Abstract
COVID19, caused by the infection of severe acute respiratory syndrome coronavirus 2 (SARSCoV2), has been a global pandemic and created unprecedented public health challenges throughout the world. Despite significant progresses in understanding the disease pathogenesis and progression, the epidemiological triad of pathogen, host, and environment remains unclear. In this paper, we develop a multiscale model to study the coupled withinhost and betweenhost dynamics of COVID19. The model includes multiple transmission routes (both humantohuman and environmenttohuman) and connects multiple scales (both the population and individual levels). A detailed analysis on the local and global dynamics of the fast system, slow system and full system shows that rich dynamics, including both forward and backward bifurcations, emerge with the coupling of viral infection and epidemiological models. Model fitting to both virological and epidemiological data facilitates the evaluation of the influence of a few infection characteristics and antiviral treatment on the spread of the disease. Our work underlines the potential role that the environment can play in the transmission of COVID19. Antiviral treatment of infected individuals can delay but cannot prevent the emergence of disease outbreaks. These results highlight the implementation of comprehensive intervention measures such as social distancing and wearing masks that aim to stop airborne transmission, combined with surface disinfection and hand hygiene that can prevent environmental transmission. The model also provides a multiscale modeling framework to study other infectious diseases when the environment can serve as a reservoir of pathogens.
Introduction
In late December 2019, highly contagious pneumonia of unknown etiology was first reported in Wuhan, China (Zhu et al. 2020). A novel strain of coronavirus was isolated from patients and later named severe acute respiratory syndrome coronavirus 2 (SARSCoV2). Since then, the Coronavirus Disease 2019 (COVID19) has spread to over 210 countries and territories, creating unprecedented public health challenges throughout the world. As of January 12, 2021, more than 91 million cases and 1.9 million deaths had been reported. Despite many theoretical, experimental and clinical studies, our current knowledge on the fundamental mechanisms of COVID19 transmission and infection remains limited.
Mathematical modeling provides a powerful theoretical means to study COVID19. A large number of mathematical and statistical models have been proposed (e.g., Kucharski et al. 2020; He et al. 2020; Li et al. 2020; Liu et al. 2020a, c; Tang et al. 2020; Weitz et al. 2020; Wu et al. 2020a, b; Zhao and Feng 2020; Hellewell et al. 2020; Chen et al. 2020; Olabode et al. 2021; Musa et al. 2022). Almost all these models are concerned with the transmission and spread of the disease (i.e., the betweenhost dynamics) at the population level. On the other hand, very little modeling effort has been devoted to the withinhost dynamics of SARSCoV2. When the coronavirus enters the human body, there are complicated interactions between the pathogen and host cells taking place, which directly shapes the disease risk and infection severity for the individual hosts and which, in a collective manner, may subsequently impact the epidemic patterns (Liu et al. 2020b; He et al. 2020). Due to the inadequate investigation of the withinhost dynamics of COVID19 and their connection to the populationlevel epidemics, there are several fundamental questions that remain unanswered or only partially answered at present; for example, how does the viral load change inside the human body, what are the shortterm and longterm interactions between SARSCoV2 and the host cells, and how does the withinhost pathogen development affect the populationlevel disease transmission and spread?
As a pilot study to address these questions, we develop a multiscale modeling framework in this paper to investigate the betweenhost and withinhost dynamics of COVID19 and their impact on each other. At the individual host level, our model characterizes the virus–cell interactions and their time evolution within the human body. At the population level, our model describes the disease transmission and spread through multiple transmission routes. Most of the betweenhost COVID19 models published thus far are based on the susceptibleexposedinfectedrecovered compartmental framework or its variants, with a focus on the direct, humantohuman transmission pathway (Chan et al. 2020). On the other hand, a recent experimental study found convincing evidences that SARSCoV2 was detectable in aerosols for up to 3 h, on copper for up to 4 h, on cardboard for up to 24 h, and on plastic and stainless steel for up to 3 days (van Doremalen et al. 2020). There are also consistent and strong evidences that the infection can spread via airborne transmission (Greenhalgh et al. 2021). All these indicate a significant risk of the indirect, environmenttohuman transmission pathway, particularly airborne and fomite transmission, for SARSCoV2. Additionally, the novel coronavirus has been found in the stool of some infected individuals (Zhang et al. 2020), which may contaminate the aquatic environment through the sewage water and add another possible route of environmental transmission for COVID19 (Yeo et al. 2020). Therefore, quantifying such indirect transmission routes in a modeling study could help us to better understand the transmission mechanisms of COVID19.
Our betweenhost model explicitly includes the environmenttohuman transmission by incorporating the concentration of SARSCoV2 in the environment, which interacts with the human hosts: susceptible individuals may be infected by contracting the environmental coronavirus, and infected individuals may shed the pathogen (through coughing, sneezing, etc.) back to the environment. In addition, we introduce the concentration of SARSCoV2 inside the human body as a withinhost variable that interacts with the host cells and that represents the individual viral load. We then bridge the withinhost and betweenhost dynamics via a twoway coupling. Since the coronavirus may enter the human body through the environment, we assume that the withinhost viral load depends on the environmental pathogen concentration. Meanwhile, since the withinhost pathogen level is directly associated with the individual symptoms and infection severity, we assume that the humantohuman transmission rates depend on the viral load within the human body.
Hence, our modeling framework incorporates multiple transmission routes (both humantohuman and environmenttohuman pathways) and connects multiple scales (both the population and individual levels). The withinhost interactions normally occur on the time scale of hours to days, which is referred to as the fast dynamics. In contrast, the betweenhost transmission and spread is on the scale of weeks, months to years, referred to as the slow dynamics. To facilitate the mathematical analysis, we will first separate the two time scales so that a thorough investigation can be conducted to each of the fast and slow systems. Then we will combine the betweenhost and withinhost systems and study the coupled dynamics. Our detailed analysis on the local and global dynamics of the fast system, the slow system and the coupled system shows that rich dynamics, including both forward and backward bifurcations, emerge with the coupling of the withinhost and betweenhost models.
The remainder of the paper is organized as follows. Section 2 is devoted to the formulation of the COVID19 model linking withinhost and betweenhost dynamics, and the derivation of the basic reproduction number for the model. We analyze the model by using the bifurcation theory and fastslow analysis in Sect. 3, and conduct numerical simulation and model validation in Sect. 4. Finally, we conclude the paper in Sect. 5 with some discussions.
Model
Let S, E, I, R denote the number of susceptible, presymptomatic infected, symptomatic infected and recovered host individuals. Let Z and V be the concentration of coronavirus in the environment and within the host, respectively. T and \(T^*\) denote the concentration of target cells and infected target cells within the host, respectively. To link the withinhost and betweenhost interactions, we extend the populationlevel model proposed in Yang and Wang (2020) with the inclusion of the withinhost dynamics (Wang et al. 2020), which takes the form
The symptomatic infected class has fully developed disease symptoms and can infect others. The presymptomatic infected class is in the incubation period; COVID19 patients do not show symptoms but are still capable of infecting others. There are some other models that include the exposed (assumed to be noninfectious), asymptomatic and symptomatic groups explicitly (Zhao and Feng 2020; Ngonghala et al. 2020; Xue et al. 2020; Tang et al. 2020). Because we will couple it with withinhost models, we keep the betweenhost model with a minimum number of variables.
The parameter \(\Lambda \) represents the generation rate of susceptible individuals, \(\mu \) is the natural death rate, \(1/\alpha \) is the period of incubation between infection and the onset of symptoms, \(\omega \) is the diseaseinduced death rate, \(\gamma \) is the rate of recovery, \(\xi _E\) and \(\xi _I\) are the respective viral release rates to the environmental reservoir from the presymptomatic infected and symptomatic infected individuals, and \(\delta \) is the viral removal rate from the environment. The functions \(\beta _E(V, E)\) and \(\beta _I(V, I)\) represent the direct, humantohuman transmission rates between presymptomatic infected and susceptible individuals, and between symptomatic infected and susceptible individuals, respectively. The function \(\beta _Z(Z)\) is the indirect, environmenttohuman transmission rate. We assume that \(\beta _E(V, E)\), \(\beta _I(V, I)\) and \(\beta _Z(Z)\) are all nonincreasing functions of E, I, and Z, respectively, because higher levels of E, I and Z would motivate stronger control measures to prevent transmission. Higher viral load can result in a higher transmission rate. Thus, we assume that \(\beta _E(V, E)\) and \(\beta _I(V, I)\) increase as V increases. For example, in the simple case, \(\beta _E(V, E)\), \(\beta _I(V, I)\) and \(\beta _Z(Z)\) can be constant, or they may take the nonlinear form (44) as shown in numerical simulations. For mathematical analysis of the multiscale model, we avoid the complexity of keeping track of the dynamics within each individual and only formulate the withinhost model for a conceptual “average” individual. The withinhost model will be fitted to the average viral load of a group of patients. The parameter b is the generation rate of target cells, \(\kappa \) is the viral infection rate, d is the death rate of target cells, q is the death rate of infected cells, p is the viral production rate, and c is the viral clearance rate. The environment can transmit virus to the individual at a rate \(\eta (Z)\). Below we summarize the assumptions on the functions.

(H1)
\(\beta _E(V,E)\), \(\beta _I(V, I)\) and \(\beta _Z(Z)\) are positive functions.

(H2)
\(\dfrac{\partial \beta _{W}(V,W)}{\partial V}> 0\), \(\dfrac{\partial \beta _{W}(V,W)}{\partial W}\le 0\) for \(W=E, I\), and \(\beta _Z'(Z)\le 0\).

(H3)
\(\beta _{E}(V,E)\) and \(\beta _{I}(V,I)\) are both concave downward (i.e., the Hessian matrices of \(\beta _{E}\) and \(\beta _{I}\) are negative semidefinite).

(H4)
\(\eta (0)=0\), \(\eta '(Z)\ge 0\) and \(\eta ''(Z)\le 0\) for \(Z\ge 0\), and \(\eta '(Z)> 0\) for \(Z>0\).
The coupled dynamics have two distinct time scales. The withinhost interactions normally occur on the time scale of hours to days, referred to as the fast dynamics. In contrast, the betweenhost transmission and spread is on the scale of weeks, months to years, referred to as the slow dynamics. These two scales are coupled by a small constant \(\epsilon \).
Model (1) has a diseasefree equilibrium solution (DES), given by \( {\mathcal {E}}_0=(S_0, E_0, I_0, R_0, Z_0, T_0, T^*_0, V_0) = (\Lambda /\mu ,0,0,0,0,b/d,0,0). \) The new infection matrix \({\mathcal {F}}\) and transmission matrix \({\mathcal {V}}\) are
and
By the nextgeneration matrix method (van den Driessche and Watmough 2002), the basic reproduction number \({\mathcal {R}}_0\) is defined as the spectral radius of \({\mathcal {F}}{\mathcal {V}}^{1}\), i.e.,
where
Let \({\mathcal {R}}_{0b}={\mathcal {R}}_{0E}+{\mathcal {R}}_{0I}+{\mathcal {R}}_{0Z}\). Here \({\mathcal {R}}_{0b}\) and \( {\mathcal {R}}_{0w}\) are the betweenhost and withinhost threshold parameters, respectively. More specifically, \({\mathcal {R}}_{0b}\) is the sum of three terms, where \({\mathcal {R}}_{0E}\) (resp. \({\mathcal {R}}_{0I}\), \({\mathcal {R}}_{0Z}\)) measures the contribution to the betweenhost basic reproduction number from direct presymptomaticinfectedtosusceptible humantohuman transmission (resp. direct symptomaticinfectedtosusceptible humantohuman transmission, indirect environmenttohuman transmission).
Analysis
Slow System
For the slow time scale t, our system can be studied using the full model (1) by setting \(\epsilon =0\), which is referred to as the slow system. In analyzing the slow system, the fast system is treated at its quasisteady state.
Since \(0<\epsilon \ll 1\), the withinhost dynamics can be studied by
Letting \(\epsilon =0\) in (5) leads to
and V is determined by
where
If \(Z=0\), it leads to the virusinfectionfree ES \((T, T^*, V)=(b/d, 0,0)\) or the coexistence state of virus and infected target cells
If \(Z>0\), it follows from (H4) that \(\eta (Z)>0\), and hence \(a_2(Z)<0\) and \(a_1^2(Z)4a_0a_2(Z)>0\). In this case, equation (7) has two zeros \(V_{\pm }=V_{\pm }(Z):=\dfrac{a_1(Z)\pm \sqrt{a_1^2(Z)4a_0a_2(Z)}}{2a_0}\) with \(V_{}<0<V_{+}\), and \(V=V_{+}(Z):=f(Z)\) is the only biological feasible equilibrium solution. The infected target cells and viral loads both persist and the corresponding ES is
Differentiating (7) with respect to Z along \(V=f(Z)\) yields
This implies that along \(V=f(Z)\)
as \(\eta '(Z)>0\) and \( f(Z)>\dfrac{a_1(Z)}{2a_0}. \) Inequality (11) shows that V is a strictly increasing function of Z. Similarly, differentiating (10) with respect to Z along \(V=f(Z)\) and using \(cV\eta (Z)=pT^*\) give us
It follows from (6), (11) and (H4) that \(T^*< b/q\), \(V'(Z)\ge 0\) and \(\eta '(Z)> 0, \eta ''(Z)\le 0\). This implies that \(f''(Z)< 0\).
Accordingly, the slow system can be written as
Equilibrium Solutions
We will analyze the equilibrium solutions (ESs) of the slow system (13). Note that the associated ES satisfies
Solving (14), we obtain
with
When \(I=0\), it leads to the diseasefree ES \((S, E, I, R, Z)=(S_0,0,0,0)\).
When \(I> 0\), substituting (15) into the second equation of (14) and solving S as a function of I at the equilibrium, we have
where \(g(I)=\beta _1(I)+\beta _2(I)+\beta _3(I)\) with
Straightforward calculation yields
and
By (H3), the Hessian matrix of \(\beta _E\) is negative semidefinite, and hence the first three terms in the bracket of the righthand side of (18) are nonpositive. By (H2) and (12), \(\frac{\partial \beta _E}{\partial V}> 0\) and \(f''(\theta _ZI)< 0\). Thus, \(\beta _1''(I)< 0.\) Likewise, one can verify that \(\beta _2''(I)< 0 \) and \( \beta _3''(I)\le 0\). Hence, \(g''(I)< 0\), and this implies that
This shows that the intersections of the curves \(\phi (I)\) and \(\psi (I)\) in \((0,\infty )\) determine the nontrivial equilibrium solutions of the slow system (13), as the nontrivial equilibrium solution is of the form
where \(I_e\) is a positive intersection of these two curves.
Since \(\phi (0)=\Lambda /\mu =S_0\) and \(\psi (0)=(\alpha +\mu )\theta _E/[\beta _E(0,0)\theta _E+\beta _I(0,0)+\beta _Z(0)\theta _Z]\), we have
Additionally, \(\phi (I)\) is a strictly decreasing line and \(\psi (I)\) is concave upward. To study the existence of the nontrivial equilibrium of (13), we consider three cases.

1.
\({\mathcal {R}}_{0b}> 1\). In this case, \(\phi (0)>\psi (0)\), these two curves have a unique intersection in \((0,\infty )\), denoted as \(I_e\).

2.
\({\mathcal {R}}_{0b}= 1\). In this case, \(\phi (0)=\psi (0)\). In view of (16), \(\phi '(0)=m\). When \(\psi '(0)\ge m\), these two curves do not intersect; when \(\psi '(0)< m\), there exists a unique intersection \(I_e \in (0,\infty )\).

3.
\({\mathcal {R}}_{0b}< 1\). In this case, \(\phi (0)<\psi (0)\) and there are three possibilities for the intersection of these two curves in \((0,\infty )\). (i) There is no intersection if \(\phi (I)<\psi (I)\) for \(I\ge 0\); (ii) They have one intersection in \((0, \infty )\) if there exists \(I^c > 0\) such that \(\phi (I^c)=\psi (I^c)\) and \(\phi '(I^c)=\psi '(I^c)= m\); (c) Otherwise, there are two intersections in \((0, \infty )\).
This result implies that system (13) has at most three biologically feasible equilibria, i.e., if \({\mathcal {R}}_{0b}>1\), this system has two equilibria: diseasefree equilibrium (DFE) and endemic equilibrium (EE); If \({\mathcal {R}}_{0b}\le 1\), it can have one, two or three equilibria (and one of them is the DFE) depending on the parameter values. This indicates that the system (13) may undergo a backward bifurcation.
Existence of Forward and Backward Bifurcations
In this section, we study the bifurcation in terms of I with \({\mathcal {R}}_{0b}\) as a bifurcation parameter. Note that \({\mathcal {R}}_{0b}\) is a multiple of \(S_0\) (i.e., \({\mathcal {R}}_{0b}=kS_0\) with \(k=[\beta _E(0,0)\theta _E+\beta _I(0,0)+\beta _Z(0)\theta _Z]/[(\alpha +\mu )\theta _E]\)). We treat \(S_0\) as an independent parameter and the rest parameters fixed. It suffices to analyze the bifurcation diagram of I with \(S_0\) as a bifurcation parameter. If \(S_0=S_0^b=1/k=\psi (0)\), then the corresponding \({\mathcal {R}}_{0b}=1\). At the equilibrium of system (13), when \(I=0\), it leads to the diseasefree equilibrium (DFE) and it follows from van den Driessche and Watmough (2002)[Theorem 2] that the DFE is locally asymptotically stable (resp. unstable) when \({\mathcal {R}}_{0b}<1\), i.e., \(S_0<S_0^b\) (resp. \({\mathcal {R}}_{0b}>1\), i.e., \(S_0>S_0^b\) ).
In the remainder of this section, we assume that \(\phi (I)\) and \(\psi (I)\) intersect in the first quadrant (i.e., system (13) has equilibrium solutions with \(I>0\)). Thus, at the equilibrium,
To compute \(dI/dS_0\), we calculate \(dS_0/dI\). Differentiating (21) with respect to I yields
By the implicit function theorem, \(\frac{dI}{dS_0}=1\big /\frac{d S_0}{d I}\) if \({d S_0}/{d I}\ne 0\). Assume that \(I\rightarrow 0^{+}\) as \(S_0\rightarrow S_{0}^b=\psi (0)\). Integrating (22) yields
In view of (19), we know \(S_0=h(I)\) as a function of I is positive and strictly concave upward.
Case I: If \(h'(0)\ge 0\) (i.e., \(\psi '(0)\ge m\)), then \(dS_0/dI=h'(I)>0\) and hence \(dI/dS_0>0\) and \(I=h^{1}(S_0)\) is a positive and strictly concave downward function for \(I>0\). In this case, system (13) exhibits a transcritical bifurcation at \(S_0=S_{0}^b\) (i.e., \({\mathcal {R}}_{0b}=1\)).
Case II: If \(h'(0)< 0\) (i.e., \(\psi '(0)< m\)), there exists a unique \(I^c>0\) such that \(h'(I^c)=0\), \(h'(I)<0\) when \(0<I<I^c\) and \(h'(I)>0\) when \(I>I^c\). Denote \(S_{0}^c=h(I^c)\) and the corresponding \({\mathcal {R}}_{0b}\) by \({\mathcal {R}}_{0b}^c\). Clearly, \(0<S_{0}^c<S_{0}^b\) and \({\mathcal {R}}_{0b}^c<1\). Moreover, \(I=h^{1}(S_0)\) has two branches, denote by \(\Phi _{1}(S_0)\) and \(\Phi _{2}(S_0)\) when \(S_0\in (S_{0}^c, S_{0}^b)\) (i.e., \({\mathcal {R}}_{0b}\in ({\mathcal {R}}_{0b}^c, 1)\)), where (a) \(0<\Phi _{1}(S_0)<\Phi _{2}(S_0)\), (b) \(\Phi _1'(S_0)<0\) and \(\Phi _2'(S_0)>0\), (c) \(\Phi _1''(S_0)<0\) and \(\Phi _2''(S_0)<0\) for \(S_0\in (S_{0}^c, S_{0}^b)\), and (d) \(\Phi _{1}(S_0^c)=\Phi _{2}(S_0^c)\). Specifically, \(\Phi _1(S_0)\) is a strictly decreasing and concave downward function, whereas \(\Phi _2(S_0)\) is a strictly increasing and concave downward function as \(S_0\in (S_{0}^c, S_{0}^b)\). On the other hand, \(I=h^{1}(S_0)\) has only one branch and is welldefined, denote as \(\Phi _3(S_0)\), when \(S_0\in [S_{0}^b, \infty )\) (i.e., \({\mathcal {R}}_{0b}>1\)). Besides, \(\Phi _3'(S_0)>0\), \(\Phi _3''(S_0)<0\) when \(S_0\in [S_{0}^b, \infty )\) and \(\Phi _2^{(n)}(S_0^b)=\Phi _3^{(n)}(S_0^b+)\) for \(n=0,1,2\). So we rename \(\Phi _3\) by \(\Phi _2\). In this case, system (13) exhibits a backward bifurcation.
The result for the occurrence of transcritical and backward bifurcations is summarized in the following lemma and illustrated in Fig. 1.
Lemma 1
System (13) undergoes a transcritical bifurcation at \(S_0=S_{0}^b\) (i.e., \({\mathcal {R}}_{0b}=1\)), which further induces a backward bifurcation if \(\psi '(0)<m\), and a forward bifurcation if \(\psi '(0)\ge m\).
Stability Analysis for Forward Bifurcation
Suppose \(S(0)+E(0)+I(0)+R(0)\le S_0\). As \(S, E, I, R\ge 0\), \(S+E+I+R\le S_0\), and \(0\le Z\le Z_{max}:=(\xi _E+\xi _I)S_0/\delta \), this yields a biologically feasible domain of system (13), which is given by
It is easy to see that \(\Omega ^s\) is positively invariant for system (13).
Assume that \(\psi '(0)\ge m\). By Lemma 1, system (13) exhibits a forward bifurcation. Together with the result in Sect. 3.1.1 leads to the following:

(1)
If \({\mathcal {R}}_{0b}\le 1\), system (13) admits a unique equilibrium solution, the DFE \({\mathcal {E}}_0^s=(\Lambda /\mu ,0,0,0,0)\), that is locally asymptomatically stable when \({\mathcal {R}}_{0b}<1\).

(2)
If \({\mathcal {R}}_{0b}> 1\), system (13) has two equilibria: the DFE \({\mathcal {E}}_0^s\) and the endemic equilibrium (EE) \({\mathcal {E}}_e^s\). Moreover, the DFE is unstable.
In what follows, a global stability analysis is performed to study the forward bifurcation. The associated result is established in the following two theorems.
Theorem 1
Suppose that \(\psi '(0)\ge m.\)

1.
Assume that either \(\eta '(0)=0\) or \(\frac{\partial }{\partial V}\beta _E(0,0)=\frac{\partial }{\partial V}\beta _I(0,0)=0\) is satisfied. If \({\mathcal {R}}_{0b}<1\) , then the DFE of system (13) is globally asymptomatically stable in \(\Omega ^s\).

2.
If \({\mathcal {R}}_{0b}>1\), the disease is uniformly persistent in the interior of \(\Omega ^s\).
Proof
When \(\psi '(0)\ge m\) and \({\mathcal {R}}_{0b}<1\), the DFE is the unique equilibrium of (13) and it is locally asymptotically stable. If either \(\eta '(0)=0\) or \(\frac{\partial }{\partial V}\beta _E(0,0)=\frac{\partial }{\partial V}\beta _I(0,0)=0\) holds, by (H1), (H3) and (12), we have
where \(f'(0)=0\) when \(\eta '(0)=0\), which is obtained from (11). Let \({\textbf {Y}}=(E, I, Z)^T\). This implies that
where
One can verify that \(u({\mathcal {V}}_b)^{1}{\mathcal {F}}_b=u{\mathcal {R}}_{0b}\) and \({\mathcal {R}}_{0b}=\rho \big (({\mathcal {V}}_b)^{1}{\mathcal {F}}_b\big )=\rho \big ({\mathcal {F}}_b({\mathcal {V}}_b)^{1}\big )\), where \(u=(\beta _E(0,0),\beta _I(0,0),\beta _Z(0))\). Consider a Lyapunov function
Differentiating L along solutions of system (13) leads to
If \({\mathcal {R}}_{0b}<1\), \(L'\le 0\). and \(L'=0\) if and only if \(u{\textbf {Y}} =0\).^{Footnote 1} This implies \(E=I=Z=0\). Using the first and the fourth equations of (13) yields \(S=S_0\) and \(R=0\). Thus, the largest invariant set where \(L'=0\) contains only one point, i.e., the DFE \({\mathcal {E}}_0^s=(S_0,0,0,0,0)\). In view of LaSalle’s Invariant Principle (Salle 1976), the DFE is globally asymptotically stable in \(\Omega ^s\) when \({\mathcal {R}}_{0b}<1\).
If \({\mathcal {R}}_{0b}>1\), we know that \({\mathcal {E}}_0\) is unstable. It follows from the persistence theory (Thieme 1993) and a similar argument as in the proof of Gao and Ruan (2011)[Proposition 3.3] that instability of \({\mathcal {E}}_0^s\) implies uniform persistence of system (13) in the interior of \(\Omega ^s\). \(\square \)
Let \({\mathcal {E}}_e^s=(S^*,E^*,I^*,R^*,Z^*)\) denote the unique endemic equilibrium when \({\mathcal {R}}_{0b}>1\). To establish the global stability of system (13) in this case, we need to make some assumptions:

(C1)
\(\left( 1\frac{\beta _E(f(Z), E)E}{\beta _E(f(Z^*), E^*)E^*}\right) \left( 1\frac{\beta _E(f(Z^*), E^*)}{\beta _E(f(Z), E)}\right) \ge 0\);

(C2)
\(\left( 1\frac{\beta _I(f(Z), I)I}{\beta _I(f(Z^*), I^*)I^*}\right) \left( 1\frac{\beta _I(f(Z^*), I^*)}{\beta _I(f(Z), I)}\right) \ge 0\);

(C3)
\(\frac{d}{dZ}(\beta _Z(Z)Z)\ge 0\)
for \(E,I,Z>0\). By the similar argument as that in Yang and Wang (2020)[Theorem 2.2], we obtain the following result. A poof is given in Appendix A for completeness.
Theorem 2
Suppose that \(\psi '(0)\ge m\) and (C1)(C3) holds. If \({\mathcal {R}}_{0b}>1\), the endemic equilibrium is globally asymptotically stable in the interior of \(\Omega ^s\).
Stability Analysis for Backward Bifurcation
In this section, we assume that \(\psi '(0)<m\). In this case, slow system (13) may exhibit a backward bifurcation according to Lemma 1. By the similar argument as that in Sect. 3.1.3, we have the following result.

(1)
If \({\mathcal {R}}_{0b}< 1\) and \({\mathcal {R}}_{0b}^c\) exists, system (13) admits up to three equilibria. The DFE \({\mathcal {E}}_0^s\) always exists and it is locally asymptomatically stable. Two EEs, \({\mathcal {E}}_t^s\) and \({\mathcal {E}}_m^s\), coexist when \({\mathcal {R}}_{0b}^c< {\mathcal {R}}_{0b}< 1\), and they merge into one when \({\mathcal {R}}_{0b}= {\mathcal {R}}_{0b}^c\). There is no positive EE when \({\mathcal {R}}_{0b}< {\mathcal {R}}_{0b}^c\).

(2)
If \({\mathcal {R}}_{0b}> 1\), system (13) has two equilibria: the DFE \({\mathcal {E}}_0^s\) and the EE \({\mathcal {E}}_e^s\). Moreover, the DFE is unstable and the EE is globally as asymptomatically stable in the interior of \(\Omega ^s\) when \({\mathcal {R}}_{0b}> 1\) and \((C1)(C3)\) holds, where the global stability of the EE is established by the same method as that in Theorem 2.
As shown in Fig. 1(a), the bifurcation diagram of I as a function of \({\mathcal {R}}_{0b}\) consists of three branches: the top, middle and bottom branches, where the bottom (resp. middle, top) branch is composed of the DFE \({\mathcal {E}}_0^s\) (resp. the EE \({\mathcal {E}}_m^s\), the EE \({\mathcal {E}}_t^s\)). Thus, it remains to investigate the stability of the middle and top branches in the case where \({\mathcal {R}}_{0b}> 1\).
Denote
Theorem 3
The middle branch of the equilibrium solutions of system (13) in the backward bifurcation diagram is unstable.
Proof
As the fourth equation (i.e., the equation for R) is decoupled from the rest of system (13), it suffices to use the following system to prove the instability of the EE \({\mathcal {E}}_m^s=(S, E, I, R, Z)\)
Linearizing system (25) at \({\mathcal {E}}_m^s=(S, E, I, R, Z)\) leads to the corresponding Jacobian matrix
It follows from the direct calculation that
where m, \(\theta _E \) and \( \theta _Z\) are defined in (17). Note that, at \({\mathcal {E}}_m^s\), \(W=\theta _W I\) for \(W=E, Z\) and
Differentiating (28) with respect to I yields
By (28), (29) and (16), \(\det (J)\) can be written as
Clearly the sign of \(\det (J)\) is determined by the sign of
On the other hand, at the EE, \(\phi (I)=\psi (I)\), i.e.,
which implies that
Differentiating both sides of (32) in terms of \(S_0\) and doing a simple algebraic manipulation, we find
Since \(G(I)>0\) and \(\dfrac{dI}{dS_0}<0\) at \({\mathcal {E}}_m^s\), \(F(I)<0\). In view of (30) and (31), \(\det (J)=\alpha \mu \delta SF(I)<0\) at \({\mathcal {E}}_m^s\). This shows that \({\mathcal {E}}_m^s\) is unstable. \(\square \)
Theorem 4
Suppose (i) \(\alpha >0\) is sufficiently small, and (ii) \(\dfrac{\partial \varPsi }{\partial Z}\le 0\) at \({\mathcal {E}}_t\). Then the top branch of the equilibrium solutions of system (13) in the backward bifurcation is locally asymptotically stable.
Proof
First, we assume that \(\alpha =0\). In this case, it is easy to verify that the characteristic equation for eigenvalues of the linearized system of (25) at \({\mathcal {E}}_t^s=(S, E, I, R, Z)\) (i.e., an equilibrium solution on the top branch) is
where \(I_4\) is the \(4\times 4\) identity matrix, \(J=(J_{ij})\) is defined in (26) and
One eigenvalue is \((\omega +\gamma + \mu )<0\). To prove the remaining three eigenvalues have negative real parts, by the Routh–Hurwitz criterion, it suffices to show that
It is obvious that \(J_{11}<0\) and \(J_{44}<0.\) Additionally, note that \(\frac{\partial }{\partial E}\beta _E(f(V),E)\le 0\), and
Thus, at \({\mathcal {E}}_t^s\)
Hence \(J_{22}<0\). This implies \(a>0\).
Along the top branch of the equilibrium solutions of system (13), \(dI/dS_0>0\). Using the same argument as in the proof of Theorem 3, one can verify that \(\det (J)>0\) at \({\mathcal {E}}_t^s\). By \(\det (J)=c(\omega +\gamma + \mu )\), we have \(c>0\).
On the other hand, it follows from direct calculation that
In view of \(\frac{\partial \varPsi }{\partial Z}\le 0\), equation (34) and the positiveness of all the model parameters involved, we have \(abc>0\), which implies that \(b>0\) as \(a>0, c>0\). This shows that \({\mathcal {E}}_t^s\) is locally asymptotically stable when \(\alpha =0\).
By the continuous dependence of the spectrum on the model parameters, the local stability of the top branch of the equilibrium solutions of system (13) remains to be true when \(\alpha >0\) is sufficiently small. \(\square \)
Fast System
In the fast time scale \(s=t/\epsilon \), the full system (1) can be written as
Letting \(\epsilon =0\) yields
which is referred to the fast system.
If \(Z=0\), the fast system (36) exhibits a forward bifurcation (Smith and Leenheer 2003). It admits at most two biologically feasible equilibria. The virusinfectionfree equilibrium \({\mathcal {E}}_{00}^f =(b/d, 0,0)\) always exists, and the coexistence of virus and infected target cells \({\mathcal {E}}_{ee}^f =\Big (\dfrac{b}{d}\dfrac{1}{{\mathcal {R}}_{0w}}, \dfrac{cd}{\kappa p}({\mathcal {R}}_{0w}1), \dfrac{d}{\kappa }({\mathcal {R}}_{0w}1)\Big )\) is determined by a threshold parameter \({\mathcal {R}}_{0w}\). More specifically,

(1)
if \({\mathcal {R}}_{0w}< 1\), there is no coexistence state, and the virusinfectionfree equilibrium \({\mathcal {E}}_{00}^f\) is globally asymptotically stable in \({\mathbb {R}}^3_{+}\).

(2)
if \({\mathcal {R}}_{0w}> 1\), the coexistence state \({\mathcal {E}}_{ee}^f\) is globally asymptotically stable in \({\mathbb {R}}^3_{+}\).
If \(Z>0\), recalling (9) and the analysis in Sect. 3.1, the fast system (36) has a unique endemic equilibrium
Accordingly, the global stability of the endemic equilibrium \({\mathcal {E}}_{e}^f\) is summarized as follows.
Theorem 5
For each \(Z>0\), the endemic equilibrium \({\mathcal {E}}_{e}^f\) of the fast system (36) is globally asymptotically stable in \({\mathbb {R}}^3_{+}\).
Proof
Consider a Lyapunov function
where the constant \(m>0\) is to be determined. It is easy to verify that \(L ^f\ge 0\) for \((T, T^*, V)\in {\mathbb {R}}^3_{+}\) and \(L ^f=0\) if and only if \((T, T^*, V)=(T_e, T_e^*, V_e)\).
For simplicity, we write \(\eta (Z)\) by \(\eta \). Differentiating \(L ^f\) along the solution of the fast system (36) gives
Use the equilibrium equations of (36) , \( b=\kappa V_eT_e+dT_e\) and \( c=\dfrac{\eta +pT^*_e}{V_e}. \) Equation (38) can be simplified as
Setting \(m=q/p\) and using \(qT_e^*=\kappa V_eT_e\) yield
It follows immediately from the arithmeticgeometric mean inequality that \(\dfrac{d L ^f}{ds}\le 0\) and \(\dfrac{d L ^f}{ds}= 0\) if and only if \((T, T_*, V)=(T_e, T_e^*, V_e)\). Thus, \({\mathcal {E}}_e^f\) is globally asymptotically stable. \(\square \)
The Full System
The time scale separation allows us to conduct a fastslow analysis to analyze the dynamics of the slow and fast system, which is useful to gain insights into the dynamics of the full system.
It is easy to see from the results in Sects. 3.1 and 3.2 that the full system (1) admits at most four biologically reasonable solutions. Let \({\mathcal {E}}_0=(\Lambda /\mu ,0,0,0,0,{b}/{d},0,0)=(S_0,0,0,0,0, T_0,0,0)\) denote the DFE, which always exists. Let
denote the boundary equilibrium solutions (bES).
In the case of \(\psi '(0)\ge m\) (i.e., the slow system exhibits a forward bifurcation), the slow system has a unique EE \((S_0m I_e, \theta _E I_e, I_e, \theta _R I_e, \theta _Z I_e)\). Hence, the full system admits the bES \({\mathcal {E}}_{b}\) and the endemic equilibrium (EE)
with \(V_{+}=f(\theta _Z I_e).\)
In the case of \(\psi '(0)< m\) (i.e., the slow system exhibits a backward bifurcation), the slow system (13) allows up to two EEs, denoted as
with \(x=m\) and t representing the middle and top branches of equilibrium solutions of the slow system, respectively. Thus, the full system admits the bES \({\mathcal {E}}_{b}\) and at most two EEs
for \(x=m,t.\)
The local stability of the DFE and bES is an immediate consequence of the results from Sects. 3.1 and 3.2, as the full system would be decoupled in those cases. The global stability of the DFE and EE is established in Theorems 6 and 7. The associated existence and stability of equilibrium solutions are summarized in Tables 1 and 2, for which DNE means “does not exist,” l.a.s. and g.a.s. are locally asymptotically stable and globally asymptotically stable, respectively, and superscript asterisk means that the result holds under certain condition and superscript n indicates that the result is verified numerically.
Let \(\eta _{max}=\max _{Z\ge 0}\{\eta (Z)\}\) which exists by (H4), \({\hat{T}}_{max}=b/\min \{d,q\}\) and \(V_{max}=(\eta _{max}+pT_{max})/c\). It is easy to verify that \( \Omega =\{(S, E, I, R, Z, T, T^*, V)\in {\mathbb {R}}^8_{+}: S+ E+I+ R\le S_0,\, Z\le Z_{max}, T+T^*\le {\hat{T}}_{max},\, V\le V_{max} \} \) is a positive invariant region for the full system (1). The global stability of the DFE \({\mathcal {E}}_0\) of our full system is established in the following result.
Theorem 6
Suppose that \(\frac{\partial \beta _E}{\partial V}(0,0)=\frac{\partial \beta _I}{\partial V}(0,0)=0\). If \({\mathcal {R}}_0<1\), then the DFE \({\mathcal {E}}_0\) of the full system (1) is globally asymptotically stable in \(\Omega \).
Proof
Note that, for all \(V\ge 0\) and \(E\ge 0\), \( \beta _E(V, E)\le \beta _E(V,0)\le \beta _E(0,0)+ \frac{\partial \beta _E}{\partial V}(0,0)V=\beta _E(0,0), \) where the first and second inequalities are obtained from (H2) and (H3), respectively, and the last equality follows from our assumption \(\frac{\partial \beta _E}{\partial V}(0,0)=0\). Similarly, one can verify that \(\beta _I(V, I)\le \beta _I(0, 0)\) for \(V\ge 0\) and \(I\ge 0\). Additionally, using \(\beta _Z'(Z)\le 0\), we have \(\beta _Z(Z)\le \beta _Z(0)\) for \(Z\ge 0\).
Denote \({\textbf {X}}=(E, I, Z, T^*, V)^T\). It follows from direct calculation that, for all \({\textbf {X}}\in \Omega \),
where \({\mathcal {F}}\) and \({\mathcal {V}}\) are defined in (2) and (3), respectively. Since both \({\mathcal {F}}\) and \({\mathcal {V}}\) are nonnegative, it follows from the Perron–Frobenius Theorem that the nonnegative matrix \({\mathcal {V}}^{1}{\mathcal {F}}\) has a nonnegative left eigenvector \(w=(w_1, w_2, w_3,w_4, w_5)\in {\mathbb {R}}^5_{+}\) associated with the eigenvalue \({\mathcal {R}}_0=\rho ({\mathcal {V}}^{1}{\mathcal {F}})=\rho ({\mathcal {F}}{\mathcal {V}}^{1})\).
Let us consider the Lyapunov function \({\mathcal {L}}=w({\mathcal {V}})^{1}{\textbf {X}}.\) Differentiate \({\mathcal {L}}\) along solutions of system (1), we have
By \({\mathcal {R}}_{0}<1\), \({\mathcal {L}}'\le 0\). Moreover, \({\mathcal {L}}'= 0\) if and only if \(w {\textbf {X}}=0\). Since at least one entry of w is positive, if \(w_1>0\), then \(E=0.\) The second equation in (40) implies that \(I=0\). Using the third equation in (40), we have \(Z=0\). By \({\mathcal {R}}_{0}=\max \{{\mathcal {R}}_{0b},{\mathcal {R}}_{0w}\}<1\), \({\mathcal {R}}_{0w}<1\) and hence the fourth and fifth equations imply the only solution to \(\kappa T_0VqT^*=pT^*cV=0\) is \(T^*=V=0\). In view of the first, fourth and sixth equations of the full system (1), \(S=S_0=\Lambda /\mu \), \(R=0\) and \(T=T_0=b/d\). Similarly, if \(w_k>0\) for some \(2\le k\le 5\), one can verify that \({\mathcal {L}}'= 0\) implies \(S=S_0\), \(T=T_0\) and \(E=I=Z=T^*=V=0\). This shows that the largest invariant set where \({\mathcal {L}}'= 0\) is the singleton \({\mathcal {E}}_0\). By LaSalle Invariance Principle, the DFE \({\mathcal {E}}_0\) is globally asymptotically stable in \(\Omega \) when \({\mathcal {R}}_{0}<1\). \(\square \)
To establish the global stability of the EE \({\mathcal {E}}_e\), denoted by \((\tilde{S}, \tilde{E},\tilde{I}, \tilde{R},\tilde{Z}, \tilde{T}, \tilde{T}^*, \tilde{V})\), we further assume that, for \(E,I,Z,V>0\),

(A1)
\(\left( 1\frac{\beta _E(V, E)E}{\beta _E(\tilde{V}, \tilde{E})\tilde{E}}\right) \left( 1\frac{\beta _E(\tilde{V}, \tilde{E})\tilde{E}V}{\beta _E(V, E)E \tilde{V}}\right) \ge 0\);

(A2)
\(\left( 1\frac{\beta _I(V, I)I}{\beta _I(\tilde{V}, \tilde{I})\tilde{I}}\right) \left( 1\frac{\beta _I(\tilde{V}, \tilde{I})\tilde{I}V}{\beta _I(V, I)I \tilde{V}}\right) \ge 0\);

(A3)
\(\frac{d}{dZ}(\beta _Z(Z)Z)\ge 0\), which is the same as (C3);

(A4)
\(\left( 1\frac{\eta (Z)}{\eta (\tilde{Z})}\right) \left( 1\frac{\eta (\tilde{Z})Z}{\eta (Z)\tilde{Z}}\right) \ge 0\).
Theorem 7
Let \(\overset{\circ }{\Omega }\) denote the interior of \(\Omega .\) and suppose that assumptions (A1)(A4) hold. If \({\mathcal {R}}_0>1\), then the EE \({\mathcal {E}}_e\) of the full system is globally asymptotically stable in \(\overset{\circ }{\Omega }\).
Proof
For simplicity, we denote \({\tilde{\beta }}_E=\beta _E(\tilde{V}, \tilde{E})\), \({\tilde{\beta }}_I=\beta _I(\tilde{V}, \tilde{I})\), \({\tilde{\beta }}_Z=\beta _Z(\tilde{Z})\), and \({\tilde{\eta }}=\eta (\tilde{Z})\). Consider the following Lyapunov function
with \(k_{S}=k_E=c_1\), \(k_I=c_2\), \(k_Z=c_3\), \(k_{T}=k_{T^*}=\epsilon \) and \(k_{V}=\epsilon m\), where \(c_i\) (\(1\le i\le 3\)) are positive constants to be be determined later and \(m=q/p\). It is easy to see that \({\mathcal {L}}\ge 0\) and \({\mathcal {L}}=0\) if and only if \(x=\tilde{x}\) for \(x=S, E, I, Z, T, T^*,V\).
Differentiating (42) along the solution of the full system (1) yields
Let \(\ell (x,\tilde{x})=\dfrac{x}{\tilde{x}}+\ln \dfrac{x}{\tilde{x}}\) for \(x>0\) and \(\tilde{x}>0\). By the similar arguments as that in the proof of Theorems 2 and 5, we find that
By (A1)(A4), (H2) and the arithmeticgeometric mean inequality,
Set
it follows from (11) that with the chosen \(c_1\), \(c_2\) and \(c_3\), the righthand side of (43) is zero. This shows that \({\mathcal {L}}'\le 0\) in \(\overset{\circ }{\Omega }\). Additionally, it is easy to verify that \({\mathcal {L}}'=0\) if and only if \(x=\tilde{x}\) for \(x=S, E, I,R, T, T^*, V\), which implies that the largest invariant set where \({\mathcal {L}}'=0\) is the singleton \({\mathcal {E}}_e\). It follows from the LaSalle Invariance Principle (Salle 1976) that \({\mathcal {E}}_e\) is globally asymptotically stable in \(\overset{\circ }{\Omega }\). \(\square \)
Model Validation and Simulation
In this section, we fit the multiscale model (1) to both the withinhost and betweenhost data of COVID19. On the basis of the fitting and parameter estimation, we evaluate the influence of a few infection characteristics and treatment on the disease dynamics. The viral load data we used to fit are the mean viral load measured by reverse transcription quantitative polymerase chain reaction (RTqPCR) from samples of posterior oropharyngeal saliva of 23 patients admitted to the Princess Margaret hospital and Queen Mary hospital in Hong Kong between Jan 22 and Feb 12, 2020 (To et al. 2020). The data of COVID19 cases are from the Johns Hopkins University Coronavirus Resource Center, which tracks COVID19 cases through a mapbased dashboard and is updated multiple times per day. We used the data of confirmed cases in Florida between March 13th and August 17th, 2020 (https://coronavirus.jhu.edu/region/us/florida).
Data Fitting
In model (1), we chose the functions \(\beta _E(V, E)\), \(\beta _I(V, I)\), \(\beta _Z(Z)\) and \(\eta (Z)\) as below. The choice of these functions was inspired by Yang and Wang (2020). We assumed that they have a base transmission rate, increase as the viral load increases, and decrease as the level of infection increases.
The initial conditions of the variables are set to S(0)=\(2\times 10^{7}\), \(E(0)=0\), \(I(0)=52\), \(R=(0)\), \(Z(0)=0\), \(T(0)=6\times 10^{4}\) cells/ml (Wang et al. 2020), \(T^*(0)=0\), \(V(0)=10^{5}\) copies/ml. We estimated parameters \(C_0\), \(C_1\), \(C_2\), \(\eta _0\), \(\kappa \), p, c, \(\epsilon \) by fitting the model to the viral load data and COVID19 case data simultaneously. The other parameters are fixed according to either the literature or our best estimates. The viral shedding rate by presymptomatic infected persons was estimated to be \(\xi _E= 2.3\) per day (Yang and Wang 2020). The viral load is higher during the early stage of infection (Wang et al. 2020). Thus, we assumed the viral shedding rate by symptomatic infected persons to be smaller and chose \(\xi _I= 1.5\) per day. The values of \(\beta _{E_0},\beta _{I_0}\) and \(\beta _{Z_0}\) are small (Yang and Wang 2020) and chosen to be on the order of \(10^{7}\) to \(10^{6}\). Because the turnover of the target cells of SARSCoV2 infection is slow (Wang et al. 2020), we assumed that both the generation and death rate of target cells (b and d) are 0.01 per day. The constant \(C_3\) in the transmission function \(\beta _Z\) is fixed to \(10^3\) and the influence of its variation on the dynamics will be investigated later.
We fit the model to both the viral load and confirmed case data simultaneously using the R programming language. The root mean square (RMS) between the model prediction and the data is calculated and parameter estimates are based on the best fitting that achieves the minimum RMS. Figure 2 shows that the model provides a good fit to both the withinhost and betweenhost data. The fitted parameters and other fixed parameters are listed in Table 3. It should be noted that the estimate of the scaleadjusting parameter \(\epsilon \) might be different for variants such as delta and omicron, compared with the wild type. However, the difference in the time scales between viral strains should be much smaller than that between the withinhost and betweenhost scales. We also note that some nonpharmaceutical control measures have been implemented during the period of data collection. These measures were not explicitly included in the model but their effect may have been reflected in the transmission rate between hosts.
The Influence of Environment, Incubation and Treatment
On the basis of the data fitting and parameter estimates, we performed numerical simulations to investigate the influence of a few factors, including the environment, incubation, and potential antiviral treatment, on the infection dynamics within individuals and the spread of the disease at the population level. Recall that \(\eta (Z)=\eta _0Z\) represents the contribution to the viral load within symptomatic infected individuals from the environment. We simulated the model with different values of \(\eta _0/\epsilon \). The predicted dynamics are very sensitive to the change of \(\eta _0/\epsilon \) (Fig. 3). For example, when \(\eta _0/\epsilon \) increases slightly from 0.01 to 0.012, the peaks of symptomatic infected I, presymptomatic infected E and environmental virus Z increase significantly and the time to reach the peaks is largely shortened (Fig. 3). In all the cases, the disease persists at the population level and the viral load within symptomatic infected individuals declines to a very low level (i.e., \({\mathcal {R}}_{0w}<1\)) but does not go to zero due to the contribution from environmental transmission. Without environmental transmission (i.e., \(\eta _0=0\)), the viral load will decline to zero quickly ( Fig. 7).
It should be noted that the final states shown in Fig. 3 are not the steady states. The simulation over a longer time interval reaching the steady state is shown in Appendix Fig. 8. We further simulated the model with different values of \(C_3\) (the constant in the environmental transmission function \(\beta _Z(Z)\)). As \(C_3\) increases to \(10^3\) (i.e., the value we fixed in the data fitting Fig. 2 and simulation in Fig. 3), the time for the system to stabilize increases (Appendix Fig. 9). This explains why under the parameter estimates from data fitting it takes a long time for the variables to reach the steady states after some oscillations, as shown in Appendix Fig. 8.
The duration of incubation is an important clinical characteristic in symptomatic infected disease surveillance, prevention and control (Lessler et al. 2009). Some studies showed that the incubation duration ranges from 1.8 to 12.8 days for COVID19 (Leung 2020; Ki 2020; Jiang et al. 2020). In Fig. 4, we simulated model (1) using different values of \(\alpha \) to investigate how the incubation duration (i.e., \(1/\alpha \)) influences the dynamics of COVID19. As the duration of incubation increases, more infections are observed and the time to reach the epidemic peak is also shortened. This is not surprising as COVID19 patients can transmit SARSCoV2 during the incubation period. This result highlights the need of identifying infected but still asymptomatic people in the early stage of infection.
Several drugs that target different aspects of COVID19 pathogenesis have been proposed. Some have been approved by the FDA while others are currently being tested in clinical trials of different stages (Sanders et al. 2020). In our model, we assumed that when COVID19 patients are treated with antiviral therapies, the viral production rate p is reduced to \((1\epsilon _t)p\), where \(\epsilon _t\) is the treatment efficacy. The simulation in Fig. 5 shows that antiviral treatment can significantly delay the spread of the disease among population. However, it does not change the final steady state. This is because the viral load declines to a very low level (see Appendix Fig. 8) even without treatment (i.e., the withinhost basic reproduction number \({\mathcal {R}}_{0w}<1\) with the parameter values obtained from data fitting). “Perfect” therapy that can completely block viral replication cannot reduce the duration of viral persistence within symptomatic infected individuals or diminish the time to recovery (see \(\epsilon _t=1\) in Appendix Fig. 10). This is different from the prediction by a model that only considered the withinhost dynamics (Wang et al. 2020). The discrepancy is due to the viral transfer from the environment to the host in the multiscale model. If the symptomatic infected individual is isolated after infection, then the individual does not have a chance to contract environmental virus. In this scenario, antiviral treatment will accelerate viral eradication and diminish the recovery time in the symptomatic infected individual (Wang et al. 2020). In summary, our simulation shows that antiviral treatment only delays the occurrence of the disease outbreak. It would not control the disease if the environmental transmission to hosts cannot be prevented.
Bistability
Table 2 shows that the model has two steady states, \({\mathcal {E}}_{0}\) and \({\mathcal {E}}_{et}\), when \({\mathcal {R}}_{0b}^c<{\mathcal {R}}_{0b}<1\), \({\mathcal {R}}_{0w}<1\) and \(\psi '(0)< m\). Although it is challenging to perform a formal mathematical analysis on the stability of the steady state \({\mathcal {E}}_{et}\) in this case, numerical simulation with different initial conditions suggests that \({\mathcal {E}}_{et}\) is locally asymptotically stable (Appendix Fig. 11). Furthermore, using initial conditions of the susceptible and symptomatic infected as an example, we identified the region in which the model converges to the diseasefree steady state \({\mathcal {E}}_{0}\) or the endemic steady state \({\mathcal {E}}_{et}\) (Fig. 6). This shows that the multiscale model exhibits a phenomenon of bistability. With lower levels of initial susceptible and symptomatic infected people, the disease is predicted to die out; when either the number of susceptible or symptomatic infected people exceeds a threshold value, the disease will persist.
Discussion
A large number of mathematical models have been developed to study the dynamics of COVID19 (for example, see Kucharski et al. 2020; Li et al. 2020; Liu et al. 2020c; Tang et al. 2020; Weitz et al. 2020; Wu et al. 2020a; Zhao and Feng 2020). However, most of the models are focused on the disease transmission at the population level. Several models have been used to study the withinhost virus dynamics (Wang et al. 2020; HernandezVargas and VelascoHernandez 2020; Gonçalves et al. 2020; Sanche et al. 2020; Goyal et al. 2020). We have not seen any models developed to investigate the coupled dynamics of COVID19. In this paper, we developed a multiscale model to study the dynamics of COVID19 at both the withinhost and betweenhost levels. In addition to the assumption that the transmission rate between hosts depends on the viral load within symptomatic infected individuals in some coupled models for other infectious diseases (Tuncer et al. 2016; Childs et al. 2019; Martcheva et al. 2015; Barfield et al. 2018; NikinBeers et al. 2018; Dorratoltaj et al. 2017; Numfor et al. 2014; Feng et al. 2015, 2012, 2013; Cen et al. 2014), we introduce a function \(\eta (Z)\) into the equation of viral load to represent the inhalation/ingestion rate of the coronavirus from the environment into the human body. This represents the contribution of the environmental reservoirs to the growth of the viral load within individual hosts. Therefore, our model provides a twoway connection between the individual viral kinetics and the populationlevel disease transmission. In comparison with the abovementioned coupled models, the transmission rate between hosts in this model was assumed to also depend on the level of infection because a higher level of infection would lead to stronger control measures and consequently reduce the transmission rate. The model is also different from that in Feng et al. (2015, 2013), in which the authors studied a coupled model specifically for an environmentallydriven infectious disease. There was no infection between infected and susceptible individuals considered in that model.
Our analysis approach for the subsystem (i.e., slow or fast subsystem) is similar to that in previous papers including (Yang and Wang 2020). Mathematical analysis of the coupled model is more challenging. We have analyzed the model using the bifurcation theory and a fastslow analysis, where the latter approach is built upon the time scale separation for the dynamical processes associated with the individual host and the environment and human population. Our detailed analysis on the local and global dynamics of the fast system, the slow system and the coupled system shows that rich dynamics, including both forward and backward bifurcations, emerge with the coupling of the withinhost and betweenhost models. Numerical simulation illustrated the bifurcations (Fig. 1). We have also numerically identified a region in which the disease either persists or dies out (Fig. 6). The emergence of bistability further suggests that the initial populations of susceptible or symptomatic infected can dictate the fate of disease transmission. These bifurcation and bistability results indicate the significance of the proposed coupled modeling approach and highlights the challenges in the prevention and control of the ongoing pandemic.
Both mathematical analysis and the simulation in Fig. 3 suggest that the environmental transmission can contribute to the COVID19 outbreak and persistence. Potential antiviral treatment can delay the disease outbreak but cannot prevent its emergence. Even when treatment significantly suppresses the viral production within the infected individual, the virus can persist in the host (Fig. 5). This may sound bizarre for a specific individual but it actually makes sense for the conceptual “average” individual considered in the model. If the disease persists at the population level, then the virus will also persist within the average individual due to environmental transmission. Although environmental transmission can contribute to viral persistence within the average host when the disease persists at the population level, it does not mean that environmental transmission plays a dominant role in the disease spread. It depends on the relative magnitude of the transmission terms \(\beta _ESE\), \(\beta _ISI\), and \(\beta _ZSZ\). Most spread of COVID19 might go through airborne humantohuman transmission (Zhang et al. 2020). Therefore, comprehensive intervention measures such as social distancing and wearing masks, combined with surface disinfection and hand hygiene that can prevent environmental transmission, should be implemented to mitigate the spread of COVID19 (Pradhan et al. 2020). Our study provides a modeling framework that can be used to evaluate the potential influence of environmental transmission on the disease dynamics.
This is the first attempt to link withinhost and betweenhost dynamics of COVID19. The validation of multiscale models is usually very challenging as it requires data from all the scales for the same individual or the same region. In this paper, we used the mean viral load of the patients from Hong Kong to represent the withinhost dynamics of symptomatic infected individuals and the accumulated cases from Florida to represent the betweenhost dynamics. Although patients in Florida may have different withinhost dynamics, we can use the above as an example to study the mutual impact of the dynamics at one level on the other. Another limitation of our multiscale model is that it assumes individual hosts have the same internal states at a time (i.e., the withinhost model is for a conceptual “average” individual). To overcome this limitation, one can couple the withinhost and betweenhost dynamics via the infection age of individuals using the nested modeling approach (Gilchrist and Sasaki 2002; Martcheva et al. 2015), which allows incorporating the staged progression nature of the disease at the population level. There are also some other potential ways to account for the individual withinhost heterogeneity in the model, such as adding a subscript i on the withinhost variables T, \(T^*\), and V for each individual i or discretizing the population with the level of the viral load as in a recent study (Lin et al. 2021). A more complex model incorporating all the individual heterogeneity will be extremely challenging, if not impossible, for mathematical analysis of the model. Lastly, we used a simple model at each scale to make it analytically more trackable. Immune response was shown to play a significant role in controlling viral replication within infected individuals (Wang et al. 2020; Goyal et al. 2020). The withinhost model can be extended to include immune responses but we speculate that it may not have a significant impact on the disease spread at the population level (note that the viral load is suppressed to a very low level in our simulations). A betweenhost model including more compartments, spatial heterogeneity, stochasticity, and vaccination status would be more realistic to study the disease spread. In a recent work (Lin et al. 2021), Lin et al. used a regionspecific model to study the COVID dynamics in a number of populous metropolitan statistical areas in the USA.
To summarize, we have developed a multiscale model to study the interaction between withinhost viral replication, the environment, and the disease spread at the population level. The analysis has generated some insights that would not be obtained if the model only considers withinhost or betweenhost dynamics. This work also provides a modeling framework for studying other infectious diseases when the environment can serve as a reservoir of pathogens.
Notes
If \(u{\textbf {Y}} =0\), then \({\textbf {Y}}=0\) and hence \(L'=0\). Conversely, suppose that \(L'=0\). Then \(0=L'\le ({\mathcal {R}}_{0b}1)u {\textbf {Y}}\) yields \(({\mathcal {R}}_{0b}1)u {\textbf {Y}}\ge 0\). Meanwhile, it follows from \({\mathcal {R}}_{0b}<1\) that \(({\mathcal {R}}_{0b}1)u {\textbf {Y}}\le 0\). Thus,
$$\begin{aligned} ({\mathcal {R}}_{0b}1)u {\textbf {Y}}= 0. \end{aligned}$$By the hypothesis, \({\mathcal {R}}_{0b}1<0\) and this implies \(u {\textbf {Y}}= 0\). We show that \(L'=0\) if and only if \(u{\textbf {Y}} =0\).
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J. Wang is supported by the NSF Grant DMS1951345 and DMS1913180. L. Rong is supported by the NSF Grant DMS1950254.
Appendices
Appendix
A Proof of Theorem 2
Proof
Consider the Lyapunov function
where \(c_1 \) and \(c_2\) are positive constants to be specified later. It is clear that \({\mathcal {L}}^s\ge 0\) and \({\mathcal {L}}^s=0\) if and only if \((S,E,I,Z)=(S^*,E^*,I^*,Z^*)\). For simplicity, we denote \(\beta _W^*=\beta (f(Z^*), W^*)\) for \(W=E, I\) and \(\beta _Z^*=\beta (Z^*)\). Differentiating \({\mathcal {L}}^s\) along solutions of system (13) in the main text and using the associated equilibrium equations to simplify, we have
By straightforward calculation, we obtain
Hence,
In view of (C1)(C3) and (H2),
Set \(c_1=\beta _I^*I^* S^*+\dfrac{\xi _I I^*\beta _Z^*Z^*S^*}{\beta _E^*E^*+\beta _I^*I^*}\) and \(c_2=\dfrac{\beta _Z^*Z^*S^*}{\beta _E^*E^*+\beta _I^*I^*}\). As \(\beta _E^*E^*+\beta _I^*I^*=\delta Z^*\), \(c_1=\beta _I^*I^* S^*+\xi _I I^*\beta _Z^*S^*/\delta \) and \(c_2=\beta _Z^*S^*/\delta \). One can verify that the righthand side of the above inequality is zero with the chosen \(c_1\) and \(c_2\). This implies \({d{\mathcal {L}}^s}/{dt}\le 0\) in the interior of \(\Omega ^s\). Furthermore, if \({d{\mathcal {L}}^s}/{dt}=0\), then
for some positive constant k. Substituting this relationship into the first two equations of system (13) in the main text leads to
which implies \(k=1\). Therefore, the only invariant set on which \({d{\mathcal {L}}^s}/{dt}=0\) contains the singleton \({\mathcal {E}}_e^s=\{S^*, E^*, I^*, R^*, Z^*\}\). It follows from the LaSalle Invariant Principle that \({\mathcal {E}}_e\) is globally asymptotically stable in the interior of \(\Omega ^s\) when \({\mathcal {R}}_0>1\). Consequently, \({\mathcal {E}}_e\) is also unique in the interior of \(\Omega ^s\). \(\square \)
B Appendix Figures
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Wang, X., Wang, S., Wang, J. et al. A Multiscale Model of COVID19 Dynamics. Bull Math Biol 84, 99 (2022). https://doi.org/10.1007/s11538022010588
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DOI: https://doi.org/10.1007/s11538022010588
Keywords
 COVID19
 Multiscale model
 Withinhost dynamics
 Betweenhost dynamics
Mathematics Subject Classification
 37N25
 34D20
 92D30