mjmsMeridian Journal of Mathematical SciencesMeridian J. Math. Sci.2471-35622471-3643Meridian Academic Press10.58301/skp6nh92Investigation of a Two-Patch Within-Host Model of Hepatitis B Viral Infection: a supplementary reportYılmazEmreVisualizationInvestigationhttps://orcid.org/0015-0000-8481-249XOseiNalediLemaireLéaSupervisionSoftwareMethodologyValidationSchool of Public Health, University of Eastfield, United KingdomSchool of Economics, Meridian National University, SingaporeDepartment of Chemistry, Sapienza Mediterranea, ItalyLindqvistGustavAcademic Editor
Chronic infection with hepatitis B virus (HBV) can lead to formation of abnormal nodular structures within the liver. To address how changes in liver anatomy affect overall virus-host dynamics, we developed within-host ordinary differential equation models of two-patch hepatitis B infection, one that assumes irreversible and one that assumes reversible movement between nodular structures. We investigated the models analytically and numerically, and determined the contribution of patch susceptibility, immune responses, and virus movement on within-patch and whole-liver virus dynamics. We explored the structural and practical identifiability of the models by implementing a differential algebra approach and the Monte Carlo approach for a specific HBV data set. We determined conditions for viral clearance, viral localization, and systemic viral infection. Our study suggests that cell susceptibility to infection within modular structures, the movement rate between patches, and the immune-mediated infected cell killing have the most influence on HBV dynamics. Our results can help inform intervention strategies.
Keywords Hepatitis B virusAsymptotic dynamicsIdentifiability analysisTwo-patch modelOrdinary differential equationsThe authors acknowledge support from the Wellcome Trust (grant 215801/Z/19/Z) and the National Institute for Health and Care Research.Introduction
Chronic hepatitis B, caused by the hepatitis B virus (HBV), remains a major health problem, affecting more than 257 million carriers globally (Schweitzer et al. 2015; Trépo et al. 2014). HBV is transmitted vertically from mother to child, through sexual contact, and after exposure with infected blood or body fluids (Guidotti and Chisari 2006). The risk of progression from acute to chronic disease is age dependent, with adult immunocompetent patients clearing the virus and infants and children developing chronic infection, followed by liver disease such as cirrhosis and hepatocellular carcinoma (Lee 1997; Locarnini et al. 2015). Chronic hepatitis B is rarely cured with available therapeutics (Dusheiko 2013; Pawlotsky 2023), hence the need for interdisciplinary studies where mathematical modeling and empirical data can merge to better describe the disease evolution following hepatitis B viral infection.
Current mathematical models describing HBV dynamics from acute to chronic infection are based on classical within-host models (Nowak et al. 1996) (with variations to account for hepatitis B virus characteristics), and are mainly validated with HBV DNA titers in the serum of infected patients and chimpanzees (Ciupe, Ribeiro, Nelson, Dusheiko, and Perelson 2007; Ciupe et al. 2014; Ciupe, Ribeiro, Nelson, and Perelson 2007; Murray et al. 2005; Dahari et al. 2009; Neumann et al. 2010; Goyal et al. 2017; Tsiang et al. 1999; Ciupe et al. 2024, 2021). Recently, multiscale mathematical models have been developed, with the aim of integrating intracellular aspects of hepatitis B replication into the cellular dynamics (Reinharz et al. 2021; Ciupe et al. 2024; Hailegiorgis et al. 2023; Shekhtman et al. 2024; Kadelka et al. 2021). They are validated using in-vitro and humanized chimeric or transgenic mice data, and have been instrumental in addressing the role of immune system (both cytolytic killing and non-cytolytic cure) on viral resolution, and the role of viral markers (such as the s- and e-antigens, and the covalently closed circular DNA) on viral persistence (Shekhtman et al. 2024; Kadelka et al. 2021; Ciupe et al. 2024).
One aspect of chronic hepatitis B that is less studied, using mathematical models, concerns the effect that hepatitis B virus has on the reshaping of the liver as the disease progresses from healthy, to fibrotic, to cirrhotic (Fig. 1A). The progression to fibrotic and cirrhotic disease is driven by repeated cycles of inflammatory responses, liver cells (hepatocytes) death, hepatocytes regeneration, and accumulation of fibrous tissue (Roehlen et al. 2020; Yuan et al. 2019). All these processes lead to the activation of collagen-producing cells, resulting in excessive accumulation of extracellular matrix, the formation of abnormal nodular structures, overall liver dysfunction, and often the need for liver transplantation (Bataller et al. 2005).
The death of hepatocytes, due to chronic hepatitis B infection, has been shown to trigger compensatory proliferation of the remaining hepatocytes (Michalopoulos and Bhushan 2021). Fibrosis and cirrhosis affect viral replication, with increased liver stiffness that promotes the progression of hepatitis B disease (Cast et al. 2019). In this paper, we investigated whether the HBV persistence is influenced by differences in hepatocyte turnover (replication and death) within the nodular structures formed during fibrosis and cirrhosis (Fig. 1B, right panel). To address this, we developed two-patch mathematical models of heterogeneous hepatitis B virus replication and used them to determine how patch-specific hepatocyte susceptibility, immune response-induced hepatocyte death, and HBV travel rates between patches influence viral persistence. The results can help elucidate how the host-virus interactions in a heterogeneous architecture influence the transition between viral resolution and viral persistence.
Proposed model of liver disease evolution following hepatitis B viral infection: (A) Liver transition from healthy to fibrosis to cirrhosis; (B) Hepatitis B virus localization during acute and chronic stages of hepatitis B viral infection. During chronic disease, liver is restructured into abnormal nodular structures separated by collagen (yellow streaks, right panel). Figure created in https://BioRender.com
Mathematical Model Development
Single-patch model of hepatitis B virus infection. The mathematical model of homogeneous hepatitis B virus infection considers the interaction between uninfected liver cells T, infected liver cells I, and hepatitis B virus V. It assumes that uninfected liver cells are produced at rate s, die at natural death rate d, and are infected at rate β β. Infected liver cells die at immune-response mediated rate δ δ (≥ d≥d) and produce new virus at rate p. Virus is cleared at rate c. The mathematical model is given by:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \frac{dT}{dt}&=s- d T -\beta T V,\\ \frac{dI}{dt}&=\beta T V -\delta I,\\ \frac{dV_1}{dt}&=p I -c V,\\ \end{aligned} \end{aligned}$$\end{document}dTdt=s-dT-βTV,dIdt=βTV-δI,dV1dt=pI-cV,with initial conditions \documentclass[12pt]{minimal}
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\begin{document}$$T_1(0)=s/d$$\end{document}T1(0)=s/d, \documentclass[12pt]{minimal}
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\begin{document}$$I(0)=I_0$$\end{document}I(0)=I0, \documentclass[12pt]{minimal}
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\begin{document}$$V(0)=V_0$$\end{document}V(0)=V0. This model was first used (in the context of hepatitis B virus infection) by Nowak et al. (1996) and utilized further in Ciupe, Ribeiro, Nelson, and Perelson (2007); Murray et al. (2005); Dahari et al. (2009); Tsiang et al. (1999); Neumann et al. (2010); Ciupe et al. (2024); K. Wang et al. (2010); Yousfi et al. (2011); Meena et al. (2023); Hews et al. (2010); Lewin et al. (2001). Its corresponding basic reproduction number (representing the average number of new viruses or infected cells generated by a single virus in a naive host) is given by\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}R_0=\frac{\beta p}{c\delta }\frac{s}{d}.\end{aligned}$$\end{document}R0=βpcδsd.It is easy to show that virus is cleared when \documentclass[12pt]{minimal}
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\begin{document}$$R_0<1$$\end{document}R0<1 and persists when \documentclass[12pt]{minimal}
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\begin{document}$$R_0>1$$\end{document}R0>1 (Ciupe and Heffernan 2017). In this study, we expanded model Eq. 1 by taking into account the anatomy of the liver during HBV-induced liver disease. It has been shown that, following hepatitis B infection, liver is restructured into abnormal nodular structures separated by collagen (Fig. 1A and B, right panels). To address differences in virus-host interactions in nodular structures, we developed a two-patch mathematical model and used it to investigate how heterogeneity in infection changes the conditions for viral clearance and viral persistence. Special attention was given to parameters describing liver injury, δ δ, and liver regeneration, s. This two-patch model is a first step in including heterogeneity, with multi-patch models being the obvious next step.
Two-patch model of hepatitis B virus infection. We expanded model Eq. 1 to account for virus dynamics in two liver nodular structures, obtaining a two-patch model of HBV infection. In each nodular structure, we modeled the interaction between uninfected liver cells \documentclass[12pt]{minimal}
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\begin{document}$$T_j$$\end{document}Tj, infected liver cells \documentclass[12pt]{minimal}
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\begin{document}$$I_j$$\end{document}Ij, and hepatitis B virus \documentclass[12pt]{minimal}
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\begin{document}$$V_j$$\end{document}Vj, with \documentclass[12pt]{minimal}
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\begin{document}$$j=\{1,2\}$$\end{document}j={1,2}. As before, we assumed that uninfected liver cells are produced at rates \documentclass[12pt]{minimal}
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\begin{document}$$s_j$$\end{document}sj (nodular structure-dependent), die at natural death rate d, and get infected at rate β β (nodular structure-independent). Infected liver cells are killed at rate δ δ and produce new virus at rate p (nodular structure-independent). Virus is cleared at rate c (nodular structure-independent). Lastly, we assumed that HBV moves between the nodular structures at rates \documentclass[12pt]{minimal}
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\begin{document}$$\phi _{ij}$$\end{document}ϕij (nodular structure-dependent), for \documentclass[12pt]{minimal}
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\begin{document}$$i,j=\{1,2\}$$\end{document}i,j={1,2} and i≠ ji≠j. The mathematical model becomes:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \frac{dT_1}{dt}&=s_1- d T_1 -\beta T_1 V_1,\\ \frac{dI_1}{dt}&=\beta T_1 V_1 -\delta I_1,\\ \frac{dV_1}{dt}&=p I_1 -c V_1+\phi _{21}V_2-\phi _{12} V_1,\\ \frac{dT_2}{dt}&=s_2- d T_2 -\beta T_2 V_2,\\ \frac{dI_2}{dt}&=\beta T_2 V_2-\delta I_2,\\ \frac{dV_2}{dt}&=p I_2 -c V_2-\phi _{21}V_2+\phi _{12} V_1,\\ \end{aligned} \end{aligned}$$\end{document}dT1dt=s1-dT1-βT1V1,dI1dt=βT1V1-δI1,dV1dt=pI1-cV1+ϕ21V2-ϕ12V1,dT2dt=s2-dT2-βT2V2,dI2dt=βT2V2-δI2,dV2dt=pI2-cV2-ϕ21V2+ϕ12V1,with initial conditions \documentclass[12pt]{minimal}
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\begin{document}$$T_1(0)=s_1/d$$\end{document}T1(0)=s1/d, \documentclass[12pt]{minimal}
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\begin{document}$$T_2(0)=s_2/d$$\end{document}T2(0)=s2/d, \documentclass[12pt]{minimal}
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\begin{document}$$V_1(0)=V_0$$\end{document}V1(0)=V0, \documentclass[12pt]{minimal}
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\begin{document}$$I_1(0)=I_0$$\end{document}I1(0)=I0, \documentclass[12pt]{minimal}
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\begin{document}$$I_2(0)=V_2(0)=0$$\end{document}I2(0)=V2(0)=0. This means that the HBV infection starts in nodular structure 1.
We considered two special cases: (i) the one-directional two-patch model, which assumes that the HBV is seeded in nodular structure 1, moves into nodular structure 2, and never returns into nodular structure 1, i.e.\documentclass[12pt]{minimal}
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\begin{document}$$\phi _{12}=\phi $$\end{document}ϕ12=ϕ and \documentclass[12pt]{minimal}
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\begin{document}$$\phi _{21}=0$$\end{document}ϕ21=0; and (ii) the two-directional two-patch model, which assumes that the HBV is seeded in nodular structure 1, and moves at the same rates between the two patches, i.e.\documentclass[12pt]{minimal}
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\begin{document}$$\phi _{12}=\phi _{21}=\phi $$\end{document}ϕ12=ϕ21=ϕ.
Results for the One-Directional Two-Patch Model
The one-directional two-patch model, which assumes that the HBV is seeded in nodular structure 1, moves into nodular structure 2, and never returns into nodular structure 1, is described by:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \frac{dT_1}{dt}&=s_1- d T_1 -\beta T_1 V_1,\\ \frac{dI_1}{dt}&=\beta T_1 V_1 -\delta I_1,\\ \frac{dV_1}{dt}&=p I_1 -c V_1-\phi V_1,\\ \frac{dT_2}{dt}&=s_2- d T_2 -\beta T_2 V_2,\\ \frac{dI_2}{dt}&=\beta T_2 V_2-\delta I_2,\\ \frac{dV_2}{dt}&=p I_2 -c V_2+\phi V_1,\\ \end{aligned} \end{aligned}$$\end{document}dT1dt=s1-dT1-βT1V1,dI1dt=βT1V1-δI1,dV1dt=pI1-cV1-ϕV1,dT2dt=s2-dT2-βT2V2,dI2dt=βT2V2-δI2,dV2dt=pI2-cV2+ϕV1,
Asymptotic Analysis Results for the One-Directional Two-Patch Model Eq. <xref rid="Equ3" ref-type="disp-formula">3</xref>
We investigated the long-term behavior of the one-directional two-patch model Eq. 3, where HBV is seeded in nodular structure 1, moves into nodular structure 2, and never returns into nodular structure 1. Model Eq. 3 has three non-negative equilibrium solutions. The infection-free equilibrium, representing viral clearance in both patches, is given by:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} E_0=(T_1^0,I_1^0,V_1^0,T_2^0,I_2^0,V_2^0) = \left( \frac{s_1}{d},0,0,\frac{s_2}{d},0,0\right) , \end{aligned}$$\end{document}E0=(T10,I10,V10,T20,I20,V20)=s1d,0,0,s2d,0,0,and is always biologically realistic. The one-patch chronic equilibrium, representing virus clearance in nodular structure 1 and persistence in nodular structure 2, is given by:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} E_1=(T_1^1,I_1^1,V_1^1,T_2^1,I_2^1,V_2^1) = \left( \frac{s_1}{d},0,0,\frac{c \delta }{\beta p},\frac{c d}{\beta p}(R_{0}^{\text {1D}}-1),\frac{d}{\beta }(R_{0}^{\text {1D}}-1)\right) , \end{aligned}$$\end{document}E1=(T11,I11,V11,T21,I21,V21)=s1d,0,0,cδβp,cdβp(R01D-1),dβ(R01D-1),where:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_{0}^{\text {1D}}=\frac{\beta p s_2}{cd\delta }.\end{aligned}$$\end{document}R01D=βps2cdδ.Equilibrium \documentclass[12pt]{minimal}
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\begin{document}$$E_1$$\end{document}E1 is biologically realistic if and only if \documentclass[12pt]{minimal}
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\begin{document}$$R_{0}^{\text {1D}}>1$$\end{document}R01D>1. Lastly, model Eq. 3 has two two-patch chronic equilibria,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} E_2&=(T_1^2,I_1^2,V_1^2,T_2^2,I_2^2,V_2^2)\\&= \left( \frac{\delta (c+\phi )}{p\beta }, \frac{d(c+\phi )}{p\beta }(R_{\text {eff}}^{\text {1D}}-1),\frac{d}{\beta }(R_{\text {eff}}^{\text {1D}}-1),\frac{cd\delta (R_0^{\text {1D}}+1) + \phi \delta d(R_{\text {eff}}^{\text {1D}}-1) - \sqrt{D}}{2\beta dp} , \right. \\ &\left. \frac{cd\delta (R_0^{\text {1D}}-1) - \phi \delta d(R_{\text {eff}}^{\text {1D}}-1) + \sqrt{D}}{2\beta p \delta },\frac{cd\delta (R_0^{\text {1D}}-1) + \phi \delta d(R_{\text {eff}}^{\text {1D}}-1) + \sqrt{D}}{2\beta c\delta } \right) ,\\ \end{aligned} \end{aligned}$$\end{document}E2=(T12,I12,V12,T22,I22,V22)=δ(c+ϕ)pβ,d(c+ϕ)pβ(Reff1D-1),dβ(Reff1D-1),cdδ(R01D+1)+ϕδd(Reff1D-1)-D2βdp,cdδ(R01D-1)-ϕδd(Reff1D-1)+D2βpδ,cdδ(R01D-1)+ϕδd(Reff1D-1)+D2βcδ,and\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} E_3&=(T_1^3,I_1^3,V_1^3,T_2^3,I_2^3,V_2^3)\\&= \left( \frac{\delta (c+\phi )}{p\beta }, \frac{d(c+\phi )}{p\beta }(R_{\text {eff}}^{\text {1D}}-1),\frac{d}{\beta }(R_{\text {eff}}^{\text {1D}}-1),\frac{cd\delta (R_0^{\text {1D}}+1) + \phi \delta d(R_{\text {eff}}^{\text {1D}}-1) + \sqrt{D}}{2\beta dp} \right. , \\ &\left. \frac{cd\delta (R_0^{\text {1D}}-1) - \phi \delta d(R_{\text {eff}}^{\text {1D}}-1) - \sqrt{D}}{2\beta p \delta },\frac{cd\delta (R_0^{\text {1D}}-1) + \phi \delta d(R_{\text {eff}}^{\text {1D}}-1)- \sqrt{D}}{2\beta c\delta }\right) .\\ \end{aligned}\end{aligned}$$\end{document}E3=(T13,I13,V13,T23,I23,V23)=δ(c+ϕ)pβ,d(c+ϕ)pβ(Reff1D-1),dβ(Reff1D-1),cdδ(R01D+1)+ϕδd(Reff1D-1)+D2βdp,cdδ(R01D-1)-ϕδd(Reff1D-1)-D2βpδ,cdδ(R01D-1)+ϕδd(Reff1D-1)-D2βcδ.where\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_{\text {eff}}^{\text {1D}}=\frac{\beta p s_1}{d\delta (c+\phi )},\end{aligned}$$\end{document}Reff1D=βps1dδ(c+ϕ),and\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\begin{aligned} D&= d^2 \delta ^2 \Big ( [c(R_0^{1D} - 1) + \phi (R_{\textrm{eff}}^{1D} - 1)]^2 + 4c\phi \,(R_{\textrm{eff}}^{1D} - 1) \Big ). \end{aligned}\end{aligned}$$\end{document}D=d2δ2([c(R01D-1)+ϕ(Reff1D-1)]2+4cϕ(Reff1D-1)).Equilibrium \documentclass[12pt]{minimal}
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\begin{document}$$E_2$$\end{document}E2 is biologically realistic if and only if \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}_{\text {eff}}^{\text {1D}}>1$$\end{document}Reff1D>1. By contrast, \documentclass[12pt]{minimal}
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\begin{document}$$E_3$$\end{document}E3 is not biologically realistic since \documentclass[12pt]{minimal}
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\begin{document}$$I_2^3$$\end{document}I23 is always negative. Next, we studied the asymptotic stability of the equilibrium solutions \documentclass[12pt]{minimal}
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\begin{document}$$E_0$$\end{document}E0, \documentclass[12pt]{minimal}
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\begin{document}$$E_1$$\end{document}E1 and \documentclass[12pt]{minimal}
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\begin{document}$$E_2$$\end{document}E2.
Similarly, let\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} P_{22}(\lambda ) = \lambda ^3 + b_1\lambda ^2 + b_2\lambda + b_3, \end{aligned}$$\end{document}P22(λ)=λ3+b1λ2+b2λ+b3,where\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} b_1&= c + \delta + d + \beta (V_2^2)\\&= c + \delta + \frac{d(R_0^{\text {1D}} + 1)}{2} + \frac{\phi \delta d(R_{\text {eff}}^{\text {1D}} - 1) + \sqrt{D}}{2c \delta },\\ b_2&= (T_2^2) \beta p + (V_2^2) \beta c + (V_2^2) \beta \delta + cd + c \delta + d \delta \\&= \frac{c + \delta }{2c \delta }\left( cd \delta (R_0^{\text {1D}} - 1) + \phi \delta d(R_{\text {eff}}^{\text {1D}} - 1) + \sqrt{D}\right) + d \delta + cd + c \delta \\&\quad + \frac{1}{2d}\left( cd\delta (R_0^{\text {1D}} + 1) + \phi \delta d(R_{\text {eff}}^{\text {1D}} - 1) - \sqrt{D}\right) ,\\ b_3&= (T_2^2) \beta dp + (V_2^2) \beta c \delta + cd \delta \\&= cd \delta (R_0^{\text {1D}} + 1) + \phi \delta d(R_{\text {eff}}^{\text {1D}} - 1), \end{aligned}$$\end{document}b1=c+δ+d+β(V22)=c+δ+d(R01D+1)2+ϕδd(Reff1D-1)+D2cδ,b2=(T22)βp+(V22)βc+(V22)βδ+cd+cδ+dδ=c+δ2cδcdδ(R01D-1)+ϕδd(Reff1D-1)+D+dδ+cd+cδ+12dcdδ(R01D+1)+ϕδd(Reff1D-1)-D,b3=(T22)βdp+(V22)βcδ+cdδ=cdδ(R01D+1)+ϕδd(Reff1D-1),and \documentclass[12pt]{minimal}
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\begin{document}$$V_2^2$$\end{document}V22 and \documentclass[12pt]{minimal}
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\begin{document}$$T_2^2$$\end{document}T22 are the target cell and virus equilibrium values for the nodular structure 2 as given by Eq. 5. Polynomial \documentclass[12pt]{minimal}
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\begin{document}$$P_{22}(\lambda )$$\end{document}P22(λ) has negative roots or roots with negative real parts when \documentclass[12pt]{minimal}
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\begin{document}$$b_1 b_2>b_3$$\end{document}b1b2>b3. Note that when \documentclass[12pt]{minimal}
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\begin{document}$$R_{\text {eff}}^{\text {1D}}>1$$\end{document}Reff1D>1, \documentclass[12pt]{minimal}
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\begin{document}$$b_i>0$$\end{document}bi>0 for all i. Moreover,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} b_1 b_2 - b_3&= \beta ^2 (c + \delta ) V_2^2 + T_2 V_2 \beta ^2 p \\&\quad + V_2 \beta (c + \delta )(c+\delta +2d) \\&\quad + T_2\beta p(c + \delta ) \\&\quad + (c+\delta )(c+d)(\delta +d), \end{aligned} \end{aligned}$$\end{document}b1b2-b3=β2(c+δ)V22+T2V2β2p+V2β(c+δ)(c+δ+2d)+T2βp(c+δ)+(c+δ)(c+d)(δ+d),is positive when \documentclass[12pt]{minimal}
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\begin{document}$$R_{\text {eff}}^{\text {1D}}>1$$\end{document}Reff1D>1. Then, by the Routh-Hurwitz condition, \documentclass[12pt]{minimal}
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\begin{document}$$P_{22}(\lambda )$$\end{document}P22(λ) has negative roots or roots with negative real part when \documentclass[12pt]{minimal}
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\begin{document}$$R_{\text {eff}}^{\text {1D}}>1$$\end{document}Reff1D>1. This guarantees the local stability analysis of the chronic equilibrium \documentclass[12pt]{minimal}
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\begin{document}$$E_2$$\end{document}E2. Conversely, when \documentclass[12pt]{minimal}
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\begin{document}$$R_{\text {eff}}^{\text {1D}}<1$$\end{document}Reff1D<1, equilibrum \documentclass[12pt]{minimal}
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\begin{document}$$E_2$$\end{document}E2 does not exist. This concludes the proof. □ □
Structural Identifiability Results for Model Eq. <xref rid="Equ3" ref-type="disp-formula">3</xref>
We investigated the structural identifiability of model Eq. 3 using the differential algebra approach and the DAISY platform (Bellu et al. 2007) (see the Materials and Methods section for further details) under two assumptions: (i) viral load is measured in both patches and the initial conditions of model Eq. 3 are either known or unknown; and (ii) susceptible liver cells are measured in both patches and the initial conditions of model Eq. 3 are either known or unknown.
Proposition 4
When measurements for the \documentclass[12pt]{minimal}
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\begin{document}$$V_2(t)$$\end{document}V2(t) variables are given, system Eq. 3 is unidentifiable under unknown initial conditions. In particular, parameters \documentclass[12pt]{minimal}
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\begin{document}$$\pi _1=\{\beta , c, d, \delta , \phi \}$$\end{document}π1={β,c,d,δ,ϕ} are globally identifiable and parameters \documentclass[12pt]{minimal}
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\begin{document}$$\pi _2=\{s_1, s_2, p\}$$\end{document}π2={s1,s2,p} are unidentifiable. If, additionally, initial conditions are known, then all parameters are globally structurally identifiable.
Identifiability analysis for model Eq. 3, performed using the DAISY software (Bellu et al. 2007). For the simulations with initial conditions, all initial conditions are known
Observe States
Initial Conditions Known
Initial Conditions Unknown
Model with \documentclass[12pt]{minimal}
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\begin{document}$$V_{2}(t)$$\end{document}V2(t) data
Numerical Results for Model Eq. <xref rid="Equ3" ref-type="disp-formula">3</xref>
As seen above, under known initial conditions and unlimited noise-free measurements of both \documentclass[12pt]{minimal}
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\begin{document}$$T_2(t)$$\end{document}T2(t)), all parameters of model Eq. 3 can be identified from data.
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\begin{document}$$s=d\times (T_1(0)+T_2(0))$$\end{document}s=d×(T1(0)+T2(0)) is susceptible to HBV infection and considered three cases for the susceptible cells ratio within the two patches \documentclass[12pt]{minimal}
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\begin{document}$$s_1:s_2 =\{10:90, 50:50, 90:10\}$$\end{document}s1:s2={10:90,50:50,90:10} (see Table 2). To create a baseline model of a chronic infection, we fitted model Eq. 3 to data from an immunosupressed mice (Zhang et al. 2023) (see Materials and Methods). This initial fit is not intended as an exhaustive representation of HBV infection, but as a starting point for model parametrization. To account for the lack of immunity, we set δ =d=0.01δ=d=0.01 /day (Ciupe, Ribeiro, Nelson, and Perelson 2007) and fixed c=4.4c=4.4 /day, as in prior studies (Murray et al. 2005). We fitted the remaining parameters \documentclass[12pt]{minimal}
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\begin{document}$$\{\beta , p, \phi \}$$\end{document}{β,p,ϕ} using the fminsearchbnd function in Matlab2021a (see Materials and Methods section for details). A summary of estimates for each case is given in Table 2 and the model dynamics are plotted in Fig. 2.
The effects of patch-susceptibility on virus dynamics within patches when viral movement is one-directional. Dynamics of (A)-(E) uninfected (solid line) and infected (dashed line) liver cells in patch 1 (red lines) and patch 2 (blue lines) and (B)-(F) total virus (black line), virus in patch 1 (red line), virus in patch 2 (blue line) versus mice data (circles) as given by model Eq. 3 in case 1: (A)-(B); case 2: (C)-(D), case 3: (E)-(F). Model parameters are given in Table 2
Model fitting resulted in similar infectivity rates among the three patch recruitment cases, i.e.\documentclass[12pt]{minimal}
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\begin{document}$$\beta =3.3\times 10^{-9}$$\end{document}β=3.3×10-9 ml/(virion × × day), \documentclass[12pt]{minimal}
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\begin{document}$$\beta =3.13\times 10^{-9}$$\end{document}β=3.13×10-9 ml/(virion × × day), for cases 1, 2 and 3, respectively. Viral production rate increased by 21%21% and 14%14% in cases 2 and 3 compared to case 1, i.e.p=1203p=1203 virion/(ml× × day) and p=1137p=1137 virion/(ml× × day), versus p=998p=998 virion/(ml× × day). The biggest difference is in the estimate of the movement rate ϕ ϕ, which was 41- and 50-times higher in cases 2 and 3, compared to case 1, ϕ =4.1ϕ=4.1 /day and ϕ =5ϕ=5 /day, versus ϕ =0.1ϕ=0.1 /day. To determine if the results can be generalized when we add noise to the data, we performed practical identifiability analysis using Monte Carlo approaches (see Materials and Methods for details). We found that all parameters are strongly practically identifiable for all cases with the exception of parameter ϕ ϕ, which is weakly practically identifiable for case 1 and case 2; and parameter p, which is weakly practically identifiable for case 1 (see Table 5 in Materials and Methods).
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\begin{document}$$R_{eff}^{1D}=23.1$$\end{document}Reff1D=23.1 for case 3), which means virus persists in both patches (see Proposition 5).
The baseline parameters obtained through data fitting provide a static representation of HBV infection, characteristic of chronic infection in an immunocompromised individual. We next generalize and perturb these baseline parameters to account for key biological features, such as immune competence and virus localization. Our goal is to determine the conditions under which the virus is cleared either locally or globally. To acomplish that, we created one-dimensional bifurcation diagrams. We used the formulas for \documentclass[12pt]{minimal}
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\begin{document}$$s_2\in (0, 500)$$\end{document}s2∈(0,500). We computed \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{1D}$$\end{document}R01D and \documentclass[12pt]{minimal}
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\begin{document}$$R_{eff}^{1D}$$\end{document}Reff1D values. As shown analytically, we find that the clearance of virus in patch 1 occurs when \documentclass[12pt]{minimal}
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\begin{document}$$R_{eff}^{1D}<1$$\end{document}Reff1D<1, regardless of \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{1D}$$\end{document}R01D value (Fig. 4B), while clearance in patch 2 (and overall) requires both \documentclass[12pt]{minimal}
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\begin{document}$$R_{eff}^{1D}<1$$\end{document}Reff1D<1 and \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{1D}<1$$\end{document}R01D<1 (Fig. 4C and A).
One-dimensional bifurcation diagrams for model Eq. 3: \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{1D}$$\end{document}R01D. The other parameters are given in Table 2 for A - Bcase 1, C - Dcase 2 and E - Fcase 3
Two-dimensional bifurcation diagrams for model Eq. 3: A.\documentclass[12pt]{minimal}
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\begin{document}$$R_{eff}^{1D}$$\end{document}Reff1D and \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{1D}$$\end{document}R01D. The other parameters are given in Table 2, case 2
Results for the Two-Directional Two-Patch Model
The two-directional two-patch model, which assumes that the HBV is seeded in nodular structure 1, and moves at the same rates between the two patches can be described by:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \frac{dT_1}{dt}&=s_1- d T_1 -\beta T_1 V_1,\\ \frac{dI_1}{dt}&=\beta T_1 V_1 -\delta I_1,\\ \frac{dV_1}{dt}&=p I_1 -c V_1-\phi V_1 + \phi V_2,\\ \frac{dT_2}{dt}&=s_2- d T_2 -\beta T_2 V_2,\\ \frac{dI_2}{dt}&=\beta T_2 V_2-\delta I_2,\\ \frac{dV_2}{dt}&=p I_2 -c V_2+\phi V_1 - \phi V_2.\\ \end{aligned} \end{aligned}$$\end{document}dT1dt=s1-dT1-βT1V1,dI1dt=βT1V1-δI1,dV1dt=pI1-cV1-ϕV1+ϕV2,dT2dt=s2-dT2-βT2V2,dI2dt=βT2V2-δI2,dV2dt=pI2-cV2+ϕV1-ϕV2.
Asymptotic Analysis Results for the Two-Directional Two-Patch Model Eq. <xref rid="Equ9" ref-type="disp-formula">9</xref>
We next investigated the long-term behavior of model Eq. 9, where HBV is seeded in nodular structure 1 and moves at the same rate ϕ ϕ between the two patches. Model Eq. 9 has at most four equilibria. The infection-free equilibrium:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} E_4=(T_1^4,I_1^4,V_1^4,T_2^4,I_2^4,V_2^4) = (\frac{s_1}{d},0,0,\frac{s_2}{d},0,0), \end{aligned}$$\end{document}E4=(T14,I14,V14,T24,I24,V24)=(s1d,0,0,s2d,0,0),represents complete viral clearance in both patches, as before. The other three potential equilibria are of the form:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}E_j=(T_1^j,I_1^j,V_1^j,T_2^j,I_2^j,V_2^j),\end{aligned}$$\end{document}Ej=(T1j,I1j,V1j,T2j,I2j,V2j),\documentclass[12pt]{minimal}
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\begin{document}$$j\in \{5,6,7\}$$\end{document}j∈{5,6,7}, where\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} I_1^j&=\frac{\phi T_1^j(-T_1^j\beta ^2p^2s_2 + T_1^j\beta c d \delta p + T_1^j\beta d \delta p \phi + \beta c \delta p s_2+\beta \delta p \phi s_2 - c^2d\delta ^2 - 2cd\delta ^2\phi )}{\delta (\beta pT_1^j - c\delta - \delta \phi )(T_1^j\beta c p + \theta _ j\beta p \phi - c^2\delta - 2c\delta \phi )},\\ V_1^j&=\frac{\phi (-T_1^j\beta ^2p^2s_2+T_1^j \beta c d \delta p + T_1^j \beta d \delta p \phi + \beta c \delta p s_2 + \beta \delta p \phi s_2 - c^2d\delta ^2 - 2cd\delta ^2\phi )}{\beta (T_1^j\beta c p + T_1^j\beta p \phi - c^2 \delta - 2c\delta \phi )(\beta pT_1^j - c\delta - \delta \phi )},\\ T_2^j&= \frac{\delta (T_1^j \beta c p + T_1^j\beta p \phi - c^2\delta - 2c\delta \phi )}{\beta p (\beta p T_1^j - c\delta -\delta \phi )},\\ I_2^j&=\frac{T_1^j\beta ^2p^2s_2 - T_1^j\beta c d \delta p - T_1^j\beta d \delta p \phi - \beta c \delta p s_2 - \beta \delta p \phi s_2 + c^2d\delta ^2 + 2cd\delta ^2\phi }{\delta \beta p(\beta p T_1^j - c\delta - \delta \phi )},\\ V_2^j&= \frac{T_1^j \beta ^2 p^2 s_2 - T_1^j \beta c d \delta p - T_1^j\beta d \delta p \phi - \beta c \delta p s_2 - \beta \delta p \phi s_2 + c^2d\delta ^2 + 2cd\delta ^2\phi }{\delta \beta (T_1^j\beta c p + T_1^j\beta p \phi - c^2\delta - 2c\delta \phi )}. \end{aligned}$$\end{document}I1j=ϕT1j(-T1jβ2p2s2+T1jβcdδp+T1jβdδpϕ+βcδps2+βδpϕs2-c2dδ2-2cdδ2ϕ)δ(βpT1j-cδ-δϕ)(T1jβcp+θjβpϕ-c2δ-2cδϕ),V1j=ϕ(-T1jβ2p2s2+T1jβcdδp+T1jβdδpϕ+βcδps2+βδpϕs2-c2dδ2-2cdδ2ϕ)β(T1jβcp+T1jβpϕ-c2δ-2cδϕ)(βpT1j-cδ-δϕ),T2j=δ(T1jβcp+T1jβpϕ-c2δ-2cδϕ)βp(βpT1j-cδ-δϕ),I2j=T1jβ2p2s2-T1jβcdδp-T1jβdδpϕ-βcδps2-βδpϕs2+c2dδ2+2cdδ2ϕδβp(βpT1j-cδ-δϕ),V2j=T1jβ2p2s2-T1jβcdδp-T1jβdδpϕ-βcδps2-βδpϕs2+c2dδ2+2cdδ2ϕδβ(T1jβcp+T1jβpϕ-c2δ-2cδϕ).Here \documentclass[12pt]{minimal}
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\begin{document}$$T_1^j$$\end{document}T1j is the root of\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \gamma _0 \left( T_1^j\right) ^3 + \gamma _1 \left( T_1^j\right) ^2 + \gamma _2 \left( T_1^j\right) +\gamma _3=0, \end{aligned}$$\end{document}γ0T1j3+γ1T1j2+γ2T1j+γ3=0,where\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \gamma _0&= \beta dp^2(\beta c+\phi ),\\ \gamma _1&= -\beta ^2cp^2s_1-\beta ^2p^2\phi s_1-\beta ^2p^2\phi s_2 - 2\beta c^2d\delta p-3\beta c d \delta p \phi ,\\ \gamma _2&= 2\beta c^2\delta p s_1 + 4\beta c \delta p \phi s_1 + \beta c \delta p \phi s_2 + \beta \delta p \phi ^2 s_1 + \beta \delta p \phi ^2 s_2 + c^3 d \delta ^2 + 2c^2 d \delta ^2 \phi ,\\ \gamma _3&= -c\delta ^2s_1(c^2+ 3c\phi +2\phi ^2). \end{aligned}$$\end{document}γ0=βdp2(βc+ϕ),γ1=-β2cp2s1-β2p2ϕs1-β2p2ϕs2-2βc2dδp-3βcdδpϕ,γ2=2βc2δps1+4βcδpϕs1+βcδpϕs2+βδpϕ2s1+βδpϕ2s2+c3dδ2+2c2dδ2ϕ,γ3=-cδ2s1(c2+3cϕ+2ϕ2).Since \documentclass[12pt]{minimal}
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\begin{document}$$\gamma _3<0$$\end{document}γ3<0, by Descartes’ rule of signs, we can have one or three \documentclass[12pt]{minimal}
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\begin{document}$$T_1^j$$\end{document}T1j positive roots. Additionally, for the chronic equilibrium \documentclass[12pt]{minimal}
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\begin{document}$$E_j$$\end{document}Ej to exist, we need to guarantee positivity of the other variables in equilibrium \documentclass[12pt]{minimal}
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\begin{document}$$E_j$$\end{document}Ej. This occurs when:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} T_1^j<\min \big \{\frac{\delta (c+\phi )}{\beta p}, \frac{c\delta (c+2\phi )}{\beta p(c+\phi )}, \frac{\delta }{\beta p}\frac{\beta p s_2(c+\phi )-cd\delta (c+2\phi )}{\beta p s_2-d\delta (c+\phi )}\big \}. \end{aligned}$$\end{document}T1j<min{δ(c+ϕ)βp,cδ(c+2ϕ)βp(c+ϕ),δβpβps2(c+ϕ)-cdδ(c+2ϕ)βps2-dδ(c+ϕ)}.
Proposition 6
The infection-free equilibrium \documentclass[12pt]{minimal}
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\begin{document}$$E_4$$\end{document}E4 is locally asymptotically stable if\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathcal {R}_0^{2D} = \frac{\beta p s_2}{d\delta }\frac{p s_1\beta + d \delta (c+\phi )}{ps_1\beta (c+\phi ) + d c\delta (c+2\phi )}<1, \end{aligned}$$\end{document}R02D=βps2dδps1β+dδ(c+ϕ)ps1β(c+ϕ)+dcδ(c+2ϕ)<1,and unstable otherwise.
Lastly, since explicit forms of chronic equilibria \documentclass[12pt]{minimal}
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\begin{document}$$E_j$$\end{document}Ej (\documentclass[12pt]{minimal}
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\begin{document}$$j={5,6,7}$$\end{document}j=5,6,7) are hard to determine, we examined their existence and asymptotic stability numerically.
Structural Identifiability Results for Model Eq. <xref rid="Equ9" ref-type="disp-formula">9</xref>
As with the one-directional two-patch model, we investigated the structural identifiability of model Eq. 9 under two assumptions: (i) viral load is measured in both patches and the initial conditions of the model Eq. 9 are either known or unknown; (ii) susceptible liver cells are measured in both patches and the initial conditions of the model Eq. 9 are either known or unknown.
Proposition 7
Given measurements for the \documentclass[12pt]{minimal}
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\begin{document}$$V_2(t)$$\end{document}V2(t) variables, system Eq. 9 is unidentifiable when initial conditions are unknown. In particular, parameters \documentclass[12pt]{minimal}
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\begin{document}$$\pi _1=\{ \beta , c, d, \delta , \}$$\end{document}π1={β,c,d,δ,} are globally structurally identifiable, and parameters \documentclass[12pt]{minimal}
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\begin{document}$$\pi _2=\{s_{1}, s_{2}, p\}$$\end{document}π2={s1,s2,p} are unidentifiable. If, additionally, initial conditions are known, then all parameters are globally structurally identifiable.
Identifiability analysis for model Eq. 9, performed using the DAISY software (Bellu et al. 2007). For the simulations with initial conditions, all initial conditions are known
Observe States
Initial Conditions Known
Initial Conditions Unknown
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\begin{document}$$V_{2}(t)$$\end{document}V2(t) data
Numerical Results for Model Eq. <xref rid="Equ9" ref-type="disp-formula">9</xref>
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\begin{document}$$T_2(t)$$\end{document}T2(t)), all parameters of model Eq. 9 can be identified from data.
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\begin{document}$$s=d\times (T_1(0)+T_2(0))$$\end{document}s=d×(T1(0)+T2(0)) is susceptible to HBV infection and considered three cases for the susceptible cells ratio within the two patches \documentclass[12pt]{minimal}
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\begin{document}$$s_1:s_2 =\{10:90, 50:50, 90:10\}$$\end{document}s1:s2={10:90,50:50,90:10} (see Table 4). As before, we set δ =d=0.01δ=d=0.01 /day (Ciupe, Ribeiro, Nelson, and Perelson 2007), fixed c=4.4c=4.4 /day and fitted parameters \documentclass[12pt]{minimal}
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\begin{document}$$\{\beta , p, \phi \}$$\end{document}{β,p,ϕ} using the fminsearchbnd function in Matlab2021a (see Materials and Methods section for details). A summary of estimates for each case is given in Table 4 and the model dynamics are plotted in Fig. 5.
The effects of patch-susceptibility on virus dynamics within patches when viral movement is two-directional. Dynamics of (A)-(E) uninfected (solid line) and infected (dashed line) liver cells in patch 1 (red lines) and patch 2 (blue lines) and (B)-(F) total virus (black line), virus in patch 1 (red line), virus in patch 2 (blue line) versus mice data (circles) as given by model Eq. 9 in case 1: (A)-(B); case 2: (C)-(D), case 3: (E)-(F). Model parameters are given in Table 4
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\begin{document}$$\beta =2.94\times 10^{-9}$$\end{document}β=2.94×10-9 ml/(virion × × day), for cases 1, 2 and 3, respectively. Moreover, viral production rates are similar among the three patch recruitment cases, i.e.p=1053p=1053 virion/(ml× × day), p=1055p=1055 virion/(ml× × day), and p=1049p=1049 virion/(ml× × day), respectively. The estimate of the movement rate is 50-times higher in case 1 and case 3, compared to case 2, i.e.ϕ =5ϕ=5 /day versus ϕ =0.1ϕ=0.1 /day. To determine if the results can be generalized when we add noise to the data, we performed practical identifiability analysis using Monte Carlo approaches (see Materials and Methods for details). We found that all parameters are strongly practically identifiable for all cases, with the exception of ϕ ϕ, which is weakly practically identifiable in all cases (see Table 6).
In all three cases we are in the \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{2D}=2.3$$\end{document}R02D=2.3, for case 3). While we know from the analytical results that this corresponds to instability of the clearance equilibrium \documentclass[12pt]{minimal}
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\begin{document}$$E_4$$\end{document}E4, we do not know if it guarantees the stability of the both-patches viral persistence equilibrium \documentclass[12pt]{minimal}
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\begin{document}$$E_5$$\end{document}E5 (note that equilibrium solutions for single-patch viral persistence do not exist for this model). We investigated these long-term results numerically by deriving one- and two-dimensional bifurcation diagrams.
As with the one-directional two-patch model, we perturb the baseline parameters obtained through data fitting to uncover general biological features, such as immune competence and virus localization. To address this, we created one-dimensional bifurcation diagrams as follows. We numerically solved system Eq. 9 for parameters in case 2 (Table 4) and \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{2D}$$\end{document}R02D. We found that both \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{2D}>1$$\end{document}R02D>1 (Fig. 6A and B). We also created two-dimensional bifurcation diagrams (using the numerical values of \documentclass[12pt]{minimal}
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\begin{document}$$\phi \in (0,10)$$\end{document}ϕ∈(0,10). We found little variability in the long-term dynamics of \documentclass[12pt]{minimal}
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\begin{document}$$V_2$$\end{document}V2, with both viruses being cleared for high infected cells killing rate δ δ and persisting for low infected cells killing rate δ δ, regardless of the movement parameter ϕ ϕ (Fig. 7). This implies that, when virus moved between patches, immune mediated removal of infected cells is mandatory for viral clearance.
One-dimensional bifurcation diagram for model Eq. 9: \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{2D}$$\end{document}R02D. The other parameters are given in Table 2 for case 2
Two-dimensional bifurcation diagram for model Eq. 9: A.\documentclass[12pt]{minimal}
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\begin{document}$$\log _{10} V_2$$\end{document}log10V2 at equilibrium versus δ δ and ϕ ϕ. The other parameters are as in Table 4, case 2
Discussion and Conclusion
In this study, we developed two within-host mathematical models of HBV infection that take into account the abnormal nodular structure of the liver following prolonged hepatitis B infection and disease (Yuan et al. 2019; Bataller et al. 2005; Roehlen et al. 2020). For simplicity, we only modeled a two-patch liver structure and assumed that: (1) each patch has a different numbers of cells susceptible to HBV infection, (2) HBV is seeded in a single-patch, (3) all other virus-host interactions are patch-agnostic. The first model, given by Eq. 3, assumes one-directional hepatitis B virus movement between patches. The second model, given by Eq. 9, assumes two-directional hepatitis B virus movement between patches. We used the models to determine differences in overall hepatitis B virus infection under viral localization (as given by the one-directional model Eq. 3) and free viral movement between patches (as given by the two-directional model Eq. 9).
We analyzed both models using asymptotic stability techniques and found that when HBV movement is one-directional, as given by model Eq. 3, equilibrium dynamics include viral clearance from both patches, viral clearance from the HBV seeding patch alone, and virus persistence in both patches. When HBV movement is two-directional, as given by model Eq. 9, however, equilibrium dynamics only include viral clearance from both patches, and virus persistence in both patches. In other words, single-patch persistence and viral containment within parts of the liver only happen when HBV movement between patches is irreversible. This can be a result of collagen production, excessive accumulation of extracellular matrix, and liver dysfunction (Bataller et al. 2005).
Before validating the models with data, we studied their structural identifiability, and found that we can identify all parameters (under unlimited data) only when viral inoculum and initial susceptible hepatocyte populations are known. This is the case regardless of the type of data used (liver cell or virus data). When these initial details are unknown, several parameters are unidentifiable. A common unidentifiable parameter between the two models Eqs. 3 and 9 is the viral production rate p, which, when known (or fixed), ensures identifiability of the remaining parameters. Since p is hard to estimate experimentally, we assumed known the initial condition of both models, and estimated p from the data.
We fitted the models to HBV DNA data from one immunocompromised, humanized mice (Zhang et al. 2023) and estimated three key parameter values for each model: the infectivity rate β β, the viral production rate p, and the HBV movement rate between patches ϕ ϕ. Since empirical data (from an immunocompromised animal) showed viral persistence following the initial expansion, we ignored the effect of immune-mediated killing rate of infected cells δ δ (which was set to uninfected cell death rate) and relaxed this assumption later on. We considered differences in susceptibility to HBV infection between patches, by varying the target cell recruitment ratio, \documentclass[12pt]{minimal}
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\begin{document}$$s_1:s_2$$\end{document}s1:s2 between 10:90, 50:50 and 90:10, while keeping the overall liver cell numbers fixed (a reasonable assumption in an immunocompromised mice (Wieland and Chisari 2005)).
We found that the estimates for the infectivity rate β β and for the viral production rate p are not affected (in a significant manner) by the target cell recruitment ratio \documentclass[12pt]{minimal}
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\begin{document}$$s_1:s_2$$\end{document}s1:s2 or by the ability of HBV to move back-and-forth between patches (\documentclass[12pt]{minimal}
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\begin{document}$$\phi _{21}=\phi _{12}=\phi $$\end{document}ϕ21=ϕ12=ϕ). The movement rate ϕ ϕ, however, is significantly impacted by the target cell recruitment ratio \documentclass[12pt]{minimal}
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\begin{document}$$s_1:s_2$$\end{document}s1:s2 and by the model. In particular, for the one-directional two-patch movement model Eq. 3, HBV chooses movement towards the non-founding patch at significantly higher rates (up to 50-times higher) when the initial patch has more susceptible cells. This outcome may be due to larger numbers of HBV being created in the original patch in the 50:50 and 90:10 cases, resulting in larger virus movement. By contrast, for the two-directional two-patch movement model Eq. 9, HBV chooses higher movement rates (up to 50-times higher) when the patches have unbalanced numbers of susceptible cells (10:90 and 90:10, respectively) and limited movement for the balanced patches (50:50). This outcome may be due to each patch having a sufficient number of target cells available for HBV infection, limiting viral diffusion.
For the one-directional model Eq. 3, the equilibrium virus levels in patch 2 exceed equilibrium virus levels in patch 1, regardless of the target cell susceptibility ratio (Fig. 2). By contrast, for the two-directional model Eq. 9, equilibrium virus levels in patch 2 exceed those in patch 1 for the 10 : 90 target cell susceptibility, but are equal to and lower than those in patch 1 for the 50 : 50 and 90 : 10 target cell susceptibility ratio (Fig. 5). Therefore, the quantitative outcomes are influenced by the ability of the virus to move one- or two-directionally between patches. Our results are not influenced by practical identifiability issues, with both models being (at least weakly) practically identifiable.
The baseline parameters obtained through data fitting in both models Eq. 3 and Eq. 9 offer a static view of HBV infection, characteristic of chronicity in an immunocompromised individual. To determine how the results change in an immune competent host, we derived one- and two-dimensional bifurcation diagrams by using δ δ (the killing rate of infected cells) as bifurcation parameter. We varied δ δ from 0.01 /day (corresponding to infected cells lifespan of 100 days) to 0.6 /day (corresponding to infected cells lifespan of 1.7 days).
For model Eq. 3 we found, as expected, that increasing δ δ results in \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{1D}$$\end{document}R01D decrease. When \documentclass[12pt]{minimal}
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\begin{document}$$R_0^{1D}$$\end{document}R01D decreases below 1, we observe viral clearance from both patches (Fig. 3, solid green lines). Interestingly, viral clearance from both patches requires that δ >0.45δ>0.45 /day (lifespan of infected cells of 2.2 days or lower) when \documentclass[12pt]{minimal}
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\begin{document}$$s_1:s_2=90:10$$\end{document}s1:s2=90:10. This suggests that higher immune responses are necessary for viral clearance when the majority of susceptible cells are in patch 2. By contrast, persistence in a single-patch outcomes (e. g. viral clearance in patch 1, and persistence in patch 2) occur for \documentclass[12pt]{minimal}
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\begin{document}$$\delta \in (0.05, 0.45)$$\end{document}δ∈(0.05,0.45) when \documentclass[12pt]{minimal}
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\begin{document}$$s_1:s_2=50:50$$\end{document}s1:s2=50:50, and not at all (in the ranges considered) when \documentclass[12pt]{minimal}
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\begin{document}$$s_1:s_2=90:10$$\end{document}s1:s2=90:10 (Fig. 3, solid red lines and Fig. 4). This suggests a limited range of immune responses lead to virus containment inside a single-patch when the majority of susceptible cells are in patch 1.
For model Eq. 9 we found that viral clearance from both patches requires that δ >0.27δ>0.27 /day (lifespan of infected cells of 3.7 days or lower) when \documentclass[12pt]{minimal}
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\begin{document}$$s_1:s_2=90:10$$\end{document}s1:s2=90:10. This suggests that higher immune responses are necessary for viral clearance when the susceptible cells in patch 1 equal or exceed the susceptible cells in patch 2 (Fig. 7).
Our study has several limitations. First, the models ignore liver proliferation following liver stress and death (Summers et al. 2003). We accounted for liver proliferation by choosing different numbers of susceptible cells in each patch. Additional data is needed to allow for increase in model complexity through the addition of a proliferation term. Second, we ignored cure of infected cells by non-cytolytic processes (Guidotti et al. 1999; Wieland et al. 2004), and intracellular viral effects (Ciupe and Conway 2024) and assumed that the most important host feature is the immune-mediated killing of infected cells, δ δ. It is known that HBV-induced liver disease is immune-mediated (Ciupe, Ribeiro, Nelson, Dusheiko, and Perelson 2007; Ciupe et al. 2014; Ciupe and Conway 2024), so more detailed models that account for immune populations and immune functions will be part of future work. Third, we used an oversimplified model for the abnormal nodular liver structure following disease, that is composed of just two patches. While this allowed for analytical tractability of our results, a multi-patch model may result in richer dynamics and even patterns of infection. Forth, we assumed either single directional movement or equal movement between patches. A bifurcation diagram of model Eq. 2 (Appendix B, Fig. 9) shows that, for \documentclass[12pt]{minimal}
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\begin{document}$$0<\Phi _{21}<\Phi $$\end{document}0<Φ21<Φ, we observe symmetry between virus levels in the two patches. At the \documentclass[12pt]{minimal}
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\begin{document}$$\Phi _{21}=0$$\end{document}Φ21=0 extreme, we recover the one-directional movement model Eq. 3; and at the \documentclass[12pt]{minimal}
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\begin{document}$$\Phi _{12}=\Phi _{21}=\Phi $$\end{document}Φ12=Φ21=Φ extreme, we recover the two-directional movement model Eq. 9. Further work is needed to uncover dynamics in the intermediate movement ranges. Lastly, to identify parameters, we had to assume that the initial conditions are known and \documentclass[12pt]{minimal}
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\begin{document}$$I_1(0)\ne 0$$\end{document}I1(0)≠0. While most within-host models assume that \documentclass[12pt]{minimal}
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\begin{document}$$I_1(0)=0$$\end{document}I1(0)=0, we bypassed the initial infection and started with one infected cell \documentclass[12pt]{minimal}
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In conclusion, we developed mathematical models that considered the formation of liver nodular structures following hepatitis B viral infection and disease and used them to investigate virus dynamics within-patches and systemically. We predicted different outcomes when viral movement between patches is either irreversible or reversible. Moreover, we found that cell susceptibility to infection within nodular structures, the movement rate between patches, and the immune-mediated infected cell killing have the most influence on the results.
Materials and MethodsStructural Identifiability Analysis
Global structural identifiability analysis determines whether a model’s parameters can be uniquely revealed given unlimited noise-free data (for reviews, see (Tuncer and Le 2018; Liyanage et al. 2024; Tuncer et al. 2025; Liyanage et al. 2025)).
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\begin{document}$$\begin{aligned}x(t) = \{T_1(t),I_1(t), V_1(t) , T_2(t), I_2(t), V_2(t)\}\in \mathbb {R}^6,\end{aligned}$$\end{document}x(t)={T1(t),I1(t),V1(t),T2(t),I2(t),V2(t)}∈R6,are the state variables at time t, that solve the ordinary differential equations governed by the rules:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}f\big (\textbf{}{x}(t),\textbf{q}\big )\in \mathbb {R}^6,\end{aligned}$$\end{document}f(x(t),q)∈R6,for models Eqs. 3 and 9. The parameter vector is:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\textbf{q}=\{s_1, s_2, \beta , d, \delta , c, p, \phi \}\in \mathbb {R}^8,\end{aligned}$$\end{document}q={s1,s2,β,d,δ,c,p,ϕ}∈R8,and the initial condition vector is:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\textbf{x}_0\in \mathbb {R}^6.\end{aligned}$$\end{document}x0∈R6.We assumed that the empirical observation represented by \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{y}(t)\in \mathbb {R}^{{4}}$$\end{document}y(t)∈R4 can be explained by models Eqs. 3 and 9 through function:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\textbf{g}\big (\textbf{x}(t),\textbf{p}\big )\in \mathbb {R}^{{6}}.\end{aligned}$$\end{document}g(x(t),p)∈R6.
Definition 1
System Eq. 12 is said to be globally structurally identifiable for parameter vector \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{q} $$\end{document}q if, for every other parameter vector \documentclass[12pt]{minimal}
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\begin{document}$$\hat{\textbf{q}},$$\end{document}q^,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}y(t,\textbf{q})=y(t,\hat{\textbf{q}}) \text { implies } \textbf{q}=\hat{\textbf{q}};\end{aligned}$$\end{document}y(t,q)=y(t,q^)impliesq=q^;it is locally structurally identifiable for parameter vector \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{q}$$\end{document}q if, for every other parameter vector \documentclass[12pt]{minimal}
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\begin{document}$$\hat{\textbf{q}}$$\end{document}q^,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}y(t,\textbf{q})=y(t,\hat{\textbf{q}}), \text { and } \hat{\textbf{q}}\in B(\textbf{q})\text { implies } \textbf{q}=\hat{\textbf{q}}, \text { where } B(\textbf{q}) \text { is a ball centered at } \textbf{q};\end{aligned}$$\end{document}y(t,q)=y(t,q^),andq^∈B(q)impliesq=q^,whereB(q)is a ball centered atq;and it is unidentifiable when at least one of its parameters fails the local identifiability test.
The methodology for determining the structural identifiability of systems of ordinary differential equations can range from the Taylor series approach (Pohjanpalo 1978), the differential algebra approach (Bellu et al. 2007; Ljung and Glad 1994), the generating series approach (Walter and Pronzato 1997), the implicit functions approach (Xia and Moog 2003), and several others. Additionally, there are several tools (platforms) that can assist in establishing whether a system of ordinary differential equations is identifiable, such as the COMBOS (Meshkat et al. 2014), the Differential Algebra for Identifiability of SYstems (DAISY, Bellu et al. (2007)), the Exact Arithmetic Rank (EAR, Anguelova et al. (2012)), the GenSSI2 (Ligon et al. 2018), the Structural Identifiability Analyser (SIAN, Hong et al. (2019)), the STRIKE-GOLDD (Villaverde et al. 2016), and the StructuralIdentifiability.jl (Dong et al. 2023). In this study, we used the differential algebra approach and the DAISY platform (Bellu et al. 2007) to determine the structural identifiability of systems Eqs. 3 and 9. The goal of the structural identifiability is to determine which parameters we can confidently estimated from unlimited observations.
Data Fitting
We used previously published longitudinal serum HBV DNA titers from one chimeric mouse with humanized liver (Zhang et al. 2023). Briefly, a urokinase-type plasminogen activator and severe combined immunodeficient (uPA/SCID) mouse was transplanted human hepatocytes (Tateno et al. 2015). Approximately 120–150 days after transplantation (when the growth and proliferation of liver cells is complete), the mouse was infected with hepatitis B virus. Serum HBV DNA was collected during the expansion, peak and persistent stage of HBV infection at days \documentclass[12pt]{minimal}
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\begin{document}$$t_{data}$$\end{document}tdata={14, 22, 33, 54, 82, 99, 120, 141, 162, 183, 197, 212} post inoculation. HBV DNA dynamics reached steady levels, similar to an acute HBV infection that becomes persistent in humans (Zhang et al. 2023).
Data fitting procedure for the two-patch models Eqs. <xref rid="Equ3" ref-type="disp-formula">3</xref> and <xref rid="Equ9" ref-type="disp-formula">9</xref>
Known parameters. We assumed that hepatocytes have a life-span of 100 days (M.-J. Wang et al. 2017), resulting in an uninfected hepatocyte death rate of d=0.01d=0.01 /day. Moreover, we assumed that \documentclass[12pt]{minimal}
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\begin{document}$$6.8\times 10^{5}$$\end{document}6.8×105 hepatocytes/ml are susceptible to HBV infection (Ciupe et al. 2024), resulting in \documentclass[12pt]{minimal}
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\begin{document}$$T(0)=T_1(0)+T_2(0)=6.8\times 10^{5}$$\end{document}T(0)=T1(0)+T2(0)=6.8×105 cells/ml and a total recruitment rate \documentclass[12pt]{minimal}
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\begin{document}$$s=d \times T(0)= 6.8\times 10^{3}$$\end{document}s=d×T(0)=6.8×103 cells/(ml× × day). HBV is cleared at rate c=4.4c=4.4 /day (Murray et al. 2005). Since we fit the model to data from immunosupressed mice, we assumed δ =d=0.01δ=d=0.01 /day.
We assumed that the HBV inoculum is seeded in patch 1, and set patch 1-specific initial virus and infected cells to \documentclass[12pt]{minimal}
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\begin{document}$$V_1(0)=10^4$$\end{document}V1(0)=104 HBV DNA/ml and \documentclass[12pt]{minimal}
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\begin{document}$$I_1(0)=I_0=1$$\end{document}I1(0)=I0=1 cells/ml (Zhang et al. 2023). Moreover, we assumed that there is no virus in patch 2 at the beginning of infection. Therefore, \documentclass[12pt]{minimal}
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\begin{document}$$I_2(0)=0$$\end{document}I2(0)=0 cells/ml and \documentclass[12pt]{minimal}
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\begin{document}$$V_2(0)=0$$\end{document}V2(0)=0 HBV DNA/ml. We split the number of uninfected hepatocytes between the two patches. \documentclass[12pt]{minimal}
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\begin{document}$$T_1(0)=T_2(0)=3.4\times 10^{5}$$\end{document}T1(0)=T2(0)=3.4×105 hepatocytes/ml, and assumed three cases for hepatocyte recruitment: (case 1) \documentclass[12pt]{minimal}
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\begin{document}$$s_1=0.1\times s=6.8\times 10^2$$\end{document}s1=0.1×s=6.8×102 cells/(ml× × day), \documentclass[12pt]{minimal}
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\begin{document}$$s_2=0.9\times s=6.12\times 10^3$$\end{document}s2=0.9×s=6.12×103 cells/(ml× × day); (case 2) \documentclass[12pt]{minimal}
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\begin{document}$$s_1=s_2=0.5\times s =3.4 \times 10^3$$\end{document}s1=s2=0.5×s=3.4×103 cells/(ml× × day); and (case 3) \documentclass[12pt]{minimal}
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\begin{document}$$s_1=0.9\times s=6.12\times 10^3$$\end{document}s1=0.9×s=6.12×103 cells/(ml× × day), \documentclass[12pt]{minimal}
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\begin{document}$$s_2=0.1\times s=6.8\times 10^2$$\end{document}s2=0.1×s=6.8×102 cells/(ml× × day).
Data fitting algorithm. The remaining parameters \documentclass[12pt]{minimal}
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\begin{document}$$\pi =\{\beta , p, \phi \}$$\end{document}π={β,p,ϕ} are assumed unknown and are estimated by minimizing the functional:\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}J(\pi )=\left( \sum _{t_{data}} \left( \log _{10} \left( V_1(t_{data}, \pi )+ V_2(t_{data}, \pi )\right) -\log _{10} V_{data} (t_{data}) \right) ^2\right) ^{1/2},\end{aligned}$$\end{document}J(π)=∑tdatalog10V1(tdata,π)+V2(tdata,π)-log10Vdata(tdata)21/2,over the parameter space π π using the built-in function fminsearchbnd in MATLAB R2021a. Parameter bounds are \documentclass[12pt]{minimal}
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\begin{document}$$10^{-10}\le \beta \le 10^{-7}$$\end{document}10-10≤β≤10-7 ml/(virion× ×day), \documentclass[12pt]{minimal}
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\begin{document}$$0.1\le \phi \le 5$$\end{document}0.1≤ϕ≤5 /day, and \documentclass[12pt]{minimal}
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\begin{document}$$0\le p\le 1500$$\end{document}0≤p≤1500 virus/(ml× ×day) for (case 1 - case 3). The initial guesses are \documentclass[12pt]{minimal}
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\begin{document}$$\beta =5\times 10^{-9}$$\end{document}β=5×10-9 ml/(virion× ×day), ϕ =0.5ϕ=0.5/day and p=100p=100 virus/(ml× ×day). The resulting values for model Eq. 3 are given in Table 2 and the dynamics of model Eq. 3 over time are given in Fig. 2. Similarly, the resulting values for model Eq. 9 are given in Table 4 and the dynamics of model Eq. 9 over time are given in Fig. 5.
Practical Identifiability Analysis
Practical identifiability is a methodology that considers the noisiness in the data set for a given model, and performs data fitting under noise considerations.
Definition 2
A model is practically identifiable if a unique parameter set can be consistently obtained through the fitting procedure to noisy data.
There are several approaches and conditions for assessing practical identifiability (Wu et al. 2008; Eisenberg et al. 2013; Roosa and Chowell 2019; Saucedo et al. 2024; Tuncer and Le 2018). In this paper, we used the Monte Carlo (MC) approach to analyze the practical identifiability of the one-directional two-patch model Eq. 3 and two-directional two-patch model Eq. 9. The Monte Carlo approach is a sampling technique that uses random numbers and probability distributions to determine the practical identifiability of a model (Metropolis and Ulam 1949). We performed MC simulations by generating M=1,000M=1,000 data sets using the true parameter set π π and adding noise to the data in increasing amounts. The MC simulations are outlined in the following steps:
Solve model Eq. 3 (Eq. 9, respectively) numerically with the true parameter vector π π to obtain the output vector \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{g}\big (\textbf{x}(t),\pi \big )$$\end{document}g(x(t),π) at discrete time points \documentclass[12pt]{minimal}
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\begin{document}$$t_{data}=\{t_i\}_{i=1}^n$$\end{document}tdata={ti}i=1n.
Generate M=1,000M=1,000 data sets with a given measurement error. We assume the error follows a normal distribution with mean 0 and variance \documentclass[12pt]{minimal}
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\begin{document}$$\sigma ^2(t)$$\end{document}σ2(t); that is, the data are described by: \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\textbf{y}_{i,j} = g\big (\textbf{x}(t_i),\pi \big )(1+\epsilon _{i,j}),\end{aligned}$$\end{document}yi,j=g(x(ti),π)(1+ϵi,j), where \documentclass[12pt]{minimal}
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\begin{document}$$\epsilon _{ij} \sim \mathcal {N}(0,\sigma )$$\end{document}ϵij∼N(0,σ) at the discrete data time points \documentclass[12pt]{minimal}
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\begin{document}$$t_{data}=\{t_i\}_{i=1}^n$$\end{document}tdata={ti}i=1n for all \documentclass[12pt]{minimal}
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\begin{document}$$j=\{1,2,...,M\}$$\end{document}j={1,2,...,M} data sets.
Estimate the parameter set \documentclass[12pt]{minimal}
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\begin{document}$$\pi _j$$\end{document}πj, by fitting model Eq. 3 (model Eq. 9, respectively) to each of the M simulated data sets. This is achieved by minimizing the difference between model Eq. 3 (model Eq. 9, respectively) output and the data generated for the specific scenario: \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} J(\pi _{j})=\big (\sum _{t_{data}} \big (\log _{10} \big (V_1(t_{data}, \pi _j)+ V_2(t_{data}, \pi _j)\big )-\log _{10} V_{data} (t_{data}) \big )^2\big )^{1/2}. \end{aligned}$$\end{document}J(πj)=(∑tdata(log10(V1(tdata,πj)+V2(tdata,πj))-log10Vdata(tdata))2)1/2.
MC approach for the one-directional two-patch model Eq. 3
case 1
case 2
case 3
ARE
β β
ϕ ϕ
p
β β
ϕ ϕ
p
β β
ϕ ϕ
p
0%
0
0
0
0
0
0
0
0
0
1%
0.3502
1.8354
4.7200
0.0542
12.5492
0.5759
0.0313
0.6557
0.4824
5%
0.4568
24.4316
6.3968
0.0700
12.7504
0.9358
0.1293
1.2374
2.0864
10%
0.5899
36.8748
8.3112
0.1391
13.2641
2.0531
0.2481
2.2095
4.0030
20%
0.7314
45.5693
10.6981
0.3188
13.1036
4.8778
0.4282
3.0876
6.9326
30%
0.9340
57.2279
13.8884
0.6018
13.3719
8.8351
0.7088
3.4158
10.8375
Identifiable?
Yes
Weakly
Weakly
Yes
Weakly
Yes
Yes
Yes
Yes
MC approach for the two-directional two-patch model Eq. 9.
case 1
case 2
case 3
ARE
β β
ϕ ϕ
p
β β
ϕ ϕ
p
β β
ϕ ϕ
p
0%
0
0
0
0
0
0
0
0
0
1%
0.0207
1.8487
0.3372
0.0242
10.3416
0.3967
0.0366
2.5045
0.5624
5%
0.2271
3.8460
3.2488
0.1183
76.3263
1.9011
0.3342
1.4423
4.6259
10%
0.4226
2.3097
5.9290
0.2775
64.4378
4.2071
0.4965
1.5473
6.8888
20%
0.6925
1.4294
9.7573
0.6096
51.1635
8.8355
0.7205
1.6928
10.0169
30%
0.8235
1.4597
11.9580
0.8160
36.3667
11.7393
0.8739
1.5403
12.5380
Identifiable?
Yes
Weakly
Yes
Yes
Weakly
Yes
Yes
Weakly
Yes
This optimization problem is solved in MATLAB R2021a using the built-in function fminsearchbnd, which is part of the Optimization Toolbox. Since fminsearchbnd is a local solver, the optimized minimum value can be influenced by the starting point. To avoid issues related to the starting value, we use the true parameter values π π as the initial parameter starting point provided to fminsearchbnd.
Calculate the average relative estimation error (ARE) for each parameter in the set π π, as follows: \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \textrm{ARE} \big (\pi ^{(k)}\big )=100 \% \times \frac{1}{M} \sum _{j=1}^{M} \frac{\left| \pi ^{(k)}- \pi _j^{(k)}\right| }{\left| \pi ^{(k)}\right| }, \end{aligned}$$\end{document}ARE(π(k))=100%×1M∑j=1Mπ(k)-πj(k)π(k), where \documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{(k)}$$\end{document}π(k) is the k-th parameter of the true parameter set π π, and \documentclass[12pt]{minimal}
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\begin{document}$$ \pi _j^{(k)}$$\end{document}πj(k) is the k-th element of \documentclass[12pt]{minimal}
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\begin{document}$$\pi _j$$\end{document}πj.
Repeat steps 1 through 4, by increasing the measurement error \documentclass[12pt]{minimal}
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\begin{document}$$\sigma = \{0, 1, 5, 10, 20, 30\}\%$$\end{document}σ={0,1,5,10,20,30}%.
The objective of this algorithm is to determine if the parameters are sensitive to gradual noise introduced into the given dataset. We used the AREs as a metric to determine the practical identifiability of model Eq. 3 (model Eq. 9, respectively) by applying the definition below (see (Heitzman-Breen et al. 2024) for details).
Definition 3
Let σ σ be the measurement error introduced to a dataset, and let the ARE be the average measurement error in the parameter \documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{(k)}$$\end{document}π(k).
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\begin{document}$$\pi ^{(k)})\le \sigma $$\end{document}π(k))≤σ, we say \documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{(k)}$$\end{document}π(k) is strongly practically identifiable.
If σ<σ< ARE(\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{(k)})\le 10 \times \sigma $$\end{document}π(k))≤10×σ, we say \documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{(k)}$$\end{document}π(k) is weakly practically identifiable.
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\begin{document}$$\pi ^{(k)})$$\end{document}π(k)), we say \documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{(k)}$$\end{document}π(k) is not practically identifiable.
We state that a model is practically identifiable if \documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{(k)}$$\end{document}π(k) is practically identifiable for all values of k.
The results for model Eq. 3 are given in Table 5.
The results for model Eq. 9 are given in Table 6.
Supporting analysis for the stability of equilibrium \documentclass[12pt]{minimal}
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\begin{document}$$E_4$$\end{document}E4
Two-dimensional bifurcation diagrams for model Eq. 2: \documentclass[12pt]{minimal}
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\begin{document}$$\log _{10} (V_1+ V_2)$$\end{document}log10(V1+V2) at equilibrium versus \documentclass[12pt]{minimal}
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\begin{document}$$\Phi _{12}$$\end{document}Φ12 and \documentclass[12pt]{minimal}
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\begin{document}$$\Phi _{21}$$\end{document}Φ21. The other parameters are given in Table 2, case 2
Appendix A Stability of Equilibrium <inline-formula id="IEq578"><alternatives><tex-math id="d33e23296">\documentclass[12pt]{minimal}
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\begin{document}$$E_4$$\end{document}</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="d33e23298"><mml:msub><mml:mi>E</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></alternatives></inline-formula>
To finalize the proof of Proposition 4.,we need to show that Fig 8\documentclass[12pt]{minimal}
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\begin{document}$$a_1 a_2 a_2>a_3^2+a_1^2 a_4$$\end{document}a1a2a2>a32+a12a4. We are including here the maple file showing that result.
Appendix B Bifurcation Diagram for the Generalized Model Eq. <xref rid="Equ2" ref-type="disp-formula">2</xref>
We are presenting a bifurcation diagram for the general model Eq. 2 where \documentclass[12pt]{minimal}
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\begin{document}$$0<\Phi _{12}<\Phi $$\end{document}0<Φ12<Φ and \documentclass[12pt]{minimal}
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\begin{document}$$0<\Phi _{21}<\Phi $$\end{document}0<Φ21<Φ. We observe symmetry between the two movements. At either extremes \documentclass[12pt]{minimal}
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\begin{document}$$\Phi _{12}=0$$\end{document}Φ12=0 or \documentclass[12pt]{minimal}
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\begin{document}$$\Phi _{21}=0$$\end{document}Φ21=0 we recover the one-directional movement model Eq. 3. When \documentclass[12pt]{minimal}
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\begin{document}$$\Phi _{12}=\Phi _{21}=\Phi $$\end{document}Φ12=Φ21=Φ, we recover the two-directional movement model Eq. 9.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
K. Castellano, O. Saucedo, S. Ciupe contributed equally to this work.
Change history
10/17/2026
The original article was revised to correct errors in the author names
Change history
10/17/2026
A Correction to this paper has been published: https://doi.org/10.1007/s11538-026-01780-7
SMC acknowledges partial support from the NIH NIGMS grant 1R01GM152743-01 and National Science Foundation grant No. 2051820. This research was enabled in part through the Virginia Tech Center for the Mathematics of Biosystems (VTCMB-033500).
Data Availability
All data necessary to replicate the results in this article are available at https://github.com/StancaCiupe/HBV-multi-patch.
DeclarationsConflicts of interest/Competing interests
We declare no conflict of interest associated with this publication.
Code availability
All code necessary to replicate the results in this article are available at https://github.com/StancaCiupe/HBV-multi-patch.
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All data supporting the findings are available in the supplementary material and in a public repository.
Demonstration content: the full text of this article is adapted from “Investigation of a Two-Patch Within-Host Model of Hepatitis B Viral Infection” by Castellano and colleagues (Bulletin of Mathematical Biology, 2025; https://doi.org/10.1007/s11538-025-01555-6), which is available under a Creative Commons license (https://creativecommons.org/licenses/by/4.0/).