(S,I,V,T,M,B,S,C)$, and the corresponding Jacobian matrix is given as:

$$J\left(E\right)=\left(\begin{array}{cccccccc}{j}_1& 0& -\beta S& 0& 0& 0& 0& 0\\ \beta V& j_2& \beta S& {\xi }_{1}I& {\xi }_{2}I& 0& 0& 0\\ 0& p{d}_i& j_3& 0& 0& 0& -{\gamma }_{2}V& {\gamma }_{1}V\\ 0& 0& 0& j_4& {\theta }_{1}C+{\theta }_{2}T& 0& 0& {\theta }_{1}M\\ 0& 0& 0& 0& -d_m& 0& 0& {\theta }_3\\ 0& 0& {\omega }_{2}B& {\omega }_{1}B& 0& j_5& 0& 0\\ 0& 0& -\xi A& 0& 0& -{\eta }_1& j_6& 0\\ 0& 0& 0& {\alpha }_1& {\alpha }_2& 0& 0& -d_c\end{array}\right),$$

where $j_1=-\beta V-d_s$, $j_2=-d_i-{\xi }_{1}T-{\xi }_{2}M$, $j_3=-d_v-{\gamma }_{1}C-{\gamma }_{2}A$, $j_4=-d_t+{\theta }_{2}M$, $j_5={\omega }_{1}T+{\omega }_{2}V-d_b$, $j_6=-d_a-\xi V$.

Next, we present five theorems to prove the local stability of the five equilibrium points (for a detailed description of the proof process, please refer to Appendix A).

Theorem 1.

If $R_0<1$ and $d_m{d}_c-{\alpha }_2{\theta }_3>0$, the disease-free equilibrium $E_0$ is locally asymptotically stable.

Theorem 2.

If $d_m{d}_c-{\alpha }_2{\theta }_3>0$ and ${\omega }_2{d}_s(R_0-1)-d_b\beta <0$, then the equilibrium $E_1$ is locally asymptotically stable.

Theorem 3.

Equilibrium $E_2$ is always unstable.

Theorem 4.

Equilibrium $E_3$ will be locally asymptotically stable if the following conditions hold:

• $F_1>0$;
• $F_1{F}_2-F_3>0$;
• $(F_1{F}_2-F_3)F_3-(F_1{F}_4-F_5)F_1>0$;
• $(F_1{F}_2-F_3)(F_3{F}_4-F_2{F}_5)-{(F_1{F}_4-F_5)}^2>0$;
• $F_5>0$.

$F_i(i=1,2,...,5)$ is given by

$$\begin{array}{cc}\hfill F_1=& b_{11}+b_{22}+b_{33}+b_{66}+b_{77};\hfill \\ \hfill F_2=& b_{33}{b}_{66}+b_{33}{b}_{77}+b_{66}{b}_{77}+b_{37}{b}_{73}+(b_{11}+b_{22})(b_{33}+b_{66}+b_{77})+b_{11}{b}_{22}\hfill \\ \hfill & -b_{23}{b}_{32};\hfill \\ \hfill F_3=& b_{33}{b}_{66}{b}_{77}+b_{63}{b}_{76}{b}_{37}+b_{66}{b}_{73}{b}_{37}+(b_{11}+b_{22})(b_{33}{b}_{66}+b_{33}{b}_{77}+b_{66}{b}_{77}\hfill \\ \hfill & +b_{37}{b}_{73})+(b_{11}+b_{22})(b_{33}+b_{66}+b_{77})-b_{32}{b}_{23}(b_{66}+b_{77})+b_{21}{b}_{13}{b}_{32}\hfill \\ \hfill & -b_{11}{b}_{32}{b}_{23};\hfill \\ \hfill F_4=& (b_{11}+b_{22})(b_{33}{b}_{66}{b}_{77}+b_{63}{b}_{76}{b}_{37}+b_{66}{b}_{73}{b}_{37})+b_{11}{b}_{22}(b_{33}{b}_{66}+b_{33}{b}_{77}\hfill \\ \hfill & +b_{66}{b}_{77}+b_{37}{b}_{73})+(b_{66}+b_{77})(b_{21}{b}_{13}{b}_{32}+b_{11}{b}_{32}{b}_{23}+b_{66}{b}_{77}{b}_{23});\hfill \\ \hfill F_5=& b_{11}{b}_{22}(b_{33}{b}_{66}{b}_{77}+b_{63}{b}_{76}{b}_{37}+b_{66}{b}_{73}{b}_{37})+b_{66}{b}_{77}(b_{21}{b}_{13}{b}_{32}+b_{11}{b}_{32}{b}_{23}),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & b_{11}=\beta V_3+d_s,b_{22}=d_i,b_{33}=d_v+{\gamma }_2{A}_3,b_{44}=d_t,b_{55}=d_m,b_{66}=d_b-{\omega }_2{V}_3;\hfill \\ \hfill & b_{77}=d_a+\xi V_3,b_{88}=d_c,b_{37}={\gamma }_2{V}_3;b_{73}=\xi A_3,b_{63}=-{\omega }_2{B}_3,b_{76}=-{\eta }_1,\hfill \\ \hfill & b_{23}=\beta S_3,b_{32}=-p{d}_i,b_{13}=\beta S_3,b_{21}=-\beta V_3,b_{58}=-{\theta }_3,b_{85}=-{\alpha }_2.\hfill \end{array}$$

Theorem 5.

Equilibrium $E_4$ is always unstable.

5. Mathematical Model with Antiviral Treatment

Mathematical models of within-host virus dynamics could not only improve our understanding of the SARS-CoV-2 replication cycle and its interactions with the immune system, but also help us explore the pharmacological effect of potential antiviral therapies [49]. Now, we consider the potential impact of antiviral treatment on virus dynamics based on model (1), following the approach presented by Zitzmann et al. [50]. The following model was used to simulate an antiviral treatment that partially or completely blocks virus production ($ϵ$). The newly infected term $p{d}_{i}I$ now takes the form $ϵp{d}_{i}I$; thus, the modified system with antiviral treatment is given by

$$\begin{array}{cc}\hfill S^{\prime }\left(t\right)& =\mathrm{\Lambda }-\beta SV-d_{s}S;\hfill \\ \hfill I^{\prime }\left(t\right)& =\beta SV-d_{i}I-{\xi }_{1}TI-{\xi }_{2}MI;\hfill \\ \hfill V^{\prime }\left(t\right)& =ϵp{d}_{i}I-d_{v}V-{\gamma }_{1}CV-{\gamma }_{2}AV;\hfill \\ \hfill T^{\prime }\left(t\right)& ={\theta }_{1}MC-d_{t}T+{\theta }_{2}MT;\hfill \\ \hfill M^{\prime }\left(t\right)& ={\theta }_{3}C-d_{m}M;\hfill \\ \hfill B^{\prime }\left(t\right)& ={\omega }_{1}BT+{\omega }_{2}BV-d_{b}B;\hfill \\ \hfill A^{\prime }\left(t\right)& ={\eta }_{1}B-d_{a}A-\xi AV;\hfill \\ \hfill C^{\prime }\left(t\right)& ={\alpha }_{1}T+{\alpha }_{2}M-d_{c}C.\hfill \end{array}$$

(3)

The parameter $ϵ$ lies between 0 and 1, and before the drug is administered, $ϵ=1$. The reproduction number under antiviral treatments, or the control reproduction number, was found to be

$$R_0^t=\rho \left(F{V}^{-1}\right)=\frac{ϵ\mathrm{\Lambda }p\beta }{d_s{d}_v}.$$

From the expression of $R_0^t$, it was noted that $R_0^t<R_0$ always holds due to the antiviral treatment. Therefore, it could be theoretically illustrated that an effective antiviral treatment could greatly reduce the transmission risk of the disease, with the system reaching a healthy state.

6. Numerical Simulations

In this section, important properties of the proposed model are investigated numerically. Throughout this section, the following set of initial conditions is used, unless stated otherwise: $S\left(0\right)=4000$, $I\left(0\right)=0.1$, $V\left(0\right)=357$, $T\left(0\right)=0.1$, $M\left(0\right)=0.1$, $B\left(0\right)=0.1$, $A\left(0\right)=0.1$, and $C\left(0\right)=0.1$. We initially justified the analytical findings by carrying out some numerical simulations. We first performed global sensitivity analysis to identify the most influential parameters with respect to the viral load (see Figure 2). It was observed that the parameters ${\omega }_2$, $d_b$, and $\beta$ had significant correlations with viral load (V). This result also reinforced the fact that the activation rate of B cells had an important influence on the viral load.

Based on the sensitivity analysis results, we chose to vary parameter $\beta =9\times {10}^{-10}$, and the other parameter values were taken from Table 1. For convenience, please refer to

Table 2. Parameter values.

Parameter	$\mathsf{\Lambda }$	$\beta$	$d_s$	$d_i$	${\xi }_1$	${\xi }_2$	p	$d_v$
Values for computing $E_0$	4000	$9\times {10}^{-10}$	$0.14$	$0.14$	$1.89\times {10}^{-6}$	$1.21\times {10}^{-7}$	10,000	$1.75$
Values for computing $E_1$	4000	$9\times {10}^{-9}$	$0.14$	$0.14$	$1.89\times {10}^{-6}$	$1.21\times {10}^{-7}$	10,000	$1.75$
Values for computing $E_3$	4000	$9\times {10}^{-7}$	$0.14$	$0.14$	$1.89\times {10}^{-6}$	$1.21\times {10}^{-7}$	10,000	$1.75$
Values for periodic solution	4000	$9\times {10}^{-7}$	$0.14$	$0.14$	$1.89\times {10}^{-6}$	$1.21\times {10}^{-7}$	10000	$1.75$
Parameter	${\gamma }_1$	${\gamma }_2$	${\theta }_1$	$d_t$	${\theta }_2$	${\theta }_3$	$d_m$	${\omega }_1$
Values for computing $E_0$	$2\times {10}^{-4}$	$2\times {10}^{-4}$	$1.43\times {10}^{-5}$	0.3	$1\times {10}^{-8}$	$1.1\times {10}^{-6}$	0.04	$1.27\times {10}^{-8}$
Values for computing $E_1$	$2\times {10}^{-4}$	$2\times {10}^{-4}$	$1.43\times {10}^{-5}$	0.3	$1\times {10}^{-8}$	$1.1\times {10}^{-6}$	0.04	$1.27\times {10}^{-8}$
Values for computing $E_3$	$2\times {10}^{-4}$	$2\times {10}^{-4}$	$1.43\times {10}^{-5}$	0.3	$1\times {10}^{-8}$	$1.1\times {10}^{-6}$	0.04	$1.27\times {10}^{-8}$
Values for periodic solution	$2\times {10}^{-4}$	$2\times {10}^{-4}$	$1.43\times {10}^{-5}$	0.3	$1\times {10}^{-8}$	$1.1\times {10}^{-6}$	0.04	$1.27\times {10}^{-8}$
Parameter	${\omega }_2$	$d_b$	${\eta }_1$	$d_a$	$\xi$	${\alpha }_1$	${\alpha }_2$	$d_c$
Values for computing $E_0$	$8.95\times {10}^{-5}$	$0.2$	0.01	0.07	$3\times {10}^{-7}$	0.0178	328	0.07
Values for computing $E_1$	$9\times {10}^{-9}$	$0.2$	0.01	0.07	$3\times {10}^{-7}$	0.0178	328	0.07
Values for computing $E_3$	$9\times {10}^{-9}$	$0.2$	0.01	0.07	$3\times {10}^{-7}$	0.0178	328	0.07
Values for periodic solution	$8.98\times {10}^{-5}$	$0.2$	0.01	0.07	$3\times {10}^{-7}$	0.0178	328	0.07

for all parameter values. According to Theorem A1, we computed the existence condition and stability of the disease-free equilibrium $E_0$ so that $R_0=0.1469<1$ and $d_m{d}_c-{\alpha }_2{\theta }_3=0.0024>0$; then, we obtained equilibrium $E_0=(2.8571\times {10}^4,0,0,0,0,0,0,0)$ (see Figure 3). We set parameters $\beta =9\times {10}^{-9}$ and ${\omega }_2=9\times {10}^{-9}$, and the other parameter values were taken from Table 1. For convenience, please refer to Table 2 for all parameter values. We computed the existence condition of equilibrium $E_1$ so that $R_0=1.4694>1$; then, we obtained equilibrium $E_1=(1.9444\times {10}^4,9.127\times {10}^3,7.302\times {10}^6,0,0,0,0,0)$ (see Figure 4). According to Theorem A2, we obtained $d_m{d}_c-{\alpha }_2{\theta }_3=0.0696>0$ and ${\omega }_2{d}_s(R_0-1)-d_b\beta =-1.2096\times {10}^{-9}<0$; hence, $E_1$ is a stable equilibrium in this parameter set. Taking $\beta =9\times {10}^{-7}$ and ${\omega }_2=9\times {10}^{-9}$, we set the other parameter values based on Table 1. For convenience, please refer to Table 2 for all parameter values. Similarly, we calculated the existence condition of equilibrium $E_3$ so that $R_0=146.9388>1+142.8571$; then, we obtained equilibrium $E_3=(198.6097,2.8373\times {10}^4,2.2222\times {10}^7,0,0,1.2627\times {10}^6,1.8744\times {10}^3,0)$ (see Figure 5) and further identified that $F_1=28.8042>0$, $F_1{F}_2-F_3=5132>0$, $(F_1{F}_2-F_3)F_3-(F_1{F}_4-F_5)F_1=1.3115\times {10}^6>0$, $(F_1{F}_2-F_3)(F_3{F}_4-F_2{F}_5)-{(F_1{F}_4-F_5)}^2=4.4913\times {10}^7>0$ and $F_5=0.1424>0$. Hence, $E_3$ is a stable equilibrium in this parameter set. Moreover, we found that system (1) showed periodic behavior for $\beta =9\times {10}^{-7}$, with the other parameter values taken from Table 1. For convenience, please refer to Table 2 for all parameter values. The solution curve exhibited periodic oscillations (see Figure 6). This may imply the influence of activating B cells.

Our further analysis revealed a relationship between the viral load and ${\omega }_2$. The results suggested that the viral load would decrease with an increasing activation rate of B cells (see Figure 7a). We were also interested in the effect of cytokines on the viral load. As shown in Figure 7b, an increase in ${\gamma }_1$ was associated with a decrease in the viral load, indicating that cytokines play a crucial role in neutralizing viral particles.

We further investigated the effect of antiviral treatment on the viral load and antibody count. In Figure 8, the viral load and antibody count are plotted for different levels of antiviral treatment. The viral load time series experienced the highest peak around the 18th day when the antiviral treatment was at a lower level ($ϵ=0.9$, blue line). However, increasing the antiviral treatment level significantly delayed the time at which the viral load reached its peak ($ϵ=0.8$ and $0.7$, red and green lines, respectively), and it could be observed that a decrease in $ϵ$ reduced the peak of the viral load. These phenomena implied the importance and necessity of interventions for SARS-CoV-2 virus infection. We could also conclude that blocking the virus production in infected cells is a more suitable target for antiviral drug development. On the other hand, the antibody count started to rise after the viral load reached its peak and then exhibited saturated behavior.

7. Discussion

Over the past three years, COVID-19 has emerged as a significant global health concern due to its severity and complex nature. Although the exact characteristics of SARS-CoV-2 remain elusive, it is presumed that our immune response plays a critical role in clearing the virus from our bodies. SARS-CoV-2-related epidemiological studies have been widespread, considering various factors at the population level. In this study, we shifted our focus to the within-host level. Methodologically, we utilized a mathematical modeling approach to investigate the within-host dynamics of SARS-CoV-2 infection. By incorporating interactions among susceptible cells, infected cells, viral particles, T cells, macrophage cells, B cells, antibodies, and cytokines, our model provides a comprehensive framework to understand the complex immune response to the virus.

We analyzed the asymptotic behavior of the model, including the existence and stability conditions for each type of equilibrium. This allowed us to gain insights into the long-term behavior of the system and understand how the virus and immune response interact over time. Specifically, we focused on the stability of the disease-free equilibrium. By analyzing its local stability, we derived the expression for the basic reproduction number, which plays a vital role in determining the long-term persistence of the SARS-CoV-2 virus. Mathematical analysis showed that if the reproduction number is less than 1 and the inequality $d_m{d}_c-{\alpha }_2{\theta }_3>0$ is satisfied, the disease-free equilibrium is stable, and the virus infection will eventually die out from the system. Conversely, if the reproduction number is greater than 1 or the inequality is not satisfied, the disease-free equilibrium is unstable, and the virus infection will persist in the system. In this paper, when we only varied the parameter $\beta$, and the values of the other parameters remained the same, as shown in Table 1, the local stability conditions for the disease-free equilibrium point $E_0$ could be satisfied. To investigate the local stability of equilibria $E_1$ and $E_3$, we conducted numerical experiments by varying parameters $\beta$ and ${\omega }_2$ to verify the stability conditions of these two equilibria, as stated in Theorems A2 and A4. All the results were thoroughly validated through numerical simulations (see Figure 4 and Figure 5).

Furthermore, through numerical simulations, we observed periodic behavior in system (1) (see Figure 6). By setting parameter $\beta =9\times {10}^{-7}$ and keeping the values of the other parameters constant, as shown in Table 1, we detected periodic solutions, indicating oscillatory patterns in the dynamics of the virus. This finding suggested that the periodic phenomenon between the virus and immune cells is a common occurrence when the human body is infected with the virus. The occurrence of periodic oscillations in the system signified two important aspects. Firstly, it suggested the presence of self-regulatory mechanisms within the system to maintain a balanced state of certain immune components. This periodic control may help limit excessive virus replication or overactive immune responses, thereby maintaining a relatively stable state during the infection process. Secondly, it may reflect the intricate interactions between immune cells and the virus, particularly in terms of the activation and suppression of immune responses. This regulation could be attributed to the complex interplay between immune cells and the dynamic replication of the virus.

The results of the global sensitivity analysis emphasized the significant influence of the parameter ${\omega }_2$ on the viral load. Consequently, enhancing the activation rate of B cells could be an effective strategy for eradicating the SARS-CoV-2 virus. Overall, this study provides valuable insights into the dynamics of SARS-CoV-2 infection within the human body. The mathematical model and analysis contribute to our understanding of the intricate relationship between the virus and the immune system, and they offer potential avenues for controlling and eliminating the virus.

Our further investigation into the impact of antiviral treatments demonstrated that interventions that block virus production from infected cells could significantly reduce the viral load. Increasing the level of antiviral treatment ($ϵ$) delayed the time at which the viral load reached its peak, indicating the potential effectiveness of antiviral drugs in mitigating the severity of the infection. The rise in antibody count following the viral load peak highlighted the importance of the adaptive immune response in neutralizing the virus.

Understanding the pathophysiology of this pandemic disease requires a comprehensive analysis, with a specific focus on the crucial role played by certain cytokines. Cytokines, a group of polypeptide signaling molecules, act through cell surface receptors to regulate a multitude of biological processes [51]. Key cytokines include those involved in adaptive immunity (e.g., IL-2 and IL-4); proinflammatory cytokines and interleukins (ILs) (e.g., interferon (IFN)-I, IL-1, IL-6, IL-17, and TNF-$\alpha$); and anti-inflammatory cytokines (e.g., IL-10). When host cells encounter stress-generating internal processes, such as microbial infection, they release cytokines as a crucial defensive response, playing a vital role in reprogramming the cell metabolism to combat the stress and protect the body [52,53]. Our model described the influence of cytokines. The results of the global sensitivity analysis demonstrated the significance of parameter ${\gamma }_1$. Figure 7b indicates that the rate of neutralization by cytokines was positively correlated with the viral load, reinforcing the direct effect of cytokines on the viral load.

One limitation of our model was its simplicity. The immune response to SARS-CoV-2 is highly complex and involves numerous factors and feedback loops. We did not consider the role of T cells and B cells related to memory, which could have an impact on long-term immunity. Future studies could refine the model by incorporating additional immune components and refining the parameter estimation. In addition, in the short time since the emergence of COVID-19, numerous studies have described abnormal levels of the following cytokines and chemokines in patients: IL-1, IL-2, IL-4, IL-6, IL-7, IL-10, IL-12, IL-13, IL-17, M-CSF, IFN-$\gamma$, MCP-1, and TNF-$\alpha$ [54,55]. The critical aspect of SARS-CoV-2 infection appears to involve the depletion of the antiviral defenses associated with the innate immune response, coupled with an increased production of inflammatory cytokines. This interplay of weakened defenses and heightened inflammation contributes to the severity of the infection [56]. It could be explored in the future studies.

8. Conclusions

In conclusion, our within-host mathematical model provides a comprehensive understanding of the dynamics of SARS-CoV-2 infection and its interaction with the host immune system. We described our model parameters with corresponding values from the existing literature in Table 1, and a schematic diagram of the model was presented in Figure 1.

We calculated the basic reproduction number $R_0$ for our model using the next-generation method. We determined the existence and stability conditions of all possible equilibria for our model. The sensitivity analysis identified key parameters influencing the viral load, highlighting the importance of B-cell activation, cytokine production, and infection rates. This information can guide the development of targeted therapeutic interventions to control viral replication and mitigate disease severity.

Furthermore, our investigation into antiviral treatments emphasized the potential of such interventions in delaying viral load peaks and reducing infection severity. Effective antiviral drugs targeting virus production from infected cells could play a critical role in managing COVID-19 infections.

Overall, our study contributes to the understanding of SARS-CoV-2 infection dynamics at the individual level, providing insights for the design of strategies to combat the virus. The findings may inform the development of targeted therapies, vaccines, and public health interventions to mitigate the impact of COVID-19 on human health and the global economy.