A simple in-host model for COVID-19 with treatments: model prediction and calibration

Abstract

In this paper, we provide a simple ODEs model with a generic nonlinear incidence rate function and incorporate two treatments, blocking the virus binding and inhibiting the virus replication to investigate the impact of calibration on model predictions for the SARS-CoV-2 infection dynamics. We derive conditions of the infection eradication for the long-term dynamics using the basic reproduction number, and complement the characterization of the dynamics at short-time using the resilience and reactivity of the virus-free equilibrium are considered to inform on the average time of recovery and sensitivity to perturbations in the initial virus free stage. Then, we calibrate the treatment model to clinical datasets for viral load in mild and severe cases and immune cells in severe cases. Based on the analysis, the model calibrated to these different datasets predicts distinct scenarios: eradication with a non reactive virus-free equilibrium, eradication with a reactive virus-free equilibrium, and failure of infection eradication. Moreover, severe cases generate richer dynamics and different outcomes with the same treatment. Calibration to different datasets can lead to diverse model predictions, but combining long- and short-term dynamics indicators allows the categorization of model predictions and determination of infection severity.

Introduction

In December 2019, a novel strain of coronavirus, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), was reported in Wuhan, China, causing coronavirus disease 2019 (COVID-19) (Hui et al. 2020; Zhu et al. 2020). The main transmission of SARS-CoV-2 is through the airborne and contact routes (Chams et al. 2020; Tay et al. 2020) and the median incubation period is around 5 days. Most symptomatic infected individuals can have a broad range of symptoms, including fever, dry cough, difficulty in breading, muscle pain, fatigue, headache, diarrhoea, and nausea (Tay et al. 2020). Severe infected individuals can develop acute respiratory distress syndrome (ARDS), which is diagnosed by difficulty in breathing and low blood oxygen level (Tay et al. 2020). Thus, the viral infection and host responses determine the disease severity and mortality in patients (Stebbing et al. 2020; Tay et al. 2020).

The SARS-CoV-2 virus expresses the spike (S) protein on its surface to bind to the receptor angiotensin-converting enzyme 2 (ACE2) expressed on the host cells to fuse to the host cell membrane and then release the viral package into the host cytoplasm (Tay et al. 2020). In lung, airway and alveolar epithelial cells, vascular endothelial cells and macrophages express this ACE2 receptor and hence are the main host cells for SARS-CoV-2. After the cell entry, the viral RNA undergoes translation and replication in the cell cytoplasm, and then the replicated viral particles are released from the infected cells to infect more cells. This process induces apoptosis resulting in alveolar damage and gas exchange impairment, which could lead to ARDS (Chams et al. 2020). When alveolar epithelial cells and alveolar macrophages detect the released viral RNA, these immune cells trigger the immune responses mediated by virus specific B and T cells to produce pro-inflammatory cytokines and chemokines that recruit monocytes and cytotoxic T lymphocytes (CTLs) into the infection site to resolve the infection (Tay et al. 2020).

In a patient with functional immune system, the immune response is able to clear the infected cells and then recedes, resulting in recovery of the patient. However, in a patient with dysfunctional immune system, the immune cells are over activated and then produce higher levels of cytokines, resulting in a cytokine storm and leading to severe symptoms (Tay et al. 2020).

Based on the infection process, one type of the treatment methods is to block the binding between the S protein and ACE2 receptor, including (1) blocking the host target ACE2 receptor, (2) injecting high concentrations of a soluble form of ACE2, or (3) monoclonal antibodies targeting the S protein (Tay et al. 2020). For instance, the neutralizing monoclonal antibody (mAB), such as Bamlanivimab, is an novel type of antiviral intervention that binds to the S protein on SARS-CoV-2 virus to prevent the cell entry. Several clinical data showed that the monotherapy of Bamlanivimab can successfully curtail viral infection, and the combination therapy of Bamlanivimab with Etesevimab or Casirivimab can further reduce the viral loads (Taylor et al. 2021). The Remdesivir, Lopinavir, Ritonavir, and Fapilavir are protease inhibitors that could also inhibit the SARS-CoV-2 infection. The Remdesivir is a novel medicine that incorporates into the nascent viral RNA chains to terminate its RNA synthesis and viral replication (Chams et al. 2020; Smith et al. 2020). Lopinavir and Ritonavir are used to treat human immunodeficiency virus (HIV) (Chams et al. 2020; Smith et al. 2020). They suppress the virus activities by binding to enzymes for virus replication, which could be used to against SARS-CoV-2 (Chams et al. 2020). Another potential agent is Fapilavir, a medicine for influenza, that can block the virus replication at the early stage of infections (Chams et al. 2020; Smith et al. 2020).

Since March 2020, myriad of between- and within-host mathematical models have been published in the context of COVID-19. In (Nath et al. 2021), some basic dynamics of a fundamental within-host model was investigated to study the SARS-Cov-2 infection. In Abuin et al. (2021), a within-host SARS-Cov-2 infection model under an antiviral effect found that early initialized antiviral therapy is essential to reducing viral load. In Sanche et al. (2021), a simple ODEs model is used to explain the relationship between the pathogenesis of COVID-19 and the inflammation level, decay rate of viral load, and amount of natural killer cells. A review of simple target cell models and their applications to different viral infections and treatments were presented in Perelson and Ke (2021). In Gonçalves et al. (2020), the authors considered a treatment target cell model with an eclipse phase to study the viral dynamics in COVID-19 patients and predict the effects of antiviral treatments. They found that Lopinavir/Ritonavir, HCQ, IFN-$\beta$-1a, and Remdesivir dramatically impact nasopharyngeal viral load kinetics if they are administered after symptom onset. In Nath et al. (2021), the authors used the basic reproduction number to characterize the global stability for a within-host ODEs model of SARS-CoV-2. Other models used fractional-order derivatives (Pandey et al. 2021; Rasheed et al. 2021; Shen et al. 2021), a more general approach in which operators depend on all of their past states not only on the current state.

More detailed within-host models are proposed to study the dynamic processes of the immune system. In Jenner et al. (2021), the authors provided a complex delay differential equations (DDEs) model of the immune response to SARS-CoV-2 with two delays to represent the eclipse time and time lag in the arrival time of CD8${}^{+}$ T cells. By fitting viral load data of SARS-Cov-2 and generating a cohort of 200 virtual patients, the authors found that virtual patients with low production rates of infected cell-derived IFN subsequently experienced highly inflammatory disease phenotypes. Moreover, they predicted that individuals with severe COVID-19 have an accelerated monocyte-to-macrophage differentiation mediated by increased IL-6 and reduced type I IFN signaling. In Wang et al. (2020), the authors created three models with bilinear incidence rate function to study the viral dynamics of SARS-CoV-2 infection and the efficacy of some treatments related to antivirial drugs. In Zhou et al. (2021), an intricate within-host model was used to numerically investigated the virus-immune interaction and treatment strategies for COVID-19 patients.

Due to the limitation in datasets for model calibration, some studies focused on the relationship among the considered datasets, model calibration, and model predictions. For instance, in Blanco-Rodríguez et al. (2021), the authors proposed six ODEs models incorporating different types of immune cells and then performed the Akaike information criterion to select the best model to capture the clinical data of CD8${}^{+}$ T cells. However, the optimized solutions did not generate consistent results for the CD4${}^{+}$ T cells or the NK cells data. This finding indicates that the model predictions highly depend on the considered dataset and model calibration. In Ke et al. (2021), the authors developed two ODEs models, one for the viral dynamics and one for the viral load and infectiousness, and incorporated two unique datasets to solve the uncertainty in parameter estimates caused by defective datasets.

This work aims to provide a simple ODEs model of SARS-CoV-2 infection without and with treatments to investigate the long- and short-term dynamics, as the model is calibrated to different clinical datasets to explore the impact on predictions. The model consists of normal healthy cells, infected cells, free virus, and immune cells with a generic nonlinear incidence rate function. We consider two types of treatments, one for blocking the binding between the viruses and host cells and one for inhibiting the replication of the viruses. For the long-term dynamics, we provide the analytical results of the existence and stability of the virus-free equilibrium $X^0$ and the positive equilibrium $X^{+}$ with respect to the reproduction number. In the infection eradication scenarios, the resilience and reactivity are used to investigate the the short-term dynamics for the recovery rate and the sensitivity to any changes in patients’ initial stages.

In the numerical investigation, we consider the treatment model with three types of incidence functions including the bilinear incidence rate, the Holling type II incidence rate, and the Beddington–DeAngelis incidence rate, and calibrate the model to clinical datasets of viral load from patients in mild and severe cases (Zheng et al. 2020) and of immune cells from patients in severe cases (Han et al. 2020). We then focus on the treatment model with the bilinear incidence function as it is found to best represent all datasets. Next, we analyze the effects of calibration on model predictions for disease outcome (success or failure in infection eradication), recovery time, and reactivity based on different datasets. For the severe cases, due to the richer dynamics, we further investigate the treatment efficacy by performing the local sensitivity analysis for small changes of efficacy and the efficacy map for the monotherapy and combination of treatments.

This paper is organized as follows. In Sect. 2, we present the ODEs model without treatments and investigate the existence and stability of equilibria with respect to the basic reproduction number. In Sect. 3, we provide the treatment model, incorporating the blocking of virus binding and inhibiting virus replication, and then extend the results of Sect. 2 to this treatment model. We also introduce the resilience and reactivity to quantify the transient behavior in Sect. 3. In Sect. 4, the treatment model with different types of incidence rate functions is calibrated to clinical data. Analysis of the relationship among the parameter sets obtained by the model calibration, disease outcome, recovery time, and reactivity, as well as the treatment efficacy are also presented in Sect. 4. We summarize our analytical and numerical findings in Sect. 5.

Control model

In this section, we first introduce a simple model for the interaction between cells and viruses with a generic nonlinear incidence rate function in the absence of treatments (i.e., the control model) to predict the long-term dynamics of the infection without any treatments. We then provide its well-posedness properties by showing the nonnegativity and boundedness of the model solution. In order to obtain conditions for infection eradication (i.e., the virus-free equilibrium) or infection persistence (i.e., positive equilibrium), we investigate how the basic reproduction number ${\mathcal{R}}_0$ affects the existence and stability of equilibria. We prove that the virus-free equilibrium is globally stable when ${\mathcal{R}}_0<1$ which provides conditions for infection eradication. We also show that a positive equilibrium exists uniquely when ${\mathcal{R}}_0>1$, and then discuss its stability which relates to the infection persistence.

Mathematical model

Our model is based on the network described in Fig. 1; the variables and parameters used in the model are listed in Table 1.

Fig. 1.

Model illustration. The binding between the surface S protein on the free SARS-CoV-2 virus (V) and the receptor ACE2 on the normal healthy host cells (S) triggers the infection process, which is presented phenomenologically by the incidence rate function f(S, V), and then transforms the normal healthy host cells into infected cells (C). This infection process can be suppressed by blocking the binding between the virus and normal healthy cells, which is shown by the function $(1-q_1)$ (Treatment 1) with the treatment efficacy $q_1$. After the cell entry, the viral RNA undergoes translation and replication in the cell cytoplasm. The replicated viral particles induce cell death of the infected cells, and then portion of the viral particles are released and become the free virus (V). The amount of the released virus can be reduced by inhibiting the virus replication, which is shown by the function $(1-q_2)$ (Treatment 2) with the treatment efficacy $q_2$. Treatments 1 and 2 are only included in the treatment model. When immune cells (T), including alveolar epithelial cells and alveolar macrophages, detect the infected cells, these immune cells trigger the immune responses to produce pro-inflammatory cytokines and chemokines that recruit monocytes and CTLs into the infection site to eliminate the infected cells

Table 1.

Variables and parameters used in models described in Fig. 1

Variable	Description	Unit
S(t)	Density of normal healthy cells at time t	$\text{Cell}/\mathrm{\mu }\text{L}$
C(t)	Density of infected cells at time t	$\text{Cell}/\mathrm{\mu }\text{L}$
V(t)	Density of free virus at time t	$\text{Copies}/\mathrm{\mu }\text{L}$
T(t)	Density of immune cells at time t	$\text{Cell}/\mathrm{\mu }\text{L}$
Parameter	Description	Unit
${\alpha }_0$	Production rate of normal healthy cells	$\text{Cell}/\mathrm{\mu }\text{L}/\text{day}$
$d_1$	Death rate of normal healthy cells	${\text{Day}}^{-1}$
${\beta }_1$	Killing rate of infected cells by immune cells	${(\text{Cell}/\mathrm{\mu }\text{L})}^{-1}/\text{day}$
$d_2$	Natural and virus-induced death infected cells	${\text{Day}}^{-1}$
${\gamma }_0$	Replication rate inside the infected cells	Copies/cell
$d_3$	Death rate of viruses	${\text{Day}}^{-1}$
${\beta }_0$	Immune response rate	${\text{Day}}^{-1}$
$d_4$	Death rate of immune cells	${\text{Day}}^{-1}$

For the model without treatment, we use the following equations to represent the dynamics of normal healthy cells S(t), infected cells C(t), free viruses V(t), and immune cells T(t).

$$\begin{array}{cc}\hfill \frac{\mathit{dS}}{\mathit{dt}}& =\underset{\text{source}}{\underbrace{{\alpha }_0}}-\underset{\text{infection}}{\underbrace{f(S,V)}}-\underset{\text{death}}{\underbrace{d_{1}S}}\hfill \end{array}$$ 1a

$$\begin{array}{cc}\hfill \frac{\mathit{dC}}{\mathit{dt}}& =\underset{\text{infection}}{\underbrace{f(S,V)}}-\underset{\text{killed by immune cells}}{\underbrace{{\beta }_{1}CT}}-\underset{\text{natural and virus-induced death}}{\underbrace{d_{2}C}}\hfill \end{array}$$ 1b

$$\begin{array}{cc}\hfill \frac{\mathit{dV}}{\mathit{dt}}& =\underset{\text{released virus from infected death cells}}{\underbrace{{\gamma }_0\times d_{2}C}}-\underset{\text{death}}{\underbrace{d_{3}V}}\hfill \end{array}$$ 1c

$$\begin{array}{cc}\hfill \frac{\mathit{dT}}{\mathit{dt}}& =\underset{\text{immune response}}{\underbrace{{\beta }_{0}C}}-\underset{\text{death}}{\underbrace{d_{4}T}}.\hfill \end{array}$$ 1d

The normal healthy cells S are daily produced at a rate ${\alpha }_0$ and die with a rate $d_1$. The infection of normal healthy cells by free viruses occurs with a rate f(S, V) and produces infected cells C, which are removed by immune cells T with a rate ${\beta }_1$ and natural and virus-induced death with a rate $d_2$. Viruses V are released from the infected dead cells with the rate ${\gamma }_0$ and cleared by the innate immune response with a rate $d_3$. The immune cells T are activated by the immune responses triggered by the infected cells with a rate ${\beta }_0$ and die with a rate $d_4$.

The infection rate of normal healthy cells by free virus f(S, V) is a generic nonlinear function satisfying the following biological feasible conditions for all $S,V>0$ (see Korobeinikov and Maini 2005)

The mathematical analysis is carried out considering a generic nonlinear incidence function and cover all functions satisfying conditions (A${}_1$)–(A${}_3$). For computational work, we will use the following three incidence rate functions f(S, V),

$$\begin{array}{c}\hfill f_1(S,V)=\beta SV,f_2(S,V)=\frac{\beta SV}{1+{\delta }_{1}V}\text{and}{f}_3(S,V)=\frac{\beta SV}{1+{\delta }_{1}V+{\delta }_{2}S}\end{array}$$ 2

with constants binding rate $\beta$ in ${(\text{copies}/\mu \text{L})}^{-1}/\text{day}$, ${\delta }_1$ in ${(\text{copies}/\mu \text{L})}^{-1}$ and ${\delta }_2$ in ${(\text{cell}/\mu \text{L})}^{-1}$. The bilinear incidence rate, $f_1$, is the mass action incidence function and describes a non saturating infection rate (Korobeinikov 2004; Smith and De Leenheer 2003). In contrast, the Holling type II incidence rate, $f_2$, assumes a saturating infection rate (Xu 2011), as well as the Beddington–DeAngelis incidence rate, $f_3$ (Huang et al. 2011).

Qualitative analysis

Before analyzing the dynamics of the model (1), the following theorem shows the well-posedness of control model (1).

Theorem 1

Model (1) considered with the initial value $(S(0),C(0),V(0),T(0))\in {\mathbb{R}}_{+}^4$ has a unique, nonnegative and bounded solution. Furthermore, the compact set

$$\begin{array}{c}\hfill \mathrm{\Gamma }=\left\{(S(t),C(t),V(t),T(t)),\in ,{\mathbb{R}}_{+}^4,:,S,(t),+,C,(t),+,V,(t),\le ,\frac{{\alpha }_0}{d},,,,T,(t),\le ,\frac{{\alpha }_0{\beta }_0}{d_{4}d}\right\}\end{array}$$

attracts all positive solutions in ${\mathbb{R}}_{+}^4$ where $d=min\{d_1,(1-{\gamma }_0)d_2,d_3\}$.

The proof can be found in “Appendix A.1”.

When there is no infection ($V=0$ and $C=0$), the normal healthy cells stabilize at the value $S^0=\frac{{\alpha }_0}{d_1}$. Hence, the virus-free state

$$\begin{array}{c}\hfill X^0=(S^0,0,0,0):=(\frac{{\alpha }_0}{d_1},0,0,0)\end{array}$$

always exists.

Notice that from (A${}_1$), $\frac{\partial f(S^0,0)}{\partial S}=0$. Then, the infected compartments C and V in the linearization system of control model (1) at $X^0$ can be written as

$$\begin{array}{c}\hfill \left[\begin{array}{c}\frac{\mathit{dC}}{\mathit{dt}}\\ \frac{\mathit{dV}}{\mathit{dt}}\end{array}\right]=\left({\mathbf{M}}_1-{\mathbf{M}}_2\right)\left[\begin{array}{c}C\\ V\end{array}\right]\end{array}$$

where

$$\begin{array}{c}\hfill {\mathbf{M}}_1=\left[\begin{array}{cc}0& \frac{\partial f(S^0,0)}{\partial V}\\ {\gamma }_0{d}_2& 0\end{array}\right],{\mathbf{M}}_2=\left[\begin{array}{cc}{d}_2& 0\\ 0& d_3\end{array}\right].\end{array}$$

The matrix ${\mathbf{M}}_1$ represents the new infections and ${\mathbf{M}}_2$ contains the transitions in and out between infected compartments. Hence, the basic reproduction number ${\mathcal{R}}_0$ is the spectral radius of the next generation matrix ${\mathbf{M}}_1{\mathbf{M}}_2^{-1}$ (Van den Driessche and Watmough 2002). Consequently,

$$\begin{array}{c}\hfill {\mathcal{R}}_0=\sqrt{\frac{{\gamma }_0}{d_3}\frac{\partial f(S^0,0)}{\partial V}}.\end{array}$$

From a biological point of view, ${\mathcal{R}}_0^2$ is the average number of viruses $({\gamma }_0\frac{\partial f(S^0,0)}{\partial V})$ produced by one virus during its average lifetime $\frac{1}{d_3}$.

We now investigate the existence and uniqueness of positive equilibrium of the control model (1) with respect to ${\mathcal{R}}_0$.

Theorem 2

When ${\mathcal{R}}_0<1$, no positive equilibrium exists.

The proof can be found in “Appendix A.2”.

Theorem 3

When ${\mathcal{R}}_0>1$, there exists a unique positive equilibrium $X^{+}=(S^{+},C^{+},V^{+},T^{+})=(S^{+},\frac{d_3}{{\gamma }_0{d}_2}{V}^{+},V^{+},\frac{{\beta }_0{d}_3}{{\gamma }_0{d}_2{d}_4}{V}^{+})$.

The proof can be found in “Appendix A.3”.

We then analyze the stability of the equilibria $X^0$ and $X^{+}$ to obtain conditions for infection eradication and persistence.

First, we analyze the stability of the virus-free equilibrium $X^0$, in the following two results.

Theorem 4

The virus-free equilibrium $X^0$ is locally asymptotically stable when ${\mathcal{R}}_0<1$ and unstable when ${\mathcal{R}}_0>1$.

The proof can be found in “Appendix A.4”.

Theorem 5

The virus-free equilibrium $X^0$ is globally asymptotically stable when ${\mathcal{R}}_0<1$.

The proof can be found in “Appendix A.5”.

Next, we analyze the stability of the positive equilibrium $X^{+}$, in the following result.

Theorem 6

When ${\mathcal{R}}_0>1$, the positive equilibrium $X^{+}=(S^{+},C^{+},V^{+},T^{+})$ is locally asymptotically stable if

$$\begin{array}{c}\hfill d_4>d_3\text{and}\frac{\partial f(S^{+},V^{+})}{\partial V}<\frac{d_3}{{\gamma }_0}\left(1,+,\frac{{\beta }_1}{d_2},T^{+}\right).\end{array}$$ 3

The proof can be found in “Appendix A.6”.

For the control model (1), we proved that the virus-free equilibrium $X^0$ is globally asymptotically stable when ${\mathcal{R}}_0<1$ and unstable when ${\mathcal{R}}_0>1$. This result indicated that, for any immune responses or strength (i.e., the initial condition of the model and values for related immune response parameters), the infection will be eradicated when ${\mathcal{R}}_0<1$. Moreover, when ${\mathcal{R}}_0>1$ and the condition (3) holds, the long-term behavior of the control model could be predicted and its solutions approach the positive equilibrium $X^{+}$, if the initial condition is close to $X^{+}$.

Treatment model

After characterizing the dynamics of the model without treatment, we add the following two classes of treatments:

• Treatment 1: Blocking the binding between the S protein and ACE2 receptor. Treatment 1 mimics the effect of neutralizing mAB such as Bamlanivimab. Bamlanivimab can be generated by B cells of convalescent patients and bind to the S protein on SARS-CoV-2 virus to block the binding between the virus and host cell and then prevent the cell entry (Taylor et al. 2021).

• Treatment 2: Inhibition of the virus replication. Treatment 2 describes the effects of HIV protease inhitors or antibacterial medicines such as Lopinavir, Ritonavir, Remdesivir, and Favipiravir. In order to inhibit the virus replication, Lopinavir and Ritonavir bind to a key enzyme for the SARS-Cov-2 virus replication (Liu and Wang 2020), Remdesivir incorporates into the viral RNA to terminate the RNA synthesis (Gordon et al. 2020), and Favipiravir suppresses the viral RNA polymerase (Sanders et al. 2020).

Mathematical model

To analyze the effects of the treatments, model (1) is extended as the following treatment model

$$\begin{array}{cc}\hfill \frac{\mathit{dS}}{\mathit{dt}}& =\underset{\text{source}}{\underbrace{{\alpha }_0}}-\underset{\text{infection}}{\underbrace{f(S,V)}}\times \underset{\text{Treatment 1: block binding}}{\underbrace{(1-q_1)}}-\underset{\text{death}}{\underbrace{d_{1}S}}\hfill \end{array}$$ 4a

$$\begin{array}{cc}\hfill \frac{\mathit{dC}}{\mathit{dt}}& =\underset{\text{infection}}{\underbrace{f(S,V)}}\times \underset{\text{Treatment 1: block binding}}{\underbrace{(1-q_1)}}-\underset{\text{killed by immune cells}}{\underbrace{{\beta }_{1}CT}}\hfill \\ \hfill & -\underset{\text{natural and virus-induced death}}{\underbrace{d_{2}C}}\hfill \end{array}$$ 4b

$$\begin{array}{cc}\hfill \frac{\mathit{dV}}{\mathit{dt}}& =\underset{\text{released virus from infected dead cells}}{\underbrace{{\gamma }_0\times d_{2}C}}\times \underset{\text{Treatment 2: inhibit replication}}{\underbrace{(1-q_2)}}\hfill \\ \hfill & -\underset{\text{death}}{\underbrace{d_{3}V}}\hfill \end{array}$$ 4c

$$\begin{array}{cc}\hfill \frac{\mathit{dT}}{\mathit{dt}}& =\underset{\text{immune response}}{\underbrace{{\beta }_{0}C}}-\underset{\text{death}}{\underbrace{d_{4}T}}\hfill \end{array}$$ 4d

where f(S, V) is a generic nonlinear incidence form satisfying (A${}_1$)–(A${}_3$) for all $S,V>0$ and $0\le q_i\le 1$ represents the efficacy of Treatments 1 and 2, respectively. Hence, $q_i=1$ represents the best treatment efficacy and $q_i=0$ accounts for no treatment.

In comparison to model (1) without treatments, the treatment term $(1-q_1)$ is incorporated in the infection function f(S, V) of model (4) to represent the reduction of infection rate f(S, V) by blocking the binding between the free virus and normal healthy cells. The treatment term $(1-q_2)$ is applied on the virus release term of V(t) equation in model (4) to describe the reduction of amount of released viruses due to the virus replication inhibition. We denote

$$\begin{array}{c}\hfill \widehat{\beta }=\beta \times (1-q_1)\text{and}{\widehat{\gamma }}_0={\gamma }_0\times (1-q_2),\end{array}$$

where $\widehat{\beta }$ and ${\widehat{\gamma }}_0$ represent the binding rate and replication rate inside the infected cells for the treatment model (4). Notice that, based on the relationship between $\beta$, ${\gamma }_0$ and $\widehat{\beta }$, ${\widehat{\gamma }}_0$, models (1) and (4) have the same structure and are the same mathematical objects.

Qualitative analysis

The virus-free state in model (4) is $X^0=(S^0,0,0,0)$ which is the same as the one in model (1). Thus, the basic reproduction number of the model (4) is

$$\begin{array}{c}\hfill {\widehat{\mathcal{R}}}_0=\sqrt{(1-q_1)\times (1-q_2)\times \frac{{\gamma }_0}{d_3}\frac{\partial f(S^0,0)}{\partial V}}=\sqrt{(1-q_1)\times (1-q_2)}\times {\mathcal{R}}_0.\end{array}$$

It is clear that ${\widehat{\mathcal{R}}}_0\le {\mathcal{R}}_0$. The following results describe the dynamics of the treatment model (4) with respect to ${\widehat{\mathcal{R}}}_0$. The proofs are omitted, since they are similar to the ones of the model (1).

Theorem 7

When ${\widehat{\mathcal{R}}}_0>1$, model (4) has a unique positive equilibrium ${\widehat{X}}^{+}:=({\widehat{S}}^{+},{\widehat{C}}^{+},{\widehat{V}}^{+},{\widehat{T}}^{+})$. When ${\widehat{\mathcal{R}}}_0<1$, no positive equilibrium exists.

The value of the positive equilibrium ${\widehat{X}}^{+}$ of model (4) depends on the treatments efficacy $q_1$ and $q_2$; moreover, ${\widehat{X}}^{+}=X^{+}$ when $q_1=q_2=0$ where $X^{+}$ is the positive equilibrium of model (1).

Theorem 8

The virus-free equilibrium $X^0$ is globally asymptotically stable in (4) when ${\widehat{\mathcal{R}}}_0<1$ and unstable when ${\widehat{\mathcal{R}}}_0>1$.

Theorem 9

When ${\widehat{\mathcal{R}}}_0>1$, the positive equilibrium ${\widehat{X}}^{+}:=({\widehat{S}}^{+},{\widehat{C}}^{+},{\widehat{V}}^{+},{\widehat{T}}^{+})$ is locally asymptotically stable if

$$\begin{array}{c}\hfill d_4>d_3\text{and}\frac{\partial f({\widehat{S}}^{+},{\widehat{V}}^{+})}{\partial V}<\frac{1}{(1-q_1)\times (1-q_2)}\times \frac{d_3}{{\gamma }_0}\times \left(1,+,\frac{{\beta }_1}{d_2},{\widehat{T}}^{+}\right).\end{array}$$ 5

In summary, the treatment model (4) has similar conclusions to the ones obtained from the control model (1). For any immune responses or strength, the infection will be eradicated by treatments when ${\widehat{\mathcal{R}}}_0<1$, whereas the virus and infected cells cannot be eliminated no matter how large the treatment efficacy is when ${\widehat{\mathcal{R}}}_0>1$.

Furthermore, the following analytical result highlights the treatment effects on the outcomes of the disease; if the treatment terms $(1-q_1)$ and $(1-q_2)$ satisfy the following conditions

$$\begin{array}{c}\hfill 1<{\mathcal{R}}_0<\frac{1}{\sqrt{(1-q_1)(1-q_2)}}\text{and}0\le (1-q_i)\le 1,i=1,2,\end{array}$$ 6

then ${\widehat{\mathcal{R}}}_0<1$ and infection will be eradicated due to ${\widehat{\mathcal{R}}}_0\le {\mathcal{R}}_0$. Note that Treatments 1 and 2 act cooperatively to ease the eradication of infection.

Resilience and reactivity

When ${\widehat{\mathcal{R}}}_0<1$, the virus-free equilibrium $X^0$ is linear stable and no positive equilibrium exists in model 4. Furthermore, recall that the virus-free equilibrium $X^0$ in this work represents the normal state of a patient before the occurrence of the infection of interest. Hence, resilience and reactivity of the virus-free equilibrium $X^0$ when the infection is eradicated (${\widehat{\mathcal{R}}}_0<1$) can be used to further inform on the dynamics of the system. The resilience represents the minimum return rate to $X^0$ in model (4); then, its inverse provides the average time to clear the infection. Reactivity, which quantifies the transient behaviour of a system in response to a perturbation from a stable equilibrium, informs on the growth of infection during a very short time period after the onset of the disease. Thus, we now discuss the minimum decay rate (resilience) and the maximal instantaneous growth rate (reactivity) of a perturbation in the linear system approximating the non-linear system 4 in the vicinity of virus-free equilibrium $X^0$ when ${\widehat{\mathcal{R}}}_0<1$(Neubert and Caswell 1997).

The resilience is defined by Neubert and Caswell (1997)

$$\begin{array}{c}\hfill \text{resilience}:=-max\left\{\text{Re},(\lambda ),:,det,(,\lambda ,I_{4\times 4},-,J,),=,0\right\}>0,\end{array}$$ 7

where J is the Jacobian matrix at the virus-free equilibrium $X^0$ in the treatment model (4),

$$\begin{array}{c}\hfill J=\left(\begin{array}{cccc}-d_1& 0& -\frac{\partial f(S^0,0)}{\partial V}\times (1-q_1)& 0\\ 0& -d_2& \frac{\partial f(S^0,0)}{\partial V}\times (1-q_1)& 0\\ 0& d_2{\gamma }_0\times (1-q_2)& -d_3& 0\\ 0& {\beta }_0& 0& -d_4\end{array}\right).\end{array}$$ 8

Consequently, the characteristic equation can be written as

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_2(\lambda ):=(\lambda +d_1)(\lambda +d_4)Q(\lambda )=0\end{array}$$ 9

where

$$\begin{array}{c}\hfill Q(\lambda ):={\lambda }^2+(d_2+d_3)\lambda +d_2{d}_3(1-{\widehat{\mathcal{R}}}_0^2).\end{array}$$ 10

The roots of the quadratic polynomial $Q(\lambda )$ in Eq. (10) are negative real roots and can be written as

$$\begin{array}{c}\hfill \frac{-\left(d_2,+,d_3\right)-\sqrt{{\left(d_2,-,d_3\right)}^2+4d_2{d}_3{\widehat{\mathcal{R}}}_0^2}}{2}<\frac{-\left(d_2,+,d_3\right)+\sqrt{{\left(d_2,-,d_3\right)}^2+4d_2{d}_3{\widehat{\mathcal{R}}}_0^2}}{2}<0.\end{array}$$

Hence, from Eqs. (9) and (7), we have

$$\begin{array}{c}\hfill \text{resilience}:=-max\left\{-,d_1,,,-,d_4,,,\frac{-\left(d_2,+,d_3\right)+\sqrt{{\left(d_2,-,d_3\right)}^2+4d_2{d}_3{\widehat{\mathcal{R}}}_0^2}}{2}\right\}>0.\\ \hfill \end{array}$$ 11

From a biological point of view, the larger the resilience is, the faster the patient recovers. The more efficient the treatments is, the larger $\frac{\left(d_2,+,d_3\right)-\sqrt{{\left(d_2,-,d_3\right)}^2+4d_2{d}_3{\widehat{\mathcal{R}}}_0^2}}{2}$ is. Thus, treatments with better efficacy might reduce the time to recover.

The reactivity is defined by

$$\begin{array}{c}\hfill \text{reactivity}:=max\left\{\lambda ,:,det,(,\lambda ,I_{4\times 4},-,H,(J),),=,0\right\},\end{array}$$ 12

with $H(J)=(J+J^T)/2$ where J is the Jacobian matrix and $J^T$ represents the transpose matrix of J. Notice that H(J) is a symmetric matrix and all its eigenvalues are real. Straightforward calculation using J in (8) leads to

13

Applying the permutation $\{1,2,3,4\}\to \{3,4,1,2\}$ on the columns of the matrix $\lambda I_{4\times 4}-H(J)$ and using the block determinant formula (Bernstein 2009, Proposition 2.8.4)

$$\begin{array}{c}\hfill det\left(\begin{array}{ccc}{H}_1& & H_2\\ H_3& & H_4\end{array}\right)=det(H_4)det(H_1-H_2{H}_4^{-1}{H}_3),\end{array}$$

we have that the eigenvalues of H(J) are the roots of the equation

$$\begin{array}{c}\hfill 4\prod _{i=1}^4\left(\lambda ,-,r_i\right)={\left(\frac{\partial f(S^0,0)}{\partial V},+,{\gamma }_0,d_2\right)}^2\left(\lambda ,+,d_1\right)\left(\lambda ,+,d_4\right)\end{array}$$ 14

where

$$\begin{array}{cc}\hfill r_{1,2}& =\frac{-(d_1+d_3)\pm \sqrt{{\left(d_1,-,d_3\right)}^2+{\left(\frac{\partial f(S^0,0)}{\partial V}\right)}^2}}{2},\hfill \\ \hfill r_{3,4}& =\frac{-(d_2+d_4)\pm \sqrt{{\left(d_2,-,d_4\right)}^2+{\beta }_0^2}}{2}.\hfill \end{array}$$

A system with a positive reactivity is sensitive to any changes in patients’ initial stages. Thus, a small change of the patient’s stage from the normal stage could cause drastic changes at the early stage of the infection before the system returns to the virus-free equilibrium $X^0$. From the biological perspective, if the system is reactive (positive reactivity), there will be a fast growth of the infection during a very short time period after the onset of the disease before infection eradication. Reactivity can be used to characterize disease severity.

Numerical investigations

In Sects. 2 and 3, we provided analytical results of the dynamics in both the control and treatment models. Now, we first carry out model calibration to clinical data from patients with different symptom severities. Due to the lack of a complete clinical dataset including all components of the model (4), we choose to calibrate the model to different types of data. We use clinical data for immune cells for patients with severe cases from Han et al. (2020) and viral loads for patients with mild and severe cases from Zheng et al. (2020) and fit them with the corresponding component in the model with treatment, separately. Since the data from Han et al. (2020), Zheng et al. (2020) involves certain types of treatments, we use the treatment model (4) to identify parameter values. We consider three types of incidence functions f(S, V) from Eq. (2) to fit the three datasets and then utilize the Akaike information criterion to select the best incidence function to represent each dataset.

We further analyze the effect of parameter values obtained by fitting the different dataset on model predictions. Finally, a local sensitivity analysis is carried out to characterize the influence of the small variation of treatment efficacy $q_1$ and $q_2$ on the dynamics of the treatment model (4) considered with parameter values obtained for the severe cases. Moreover, when the infection cannot be eradicated, we generate efficacy maps of the percentage of infected cells in $q_1{q}_2$-plane to investigate effect of the large variation of $q_1$ and $q_2$ on the disease severity.

Clinical data

For model calibration, two types of clinical data are considered: viral load (Zheng et al. 2020) and immune cells (Han et al. 2020) dynamics. The data are grabbed from the figures within the references (Han et al. 2020; Zheng et al. 2020) by using the MATLAB package Grabit.

In Zheng et al. (2020, Fig. 3) provided clinical data of viral loads of SARS-CoV-2 in patients from respiratory samples. Two datasets are considered to calibrate the model solution V; the first dataset is from patients with mild cases (dataset-V-1 in Table 2) and the second from patients with severe cases (dataset-V-2 in Table 2). For mild cases (dataset-V-1, red dots in the first row in Fig. 2), the viral load in respiratory samples peaks in the second week, then decreases and stabilizes to a low value. In severe cases (dataset-V-2, red dots in the second row in Fig. 2), the viral load oscillates around large amounts during the four weeks after the disease onset.

Table 2.

Datasets used in the study

Viral load data	Corresponding case	Figure	References
Dataset-V-1	Mild symptom case	1st row of Fig. 2	Zheng et al. (2020, Fig. 3)
Dataset-V-2	Severe symptom cases	2nd row of Fig. 2	Zheng et al. (2020, Fig. 3)
Immune cells data	Corresponding case	Figure	Reference
Dataset-T	Severe symptom case	3rd row of Fig. 2	Han et al. (2020, Figs. 3D, 4D)

The first column provides the name of the dataset corresponding to the cases shown in the second column, which are displayed as the red dots in the figures listed in the third column. The last column shows the source of the original datasets. The datasets $\{$dataset-V-1, dataset-V-2$\}$ and $\{$dataset-T$\}$ are for viral load and immune cells, respectively

Fig. 2.

Calibration of model (4). Each column corresponds to model (4) considered with one of the incidence functions $f_i$ defined in Eq. (2). A–C (resp. D–F) Calibration results to viral load data from Zheng et al. (2020), dataset-V-1 (resp. dataset-V-2). G–I Calibration results to immune cells data (dataset-T), which are means of the sum of monocytes, CD4${}^{+}$ and CD8${}^{+}$ T cells in Han et al. (2020, Figs. 3D, 4D). For each panel, the red dots are the clinical data, the best fit of model (4) obtained with the optimal parameter set $P_{\mathit{best}}$ is the green curve, and the gray curves correspond to optimized parameter sets with a normalized error (17) less than 1.2 ($20\%$ from the best fit). For information, the optimal parameter set values $P_{\mathit{best}}$ for each dataset are listed in Tables 3, 4 and 5 in “Appendix B” (color figure online)

For clinical data of immune cells, the sum of monocytes, CD4${}^{+}$ and CD8${}^{+}$ T cells counts in Han et al. (2020, Figs. 3D, 4D) for severe cases is used to calibrate the model solution T. The amount of immune cells increases over time (dataset-T in Table 2, red dots in the third row in Fig. 2).

Model calibration and selection

For each considered dataset (Table 2), the model (4) solutions obtained with $f_1$, $f_2$ and $f_3$ are calibrated to clinical data by minimizing the following objective function

$$\begin{array}{c}\hfill \text{RSS}(P)=\sum _{i=1}^N{\left(Z(t_i;P)-Z_i\right)}^2,Z\in \{V,T\},\end{array}$$ 15

where $Z(t_i,P)$ is the predicted appropriate value at time $t_i$ of the model (4) obtained with parameter set P and $Z_i$ is the observed appropriate value at time $t_i$, N is the number of observations in considered clinical data.

The optimization problem is solved using the differential evolution algorithm (Suganthan 2012) implemented in MATLAB, the genetic algorithm coded in the R package GA (Scrucca 2013), and the commands ParametricNDSolveValue and FindFit with the option NMinimize in Wolfram Mathematica. Due to the stochastic nature of optimization methods used, we run 150 optimizations for each dataset and incidence function $f_i$ with $i\in \{1,2,3\}$ and collect one optimal parameter set $P_{\mathit{best}}$ such that

$$\begin{array}{c}\hfill \text{RSS}(P_{\mathit{best}})=\underset{P}{min}(\text{RSS}(P))\end{array}$$ 16

for each dataset and incidence function. Furthermore, to compare the 150 parameter sets P we consider the normalized root mean square error (RMSE)

$$\begin{array}{c}\hfill \text{Normalized RMSE}=\frac{\sqrt{RSS(P)}}{\sqrt{RSS(P_{\mathit{Best}})}}.\end{array}$$ 17

For calibration results, we focus on the dynamics of the model (4) with the optimized parameter sets satisfying normalized RMSE$<1.2$ for each dataset and incidence function $f_i$.

Finally, for each dataset, the Akaike Information Criterion (AIC) of model (4) considered with the functions $f_i$ with $i\in \{1,2,3\}$ is computed (Portet 2020)

$$\begin{array}{c}\hfill {\text{AIC}}_i=Nln\left(\frac{\text{RSS}(P_{\mathit{best}})}{N}\right)+2K_i,\end{array}$$ 18

where $K_i$ is the number of parameters in model considered with $f_i$ plus one. For a small number of observations, the corrected AIC (AICc) is used

$$\begin{array}{c}\hfill {\text{AICc}}_i={\text{AIC}}_i+\frac{2K_i(K_i+1)}{N-K_i-1}.\end{array}$$ 19

For a dataset, the model with the minimum AIC or AICc value is the best model to represent the data. Furthermore, Akaike weights of model considered with $f_i$ are computed

$$\begin{array}{c}\hfill {\omega }_i=\frac{exp\left(\frac{-{\mathrm{\Psi }}_i}{2}\right)}{\sum _{j=1}^{3}exp\left(\frac{-{\mathrm{\Psi }}_j}{2}\right)},\end{array}$$

where ${\mathrm{\Psi }}_i={\text{AIC}}_i-\underset{i}{min}{\text{AIC}}_i$. The smaller the weight ${\omega }_i$ is, the less plausible the model with $f_i$ is; the model considered with $f_i$ is selected as the best model to represent a dataset if ${\omega }_i>0.9$.

Calibration to viral load clinical data

For the mild case in Zheng et al. (2020) (dataset-V-1), the optimized solutions of the model (4) considered with $f_2$ fail to capture the high peak during the second week after the disease onset observed in clinical data (Fig. 2B). However, the clinical data are well represented when considering $f_1$ or $f_3$. In severe cases (dataset-V-2), none of the optimized solutions of the model (4) obtained with neither $f_1$, $f_2$ nor $f_3$ captures the oscillatory dynamics of clinical data; however, all optimized solutions capture the average trend of data. Optimized solutions increase in the first week to reach the median values of data and then the viral load maintains at high values during the third and fourth weeks after the disease onset (Fig. 2D–F).

For both virus load datasets, model (4) with $f_1$ is found to be the best model to represent the data (Fig. 3). Using previous analytical results, we then compute ${\widehat{\mathcal{R}}}_0$ for model (4) considered with $f_1$ and values of the optimal parameter sets $P_{\mathit{best}}$ obtained to best present the mild and severe cases. For both cases, the model predicts the eradication of infection as ${\widehat{\mathcal{R}}}_0$ is equal to 0.045 for mild cases and 0.280 for severe cases. As for both mild and severe cases ${\widehat{\mathcal{R}}}_0<1$, resilience and reactivity are also computed. The model predicts a faster recovery in the mild cases than in severe cases; the resilience for mild cases is about four times larger than for severe cases (resilience is equal to 5.9 (resp. 1.4) when calibrating to dataset-V-1 (resp. dataset-V-2)). Furthermore, the system is found to be reactive only when calibrated to severe cases; reactivity is positive (resp. negative) and equal to 0.013810 (resp. $-0.00037341$) for severe (resp. mild) cases. This is illustrated in Fig. 2A, D; at the onset of the disease, for severe cases there is an increase of the virus load that does not occur for mild cases. Parameter values of $P_{\mathit{best}}$ obtained with dataset-V-2 make model (4) considered with $f_1$ sensitive to changes of patients’ initial stages.

Fig. 3.

Model selection results. Akaike weights of model (4) considered with functions $f_1$, $f_2$ and $f_3$ for three dataset displayed in Fig. 2. The orange, brown, and blue bars are the Akaike weights of model (4) with functions $f_1$, $f_2$ and $f_3$, respectively. The green line marks 0.9 to indicate the best model for each dataset when the corresponding Akaike weight passes this line. The AIC for viral load datasets is calculated using Eq. (18) and for immune cells dataset using Eq. (19) (color figure online)

Calibration to immune cells clinical data

Optimized solutions T of model (4) considered with any $f_i$ represent closely the dynamics of immune cells in a severe case from dataset-T (Fig. 2G–I). For this dataset and each $f_i$, we again consider the optimal parameter set $P_{\mathit{best}}$ corresponding to green curves in Fig. 2G–I, and compute ${\widehat{\mathcal{R}}}_0$, which is found to be 3.824 for $f_1$, 62.633 for $f_2$ and 3887.86 for $f_3$. With any of the three forms for the incidence function $f_i$, the model predicts the failure in infection eradication in the severe cases. Using the Akaike Information Criterion, model (4) with $f_1$ is again selected as the best model to represent the immune cells dynamics of dataset-T; note that $f_1$ is the simplest model in term of number of parameters. Thus,in the next sections, we only consider the model (4) with the function $f_1$.

Effects of calibration on model predictions

In this section, we focus on the parameter sets with normalized RMSE less than 1.2 (parameter sets yielding solutions in Fig. 2 with an error less than $20\%$ from the optimal set $P_{\mathit{best}}$) to investigate the impact of parameter values obtained when calibrating the model to the different datasets to model predictions including disease outcome, recovery time, and reactivity.

First, repartition of parameter values in 3-dimensional parameter spaces are shown in A–C panels of Fig. 4 for the three datasets. Level of value clustering gives an indication on how the 9 parameters of the model with $f_1$ are identifiable depending on the dataset considered. Parameters appear to be the most identifiable when considering dataset-V-1 (Fig. 4I(A–C)); in particular, values of parameter sets with lowest errors are tightly clustered in ${\alpha }_0\widehat{\beta }\widehat{{\gamma }_0}-$space (A in Fig. 4). With dataset-V-2, parameter values providing similar calibration errors are loosely dispersed in the parameter space as shown in Fig. 4II(A–C) indicating parameters are the least identifiable when considering dataset-V-2.

Fig. 4.

Repartition of parameter values. Parameters obtained when calibrating model (4) with $f_1$ to dataset-V-1 (I), dataset-V-2 (II) and dataset-T (III). A–C Dots represent 150 datasets obtained by minizing Eq. (15) in 3D-spaces of parameters in log-scales. Colored dots correspond to parameter sets with normalized RMSE lower than 1.2 (with 1 corresponding to $P_{\mathit{best}}$ and brightest green) yielding curves displayed in Fig. 2. D–E Values of ${\widehat{\mathcal{R}}}_0$ for the 150 datasets with ${\widehat{\mathcal{R}}}_0<1$ in blue when the model predicts the infection eradication and otherwise ${\widehat{\mathcal{R}}}_0>1$ in red. F Average time to recovery (1/Resilience) when applicable ${\widehat{\mathcal{R}}}_0<1$ for the 150 parameter sets (white dots when not applicable, i.e., ${\widehat{\mathcal{R}}}_0\le 1$). G Reactivity of model when considered with the 150 parameter sets for which ${\widehat{\mathcal{R}}}_0<1$: positive reactivity in red and non-negative reactivity in blue (white dots when not applicable, i.e., ${\widehat{\mathcal{R}}}_0\le 1$). D–G Bright colors indicate parameter sets with normalized RMSE lower than 1.2 (22 parameter sets for dataset-V1, 145 for dataset-V2 and 13 for dataset-T), light colors are parameter sets with normalized RMSE$>1.2$ (color figure online)

Second, values of ${\widehat{\mathcal{R}}}_0$, which inform on the disease outcome, are computed for the function $f_1$ (panels D-E in Fig. 4)

$$\begin{array}{c}\hfill {\widehat{\mathcal{R}}}_0=\sqrt{(1-q_1)\times (1-q_2)\times \frac{{\gamma }_0}{d_3}\frac{\partial f_1(S^0,0)}{\partial V}}=\sqrt{\frac{{\alpha }_0\widehat{\beta }{\widehat{\gamma }}_0}{d_1{d}_3}}.\end{array}$$ 20

For the mild case, even if $d_3$ values appear to be loosely clustered in comparison to ${\alpha }_0,\widehat{\beta },{\widehat{\gamma }}_0$, and $d_1$, model (4) with parameter sets (with lowest errors) induced by the dataset-V-1 provides consistent predictions of infection eradication (i.e., ${\widehat{\mathcal{R}}}_0<1$, blue dots in Fig. 4I(D, E)). Instead, for the severe case dataset-V-2 (Fig. 4II(D, E)), model (4) predicts both disease outcomes depending on parameter values. The parameter sets with relatively smaller values (resp. larger values) of ${\alpha }_0$ or $\widehat{\beta }$ or ${\widehat{\gamma }}_0$ (Fig. 4II(D)) or with relatively larger values (resp. smaller values) of $d_1$ (Fig. 4II(E)) induce a smaller value (resp. larger value) of ${\widehat{\mathcal{R}}}_0$ and hence lead to infection eradication (resp. failure in infection eradication). However, note that the lowest error parameter set leads to eradication condition. The parameter sets induced by the severe case dataset-T mainly generate large values of ${\widehat{\mathcal{R}}}_0$ resulting in failure in infection eradication (in Fig. 4III(D, E)).

Then, the recovery time is computed for parameter sets predicting an infection eradication (panels F in Fig. 4). The recovery time is in the range of [0.070, 0.194] for the mild case dataset-V-1, [0.024, 4.163] for the severe case dataset-V-2, and [0.460, 0.758] for the severe case dataset-T. Thus, patients in mild cases are found to recover faster than patients in severe cases.

Finally, the reactivity for the parameter sets satisfying the eradication condition is computed (panels G in Fig. 4). Comparing the results between the mild cases dataset-V-1 and severe cases dataset-V-2, the parameter sets of the severe cases generate a higher proportion of reactive equilibrum than the mild cases; with dataset-V-2, for almost all parameter sets, the virus-free equilibrium has a positive reactivity. Additionally, for the other severe case, dataset-T, only two parameter sets induce infection eradication and both generate reactive virus-free equilibrium. Thus, model (4) considered with parameter sets from the severe cases is sensitive to any changes in patients’ initial stages.

Calibrating model (4) to three dataset provides us with three different situations: eradication with a non reactive virus-free equilibrium (mild cases, dataset-V1), eradication with a reactive virus-free equilibrium (severe cases, dataset-V2) and failure of infection eradication (severe cases, dataset-T). The model predictions from dataset-V-1 are the most consistent with respect to variations in treatment related parameters $\widehat{\beta }$ and ${\widehat{\gamma }}_0$ and exhibit the simplest dynamics. Therefore, to further investigate how treatments affect the dynamics of the model we only focus on the severe situations related to dataset-V-2 and dataset-T providing richer dynamics.

Efficacy of treatments

To prevent the COVID-19 progression and its longer-term complications, interventions are timely and early stage interventions have been shown to be essential (Prescott and Rice 2020; Sterne et al. 2020). Thus, we study the influence of small variations of treatments on the model (4) solution by using a local sensitivity analysis.

The sensitivity of the model (4) solutions to parameters is given by the partial derivative of $\mathbf{X}={(S,C,V,T)}^T$ with respect to any parameter p of model (4), denoted by ${\mathbf{X}}_p=\frac{\partial \mathbf{X}}{\partial p}$. Define $\mathbf{f}(S,C,V,T):=\frac{d\mathbf{X}}{\mathit{dt}}$, then by the Chain Rule and Clairaut’s Theorem, we have

$$\begin{array}{c}\hfill \frac{d{\mathbf{X}}_p}{\mathit{dt}}=\frac{d\mathbf{f}}{d\mathbf{X}}{\mathbf{X}}_p+\frac{\partial \mathbf{f}}{\partial p},{\mathbf{X}}_p(0)=\frac{\partial {\mathbf{X}}_0(p)}{\partial p}.\end{array}$$ 21

The logarithmic sensitivity solution $p\frac{{\mathbf{X}}_p}{\mathbf{X}}$ describes the percentage changes in the solutions induced by positive perturbations of the parameters. The detailed method is given in Bortz and Nelson (2004). We use the command NDSolve in Wolfram Mathematica to obtain the numerical solutions of the model (21).

Sensitivities to treatment efficacy $q_1$ and $q_2$ are computed with parameter values $P_{\mathit{best}}$ obtained by fitting dataset-V-2 and dataset-T (see values in Table 3). We take $q_i=0.1$ to represent inefficient treatments. Then, using $\widehat{\beta }=\beta (1-q_1)$ and ${\widehat{\gamma }}_0={\gamma }_0(1-q_2)$ with values from Table 3 , we calculate the binding rate $\beta$ and viral production rate ${\gamma }_0$ and evaluate $q_1\frac{{\mathbf{X}}_{q_1}}{\mathbf{X}}$ and $q_2\frac{{\mathbf{X}}_{q_2}}{\mathbf{X}}$.

Table 3.

$P_{\mathit{best}}$ for the best fitting result with $f_1$ in Fig. 2

Parameter	Unit	Figure 2A	Figure 2D	Figure 2G
${\alpha }_0$	Cell/$\mathrm{\mu }$L/day	0.0078	2.8104	0.3093
$\widehat{\beta }$	${(\text{Copies}/\mu \text{L})}^{-1}/\text{day}$	3.0240$\times {10}^{-4}$	1.2550$\times {10}^{-3}$	$1.5300\times {10}^{-6}$
$d_1$	Day${}^{-1}$	3.7590$\times {10}^{-4}$	0.4706	$5.1316\times {10}^{-4}$
${\beta }_1$	Cell/$\mathrm{\mu }$L/day	5.6308	3.5313	$1.7100\times {10}^{-6}$
$d_2$	Day${}^{-1}$	2.0672	1.3649	2.2595
${\widehat{\gamma }}_0$	Dimensionless	1.9380	0.3798	0.6231
$d_3$	Day${}^{-1}$	5.8537	0.0364	$3.9300\times {10}^{-5}$
${\beta }_0$	Day${}^{-1}$	5.8000 $\times {10}^{-5}$	7.9797$\times {10}^{-4}$	7.6082
$d_4$	Day${}^{-1}$	0.5946	1.1150	0.0184

For the dataset-V-2, Treatments 1 and 2 have the similar effects on healthy, infected and immune cells while they have opposite effects in the long run on the virus load. A small increase in neither treatment efficacy has a negative effect on the infected cells during the early stage (i.e., up to 5 days) followed by a positive effect in [5, 15] days, and then that changes back to a negative effect thereafter. Note that for both treatments, sensitivities of infected and immune cells have similar dynamics; during the first two weeks, the sensitivity of the immune cell has lower magnitudes and is slightly delayed in time with respect to the infected cells one (Fig. 5A, C). This finding suggests that both treatments could trigger an aggravation of infection during the early stage, although they reduce the infection thereafter. However, Treatments 1 and 2 have different effects on the viral load. Treatment 1 has a negative effect on the viral load during the early stage and then switches to a positive effect thereafter, suggesting that blocking virus binding can only reduce the viral load during the early stage. On the other hand, Treatment 2 always has a negative effect on the viral load. Thus, in the dataset-V-2, inhibiting virus replication is more efficient in reducing virus load than blocking virus binding.

Fig. 5.

Logarithmic sensitivities to treatment efficacy for severe cases. The first (resp. second) column shows results for parameter values estimated with dataset-V-2 (resp. dataset-T). The horizontal axis corresponds to time in days, while the vertical axis describes $q_1\times X_{q_1}/X$ and $q_2\times X_{q_2}/X$ for the first and second rows, respectively. The positive value (resp. negative value) of $p_i\times X_{p_i}(t)/X(t)$ indicates a positive correlation (resp. negative correlation) between the variable and the parameter $p_i$ at time t, so X increases (resp. decreases) at time t as the parameter $p_i$ increases

When considering parameter values related to dataset-T, Treatments 1 and 2 have similar effects on all variables of the model (4) (Fig. 5B, D). First, for a long period of time (about two months) in comparison to the dynamics observed with dataset-V-2, a small positive perturbation in $q_1$ or $q_2$ yields a decrease in the amounts of infected cells, immune cells, and viral load. Then, Treatments 1 and 2 have positive effect on the amount of infected cells, which triggers a delayed increase of immune response, and the negative effect on virus load fades. After 100 days, a small increase in $q_1$ or $q_2$ appears to cause a huge increase in the amount of infected cells. This might suggest the building-up of a resistance of infection to both treatments, which does not appear in the other severe case dataset-V-2.

We have different conclusions when we consider model (4) with parameter values estimated from the two severe cases. For the dataset-V-2, Treatment 2 has higher reductions in the viral load and the amount of infected cells than Treatment 1, and there is no resistance appears in both treatments. However, for the dataset-T, although both treatments initially reduce the amounts of infected cells and viruses, they undergo resistance of the infection in long-term dynamics. Even though both represent severe cases, the dataset-V-2 and dataset-T describe different observation (virus load and immune cells) and are collected from different patient samples and different studies (Zheng et al. 2020, Fig. 3; Han et al. 2020, Figs. 3D, 4D), that may cause differences in model predictions.

Next, we further investigate how Treatments 1 and 2 reduce the percentage of infected cells when there is no infection eradication; in other words, when the positive equilibrium ${\widehat{X}}^{+}$ exists and is locally asymptotically stable, i.e. when ${\widehat{\mathcal{R}}}_0>1$ (Theorem 7). We consider the parameter sets $P_v$ and $P_t$ ($=P_{\mathit{best}}$) with the smallest normalized RMSE giving ${\widehat{\mathcal{R}}}_0>1$ for the dataset-V-2 and dataset-T, respectively, and calculate ${\widehat{X}}^{+}$ by using the numerical solutions of the model (4).

Figure 6 shows the percentage of infected cells ${\widehat{C}}^{+}/({\widehat{S}}^{+}+{\widehat{C}}^{+})\times 100\%$ for the positive equilibrium ${\widehat{X}}^{+}$ for the model (4) with parameter sets $P_v$ (panel A) and $P_t$ (panel B), as both treatment efficacies are varying $0\le q_i\le 1$ for $i=1,2$. For both severe cases, when the efficacy of one treatment is fixed, the percentage of infected cells decreases as the efficacy of the other treatment increases (Fig. 6). Moreover, the combination of Treatments 1 and 2 allows a better reduction of the percentage of infected cells than the monotherapy of Treatment 1 or 2. Additionally, the reduction rate is slightly faster in the case of dataset-V-2 than the case of dataset-T, suggesting that a patient’s condition related to dataset-V-2 may have a better treatment outcome in the combination of treatments than the patient’s condition related to dataset-T. These results are not only consistent with Fig. 5 for the small variations of $q_i$, but also agree with the finding from Fig. 4II(D, E) that patients related to dataset-V-2 may have a better chance to reach the infection eradication than the patients related to dataset-T. Moreover, the finding here also provides a guide for a dosage protocol of treatments to reduce the percentage of infected cells.

Fig. 6.

Percentage of infected cells in ${\widehat{X}}^{+}$. The efficacy map of the percentage of infected cells ${\widehat{C}}^{+}/({\widehat{S}}^{+}+{\widehat{C}}^{+})\times 100\%$ in the positive equilibrium ${\widehat{X}}^{+}$ for the model (4) with $f_1$. Parameter values in A and B are taken from the parameter sets $P_v$ (parameter set for dataset-V-2 with the smallest normalized RMSE giving ${\widehat{\mathcal{R}}}_0>1$) and $P_t=P_{\mathit{best}}$ for dataset-T, respectively. The horizontal and vertical axes are the treatment efficacy $q_1$ and $q_2$, respectively, with $q_i=1$ for the best efficacy and $q_i=0$ for no treatment

Conclusion and discussion

In this work, we have investigated the dynamics of SARS-CoV-2 infection using a simple ODEs model with a generic nonlinear incidence rate. We incorporated two types of treatments, blocking the virus binding and inhibiting the virus replication, into the model. We have analytically studied the long-term dynamics of the proposed models with and without treatments. For the control model (1), we characterized the existence and stability of the virus-free equilibrium $X^0$ for the infection eradication and the positive equilibrium $X^{+}$ for the infection persistence using the basic reproduction number. Next, we extended the analytical results to the treatment model (4) to obtain criteria on treatments to guarantee the infection eradication (i.e., ${\widehat{\mathcal{R}}}_0<1$) or predict the failure in infection eradication (i.e., ${\widehat{\mathcal{R}}}_0>1$). Then, for the treatment model (4), we introduced the resilience and reactivity under the infection eradication scenario (i.e., ${\widehat{\mathcal{R}}}_0<1$) to quantify the average time of recovery and the maximal instantaneous growth rate of a perturbation in the short-term dynamics. Thus, for the practical application, when a parameter set for patients is obtained by data fitting, mathematical analysis results can be used to predict whether the patients will recover, with which recovery rate and how sensitive the patients react to perturbations from an infection free initial stage.

We have calibrated the treatment model (4) with different types of clinical datasets for viral load in the mild and severe cases from Zheng et al. (2020) and immune cells in severe cases from Han et al. (2020). Note that the three datasets considered not only differ in terms of the observed variables (virus load or immune cells) but also in terms of the time scales they represent. Dataset-V-1 appears to describe a whole dynamics up to a stabilization, whereas dataset-V-2 exhibits oscillations with damping and dataset-T has an increasing trend. We have considered three types of incidence rate function: bilinear incidence rate $f_1$, Holling type II incidence rate $f_2$ and Beddington–DeAngelis incidence rate $f_3$. The Akaike information criterion indicates that the treatment model (4) with the bilinear incidence rate $f_1$, which is the simplest model, best represents all considered clinical datasets. Thus, we used $f_1$ to study the model predictions. Furthermore, due to the rich dynamics in both severe cases, we investigated how the treatment efficacy affects the model predictions in the severe cases when there is no infection eradication by performing a local sensitivity analysis and computing the efficacy map of treatments.

Using the three datasets, we analyzed the impact of calibration on predictions by considering the parameter sets of the optimal solution and optimized solutions with lowest errors (namely, with normalized RMSE less than 1.2).

• Calibrating to the mild cases from dataset-V-1: The optimal solution predicts eradication with a non reactive virus-free equilibrium. When considering dataset-V-1, parameters are the most identifiable; parameter values are tightly clustered in ${\alpha }_0\widehat{\beta }{\widehat{\gamma }}_0-$ and $d_1{d}_3-$spaces resulting in the consistent infection eradication outcome for all considered optimized solutions. In particular, predictions from dataset-V-1 are the most consistent with respect to variations in treatment related parameters $\widehat{\beta }$ and ${\widehat{\gamma }}_0$. Optimized solutions predict a fast recovery and have a small proportion of reactive equilibrium. Hence, the most disease outcome generated with the mild cases dataset-V-1 is the infection eradication with no severe symptoms.

• Calibrating to the severe cases from dataset-V-2: The optimal solution predicts eradication with a reactive virus-free equilibrium. When considering dataset-V-2, parameters are the least identifiable; parameter values are loosely dispersed in the parameter space inducing non-homogeneous predictions of eradication and no eradication and a wide range of recovery time. However, the dependency on parameters (spatial segregation of red and blue dots in Fig. 4II(D, E)) is noticeable. Optimized solutions have reactive virus-free equilibrium. For treatments efficacy, when no infection eradication, inhibiting virus replication is found to be more efficient in reducing virus load than blocking virus binding. This finding suggests that the disease outcome generated with the severe cases dataset-V-2 is the infection eradication but severe symptoms might occur depending on initial virus load or immune condition of patients and treatment efficacy might also play a role.

• Calibrating to the severe cases from dataset-T: The optimal solution predicts a failure of infection eradication. With dataset-T, only a few parameter sets have lowest errors and their values are loosely dispersed in the parameter space; however, the failure in infection eradication is consistently predicted in optimized solutions. Values of treatment related parameters $\widehat{\beta }$ and ${\widehat{\gamma }}_0$ cover a large range, suggesting that regardless of the treatment efficacy the infection fails to be eradicated. Moreover, local sensitivity analysis suggests the build-up of a resistance to treatments in the long-run. In a few infection eradication situations, the virus-free equilibrium is reactive. Thus, the most disease outcome generated with the severe cases dataset-T is the failure in infection eradication, regardless of the treatment efficacy.

Additionally, the recovery time is shorter for the mild cases (dataset-V-1) than for the severe cases (dataset-V-2 and dataset-T); and a higher proportion of reactive virus-free equilibrium appears in the severe cases. This observation suggests that the resilience and reactivity could be used to determine the severity of infection.

Moreover, for both severe cases when the infection fails to be eradicated, both treatments have similar effects on the normal, infected, and immune cells, but inhibiting virus replication induces higher reductions in the viral load than blocking virus binding. Moreover, neither treatments lead to resistance of the infection for the dataset-V-2, whereas the resistance appears in long-term dynamics for the dataset-T. In both cases, the combination of treatments induces a better reduction rate than the monotherapy. Moreover, the reduction rates of monotherapy and combination of treatments are slightly faster in dataset-V-2 than in dataset-T. Therefore, even though both represented severe cases, the dataset-V-2 and dataset-T still result in different treatment predictions. A patient’s condition related to dataset-V-2 could have a better treatment outcome without resistance comparing to the patient’s condition related to dataset-T.

Overall, our work provided methodologies to analyze the long-term and short-term dynamics of SARS-CoV-2 infection with respect to the reproduction number, resilience and reactivity. Combining analytical results, model calibration and numerical investigation, we assessed parameter identifiability with respect to data (practical identifiability), categorized the model predictions and determined the infection severity by using the recovery time and reactivity. Even though the considered model is simple, it can accommodate diverse dynamics and be used as a tool to further analyze data.

Appendix A: Proofs of theorems

In this “Appendix”, we provide a proof for each theorem of the manuscript.

A.1 Proof of Theorem 1

Proof

First, we show that the solution of model (1) is unique and nonnegative with the initial value $(S(0),C(0),V(0),T(0))\in {\mathbb{R}}_{+}^4$.

Notice that model (1) has a unique solution through any initial condition $(S(0),C(0),V(0),T(0))\in {\mathbb{R}}_{+}^4$ due to the smoothness of the function defined by the right-hand side of (1) in any open region of ${\mathbb{R}}^4$ (Kuznetsov 2013, Theorem 1.4). For $Z\in \{S,C,V,T\}$, we have

$$\begin{array}{c}\hfill {\left(\frac{\mathit{dZ}}{\mathit{dt}}\right|}_{Z(0)=0}\ge 0.\end{array}$$

Thus, the model (1) admits a nonnegative solution when $(S(0),C(0),V(0),T(0))\in {\mathbb{R}}_{+}^4$ with the maximal interval of existence $[0,\zeta )$ for some $\zeta >0$, see (Smith 1995, Theorem 5.2.1).

Now, we show the boundedness of solution of the model (1). From Eqs. (1a)–(1c), we have

$$\begin{array}{c}\hfill \frac{d}{\mathit{dt}}(S+C+V)={\alpha }_0-d(S+C+V)\end{array}$$

where $d=min\{d_1,(1-{\gamma }_0)d_2,d_3\}$. The equation $\frac{\mathit{du}}{\mathit{dt}}={\alpha }_0-du$ admits a globally asymptotically stable equilibrium $\frac{{\alpha }_0}{d}$. Consequently, by the comparison principle (Smith and Waltman 1995, Theorem B.1), we have

$$\begin{array}{c}\hfill \underset{t\to \infty }{limsup}\left(S(t)+C(t)+V(t)\right)\le \frac{{\alpha }_0}{d}.\end{array}$$ A.1

Since $C(t)\le \frac{{\alpha }_0}{d}$, Eq. (1d) gives

$$\begin{array}{c}\hfill \frac{\mathit{dT}}{\mathit{dt}}\le \frac{{\alpha }_0{\beta }_0}{d}-d_{4}T.\end{array}$$

Hence,

$$\begin{array}{c}\hfill \underset{t\to \infty }{limsup}T(t)\le \frac{{\alpha }_0{\beta }_0}{d_{4}d}.\end{array}$$ A.2

Thus, S(t), C(t), V(t) and T(t) are ultimately bounded, and hence, $\zeta =\infty$. From Eqs. (A.1) and (A.2), we have the global attractivity of $\mathrm{\Gamma }$. This proves the theorem. $\square$

A.2 Proof of Theorem 2

Proof

First, notice that any positive equilibrium of model (1) satisfies $T={\beta }_{0}C/d_4$, $C=\frac{d_3}{{\gamma }_0{d}_2}V$, $f(S,V)={\alpha }_0-d_{1}S$ and

$$\begin{array}{c}\hfill G(S,V):=f(S,V)-g_2{V}^2-g_{1}V=0\end{array}$$ A.3

where $g_1=\frac{d_3}{{\gamma }_0}$ and $g_2=\frac{d_3^2{\beta }_0{\beta }_1}{d_4{d}_2^2{\gamma }_0^2}$. Hence, to find all positive equilibria of model (1), it suffices to solve $G(S,V)=0$ and

$$\begin{array}{c}\hfill {\alpha }_0-d_{1}S-g_2{V}^2-g_{1}V=0.\end{array}$$ A.4

Assume the contrary, there exists a positive equilibrium $X^{+}=(S^{+},C^{+},T^{+})$ when ${\mathcal{R}}_0<1$, then $X^{+}$ satisfies Eq. (A.4). Thus, ${\alpha }_0-d_1{S}^{+}>0$, and hence, $S^{+}<S^0$, that is, $S^{+}\in (0,S^0)$.

Claim 1. $G(S,V)<0$ for any $S\in (0,S^0)$ and $V>0$.

From Eq. (A.3), we get

$$\begin{array}{c}\hfill \frac{\partial G}{\partial V}=\frac{\partial f}{\partial V}-2g_{2}V-g_1.\end{array}$$

Thus, with (A${}_3$), we have

$$\begin{array}{c}\hfill \frac{{\partial }^{2}G}{\partial V^2}=\frac{{\partial }^{2}f}{\partial V^2}-2g_2<\frac{{\partial }^{2}f}{\partial V^2}\le 0.\end{array}$$

When $S=S^0$ and ${\mathcal{R}}_0<1$, we have

$$\begin{array}{c}\hfill \frac{\partial G(S^0,0)}{\partial V}=g_1({\mathcal{R}}_0^2-1)<0.\end{array}$$

From (A${}_3$), we know that $\frac{\partial G}{\partial V}$ decreases for all $V>0$. Thus,

$$\begin{array}{c}\hfill \frac{\partial G(S^0,V)}{\partial V}<\frac{\partial g(S^0,0)}{\partial V}<0,\forall V>0.\end{array}$$

Consequently, $G(S^0,V)$ decreases for all $V>0$. Notice that $f(S^0,0)=0$, hence, $G(S^0,V)<0$ all $V>0$. It is clear $\frac{\partial G}{\partial S}=\frac{\partial f}{\partial S}>0$. Then for any $S\in (0,S^0)$ and $V>0$, $G(S,V)<G(S^0,V)<0$. This proves the claim.

From Claim 1, Eq. (A.3) does not have a solution, and hence, no positive equilibrium exists. This proves the theorem. $\square$

A.3 Proof of Theorem 3

Proof

Recall that any positive equilibrium satisfies Eqs. (A.3) and (A.4). From (A${}_2$), we know that $\frac{\partial f}{\partial S}>0$, hence, the implicit function theorem guarantees the existence of a one-to-one function $h_1$ such that

$$\begin{array}{c}\hfill S=h_1(V)\end{array}$$

for all S and V satisfy Eq. (A.3). Taking derivation with respect to V in Eq. (A.3) leads to

$$\begin{array}{c}\hfill \frac{d{h}_1}{\mathit{dV}}=\frac{\mathit{dS}}{\mathit{dV}}=\frac{2g_{2}V+g_1-\frac{\partial f}{\partial V}}{\frac{\partial f}{\partial V}}.\end{array}$$

Claim 2. $\frac{d{h}_1}{\mathit{dV}}\ge 0$ for all $V\ge 0$.

For any S and V satisfy Eq. (A.3), we have

$$\begin{array}{c}\hfill \widehat{f}(S,V):=f(S,V)-g_2{V}^2=g_{1}V.\end{array}$$

Thus, for all $V\ge 0$,

$$\begin{array}{c}\hfill \frac{d{h}_1}{\mathit{dV}}\ge 0\iff \frac{\partial \widehat{f}}{\partial V}\le g_1.\end{array}$$

Assume the contrary, $\frac{\partial \widehat{f}}{\partial V}>g_1$. Then by the mean value theorem, we have

$$\begin{array}{c}\hfill \frac{\partial \widehat{f}(S,V_0)}{\partial V}=\frac{\widehat{f}(S,V_0)-\widehat{f}(S,0)}{V_0}=\frac{\widehat{f}(S,V_0)}{V_0}=g_1,\text{for some}{V}_0\in (0,V).\end{array}$$

Since $\frac{\partial \widehat{f}}{\partial V}>g_1$, applying the mean value theorem again leads to

$$\begin{array}{c}\hfill \frac{{\partial }^2\widehat{f}(S,V_1)}{\partial V^2}=\frac{\frac{\partial \widehat{f}(S,V)}{\partial V}-\frac{\partial \widehat{f}(S,V_0)}{\partial V}}{V-V_0}=\frac{\frac{\partial \widehat{f}(S,V)}{\partial V}-g_1}{V-V_0}>0.\end{array}$$

Consequently, $\frac{{\partial }^{2}f(S,V_1)}{\partial V^2}>0$ which contradicts (A${}_3$). Thus, $\frac{\partial \widehat{f}}{\partial V}\le g_1$, and hence, $\frac{d{h}_1}{\mathit{dV}}\ge 0$. This proves the claim.

From Eq. (A.4), we have

$$\begin{array}{c}\hfill h_2(V):=\frac{{\alpha }_0}{d_1}-\frac{g_2}{d_1}{V}^2-\frac{g_1}{d_1}V=S.\end{array}$$

It is clear that $h_2$ decreases when $V>0$ and $h_2(0)=S^0$.

Claim 3. Let $h_1(0)=\tilde{S}$. When ${\mathcal{R}}_0>1$, then $\tilde{S}<S^0$.

When ${\mathcal{R}}_0>1$, then ${\mathcal{R}}_0^2>1$. Hence, $\frac{\partial f(S^0,0)}{\partial V}>\frac{d_3}{{\gamma }_0}=g_1$. From Eq. (A.3), we have $\frac{\partial f(\tilde{S},0)}{\partial V}=g_1$. Thus,

$$\begin{array}{c}\hfill \frac{\partial f(\tilde{S},0)}{\partial V}<\frac{\partial f(S^0,0)}{\partial V}.\end{array}$$

From (A${}_1$) and (A${}_2$), we have $\frac{\partial f(S,0)}{\partial V}\ge 0$ for all $S>0$. Thus, $\tilde{S}<S^0$. This proves the claim.

From the above arguments, the two functions $h_1$ and $h_2$ intersect at one point $(S^{+},V^{+})$, see Fig. 7. Consequently, $C^{+}=\frac{d_3}{{\gamma }_0{d}_2}{V}^{+}$ and $T^{+}={\beta }_0{C}^{+}/d_4$. Hence, model (1) has a unique positive equilibrium $X^{+}=(S^{+},C^{+},V^{+},T^{+})$. This proves the theorem. $\square$

Fig. 7.

Intersection of the functions $h_1(V)$ and $h_2(V)$. The green dashed curve is $h_1(V)$ and the red sold curve is $h_2(V)$ (color figure online)

A.4 Proof of Theorem 4

Proof

At $X^0$, the corresponding characteristic equation is

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_0(\lambda ):=(\lambda +d_1)(\lambda +d_4)({\lambda }^2+(d_2+d_3)\lambda +d_2{d}_3(1-{\mathcal{R}}_0^2))=0\end{array}$$ A.5

Since the roots $\lambda =-d_1$ and $\lambda =-d_4$ are negative, the stability is determined by the roots of the quadratic polynomial in Eq. (A.5). Notice that $d_2+d_3>0$ and $d_2{d}_3(1-{\mathcal{R}}_0^2)>0$ when ${\mathcal{R}}_0<1$. Thus, by the Routh–Hurwitz stability criterion (Levine 2018), all roots of the quadratic polynomial have negative real part, and hence, the virus-free equilibrium $X^0$ is locally asymptotically stable when ${\mathcal{R}}_0<1$.

When ${\mathcal{R}}_0>1$, ${\mathrm{\Delta }}_0(0)=d_1{d}_2{d}_3{d}_4(1-{\mathcal{R}}_0^2)<0$. Since ${lim}_{\lambda \to \infty }{\mathrm{\Delta }}_0(\lambda )\to \infty$, there exists $\overline{\lambda }>0$ such that ${\mathrm{\Delta }}_0(\overline{\lambda })=0$. Thus, $X^0$ is unstable. This proves the theorem. $\square$

A.5 Proof of Theorem 5

Proof

Let

$$\begin{array}{cc}\hfill Z^{\infty }& =\underset{t\to \infty }{lim\; sup}Z(t),Z_{\infty }=\underset{t\to \infty }{lim\; inf}Z(t),Z\in \{S,C,V,T\}\hfill \end{array}$$

From Theorem 1, we know that $0\le Z_{\infty }\le Z^{\infty }<\infty$. We use the method of fluctuations (see e.g. Hirsch et al. 1985; Thieme 2003) to prove $S_{\infty }=S^{\infty }=S^0$, $C_{\infty }=C^{\infty }=0$, $V_{\infty }=V^{\infty }=0$ and $T_{\infty }=T^{\infty }=0$, and hence,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}(S(t),C(t),V(t),T(t))=(S^0,0,0,0).\end{array}$$

By Hirsch et al. (1985, Lemma 4.2), there exist eight sequences ${\kappa }_n^Z\to \infty$ and ${\xi }_n^Z\to \infty$ as $n\to \infty$, such that

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}Z({\kappa }_n^Z)& =Z^{\infty },\underset{n\to \infty }{lim}Z({\xi }_n^Z)=Z^{\infty },{\left(\frac{\mathit{dZ}}{\mathit{dt}}\right|}_{t={\kappa }_n^Z}={\left(\frac{\mathit{dZ}}{\mathit{dt}}\right|}_{t={\xi }_n^Z}=0,\forall n\ge 1;\hfill \end{array}$$

Let $n\ge 1$ and $n\to \infty$. From Eq. (1a), we have

$$\begin{array}{c}\hfill 0={\left(\frac{\mathit{dS}}{\mathit{dt}}\right|}_{t={\kappa }_n^S}={\alpha }_0-f\left(S^{\infty },V\left({\kappa }_n^S\right)\right)-d_1{S}^{\infty }\le {\alpha }_0-d_1{S}^{\infty }.\end{array}$$

Thus,

$$\begin{array}{c}\hfill S^{\infty }\le S^0=\frac{{\alpha }_0}{d_1}.\end{array}$$ A.6

From Eq. (1b) and the inequality (A.6), we have

$$\begin{array}{cc}\hfill 0={\left(\frac{\mathit{dC}}{\mathit{dt}}\right|}_{t={\kappa }_n^C}=& f\left(S\left({\kappa }_n^C\right),V\left({\kappa }_n^C\right)\right)-{\beta }_1{C}^{\infty }T\left({\kappa }_n^C\right)-d_2{C}^{\infty }\hfill \\ \hfill \le & f\left(S^0,V^{\infty }\right)-d_2{C}^{\infty }\hfill \end{array}$$ A.7

It follow from (A${}_1$) that the Taylor series expansion of $f\left(S^0,V^{\infty }\right)$ is

$$\begin{array}{c}\hfill f\left(S^0,V^{\infty }\right)=\frac{\partial f(S^0,0)}{\partial V}{V}^{\infty }+\frac{{\partial }^{2}f(S^0,\eta )}{\partial V^2}{V^{\infty }}^2,\text{for some}\eta \in (0,V^{\infty })\end{array}$$

Thus, from (A${}_3$), we have

$$\begin{array}{c}\hfill f\left(S^0,V^{\infty }\right)\le \frac{\partial f(S^0,0)}{\partial V}{V}^{\infty }\end{array}$$

Consequently, from the inequality (A.7), we have

$$\begin{array}{c}\hfill 0\le \frac{\partial f(S^0,0)}{\partial V}{V}^{\infty }-d_2{C}^{\infty }.\end{array}$$ A.8

From Eq. (1c), we have

$$\begin{array}{c}\hfill 0={\left(\frac{\mathit{dV}}{\mathit{dt}}\right|}_{t={\kappa }_n^V}={\gamma }_0{d}_{2}C\left({\kappa }_n^V\right)-d_3{V}^{\infty }\le {\gamma }_0{d}_2{C}^{\infty }-d_3{V}^{\infty }.\end{array}$$

Hence, $V^{\infty }\le \frac{{\gamma }_0{d}_2}{d_3}{C}^{\infty }$. Then, it follows from the inequality (A.8) that

$$\begin{array}{c}\hfill 0\le d_2{C}^{\infty }({\mathcal{R}}_0^2-1).\end{array}$$

When ${\mathcal{R}}_0<1$, ${\mathcal{R}}_0^2<1$. Thus, $C^{\infty }=0$ because $C(t)\ge 0$, and hence, $C_{\infty }=C^{\infty }=0$. Similarly, $V_{\infty }=V^{\infty }=0$.

From Eq. (1d), we have

$$\begin{array}{c}\hfill 0={\left(\frac{\mathit{dT}}{\mathit{dt}}\right|}_{t={\kappa }_n^T}={\beta }_{0}C\left({\kappa }_n^T\right)-d_4{T}^{\infty }\le {\beta }_0{C}^{\infty }-d_4{T}^{\infty }.\end{array}$$

Since $C^{\infty }=0$, we have $0\le -d_4{T}^{\infty }$. Thus, $T^{\infty }\le 0$, and hence, $T^{\infty }=0$. Therefore, $T_{\infty }=T^{\infty }=0$.

To prove that $S_{\infty }=S^{\infty }=S^0$. Notice that Eq. (1a) gives

$$\begin{array}{c}\hfill 0={\left(\frac{\mathit{dS}}{\mathit{dt}}\right|}_{t={\xi }_n^S}={\alpha }_0-f\left(S_{\infty },V\left({\xi }_n^S\right)\right)-d_1{S}_{\infty }\ge {\alpha }_0-f\left(S_{\infty },V^{\infty }\right)-d_1{S}^{\infty }.\end{array}$$

Since $V^{\infty }=0$, it follows from (A${}_1$) that

$$\begin{array}{c}\hfill S_{\infty }\ge S^0=\frac{{\alpha }_0}{d_1}.\end{array}$$ A.9

From the inequalities (A.6) and (A.9), we have, $S^{\infty }\le S^0\le S_{\infty }$. Thus, $S^{\infty }=S_{\infty }=S^0$ because $0\le S_{\infty }\le S^{\infty }$.

Hence, the virus-free equilibrium $X^0$ is globally attractive, that is,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}(S(t),C(t),V(t),T(t))=(S^0,0,0,0).\end{array}$$

Together with the local stability of $X^0$ established in Theorem 4, confirms the global asymptotic stability of $X^0$. This proves the theorem. $\square$

A.6 Proof of Theorem 6

Proof

By Theorem 3, there exists a unique positive equilibrium $X^{+}$ when ${\mathcal{R}}_0>1$. Consequently, the corresponding characteristic equation is

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_1(\lambda )={\lambda }^4+b_3{\lambda }^3+b_2{\lambda }^2+b_1\lambda +b_0\end{array}$$

where

$$\begin{array}{cc}\hfill b_0=& d_1{d}_2{d}_4{\gamma }_0\left(\frac{d_3}{{\gamma }_0}+2\frac{{\beta }_1{d}_3}{{\gamma }_0{d}_2}{T}^{+}-\frac{\partial f(S^{+},V^{+})}{\partial V}\right)\hfill \\ \hfill & +d_3{d}_4\frac{\partial f(S^{+},V^{+})}{\partial S}\left(d_2+2{\beta }_1{T}^{+}\right)\hfill \\ \hfill b_1=& {\gamma }_0{d}_2\left(d_1+d_4\right)\left(\frac{d_3}{{\gamma }_0}+\frac{{\beta }_1{d}_3}{{\gamma }_0{d}_2}{T}^{+}-\frac{\partial f(S^{+},V^{+})}{\partial V}\right)+d_4{\beta }_1{T}^{+}\left(2d_1+d_3\right)\hfill \\ \hfill & +\frac{\partial f(S^{+},V^{+})}{\partial S}\left(d_2{d}_3+d_4\left(d_2+d_3\right)+{\beta }_1{T}^{+}\left(d_3+2d_4\right)\right)+d_1{d}_4\left(d_2+d_3\right)\hfill \\ \hfill b_2=& d_2{\gamma }_0\left(\frac{d_3}{{\gamma }_0}+\frac{{\beta }_1{d}_3}{{\gamma }_0{d}_2}{T}^{+}-\frac{\partial f(S^{+},V^{+})}{\partial V}\right)+d_4\left(d_2+d_3+2{\beta }_1{T}^{+}\right)\hfill \\ \hfill & +\left(\frac{\partial f(S^{+},V^{+})}{\partial S}+d_1\right)\left(d_1+d_2+d_4+{\beta }_1{T}^{+}\right)\hfill \\ \hfill b_3=& d_1+d_2+d_3+d_4+{\beta }_1{T}^{+}+\frac{\partial f(S^{+},V^{+})}{\partial S}.\hfill \end{array}$$

It follows from the condition (3) that $b_i>0$, $i=0,1,2,3$. Furthermore $b_1^2+b_3^2{b}_0>0$ and

$$\begin{array}{cc}\hfill b_1{b}_2{b}_3-b_1^2-b_3^2{b}_0=& 2d_4[{\beta }_1^2{(T^{+})}^2(2(d_4^2-d_3^2)+d_3{d}_4+d_2(d_3+4d_4))\hfill \\ \hfill & +d_2{\beta }_1{T}^{+}((d_4^2-d_3^2)+3d_3{d}_4+d_4^2)]\frac{\partial f(S^{+},V^{+})}{\partial S}\hfill \\ \hfill & +\sum _{i=1}^6{A}_i>0\hfill \end{array}$$ A.10

where $A_i$, $i=1,\dots ,6$, are positive when condition (3) holds. The expression of $A_i$, $i=1,\dots ,6$, is given in the “Appendix A.7”. Consequently, by the Routh–Hurwitz stability criterion (Levine 2018), the positive equilibrium $X^{+}$ is locally asymptotically stable when the condition (3) holds. This proves the theorem. $\square$

A.7 Terms of $A_i$ in Eq. (A.10)

The expression of $A_i$, $i=1,\dots ,6$, in Eq. (A.10).

$$\begin{array}{cc}\hfill A_1=& d_1(d_2+d_3)(d_1+d_2+d_3)d_4(d_1+d_4)(d_2+d_3+d_4)\hfill \\ \hfill & +((d_2^2+d_2{d}_3+d_3^2)d_4^2(d_2+d_3+d_4)+d_1^2[3d_3{d}_4(d_3+d_4)+d_2^2(d_3+3d_4)\hfill \\ \hfill & +d_2(d_3^2+6d_3{d}_4+3d_4^2)]+d_1[d_2^3(d_3+3d_4)+d_3{d}_4(2d_3^2+4d_3{d}_4+d_4^2)\hfill \\ \hfill & +d_2^2(2d_3^2+7d_3{d}_4+4d_4^2)+d_2(d_3^3+6d_3^2{d}_4+7d_3{d}_4^2+d_4^3)])\frac{\partial f(S^{+},V^{+})}{\partial S}\hfill \\ \hfill & +(2d_2^3(d_3+d_4)+2d_2^2{(d_3+d_4)}^2+d_3^2{d}_4(d_3+2d_4)+d_2{d}_3(d_3^2+3d_3{d}_4+2d_4^2)\hfill \\ \hfill & +d_1[3d_3{d}_4(d_3+d_4)+d_2^2(2d_3+3d_4)+d_2(2d_3^2+6d_3{d}_4+3d_4^2)]){\frac{\partial f(S^{+},V^{+})}{\partial S}}^2\hfill \\ \hfill & +(d_2+d3)(d_2+d_4)(d_3+d_4){\frac{\partial f(S^{+},V^{+})}{\partial S}}^3\hfill \\ \hfill A_2=& {\gamma }_0(d_2(d_2+d_3)(d_1+d_4)(d_1^2+d_1(d_2+d_3)+d_4(d_2+d_3+d_4))\hfill \\ \hfill & +d_2[d_1(d_2+d_3)+d_4(2d_3+d_4)+d_2(d_3+2d_4)]{\frac{\partial f(S^{+},V^{+})}{\partial S}}^2\hfill \\ \hfill & +[d_2(2d_1^2(d_2+d_3)+2d_3{d}_4(d_3+d_4)+d_2^2(d_3+3d_4)\hfill \\ \hfill & +d_1[2d_2^2+2d_2{d}_3+d_3^2+3d_2{d}_4\hfill \\ \hfill & +3d_3{d}_4]+d_2[d_3^2+4d_3{d}_4+2d_4^2]]\frac{\partial f(S^{+},V^{+})}{\partial S})\left(\frac{d_3}{{\gamma }_0}+\frac{{\beta }_1{d}_3}{{\gamma }_0{d}_2}{T}^{+}-\frac{\partial f(S^{+},V^{+})}{\partial V}\right)\hfill \\ \hfill & +{\gamma }_0^2{d}_2^2(d_1+d_4)[d_2+d_3+\frac{\partial f(S^{+},V^{+})}{\partial S}]{\left(\frac{d_3}{{\gamma }_0}+\frac{{\beta }_1{d}_3}{{\gamma }_0{d}_2}{T}^{+}-\frac{\partial f(S^{+},V^{+})}{\partial V}\right)}^2\hfill \\ \hfill A_3=& d_4(d_3{d}_4(d_2+d_3)(d_2+d_3+d_4)+d_1{d}_4(d_2+d_3)(5d_2+2d_3+4d_4)+\hfill \\ \hfill & d_1^2[4d_2^2+7d_2{d}_3+3d_3^2+6d_2{d}_4+5d_3{d}_4+2d_4^2]+d_1^3[3d_2+2(d_3+d_4)]){\beta }_1{T}^{+}\hfill \\ \hfill & +(d_2^2(4d_3+6d_4)+d_3(d_3^2+d_3{d}_4+3d_4^2)+d_2(2d_3^2+5d_3{d}_4+6d_4^2)\hfill \\ \hfill & +d_1(4d_2{d}_3+2d_3^2+9d_2{d}_4+6d_3{d}_4+6d_4^2)){\frac{\partial f(S^{+},V^{+})}{\partial S}}^2{\beta }_1{T}^{+}\hfill \\ \hfill & +(2d_2{d}_3+d_3^2+3d_2{d}_4+2d_3{d}_4+2d_4^2){\frac{\partial f(S^{+},V^{+})}{\partial S}}^3{\beta }_1{T}^{+}\hfill \\ \hfill A_4=& {\gamma }_0{\beta }_1{T}^{+}[d_2(d_1^3+d_1^2(2d_2+2d_3+d_4)+d_1{d}_4(6d_2+3d_3+2d_4)+d_4(d_3^2+2d_4^2\hfill \\ \hfill & +d_2(d_3+3d_4)))+d_2\frac{\partial f(S^{+},V^{+})}{\partial S}[2d_1^2+2d_2{d}_3+d_3^2+5d_2{d}_4+4d_3{d}_4+2d_4^2\hfill \\ \hfill & +2d_1(d_2+2d_4)]+d_2(d_1+d_3+3d_4){\frac{\partial f(S^{+},V^{+})}{\partial S}}^2](\frac{d_3}{{\gamma }_0}+\frac{{\beta }_1{d}_3}{{\gamma }_0{d}_2}{T}^{+}\hfill \\ \hfill & -\frac{\partial f(S^{+},V^{+})}{\partial V})+{\gamma }_0^2{\beta }_1^2{(T^{+})}^2{d}_2^2(d_1+d_4){\left(\frac{d_3}{{\gamma }_0}+\frac{{\beta }_1{d}_3}{{\gamma }_0{d}_2}{T}^{+}-\frac{\partial f(S^{+},V^{+})}{\partial V}\right)}^2\hfill \\ \hfill A_5=& {(T^{+})}^2(d_4[2d_1^3+d_3{d}_4(3d_2+2d_4)+2d_1{d}_4(4d_2+d_3+2d_4)+d_1^2(5d_2+4(d_3+d_4))]\hfill \\ \hfill & [3d_2{d}_3+5d_2{d}_4+4d_4^2+2d_1(d_3+3d_4)]{\frac{\partial f(S^{+},V^{+})}{\partial S}}^2+(d_3+2d_4){\frac{\partial f(S^{+},V^{+})}{\partial S}}^3){\beta }_1^2\hfill \\ \hfill & [2d_4(d_1^2+2d_1{d}_4+d_3{d}_4)+(d_1+d_4)(d_3+4d_4)\frac{\partial f(S^{+},V^{+})}{\partial S}\hfill \\ \hfill & +(d_3+2d_4){\frac{\partial f(S^{+},V^{+})}{\partial S}}^2]{\beta }_1^3{(T^{+})}^3\hfill \\ \hfill A_6=& {\beta }_1{T}^{+}(d_1^2[2d_2{d}_3+d_3^2+9d_2{d}_4+6d_3{d}_4+6d_4^2]+d_1(d_3^3+4d_3^2{d}_4+9d_3{d}_4^2+2d_4^3\hfill \\ \hfill & +d_2^2(3d_3+10d_4)+4d_2(d_3^2+3d_3{d}_4+3d_4^2))+d_4[d_3{d}_4(d_3+2d_4)\hfill \\ \hfill & +d_2^2(2d_3+5d_4)]+{\beta }_1{T}^{+}[d_1^2(d_3+6d_4)+d_1(3d_2{d}_3+2d_3^2+10d_2{d}_4\hfill \\ \hfill & +4d_3{d}_4+8d_4^2)])\frac{\partial f(S^{+},V^{+})}{\partial S}+{\gamma }_0{\beta }_1^2{(T^{+})}^2[d_2[d_1^2+4d_1{d}_4+d_4(d_3+2d_4)]\hfill \\ \hfill & +d_2(d_1+d_3+3d_4)\frac{\partial f(S^{+},V^{+})}{\partial S}]\left(\frac{d_3}{{\gamma }_0}+\frac{{\beta }_1{d}_3}{{\gamma }_0{d}_2}{T}^{+}-\frac{\partial f(S^{+},V^{+})}{\partial V}\right).\hfill \end{array}$$

Appendix B: Parameter values

Values of $P_{\mathit{best}}$ obtained by calibrating model (4) to each dataset and incidence function can be found in Tables 3, 4 and 5. These values are used in Sect. 4.

Table 4.

$P_{\mathit{best}}$ for the best fitting result with $f_2$ in Fig. 2

Parameter	Unit	Figure 2B	Figure 2E	Figure 2H
${\alpha }_0$	Cell/$\mathrm{\mu }$L/day	0.2337	8.3343	1.2856
$\widehat{\beta }$	${(\text{Copies}/\mu \text{L})}^{-1}/\text{day}$	2.0064	0.9533	$1.0000\times {10}^{-6}$
$d_1$	Day${}^{-1}$	0.0550	0.8823	$3.6824\times {10}^{-4}$
${\beta }_1$	Cell/$\mathrm{\mu }$L/day	14.8095	1.9328	$1.0000\times {10}^{-6}$
$d_2$	Day${}^{-1}$	0.9393	1.4381	1.2464
${\widehat{\gamma }}_0$	Dimensionless	3.7297	0.5225	1.2360
$d_3$	Day${}^{-1}$	0.6383	0.0260	$1.1000\times {10}^{-6}$
${\beta }_0$	Day${}^{-1}$	1.0000$\times {10}^{-6}$	3.84327$\times {10}^{-4}$	4.0180
$d_4$	Day${}^{-1}$	0.9146	1.2254	$1.0000\times {10}^{-6}$
${\delta }_1$	${(\text{Copies}/\mu \text{L})}^{-1}$	4.9502	3.1320	$1.0000\times {10}^{-6}$

Table 5.

$P_{\mathit{best}}$ for the best fitting result with $f_3$ in Fig. 2

Parameter	Unit	Figure 2C	Figure 2F	Figure 2I
${\alpha }_0$	Cell/$\mathrm{\mu }$L/day	3.5514	1.6650	0.0403
$\widehat{\beta }$	${(\text{Copies}/\mu \text{L})}^{-1}/\text{day}$	10.1249	3.1125	1.8459
$d_1$	Day${}^{-1}$	0.6595	0.9011	$2.7451\times {10}^{-3}$
${\beta }_1$	Cell/$\mathrm{\mu }$L/day	10.4628	1.5283	0.1263
$d_2$	Day${}^{-1}$	88.9705	2.2783	1.2755
${\widehat{\gamma }}_0$	Dimensionless	732.6705	4.4756	1.8382
$d_3$	Day${}^{-1}$	7.7014	0.0452	$1.0000\times {10}^{-6}$
${\beta }_0$	Day${}^{-1}$	0.4973	0.3270	1.9595
$d_4$	Day${}^{-1}$	0.9887	0.4995	$3.3715\times {10}^{-4}$
${\delta }_1$	${(\text{copies}/\mu \text{L})}^{-1}$	0.0308	0.1615	$4.7100\times {10}^{-5}$
${\delta }_2$	${(\text{Cell}/\mu \text{L})}^{-1}$	640.8021	1.2630	0.1564

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.