Cytoplasmic Recycling of rcDNA-containing Capsids Enhances HBV Infection

Hepatitis B virus (HBV) infection is a deadly liver disease. The main aim of this work is to explore the role of cytoplasmic recycling of rcDNA-containing capsids in the hepatitis B virus (HBV) infection. To this purpose, considering the recycling of capsids, a noval mathematical model is proposed in order to understand the dynamics of this viral infection in a better way. Through a rigorous comparison with experimental data obtained from four chimpanzees, the proposed model exhibits a robust alignment with the dynamics of infection. The eﬀects of three parameters (recycling rate, virus production rate, and volume fraction of newly produced capsids) are examined, revealing an interesting observation: the inclusion of recycling reverses the inﬂuence of both virus production rate and the volume fraction of newly produced capsids in infection. A comprehensive global sensitivity analysis is conducted to identify the most positively as well as negatively sensitive parameters for each compartment in the model. In conclusion, this study underscores that the accumulation of rcDNA-containing capsids within the infected hepatocyte is a key factor contributing to the exacerbation of the disease. In addition, another major ﬁnding of our study is that due to recycling of capsids, the number of released viruses increases in spite of low virus production rate. In other words, the recycling of capsids acts as a positive feedback loop in the viral infection.


Introduction
Hepatitis B virus (HBV) causes a deadly liver disease hepatitis B. As reported by the World Health Organization (WHO), around 296 million people worldwide suffer from the chronic HBV infection, where 1.5 million new cases were found each year.It was estimated that 820 000 people died due to hepatitis B virus infection in the year 2019 alone, most of them had either cirrhosis or hepatocellular carcinoma (HCC) [1].Naturally, there are two types of HBV infection: acute and chronic.In the acute phase of infection, HBV DNA copies may reach as high as 10 10 copies/ml.Acute infection typically lasts a few weeks and are eventually cured as a result of immune response [2,3].As for the adult population, there is a clearance rate of 85-95% in acute infection.However, chronic infections can last for many years and may result in terrible diseases, such as liver cirrhosis and HCC [4].Chronic infection generally is a life-long incurable condition which affects the personal impacts of the patient such as stigma-discrimination, anxiety about disease progression, and long-term health care costs etc.According to the literature and clinical findings, horizontal and vertical transmission of this virus are two main modes of transmission to human populations.Blood transfusions, unprotected sex, reusing syringes and blades, and the reuse of medical equipment during surgery are examples of horizontal transmission.Although, a complete understanding about the mechanisms of HBV persistence remains elusive so far.Consequently, hepatitis B continues to pose a significant health threat to populations throughout the World.
The life cycle of HBV is unique and quite complex.At the onset of the infection, virion particles first interact with heparan sulfate proteoglycans (HSPGs) receptor in a non-specific and low-affinity manner, and then it binds to NA(+)-taurocholate co-transporting polypeptide (NTCP) receptor with high affinity and enters into the hepatocytes through clathrin-mediated endocytosis [5] and releases its relaxed circular DNA (rcDNA)-containing capsids into cytoplasm.NTCP, localized predominantly in the liver, plays a key role in the uptake of bile salts into hepatocytes.It has been identified as a crucial entry receptor for both HBV and hepatitis D virus (HDV) in 2012 [6].The incoming nucleocapsid in the cytoplasm is steered toward the nucleus along with the microtubules [7].Subsequently, it uncoats its rcDNA which is then delivered into the nucleus through the nuclear pore complex in an importin-dependent manner.In the nucleus, HBV genomic DNA is modified by cellular factors and is converted to covalently closed circular DNA (cccDNA) by the host DNA repair mechanism [8,9].These cccDNAs not only withstand antiviral interventions but also elude innate antiviral surveillance.Although it is unclear how and where cccDNA is maintained in the nucleus.cccDNA resides episomally, but it is inherently stable, and functions as a template for viral RNA over the long term.RNA polymerase IIs (RNA Poly IIs) use the cccDNA as a template to produce viral RNAs, such as pgRNA, L, S, PreC, and X mRNAs.These viral proteins are synthesized by ribosomes through the translation of mRNAs.Polymerase and pgRNA form a 1:1 complex called a Ribonucleoprotein (RNP) complex [10].The encapsulation of this RNP complex by C proteins leads to the formation of new pgRNA-containing nucleocapsids, commonly referred to as immature nucleocapsids.The pgRNA is reverse-transcribed by polymerase [11], resulting in the immature nucleocapsid being converted into a rcDNA-containing nucleocapsid, known as the mature nucleocapsid [12].This newly produced rcDNA-containing nucleocapsids either return to the nucleus and maintain the cccDNA pool or are enveloped by surface protein and release as complete virus from the infected hepatocytes [14].A portion of rcDNA-containing capsids is transported again into the nucleus to increase the amount of supercoiled cccDNA.It is known as recycling of rcDNA-containing capsids [10] and is considered as a significant factor in the intracellular dynamics of HBV replication.Besides the virus, the infected cell releases two non-infectious viral-like particles, known as sub-viral particles, which take on the forms of both spherical and filamentous structures.
In the Literature, there are several viral dynamic models [16][17][18][19][20][21] have been proposed over the last two decades.Some of these are associated with HBV infection.These models are useful in understanding the pathogenesis of infection as well as devising better treatment protocols.In the year 1996, Nowak et al. [22] analyzed a basic HBV infection model comprised of three compartments: uninfected hepatocytes, infected hepatocytes, and virions for the first time.Wodraz et al. [23] improved the basic model by including the effects of cytotoxic T cells and antibodies.By putting a standard incidence function in the place of mass action term of the uninfected hepatocytes and viruses, Min et al. [26] modified the classic viral infection model [22].According to them the mass action term is not rational for the HBV infection as it implies that someone with a smaller liver is unlikely to be infected.In addition to the standard incidence function, the time delay during the production process of the virus was also taken into account by Gourley et al. [28].Rather than using constant growth terms, Hews et al. [29] introduced a modified model that considers the logistic growth of uninfected hepatocytes.Eikenberry et al. [25] presented a delay model on HBV infection when uninfected hepatocytes proliferated logistically.Haung et al. [30] proposed another HBV dynamics model using different incidence function, called Beddington-DeAngelis type incidence function.Through a comprehensive investigation, Yu Ji et al. [31] first showed that the effects of immune response is not constant, and it follows a periodic function.The immune-mediated cure of infected hepatocytes is an important factor in combating viral infections.Incorporating the effects of cure rate of infected hepatocytes, Wang et al. [24] extended the model proposed by Min et al. [26].Fatehi et al. [32] determined that in HBV infection, NK cell takes a significant role in apoptosis as it kills the infected cell by producing the perforin and granzymes.Using a mathematical model that incorporates intracellular components of HBV infection, Murray et al. [11] studied the infection dynamics and clearance of viruses in three chimpanzees who were acutely infected.Murray et al. [35] proposed another mathematical model with three compartments and measured the half-life of HBV as approximately four hours.Manna and Chakrabraty [37] modified this model (proposed by Murray et al. [35]) by considering the uninfected hepatocytes and subsequently studied the effects of delay in intracellular process.Considering the impacts of antibodies and CTLs, Danane et al. [38] extended the model proposed by Manna and Chakrabraty [37], including immune response of CTLs and examined role of optimal therapy in controlling viral replication.Using an average incidence rate, Guo et al. [39] established the global stability of a delayeddiffusive HBV infection model.Fatehi et al. [10] built up an intracellular model of HBV infection and compared various kinds of therapeutic strategies that can be applied in future.The age of an infected cells serves as a key biological indicator, providing valuable insights into the dynamics of virus behavior.Recently, Liu et al. [40] proposed an age-structured model of HBV that treated HBV capsids as a separate compartment.Beyond the studies mentioned above, a broad range of literature exists, examining various aspects of HBV infection.Although, to the best of our knowledge, none of the aforementioned studies addressed the role of capsid recycling in the infection.
In some recent biological studies [12,14,41], it is observed that the severity of HBV infection are greatly affected by the recycling of rcDNA-containing capsids.Recently, Thio et al. [15] established the fact that HBV maintains a consistent reservoir of cccDNA through the intracellular recycling of HBV genomes.Nonetheless, the clear concept about how this recycling of HBV capsids contributes to the virus replication is lacking and poorly understood.One of the reason could be the increase in number of viruses in the liver, since a portion of newly produced capsids are eventually reused as core particles providing an additional source of supercoiled cccDNA.In order to control HBV transmission in host, having a clear knowledge on recycling of capsids is extremely important and exceedingly necessary.Although some mathematical studies have been conducted on HBV transmission in a host, only a few of them considered the rcDNA-containing capsids as a separate compartment.However, these models failed to capture the actual dynamics of HBV infection.The main reason for this failure could be ignoring the recycling effects of capsids.In this study, an improved mathematical model incorporating the recycling effects of HBV capsids is proposed for the first time.This model is expected to reveal the HBV intracellular dynamics more realistically compared to already available models.In a nutshell, we have mainly concetrated on the following things: 1.The effects of volume fraction of capsids on the disease dynamics.2. The effects of recycling of capsids in the HBV infection.3. The effects of capsids production rate on infection.4. Global sensitivity analysis (GSA) of model parameters.
-Long-term temporal behavior of parameter

Model formulation
The persistence of HBV infection for a long period in patients depends on the ability to subtend of cccDNA in the infected hepatocytes.cccDNA plays a central role in disease progression.There are two main sources of cccDNA: (i) rcDNA-containing capsids produced directly from the viruses incoming from extracellular space, and (ii) rcDNA-containing capsids produced within the hepatocytes due to recycling [10,12,41].The recycling of capsids is not a continuous intracellular process.Depending on the availability of viral surface proteins (L, M, S), a portion of newly produced rcDNA-containing capsids goes back to the nucleus to amplify the pool of cccDNA [13].So, the volume fraction of capsids (α) responsible for virus production is generally a function of the surface protein (L, M, S) i.e. α = α(surface proteins).However, a fixed estimated value of α is used throughout the study for simulation purposes.Accordingly, the HBV infection dynamics is described by the following system of ordinary differential equations.The temporal change of each compartment is calculated based on the mass-action principle.
Susceptible hepatocyte : Virus: Here, X(t), Y (t), D(t) and V (t) denote the numbers of susceptible hepatocytes, infected hepatocytes, rcDNAcontaining capsids, and free viruses respectively.All the parameters µ, k, δ, a, α, β, γ and c are non-negative real numbers.In this model, λ is assumed to be constant growth rate of susceptible hepatocytes, and µ is their natural death rate.The usual death rate of infected hepatocytes, as well as capsids is δ since capsids are within infected hepatocytes , so they will be removed at the same rate with infected hepatocytes.The parameter k describes rate at which the susceptible hepatocytes are infected by the viruses.HBV capsids are produced at the rate a from infected hepatocytes , and β denotes the production rate of new viruses.Here, c is the virus clearance rate, and capsids reproduce themselves by recycling at rate γ.In Figure 1, the diagrammatic representation of the system (1) is shown.
This paper is organized as follows.In Section 3, we prove that proposed model is well-posed by showing the existence, uniqueness, non-negativity, and boundedness of the solutions of the system (1).In Section 4, the proposed model is validated by experimental data.We determine the existence of all steady states and establish the global stabilities of equilibria by constructing appropriate Lyapunov functionals and using Lyapunov-LaSalle invariance principle in Section 5. Bifurcation analysis is performed in Section 6.On the other hand, Section 7 discusses the effects of some model parameters from different angles.We study the global sensitivity analysis of model parameters in section 8. Finally, a brief conclusion is provided in Section 9.

Properties of the solutions
Here, the existence and uniqueness of the solution for the system (1) are discussed.In order to ensure the feasibility of the model from biological point of view, it is crucial to show that all solutions remain non-negative and bounded across all non-negative initial conditions.The rationale behind this is that the number of cells or viruses must not drop below zero or exhibit limitless growth after the time of infection.Accordingly, the subsequent discussion confirms the non-negativity and boundedness of the solution.

Existence and uniqueness of the solutions
Each function of the right hand side of system (1) is polynomial functions of four variables X(t), Y (t), D(t), V (t).So, each function is continuous and satisfies Lipschitz's condition on any closed interval [0, η], η ∈ R + , set of all positive real numbers.Therefore, the solution of the system (1) exists and is unique.
The proof of this theorem is presented in Appendix A.
So, X(t) and Y (t) are bounded for all t > 0. Now, from the third equation of the system (1), one can get Therefore, D(t) is bounded for all t > 0. Using the boundedness of D(t), from the last equation of the system (1) one can find Hence, V (t) is bounded for all t > 0. Therefore, all the population (X(t), Consequently, one can also observe the closed, bounded, and positively invariant set as follows: If γ meets this condition, then D(t) becomes always bounded.Otherwise, D(t) diverges to positive infinite i.e. the severity of infection increases significantly and situation of the patient becomes worse and worse with time.This relationship between these four parameters is very important when treating HBV patients.

Parameter estimation and model calibration
In order to enhance the realism of viral dynamics and to improve the reliability of our robust predictions, the proposed model is calibrated with the experimental data of Asabe et al. [36].In their study, Asabe et al. examined the effects of varying viral inoculum sizes on the kinetics of viral spread and immune system in a cohort of nine young, healthy, HBV-seronegative chimpanzees.It was observed that the viral load of six out of nine chimpanzees reached a peak level of 2 × 10 10 within three weeks from the time of inoculation.Later, the virion load decreased within 15 weeks and reached below the detection level.On the other hand, that experiment also documented that the remaining three chimpanzees developed chronic infection.Out of these nine chimpanzees, the concentration profiles of HBV DNA of two chimpanzees are considered to estimate the model parameters and validate our model.They were (i) Ch1603, and (ii) Ch1616.In each case, the model parameters are estimated by minimizing the sum-of-squares error (SSE) which is given by where P i and p(i) denote the experimental data and model solution, respectively.In Figure 2, the experimental data and solutions of the proposed model are compared.It is observed that for both chimpanzees, model solutions agree well with the experimental data i.e. infection dynamics are captured very well by this proposed model.
Based on the existing literature, a huge number (300 million) of people are living with a chronic HBV infection.Approximately, more than 90% of infected infants with this virus develop chronic infection [33].Moreover, chronic HBV infection is frequently a lifelong and incurable condition, leading to various personal consequences, including anxiety regarding disease progression, stigma, discrimination, and long-term healthcare costs [34].Due to this, our interest lies in exploring the dynamics of infection within the context of chronic situations.To this purpose, the average of estimated values of parameters for the chimpanzees Ch1603 and Ch1616, are enlisted in Table 1.These estimated values of parameters are used to facilitate further analysis and numerical simulations, such as the effects of some model parameters, sensitivity analysis.HBV DNA

Ch1603
Fig. 2 Experimental validation of the model.The experimental data is represented by red circles, while the solid blue line corresponds to the numerical solution of the system (1).

Existence and stability of equilibria
Before proceeding to the detailed study, it is mentioned that the condition R s > 0 will be used throughout the further study.

Existence of equilibria
In order to evaluate the equilibrium points of the system (1), one need to consider the zero growth isoclines and their points of intersection.Thus, the equilibrium points of the system (1) are found by solving the system of equations It can be shown that the system (1) possesses two sets of equilibrium points.
(i) The uninfected steady-state or disease-free equilibrium (E u ): The uninfected steady-state or disease-free equilibrium always exists, and it is denoted by E u = λ µ , 0, 0, 0 .
In case of viral infection, the basic reproduction number is the number of secondary infective cells produced by a single infective cell, which is introduced into a fully susceptible population [43].The first equation of the system (1) is for the uninfected class and the last three equations are meant for the infected class.The next-generation approach [44,45] is used to determine the basic reproduction number.The basic reproduction number of the system (1) is denoted by R 0 and it is given by R (ii) The infected steady-state or endemic equilibrium (E i ): Mathematically, infected steady-state or endemic equilibrium points exists always.But biologically, the existence of this steady state depends on the basic reproduction number (R 0 ) and R s (which is defined earlier).When R 0 > 1, and R s > 0, the endemic equilibrium (E i ) occurs and is given by , where

Global stability analysis of equilibria
In order to prove the global stability of equilibria, Theorem 7.1 of the book [46] is used.Two suitable Lyapunuv functions are defined as follows: Both functions are radially unbounded and globally positively definite.Therefore, the choices of Lyapunov functions are appropriate.
Theorem 3 The disease-free equilibrium point E u is globally asymptotically stable if R 0 ≤ 1.
Proof To prove the asymptotic global stability of disease-free equilibrium point, the first Lyapunov function L 1 (t) (defined above) is considered.
From the article of Kajiwara et al. [51], it follows that 2 − The solution of the equation dL 1 (t) dt = 0 is λ µ , 0, 0, 0 only, which is the equilibrium point E u .So, based on the Lyapunov-LaSalle invariance principle [46], the disease-free equilibrium point, E u is globally asymptotically stable whenever R 0 ≤ 1.
Theorem 4 The endemic equilibrium E i is globally asymptotically stable if R 0 > 1.
Proof We are approaching the problem by taking into account the second Lyapunov function L 2 (t).
where R s is defined above.Differentiating (3) with respect to t, we have Clearly, the first term of the above equation is negative unless X(t) = X 1 .To prove the negativity of the second term, define It is clear that x i ≥ 0 for i = 1, 2, 3, 4 and x 1 x 2 x 3 x 4 = 1.Applying A.M ≥ G.M , Hence, It is noticed that the solution of the equation , which is the equilibrium point E i .So, the endemic equilibrium point is globally asymptotically stable whenever R 0 > 1 based on the Lyapunov-LaSalle invariance principle.

Bifurcation Analysis
In this section, the bifurcation analysis of the proposed model (1) is performed.Stability criteria of E u indicates that E u will be stable if µ > akαβλ R s cδ (= µ * ), otherwise it will be unstable i.e. the stability of E u changes as µ crosses the threshold value µ = µ * .On the other hand, the endemic equilibrium E i (although E i is not feasible in the context of biology) becomes unstable if µ > µ * .Thus, the equilibrium points coincide and exchange their stability which leads to the transcrical bifucation of the system (1) around the point E u at µ = µ * with natural death rate parameter µ as the bifurcation parameter.It is seen that the Jacobian matrix which is calculated at has one zero eigenvalue when µ = µ * .In order to verify the existence of transcritical bifurcation analytically, Sotomayor's theorem [47] is applied on the system (1) around the disease-free equilibrium point E u when µ = µ * .The R.H.S of system (1) can be represented in vector form as Differentiating the function f (X, Y, D, V ) partially with respect to µ, it is obtained that f µ (X, Y, D, V ) = (−X 0 0 0) T .The Jacobian matrix (4) and its transpose matrix have an eigenvalue ξ = 0 with eigenvectors Now, at E u we have verified the following transvesality conditions: All the notations used here are same as in the book of Lawrence Perko [47].Therefore, all three conditions of Sotomayor's theorem hold and the system (1) undergoes transcritical bifurcation at E u as the natural death rate of uninfected hepatocytes µ, crosses the threshold value µ * .
Remark 2 Based on the theoretical results, it is seen that R 0 and R s are two crucial threshold numbers governing the dynamics of HBV infection.

Numerical simulation
Two new parameters are introduced into our model: the volume fraction of capsids in favor of virus production (α) and the recycling rate (γ) of newly produced capsids.In this section, the changes in infection dynamics with the change of these two parameters and virus production rate (β) are depicted through numerical simulation.
It is also illustrated how the recycling of capsids reverses the impact of volume fraction and virus production rate on the infection.The system (1) is solved with help of well-known fourth-order Runge-Kutta method and the results are plotted graphically.The entire numerical experiment is carried out in case of chronic infection.
The value of parameters are taken from Table 1.

Experiment-1: Effects of volume fraction (α) of rcDNA-containing capsids on infection dynamics
The effects of volume fraction of capsids (α) on the system (1) are examined for two distinct cases.In the first case, impacts of recycling of capsids on the system (1) are ignored, whereas in the second case, it is considered.In Figure 3, effects of α in the absence of recycling of capsids are shown.The numerical simulation is performed for six values of α, namely, α = 0.5, 0.6, 0.7, 0.8, 0.9 & 1.0, keeping the other parameters fixed.Both the condition R s > 0 and R 0 > 1 are satisfied for every value of α i.e. the system converges to corresponding endemic equilibrium point.Figure 3 shows that the stability level of uninfected hepatocytes and HBV capsids decline, but infected hepatocytes and viruses progressively increase while α increases.
However, upon incorporating the recycling effects of capsids into the model, opposite patterns are observed for uninfected, infected hepatocytes, and viruses in Figure 4.There is no change observed in the trend of capsids compartment, but the stability level significantly increases for the same value of volume fraction.Consequently, when α = 1.0, it is clear that the concentration level of uninfected hepatocytes is at a highest level while that of infected hepatocytes, HBV capsids, and viruses get stabilized at a lowest level.Thus, the results for these two cases are significantly different.The low value of volume fraction of capsids (α) implies that a less number of capsids can produce new viruses, and a large number of capsids get accumulated inside the hepatocytes.Therefore, accumulation of core particles (capsids) within the infected hepatocytes can be a cause of severe infection rather than the rapidly release of viruses.Fig. 3 The effects of volume fraction of capsid (α) on the dynamical system (1) in absence of recycling of capsids (γ = 0).
Experiment-2: Effects of recycling rate (γ) In Figure 5, we present the impact of recycling rate (γ) on all the four populations of the system (1).The parameter values are same as in Table 1 except γ.Six different values of γ (= 0.5, 1.0, 1.5, 2.0, 2.5, 3.0) are chosen for simulation in such a way that the criteria for boundedness of solution R s > 0 is satisfied.For these values of parameters , the associated value of R 0 exceed unity indicating chronic infection.The main observation obtained from this experiment that when recycling of capsids is not considered, system asymptotically achieves its stability.Concentration level of uninfected hepatocytes drops when γ rises.On the other side, the concentration level of infected hepatocytes increases with the increase in γ and attains its minimum value when γ = 0.A similar scenario is observed for capsids and virions.It is also observed that the peak level of infected, capsid and virus compartment becomes smaller as γ decreases i.e. the critical phase of the patient diminishes and disappears more rapidly.Biologically, one can see that for high value of γ, the situation becomes more critical for the patient, and it is difficult to be cured.Sometime, the concentration of uninfected hepatocytes falls below the biologically plausible threshold (20% of the original hepatocyte) as defined by Goyal et al. [53].Therefore, this experiment underscores the importance of incorporation of capsid recycling into the model for a more realistic understanding about HBV infection dynamics.Fig. 4 The effects of volume fraction of capsid (α) on the dynamical system (1) when the effects of recycling of capsids is considered (γ > 0).

Experiment-3: Effects of capsids to virus production rate (β)
For different values of capsids to virus production rate, the variations in the dynamics of uninfected hepatocytes, infected hepatocytes, capsids, and viruses are shown in Figure 6 (without recycling) and Figure 7 (with recycling).Six values of β (=0.6, 0.7, 0.8, 0.9, 1.0, 1.1) are considered keeping other parameters fixed as shown in Table 1.For each value of β, the inequalities R s > 0 and R 0 > 0 hold.Similarly as Experiment-1, in this instance we also consider two cases: without and with recycling.In both cases, the effects of β on infection dynamics closely resemble to the impacts of α observed in Experiment-1.Figure 7 demonstrates that low value of β makes things worse for the sufferer, thus making it difficult to cure.Small virion release rates (β) in the HBV replication process may be a risk factor for chronic hepatitis exacerbation over time.So, this discussion underlines the importance of recycling of capsids in cases of HBV infection.

Observation:
Based on the results obtained from these three experiments, it is observed that the recycling of capsids is appeared to be an influential factor in progression of the infection.Recycling of capsids enhances the number of viruses amplifying the pool of cccDNA.As a result, infection becomes severer.In a nutshell, recycling of capsids acts as a positive feedback loop in case of HBV infection.Through the study of intracellular HBV dynamics, Jun Nakabayashi [48] concluded that recycling of capsids can contribute to the accumulation of core particles, indicating a risk factor for the exacerbation of the disease.Recently, Sakaguchi [49] noted that HBV infection can be reduced substantially by disrupting the capsid recycling which implies that the prompt import of newly generated capsids creates a more robust infection.Ko et al. [50] established that the stability of the cccDNA pool in HBV is maintained through the internal recycling of HBV genomes i.e. recycling emerges as a pivotal factor in the persistence of HBV infection.
In the absence of capsid recycling, uninfected hepatocytes reach a steady state with the highest value for a low value of volume fraction or of virus production rate (refer to Figure 3 and Figure 6).However, this observation does not accurately reflect the actual dynamics of the infection.Consequently, Figure 3 and Figure 6 fail to display the true roles of α and β in the infection dynamics.On the other hand, when recycling of capsids is considered, a low value of volume fraction or of virus production rate results in a substantial increase Fig. 5 The effects of recycling rate of capsid (γ) on the dynamical system (1).
in the number of capsids through recycling loop, thereby amplifying the production of new viruses.In this case, Figure 4 and Figure 7 provide a more accurate representation of the underlying biological dynamics.
The obtained results due to the inclusion of capsid recycling also corroborate with the conclusions shown in previous biological studies [48][49][50].Incorporation of capsid recycling enriches the virus dynamics and revels the fact that recycling of capsid acts as a positive feedback loop in case of HBV infection.The nature of HBV infection found in the existing dynamics models [37,38,40,42] which did not consider the capsid recycling, is not similar to the nature of HBV infection observed in the biological studies [48][49][50].In contrast, the present study offers a more comprehensive depiction of the HBV dynamics and close align with the actual dynamics of infection.This is the novelty of the present study.Therefore, it is very important to consider the effects of recycling of capsids while treating the infection and proposing any new antiviral therapy.

Global sensitivity analysis of parameters
The accuracy of the results of a mathematical model related to some biological phenomena often becomes poor because of uncertainties in experimental data which are utilized in the estimation of model parameters.
Recently, many authors study the effects of single parameter keeping all others parameters fixed at their estimated values.This type of sensitivity analysis is called local sensitivity analysis.But local sensitivity analysis doesn't provide the proper information of uncertainty and sensitivity of the parameters.In order to find out the contributions of each model parameters universally in HBV infection dynamics, the global sensitivity analysis is performed using the technique "Latin hypercube sampling-partial rank correlation coefficient" (LHS-PRCC) described by Marino et al. [54].Fig. 6 The effects of virus production rate (β) in the absence of recycling of capsids (γ = 0) on the dynamical system (1).
Fig. 7 The effects of virus production rate (β) on the dynamical system (1) when effects of recycling of capsids is considered (γ > 0).
distribution is chosen for all parameters depending on a priori information and existing data.The model is then simulated iteratively over all p-tuples parameter pairs.It is recommended that the sample size N be at least (p + 1), but it is better to take a larger sample size to ensure the desired accuracy of the results .Here, the sample size is set to 1000.
The correlation coefficient (CC) measures the strength of a linear relationship between the inputs and the outputs.The CC is calculated between the input variable (X) and output variable (Y ) as follows: where X and Ȳ represent the sample means of X and Y respectively and r ∈ [−1, 1].In case of raw data of X and Y , the coefficient r is known as sample or Pearson correlation coefficient.The CC (r) is called Spearman or rank correlation coefficient if the data are rank-transformed.In this study, the model parameters (λ, k, µ, δ, a, γ, α, β, c) and model variables (X, Y, D, V ) are considered as input and output data, respectively .By using LHS-PRCC, one can derive insightful conclusions about how the model parameters influence on the outputs of a system.

Scatter plots: The monotonic relationship between input and output variables
Besides the improvement and generalization of a dynamical system, it has attracted the attention of many researchers to know how the outputs are affected if the parameters' values vary in a reasonable range.In the practical field of application especially in virus dynamics model, it is very important and essential to study the sensitivity of parameters.In such cases, PRCC values can provide useful information.Based on the PRCC values, one can identify which set of parameters is the most significant for achieving some specific goals such as control or regulatory mechanisms, reduce viral load, increase immune response, proposing any new therapy and optimization of drug usage.In order to analyze the sensitivity of parameters, the baseline values are taken from Table 1.To this purpose, we systematically vary all parameters within the range from 80% to 120% of their base values.The algorithm of global sensitivity analysis using LHS-PRCC method is displayed in Figure 8 in the form of a flow chart.Simulation results of the proposed model (1) are visualized by scatter plots on Figure 12 -Figure 15.The PRCC values of all parameters are calculated at day 1400 with respect to the dependent variables.The positive correlation of a compartment to a model parameter (PRCC value positive) ensures that if the value of this parameter increases individually or simultaneously, the concentration of the compartment increases accordingly.On the other hand, negative correlation (PRCC value negative) tells us the opposite aspects.Parameters with p-values exceeding 0.01 are considered to have no impact on the infection.Based on the PRCC values, the model parameters are arranged in descending order for the uninfected, infected, and virus classes as follows: -Uninfected hepatocyte: α, δ, c, β, λ, µ, a, k, γ.
In Figure 9, the PRCC values of each parameter against the model variables (X, Y, D, V ) are demonstrated where bars show the PRCC of each mentioned parameters at t = 1400 days.

Long-term behavior of the sequence of PRCC values
Initially, we study the GSA for the outputs at t = 1400 days.However, focusing on GSA at a single time point doesn't provide a comprehensive understanding of a parameter's behavior over the entire infection period.
In order to gain a holistic understanding about the behavior of a parameter throughout the whole time span, we perform GSA for each parameter at each time point.Figure 10 illustrates how the PRCC value of each parameter change across all compartments over time.The changing tendency of a specific parameter on different compartments reveals significant variations.Figure 10 also indicates that the sequence of PRCC values of each parameter across all four compartments i.e. sensitiveness of each parameter is convergent, ensuring the robustness and stability of the sensitivity analysis.Based on the long-term tendency of PRCC value, we note that at the onset of the experiment, certain parameters (e.g.λ and µ for uninfected hepatocytes) exhibit heightened sensitivity, but over time, their significance diminishes.Similarly, some other parameters (e.g.α, k for uninfected hepatocytes) that initially appear insignificant but become sensitive after a certain time.
Global sensitivity analysis discloses numerous new and remarkable findings, which are outlined as follows: 1.The parameters α, λ, γ are identified as the most positively sensitive parameters for uninfected hepatocytes, infected hepatocytes, and the viruses, respectively.On the other hand, the parameters γ, δ, α are found to be the most negatively sensitive parameters for the same.2. In chronic infection, it has been observed that the virus production rate (β) has a relatively less influence on the overall infection.3. The recycling rate (γ) is the most negatively sensitive parameter for uninfected liver cells and most positively sensitive parameter for virus compartment.This implies that if the value of γ increases, the number of virus increases whereas the number of uninfected hepatocytes decreases simultaneously.Therefore, GSA also proves that the recycling of capsids acts as a positive feedback loop in the context of infection.

Conclusions
In this study, HBV infection dynamics is modeled based on the biological findings.In order to describe this viral infection in a more realistic way, the recycling effects of capsids are incorporated into this model.The inclusion of recycling effects leads to a notable paradigm shift in the outcomes of the proposed model.The non-negativity and boundedness of the solutions for any non-negative initial condition establish the feasibility of the system.The stability analysis of the system indicates that both the equilibrium points are globally asymptotically stable under some conditions (disease-free: R 0 < 1, endemic: R 0 > 1) i.e. the patient will either achieve a full recovery, or the viral infection will persist at a stable level for the rest of the life.Upon comparing the model solution with the experimental data collected from two chimpanzees [36], it is seen that the model solution agrees well with the experimental data.Hence, the proposed model effectively can capture and represent the intricate dynamics of HBV infection, making it a more realistic and reliable tool for studying this disease.In addition, it is further observed that due to recycling, the viral load increases considerably.The long-term behavior of parameters are also discussed through global sensitivity analysis by applying the LHS-PRCC method.
From the simulated results, the following findings are observed.
1. Most of the mathematical models on HBV infection developed so far, underestimate the production of virions and suppress the production of capsids as the recycling effects of capsids was ignored.Consequently, these models fail to capture the actual dynamics of HBV infection, whereas, the proposed model shows a more realistic production of virions and the actual dynamics of the infection.2. Recycling rate of capsids (γ) is one of the deciding parameters to determine the severity of infection.So, it is very important to pay attention to this kind of parameter while proposing any strategy to control this disease.3.This study analyses the effects of volume fraction of capsids (α) on disease dynamics probably for the first time.It is found that the inclusion of recycling of capsids reverses the effects of volume fraction on the infection.This is a striking outcome of the present study that changes the usual understandings of viral dynamics.So, volume fraction of capsids emerges as a viable member in the set of disease-controlling parameters.4. It is also observed that the number of released viruses increases in spite of low virus production rate due to recycling of capsids.Though this result appears to be contradictory to the known fact, but our study has clearly explained this new findings.In order to gain deeper insights about this infection, the emergence of this unusual behavior becomes very important.On the other word, the recycling of capsids acts as a positive feedback loop in this viral infection.5. Based on the partial rank correlation coefficient values, the global sensitivity analysis unequivocally identifies that the disease progression is highly influenced by the volume fraction of capsids as well as the recycling rate.These findings highlight the pivotal role of these factors in shaping the dynamics of the disease and warrant further attention in future studies.6.The strong concurrence between the model solution and the experimental data substantiates that the proposed model is biologically reliable.
As mentioned above, our model provides a theoretical backbone of the mechanism causing the exacerbation during the chronic HBV infection.This is a new and relatively simple mathematical model that can describe the infection dynamics more accurately.Using these new findings, this model can be applied to a variety of clinical trials and for the formulation of new drugs.

8. 1
Latin hypercube sampling (LHS)-Partial rank correlation coefficient (PRCC)Latin hypercube sampling is one kind of Monte Carlo class of sampling methods.In 1979, McKay et al.[56] first introduced this sampling method.With the help of LHS, sample inputs of the model are arranged within a "hypercube of dimensions p", where p represents the number of model parameters.For our proposed model(1), the number of model parameters (p) is equal to 9. A probability density function (pdf) is employed for sampling parameter values based on parameter ranges partitioned into intervals.In this study, the uniform

Fig. 8
Fig. 8 Working steps of global sensitivity analysis are shown concisely in this diagram.

Fig. 9
Fig. 9 PRCC values of parameters corresponding to uninfected, infected hepatocyte, capsid and virus classes at t = 1400 day are plotted.Here, MNSP: Most Negatively Sensitive Parameter; MPSP: Most Positively Sensitive Parameter.

Fig. 10
Fig. 10 PRCC value of each parameter over time.

Table 1
Estimation of parameters.

Table 2
4. The volume fraction of capsids is identified as the most positively sensitive parameter for the uninfected compartment.That means if less number of capsids involved in producing new virions and a larger number of capsids undergo in recycling, then, as a result, this would make the infection more severe.PRCC value of parameters.