Stochastic Formulation of Multiscale Model of Hepatitis B Viral Infections

Abstract

The study investigates and analyzes certain qualitative properties of a stochastic dynamical multiscale model for hepatitis B viral infection. By formulating appropriate stochastic Lyapunov functions, the study derives sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions of the multiscale model. Additionally, the study establishes conditions under which the virus can be eradicated from the population. The findings indicate that low-intensity white noise guarantees a unique ergodic stationary distribution, while higher noise levels can result in viral extinction.

1. Introduction

Viral infections propagate by overcoming multiple biological barriers and transitioning across different levels of biological organization. Hepatitis B virus targets hepatocytes, the main functional cells of the liver, causing both acute and chronic diseases [1]. Some studies [2,3,4,5,6] have conducted research on viral hepatitis and multiscale modeling of infectious diseases. However, stochastic modeling plays a crucial role in capturing the inherent randomness and uncertainty in such processes by incorporating probabilistic elements, thereby providing a more realistic representation of real world dynamics. Since real-life biological phenomena involve intrinsic variability and stochasticity, stochastic models, especially those that account for random viral contacts during latent and infectious periods, are considered to comprise a more appropriate modeling framework. It is well established that biological systems are influenced by ambient white noise [7]. Despite the extensive scientific research on the dynamics of hepatic viral infections, previous studies have primarily employed ordinary differential equations (ODEs). While ODE models offer a useful approximation of average system behavior, the models inherently assume continuous and deterministic evolution of populations. As a result, ODE models fail to capture critical stochastic phenomena such as extinction events and rare events that drive transitions between disease states. It is noteworthy that stochastic models can account for these probabilistic outcomes, particularly in systems with low copy numbers of virions or infected cells, where random fluctuations significantly influence the overall dynamics. As such, incorporating stochasticity allows for a more comprehensive and accurate representation of HBV dynamics, particularly during the early stages of infection or under therapeutic interventions where population sizes may be small [8].

As a result, this study presents a stochastic differential equation (SDE) formulation of a simplified nested multiscale model at the cellular level, serving as a modification of the model system (2.8) proposed in [9]. The reasons for adopting stochastic modeling in the study of hepatic viral infection include biological complexity at the cellular level, specifically the presence of surface receptors on infected cells that mediate the entry and release of nucleocapsids into the cytoplasm. Stochastic differential equations (SDEs) allow for the representation of such systems by generating a distribution of possible outcomes, in contrast to deterministic models, which yield a single expected trajectory. This approach is significant because it examines how noise affects the simplified nested multiscale model system (2.8) presented in [9]. Furthermore, this study incorporates stochastic parameters into the model system (2.8) from [9] and analyzes the resulting stochastic version of the modified mathematical model. Various techniques exist for introducing stochasticity into a system; here, we adopt the approach of parameter perturbation using white noise. This noise is assumed to be proportional to the state variables’ susceptible cells ${\displaystyle \left(S_C\right)}$, infected cells ${\displaystyle \left(I_C\right)}$, and community viral load ${\displaystyle \left(V_C\right)}$, and it influences the corresponding derivatives, ${\displaystyle S_C^{’},I_C^{’}}$, and ${\displaystyle V_C^{’}}$. The stochastic system is formulated as shown in model system (1):

$$\begin{array}{ccc}\hfill d{S}_C& =& [{\Lambda }_C-{\beta }_C{V}_C{S}_C-{\mu }_C{S}_C]dt+{\sigma }_1{S}_{C}d{B}_1\left(t\right),\hfill \\ \hfill d{I}_C& =& [{\beta }_C{V}_C{S}_C-({\mu }_C+d_C)I_C]dt+{\sigma }_2{I}_{C}d{B}_2\left(t\right),\hfill \\ \hfill d{V}_C& =& [V_s{r}_c{I}_C-{\sigma }_C{V}_C]dt+{\sigma }_3{V}_{C}d{B}_3\left(t\right),\hfill \\ \hfill d{h}_r& =& [{\Lambda }_r-{\eta }_r{h}_r{V}_s-{\delta }_r{h}_r]ds+{\sigma }_4{h}_{r}d{B}_4\left(s\right),\hfill \\ \hfill d{h}_t& =& [{\eta }_r{h}_r{V}_s-({\alpha }_t+{\mu }_t+{\delta }_t)h_t]ds+{\sigma }_5{h}_{t}d{B}_5\left(s\right),\hfill \\ \hfill d{h}_c& =& [{\alpha }_t{h}_t-({\alpha }_c+{\rho }_c)h_c]ds+{\sigma }_6{h}_{c}d{B}_6\left(s\right),\hfill \\ \hfill d{h}_i& =& [{\mu }_t{h}_t-({\alpha }_i+{\delta }_i)h_i]ds+{\sigma }_7{h}_{i}d{B}_7\left(s\right),\hfill \\ \hfill d{V}_s& =& [N_c{\alpha }_c{h}_c+N_i{\alpha }_i{h}_i-(r_c+r_i)V_s]ds+{\sigma }_8{V}_{s}d{B}_8\left(s\right).\hfill \end{array}$$

(1)

The stochastic nested multiscale model (1) can be simplified as presented in model system (2.8) of [9], where ${\displaystyle N_s}$ is a composite parameter used to upscale the within-cell parameters to the between-cell parameters and variables. The simplified model is shown in Equation (2):

$$\begin{array}{ccc}\hfill d{S}_C& =& [{\Lambda }_C-{\beta }_C{V}_C{S}_C-{\mu }_C{S}_C]dt+{\sigma }_1{S}_{C}d{B}_1\left(t\right),\hfill \\ \hfill d{I}_C& =& [{\beta }_C{V}_C{S}_C-({\mu }_C+d_C)I_C]dt+{\sigma }_2{I}_{C}d{B}_2\left(t\right),\hfill \\ \hfill d{V}_C& =& [N_s{r}_c{I}_C-{\sigma }_C{V}_C]dt+{\sigma }_3{V}_{C}d{B}_3\left(t\right).\hfill \end{array}$$

(2)

where ${\displaystyle B_i\left(t\right),i=1,2,3}$ are mutually independent standard Brownian motions defined on a complete probability space ${\displaystyle (\Omega ,\mathcal{F},\mathbb{P})}$, a complete probability space with filtration ${\displaystyle {\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}}$, which is increasing and right-continuous. while ${\displaystyle {\mathcal{F}}_0}$ contains all ${\displaystyle \mathbb{P}}$-null sets and ${\displaystyle {\sigma }_i>0,i=1,2,3}$ denote the intensities of the white noises [10]. Let ${\displaystyle B_i\left(t\right),(i=1,2,3)}$ be scalar Brownian motions defined on a probability space. The importance of filtration tells us the information to be available at the future time ${\displaystyle t}$. The main contribution and innovation is to study the impact of stochastic noise on the dynamics of the nested multiscale model at the cell level of biological organization. The paper is organized as follows. Section 2 presents the preliminaries necessary for the model description. In Section 3, we emphasize the existence and uniqueness of the global positive solution. Section 4 discusses the stationary distribution and ergodicity. Virus eradication is addressed in Section 5. Section 6 concludes the study.

2. Preliminaries for the Model Description

In this section, the study presents some known and useful results, including definitions, lemmas, theorems, and basic concepts related to stochastic processes, which are essential for analyzing system (2) [11,12,13].

Let ${\displaystyle (\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P})}$ be a complete probability space equipped with a filtration ${\displaystyle {\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}}$ that satisfies the usual conditions; that is, the filtration is increasing and right-continuous, and ${\displaystyle {\mathcal{F}}_0}$ contains all ${\displaystyle \mathbb{P}}$ sets. Consider the n-dimensional stochastic differential equations (SDEs)

$$dx\left(t\right)=f\left(x\right(t),t)dt+g\left(x\right(t),t)dB\left(t\right),$$

(3)

with initial value ${\displaystyle x\left(t_0\right)=x_0\in {\mathbb{R}}^n,x\in {\mathbb{R}}^n}$ (with ${\displaystyle x\ge 0}$), and ${\displaystyle B\left(t\right)}$ is the ${\displaystyle m}$-dimensional standard Brownian motion defined on ${\displaystyle (\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P})}$. Let ${\displaystyle C^{2,1}({\mathbb{R}}^n\times [t_0,\infty ];{\mathbb{R}}_{+})}$ be the family of all non-negative functions ${\displaystyle V(x,t)}$ defined on ${\displaystyle {\mathbb{R}}^n\times [t_0,\infty ]}$ such that ${\displaystyle V(x,t)}$ is continuously twice-differentiable in ${\displaystyle x}$ and once in ${\displaystyle t}$. Then, the differential operator ${\displaystyle \mathcal{L}}$ of (3) is defined by

$$\mathcal{L} = {\displaystyle \frac{\partial }{\partial t}}+\sum _{i=1}^n{f}_i(x,t){\displaystyle \frac{\partial }{\partial x_i}}+{\displaystyle \frac{1}{2}}\sum _{i,j=1}^n{\left[g^T(x,t)g(x,t)\right]}_{ij}{\displaystyle \frac{{\partial }^2}{\partial x_i\partial x_j}}.$$

If ${\displaystyle \mathcal{L}}$ acts on ${\displaystyle V\in C^{2,1}({\mathbb{R}}^n\times [t_0,\infty ];{\mathbb{R}}_{+})}$, then

$$\mathcal{L}V(x,t)=V_t+V_{x}f+{\displaystyle \frac{1}{2}}trace\left[g^T{V}_{xx}g\right],$$

where ${\displaystyle V_t=\frac{\partial V}{\partial t}}$, ${\displaystyle V_x=(\frac{\partial V}{\partial x_1},\frac{\partial V}{\partial x_2},...,\frac{\partial V}{\partial x_n})}$ and ${\displaystyle V_{xx}={\left(\frac{{\partial }^{2}V}{\partial x_i\partial x_j}\right)}_{n\times n}}$.

By applying Itô’s formula, as demonstrated in [11,13], for ${\displaystyle x\in {\mathbb{R}}^n}$, we obtain

$$dV(x\left(t\right),t)=\mathcal{L}(x\left(t\right),t)dt+V_{x}gdB\left(t\right).$$

(4)

The subsequent notations are also presented:

${\displaystyle {\mathrm{R}}_{+}^d=\{x=(x_1,\dots ,x_d)\in {\mathrm{R}}^d:x_i>0,1 \le i \le d\}}$ and ${\displaystyle {\overline{\mathrm{R}}}_{+}^d=\{x=(x_1,\dots ,x_d)\in {\mathrm{R}}^d:x_i}$${\displaystyle \ge 0,1 \le i \le d\}}$, ${\displaystyle a\wedge b=min\{a,b\},a\vee b=max\{a,b\},X=\frac{1}{t}{\int }_0^{t}X\left(s\right)ds}$, where ${\displaystyle X\in {\mathbb{L}}^1({\overline{\mathrm{R}}}_{+}, {\overline{\mathrm{R}}}_{+})}$ is an integrable function.

Definition 1.

If the transition probability function ${\displaystyle \mathbb{P}(s,x,t,A)}$ is time-homogeneous, meaning that the function ${\displaystyle \mathbb{P}(s,x,t+s,A)}$ is independent of ${\displaystyle 0 \le s \le t}$, where ${\displaystyle x\in {\mathbb{R}}^n,A\in \mathrm{B}}$ and ${\displaystyle \mathrm{B}}$ denotes the ${\displaystyle \sigma }$-algebra of Borel sets in ${\displaystyle {\mathbb{R}}^n}$, then the corresponding Markov process is also called time-homogeneous.

Let ${\displaystyle X\left(t\right)}$ be a regular time-homogeneous Markov ${\displaystyle \{E_x\equiv {\mathbb{R}}^l\}}$ process in a ${\displaystyle 1}$-dimensional Euclidean space, ${\displaystyle E_l}$, described by the following SDE:

$$dX\left(t\right)=b\left(X\right)dt+\sum _{r=1}^k{g}_r\left(X\right)d{B}_r\left(t\right).$$

The diffusion matrix is defined as

$$A\left(x\right)=\left(a_{ij}\left(x\right)\right), a_{ij}\left(x\right)=\sum _{r=1}^k{g}_r^i\left(x\right)g_r^j\left(x\right).$$

Then, we have the following result, as demonstrated in [13]:

Lemma 1.

The Markov process ${\displaystyle X\left(t\right)}$ possesses a unique ergodic stationary distribution ${\displaystyle \pi (.)}$ if there exists a bounded open domain ${\displaystyle Z\subset {\mathbb{R}}^d}$ with a smooth boundary ${\displaystyle \Gamma }$, characterized by the following properties:

Ellipticity condition (${\displaystyle A_1}$): For all ${\displaystyle \forall x\in Z}$ and ${\displaystyle \xi \in {\mathbb{R}}^l}$, there exists a positive number ${\displaystyle M}$ such that the diffusion matrix ${\displaystyle A\left(x\right)}$ is strictly positive-definite, meaning that

$$\sum _{i,j=1}^l{a}_{ij}\left(x\right){\xi }_i{\xi }_j⩾M{\left|\xi \right|}^2,x\in Z,\xi \in {\mathbb{R}}^l.$$

Lyapunov condition (${\displaystyle A_2}$): there exists a non-negative ${\displaystyle C^2}$- function ${\displaystyle V}$ such that ${\displaystyle \mathcal{L}V}$ is negative for any ${\displaystyle {\mathbb{R}}^l\setminus Z}$.

Lemma 2.

Let ${\displaystyle (S_C\left(t\right), I_C\left(t\right), V_C\left(t\right))}$ be the solution to the model (2) with value ${\displaystyle (S_C\left(0\right), I_C\left(0\right), }$${\displaystyle V_C\left(0\right))\in R_{+}^3}$. Then, ${\displaystyle \underset{t⟶\infty }{lim}\frac{S_C\left(t\right)+I_C\left(t\right)+V_C\left(t\right)}{t}=0}$ a.s. Moreover, ${\displaystyle \underset{t⟶\infty }{lim}\frac{S_C\left(t\right)}{t}=0}$, ${\displaystyle \underset{t⟶\infty }{lim}\frac{I_C\left(t\right)}{t}=0}$, ${\displaystyle \underset{t⟶\infty }{lim}\frac{V_C\left(t\right)}{t}=0}$ a.s.

Lemma 3.

Assume that ${\displaystyle {\mu }_C>\frac{{\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2}{2}}$. Let ${\displaystyle (S_C\left(t\right), I_C\left(t\right), V_C\left(t\right))}$ be the solution to the model (1.2) with value ${\displaystyle (S_C\left(0\right), I_C\left(0\right), V_C\left(0\right))\in R_{+}^3}$. Then, ${\displaystyle \underset{t⟶\infty }{lim}\frac{{\int }_0^t{S}_C\left(s\right)d{B}_1\left(s\right)}{t}=0}$, ${\displaystyle \underset{t⟶\infty }{lim}\frac{{\int }_0^t{I}_C\left(s\right)d{B}_2\left(s\right)}{t}=0}$, ${\displaystyle \underset{t⟶\infty }{lim}\frac{{\int }_0^t{V}_C\left(s\right)d{B}_3\left(s\right)}{t}=0}$${\displaystyle a.s.}$

3. Existence and Uniqueness of Global Positive Solutions

To analyze the dynamical behavior of system (2), it is essential to first verify that the system admits a positive and global solution. Since the model variables represent subpopulations, it is biologically reasonable to require that their values remain positive for all time. A key preliminary result is the existence and uniqueness of such a global positive solution, which serves as a foundational requirement for studying the system’s long-term behavior.

Theorem 1.

For any initial value ${\displaystyle (S_C\left(0\right), I_C\left(0\right), V_C\left(0\right))\in R_{+}^3}$, there is a unique positive solution ${\displaystyle S_C\left(t\right), I_C\left(t\right), V_C\left(t\right)}$ of system (1.2) on ${\displaystyle t⩾0}$, and the solution will remain in ${\displaystyle R_{+}^3}$ with probability one almost surely ${\displaystyle (a.s.)}$.

Proof. 

Since the coefficients of (2) satisfy the local Lipschitz condition, there exists a unique local solution for any value ${\displaystyle (S_C\left(0\right), I_C\left(0\right), V_C\left(0\right))\in R_{+}^3}$. ${\displaystyle (I_C\left(t\right), V_C\left(t\right), S_C\left(t\right))}$ on ${\displaystyle [0,{\tau }_e]}$, with the explosion time denoted by ${\displaystyle {\tau }_e}$. For this solution to be global, it is expected that ${\displaystyle {\tau }_e=\infty }$${\displaystyle a.s.}$ Let ${\displaystyle z_0}$ be sufficiently large such that ${\displaystyle S_C\left(0\right), I_C\left(0\right)}$ and ${\displaystyle V_C\left(0\right)}$ all lie within the interval ${\displaystyle [\frac{1}{z_0},z_0]}$. For each integer ${\displaystyle z⩾z_0}$, the following is defined as stopping time:

$${\tau }_z=inf\{t\in [0,{\tau }_e]:min\{S_C\left(t\right),I_C\left(t\right),V_C\left(t\right)\}⩽{\displaystyle \frac{1}{z}}ormax\{S_C\left(t\right),I_C\left(t\right),V_C\left(t\right)\}⩾z\}.$$

With ${\displaystyle \varnothing }$ denoting the empty set as usual, we set ${\displaystyle \inf \varnothing =\infty }$. It is obvious that ${\displaystyle {\tau }_z}$ is increasing as ${\displaystyle z⟶\infty }$. Then, two statements can be proven to hold ${\displaystyle a.s.}$

Firstly, let ${\displaystyle {\tau }_{\infty }=\underset{z⟶\infty }{lim}{\tau }_z}$ so that ${\displaystyle {\tau }_{\infty }={\tau }_e}$${\displaystyle a.s.}$ Next, there is the need to prove that ${\displaystyle {\tau }_{\infty }=\infty }$${\displaystyle a.s.}$ so that ${\displaystyle S_C\left(t\right), I_C\left(t\right), V_C\left(t\right)\in R_{+}^3}$${\displaystyle a.s.}$ for ${\displaystyle t\ge 0}$. If this claim is untrue, then ${\displaystyle \mathbb{P}\{{\tau }_z\in \le \tilde{T}\}>ϵ}$ is true for two constants, ${\displaystyle \tilde{T}>0}$ and ${\displaystyle ϵ\in (0,1)}$. Hence, there is an integer ${\displaystyle z_1\ge z_0}$ such that

$$\mathbb{P}\{{\tau }_z \le \tilde{T}\}\ge ϵ\forall z\ge z_1.$$

(5)

Define a ${\displaystyle C^2}$- function ${\displaystyle U:R_{+}^3⟶R_{+}}$ by

$$U(S_C, I_C, V_C)=(S_C-1-ln{S}_C)+(I_C-1-ln{I}_C)+(V_C-1-ln{V}_C),$$

$$=(S_C+I_C+V_C)-(ln{S}_C+ln{I}_C+ln{V}_C)-3.$$

where ${\displaystyle a}$ is the constant that will be found later. From ${\displaystyle u-1-lnu\ge 0,\forall u>0}$, it is evident that this function is non-negative.

Applying Itô’s formula to ${\displaystyle U}$ along the trajectories of the system, this is obtained as follows:

$$dU(S_C, I_C, V_C)=\mathcal{L}U(S_C, I_C, V_C)dt+{\sigma }_1(S_C-1)d{B}_1\left(t\right)+{\sigma }_2(I_C-1)d{B}_2\left(t\right)+{\sigma }_3(V_C-1)d{B}_3\left(t\right).$$

where ${\displaystyle \mathcal{L}U:R_{+}^3⟶R_{+}}$ is defined by

$$\begin{array}{cc}\hfill \mathcal{L}U=& (1-{\displaystyle \frac{1}{S_C}})[{\Lambda }_C-{\beta }_C{V}_C{S}_C-{\mu }_C{S}_C]+(1-{\displaystyle \frac{1}{I_C}})[{\beta }_C{V}_C{S}_C-({\mu }_C+d_C)I_C]\hfill \\ & +(1-{\displaystyle \frac{1}{V_C}})[N_s{r}_c{I}_C-{\sigma }_C{V}_C]+{\displaystyle \frac{{\sigma }_1^2}{2}}+{\displaystyle \frac{{\sigma }_2^2}{2}}+{\displaystyle \frac{{\sigma }_3^2}{2}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill =& {\Lambda }_C+2{\mu }_C+{\sigma }_C+d_C+{\displaystyle \frac{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2}{2}}-{\mu }_C{S}_C-({\mu }_C+d_C)I_C-{\sigma }_C{V}_C\hfill \\ & -{\displaystyle \frac{{\Lambda }_C}{S_C}}+{\beta }_C{V}_C+N_s{r}_c{I}_C-{\displaystyle \frac{{\beta }_C{V}_C{S}_C}{I_C}}-{\displaystyle \frac{N_s{r}_c{I}_C}{V_C}},\hfill \end{array}$$

$$\begin{array}{c}\mathcal{L}U \le {\Lambda }_C+2{\mu }_C+{\sigma }_C+d_C+{\displaystyle \frac{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2}{2}}:=K.\end{array}$$

where ${\displaystyle K}$ is a positive constant. Hence,

$$\begin{array}{c}\hfill dU(S_C, I_C, V_C) \le Kdt+{\sigma }_1(S_C-1)d{B}_1\left(t\right)+{\sigma }_2(I_C-1)d{B}_2\left(t\right)+{\sigma }_3(V_C-1)d{B}_3\left(t\right).\end{array}$$

(6)

Taking the expectation and integrating both sides of (6) from ${\displaystyle 0}$ to ${\displaystyle {\tau }_z\wedge \tilde{T}=min\{{\tau }_z, \tilde{T}\}}$ yields

$$\mathbb{E}U(S_C({\tau }_z\wedge \tilde{T}), I_C({\tau }_z\wedge \tilde{T}), V_C({\tau }_z\wedge \tilde{T})) \le U(S_C\left(0\right), I_C\left(0\right), V_C\left(0\right))+\mathbb{E}\left[{\int }_0^{{\tau }_z\wedge \tilde{T}}Kds\right],$$

$$\le U(S_C\left(0\right), I_C\left(0\right), V_C\left(0\right))+K\mathbb{E}({\tau }_z\wedge \tilde{T}).$$

Hence,

$$\mathbb{E}U(S_C({\tau }_z\wedge \tilde{T}), I_C({\tau }_z\wedge \tilde{T}), V_C({\tau }_z\wedge \tilde{T})) \le U(S_C\left(0\right), I_C\left(0\right), V_C\left(0\right))+K\tilde{T}.$$

(7)

Now, let ${\displaystyle {\Omega }_z=\{\omega \in :{\tau }_z={\tau }_z\left(\omega \right) \le \tilde{T}\}}$ for ${\displaystyle z=z_1}$, and, according to (5), there is ${\displaystyle \mathcal{P}({\Omega }_z\ge ϵ)}$. Since for every ${\displaystyle \omega \in {\Omega }_z}$ there exists ${\displaystyle S_C({\tau }_z,\Omega )}$ or ${\displaystyle I_C({\tau }_z,\Omega )}$ or ${\displaystyle V_C({\tau }_z,\Omega )}$= ${\displaystyle z}$ or ${\displaystyle \frac{1}{z}}$, it follows that ${\displaystyle S_C({\tau }_z,\Omega ), I_C({\tau }_z,\Omega ), V_C({\tau }_z,\Omega )}$ is no less than either

$${\displaystyle \frac{1}{z}}-1-ln{\displaystyle \frac{1}{z}}={\displaystyle \frac{1}{z}}-1+lnz,$$

Thus,

$$U(S_C({\tau }_z,\Omega ), I_C({\tau }_z,\Omega ), V_C({\tau }_z,\Omega ))\ge (z-1-lnz)\wedge ({\displaystyle \frac{1}{z}}-1-ln{\displaystyle \frac{1}{z}}).$$

It also follows from (7) that

$$U(S_C\left(0\right), I_C\left(0\right), V_C\left(0\right))+K\widetilde{S_C}\ge \mathbb{E}U({|}_{\Omega }{S}_C({\tau }_z\wedge \widetilde{S_C}), I_C({\tau }_z\wedge \widetilde{S_C}),,V_C({\tau }_z\wedge \widetilde{S_C})),$$

$$\ge ϵ(z-1-lnz)\wedge ({\displaystyle \frac{1}{z}}-1-ln{\displaystyle \frac{1}{z}}).$$

where ${\displaystyle {|}_{\Omega }}$ stands for indicator function of ${\displaystyle {\Omega }_z}$. Letting ${\displaystyle z⟶\infty }$ leads to contradiction

$$\infty >U(S_C\left(0\right), I_C\left(0\right), V_C\left(0\right))+K\tilde{T}=\infty .$$

Hence, it must have ${\displaystyle {\tau }_{\infty }=\infty }$${\displaystyle a.s.}$ The proof is now concluded. The solution ${\displaystyle (S_C, I_C, V_C)}$ is globally positive ${\displaystyle a.s.}$ ${\displaystyle \square }$

4. Stationary Distribution and Ergodicity (Virus Persistence)

In deterministic systems, the long-term behavior is typically addressed by demonstrating that the endemic equilibrium of the model is either a global attractor or globally asymptotically stable. However, in the stochastic system (2), an endemic equilibrium does not exist. Instead, it is shown that a stationary distribution exists under specific conditions related to the reproduction number and model parameters. This implies that the disease will persist in the population, as discussed in [14]. The criteria established in this section guarantee the existence and uniqueness of an ergodic stationary distribution for system (2), indicating that the viral load ${\displaystyle N_s}$ will be sustained over time. The following theorem formalizes this result.

Theorem 2.

For any initial value ${\displaystyle (S_C\left(0\right), I_C\left(0\right), V_C\left(0\right))\in R_{+}^3}$, the system (2) has a unique ergodic stationary distribution ${\displaystyle \pi (.)}$ if and only if ${\displaystyle R_0^s>1}$.

Proof. 

To prove the theorem above, it is necessary to ensure that both conditions stated in Lemma 1 are satisfied. The first condition requires that the system (2) be expressed in matrix form, which is given as follows:

$$\left(\begin{array}{c}d{S}_C\\ \\ d{I}_C\\ \\ d{V}_C\end{array}\right)=\left(\begin{array}{c}{\Lambda }_C-{\beta }_C{V}_C{S}_C-{\mu }_C{S}_C\\ \\ {\beta }_C{V}_C{S}_C-({\mu }_C+d_C)I_C\\ \\ N_s{r}_c{I}_C-{\sigma }_C{V}_C\end{array}\right)+{\left(\begin{array}{ccc}{\sigma }_1{S}_C& 0& 0\\ \\ 0& {\sigma }_2{I}_C& 0\\ \\ 0& 0& {\sigma }_3{V}_C\end{array}\right)}^T\left(\begin{array}{c}d{B}_1\left(t\right)\\ \\ d{B}_2\left(t\right)\\ \\ d{B}_3\left(t\right)\end{array}\right),$$

The diffusion matrix of the system (2) is given as

$$A=\left(a_{ij}\right)=g^i(S_C, I_C, V_C)g^j(S_C, I_C, V_C)=\left(\begin{array}{ccc}{\sigma }_1^2{S}_C^2& 0& 0\\ \\ 0& {\sigma }_2^2{I}_C^2& 0\\ \\ 0& 0& {\sigma }_3^2{V}_C^2\end{array}\right).$$

Now, this outcome is chosen:

$$M=\underset{(S_C,I_C,V_C)\in D\subset {\mathrm{R}}_{+}^3}{min}\{{\sigma }_1^2{S}_C^2, {\sigma }_2^2{I}_C^2, {\sigma }_3^2{V}_C^2\}.$$

Hence, there is

$$\sum _{i,j=1}^3{a}_{ij}(S_C, I_C, V_C){\xi }_i{\xi }_j={\sigma }_1^2{S}_C^2{\xi }_1^2+{\sigma }_2^2{I}_C^2{\xi }_2^2+{\sigma }_3^2{V}_C^2{\xi }_3^2⩾M{\left|\xi \right|}^2.$$

where ${\displaystyle (S_C, I_C, V_C)\in \overline{D}, \xi =({\xi }_1, {\xi }_2, {\xi }_3)\in {\mathrm{R}}_{+}^3}$ and ${\displaystyle \overline{D}=[k,\frac{1}{k}]\times [k,\frac{1}{k}]\times [k,\frac{1}{k}]}$. Therefore, the condition ${\displaystyle A_1}$ in Lemma 1 holds.

To verify the second condition (${\displaystyle A_2}$) in Lemma 1, we first derive the stochastic reproduction number and subsequently prove that condition ${\displaystyle A_2}$ holds. □

4.1. Reproduction Number for Stochastic Model

The stochastic reproduction number ${\displaystyle R_0^s}$ is defined as the average number of new infections generated by one infected individual introduced into a fully susceptible population considering the effects of stochastic perturbations such as white noise, demographic randomness, and so on. In this subsection, Itô’s formula is applied with a twice-differentiable function, choosing ${\displaystyle f(t,X(t\left)\right)=lnX\left(t\right)}$, to obtain the reproduction number for the stochastic system. The associated Taylor series expansion is expressed as

$$df(t, X\left(t\right))={\displaystyle \frac{\partial f}{\partial t}}+{\displaystyle \frac{\partial f}{\partial X\left(t\right)}}dX\left(t\right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^{2}f}{\partial X^2\left(t\right)}}{\left(dX\left(t\right)\right)}^2+{\displaystyle \frac{{\partial }^{2}f}{\partial t\partial X\left(t\right)}}dtdX\left(t\right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^{2}f}{\partial t^2}}{\left(dt\right)}^2.$$

(8)

It must also be noted that

$${\displaystyle \frac{\partial f}{\partial t}}=0,{\displaystyle \frac{\partial f}{\partial X\left(t\right)}}={\displaystyle \frac{1}{X\left(t\right)}},{\displaystyle \frac{{\partial }^{2}f}{\partial X^2\left(t\right)}}=-{\displaystyle \frac{1}{X^2\left(t\right)}},{\displaystyle \frac{{\partial }^{2}f}{\partial t\partial X}}={\displaystyle \frac{{\partial }^{2}f}{\partial t^2}}=0.$$

So, system (2) becomes

$$\begin{array}{ccc}\hfill ln{S}_C& =& [{\displaystyle \frac{{\Lambda }_C}{S_C}}-{\beta }_C{V}_C-({\mu }_C+{\displaystyle \frac{1}{2}}{\sigma }_1^2)]dt+{\sigma }_{1}d{B}_1\left(t\right),\hfill \\ \hfill ln{I}_C& =& [{\displaystyle \frac{{\beta }_C{S}_C{V}_C}{I_C}}-({\mu }_C+d_C+{\displaystyle \frac{1}{2}}{\sigma }_2^2)]dt+{\sigma }_{2}d{B}_2\left(t\right),\hfill \\ \hfill ln{V}_C& =& [{\displaystyle \frac{N_s{r}_c{I}_C}{V_C}}-({\sigma }_C+{\displaystyle \frac{1}{2}}{\sigma }_3^2)]dt+{\sigma }_{3}d{B}_3\left(t\right).\hfill \end{array}$$

(9)

By applying the next-generation matrix approach [15] to Equation (9), the stochastic reproduction number was derived as follows:

$$F=\left(\begin{array}{cc}0& {\displaystyle \frac{{\beta }_C{\Lambda }_C}{({\mu }_C+\frac{1}{2}{\sigma }_1^2)}}\\ 0& 0\end{array}\right),$$

$$V=\left(\begin{array}{cc}({\mu }_C+d_C+{\displaystyle \frac{1}{2}}{\sigma }_2^2)& 0\\ -N_s{r}_c& ({\sigma }_C+{\displaystyle \frac{1}{2}}{\sigma }_3^2)\end{array}\right),$$

$$detV=({\mu }_C+d_C+{\displaystyle \frac{1}{2}}{\sigma }_2^2)({\sigma }_C+{\displaystyle \frac{1}{2}}{\sigma }_3^2),$$

$$V^{-1}=\left(\begin{array}{cc}{\displaystyle \frac{1}{({\mu }_C+d_C+\frac{1}{2}{\sigma }_2^2)}}& 0\\ {\displaystyle \frac{N_s{r}_c}{({\mu }_C+d_C+\frac{1}{2}{\sigma }_2^2)({\sigma }_C+\frac{1}{2}{\sigma }_3^2)}}& {\displaystyle \frac{1}{({\sigma }_C+\frac{1}{2}{\sigma }_3^2)}}\end{array}\right),$$

$$F{V}^{-1}=\left(\begin{array}{cc}{\displaystyle \frac{N_s{r}_c{\beta }_C{\Lambda }_C}{({\mu }_C+\frac{1}{2}{\sigma }_1^2)({\mu }_C+d_C+\frac{1}{2}{\sigma }_2^2)({\sigma }_C+\frac{1}{2}{\sigma }_3^2)}}& {\displaystyle \frac{{\beta }_C{\Lambda }_C}{({\mu }_C+\frac{1}{2}{\sigma }_1^2)({\sigma }_C+\frac{1}{2}{\sigma }_3^2)}}\\ \\ 0& 0\end{array}\right).$$

Then, stochastic reproduction number denoted as ${\displaystyle R_0^s}$ is given below:

$$R_0^s={\displaystyle \frac{N_s{r}_c{\beta }_C{\Lambda }_C}{({\mu }_C+\frac{1}{2}{\sigma }_1^2)({\mu }_C+d_C+\frac{1}{2}{\sigma }_2^2)({\sigma }_C+\frac{1}{2}{\sigma }_3^2)}}.$$

(10)

Substituting the expression for ${\displaystyle N_s}$, derived in Appendix A, into the reproduction number of the multiscale model in Equation (10) yields the following result:

$$R_0^s={\displaystyle \frac{({\delta }_r+\frac{1}{2}{\sigma }_4^2)r_c{\beta }_C{\Lambda }_C(R{e}_0^s-1)}{{\eta }_r({\mu }_C+\frac{1}{2}{\sigma }_1^2)({\mu }_C+d_C+\frac{1}{2}{\sigma }_2^2)({\sigma }_C+\frac{1}{2}{\sigma }_3^2)}}.$$

$$\begin{array}{cc}\hfill R_0^s=& {\displaystyle \frac{({\delta }_r+\frac{1}{2}{\sigma }_4^2)r_c{\beta }_C{\Lambda }_C}{{\eta }_r({\mu }_C+\frac{1}{2}{\sigma }_1^2)({\mu }_C+d_C+\frac{1}{2}{\sigma }_2^2)({\sigma }_C+\frac{1}{2}{\sigma }_3^2)}}\hfill \\ & \left[{\displaystyle \frac{{\eta }_r{\Lambda }_r[N_c{\alpha }_c{\alpha }_t({\alpha }_i+{\delta }_i+\frac{1}{2}{\sigma }_7^2)+N_i{\alpha }_i{\mu }_t({\alpha }_c+{\rho }_c+\frac{1}{2}{\sigma }_6^2)]}{({\delta }_r+\frac{1}{2}{\sigma }_4^2)({\alpha }_t+{\mu }_t+{\delta }_t+\frac{1}{2}{\sigma }_5^2)({\alpha }_c+{\rho }_c+\frac{1}{2}{\sigma }_6^2)({\alpha }_i+{\delta }_i+\frac{1}{2}{\sigma }_7^2)(r_c+r_i+\frac{1}{2}{\sigma }_8^2)}}-1\right].\hfill \end{array}$$

(11)

4.2. The Second Condition of Lemma 1

It is necessary to construct a ${\displaystyle C^2}$- function ${\displaystyle \overline{V}:R_{+}^3⟶R}$ as follows:

$$\overline{V}(S_C, I_C, V_C)=V(S_C, I_C, V_C)-V(S_C\left(0\right),I_C\left(0\right),V_C\left(0\right)),$$

$$=M{V}_1+V_2+V_3+V_4+V_5-V(S_C\left(0\right),I_C\left(0\right),V_C\left(0\right)).$$

It is important to note that ${\displaystyle V(S_C, I_C, V_C)}$ is not only continuous but also approaches infinity as ${\displaystyle (S_C, I_C, V_C)}$ approaches the boundary of ${\displaystyle R_{+}^3}$. Since it must be bounded from below, it does so at a point in the interior of ${\displaystyle R_{+}^3}$ called ${\displaystyle (S_C\left(0\right), I_C\left(0\right), V_C\left(0\right))}$. We now construct ${\displaystyle V:R_{+}^3⟶R}$ in the form

$$V(S_C, I_C, V_C)=M{V}_1(S_C, I_C, V_C)+V_2\left(S_C\right)+V_3\left(I_C\right)+V_4\left(V_C\right)+V_5(S_C, I_C, V_C).$$

where ${\displaystyle V_1=-ln{S}_C-a_{1}ln{I}_C-a_{2}ln{V}_C}$, ${\displaystyle V_2=-ln{S}_C}$, ${\displaystyle V_3=-ln{I}_C}$, ${\displaystyle V_4=-ln{V}_C}$, ${\displaystyle V_5=\frac{1}{\theta +1}{(S_C+I_C+V_C)}^{\theta +1}}$, where ${\displaystyle \theta }$ is a sufficiently small constant satisfying ${\displaystyle 0<\theta <\frac{2{\mu }_C}{{\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2}}$ and ${\displaystyle a_i^{’}s}$, ${\displaystyle i=1,2}$ in ${\displaystyle V_1}$ are the positive constants to be determined. Also, each ${\displaystyle V_i,i=1,\dots ,5}$ is uniquely obtained, and ${\displaystyle M>0}$ is sufficiently large number satisfying the following condition:

$$-M\Psi +B \le -2$$

(12)

Applying the Itô lemma to ${\displaystyle V_1}$, the following was obtained:

$$\begin{array}{cc}\hfill \mathcal{L}{V}_1=& -{\displaystyle \frac{{\Lambda }_C}{S_C}}+{\beta }_C{\Lambda }_C+({\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}})-a_1{\displaystyle \frac{{\beta }_C{S}_C{V}_C}{I_C}}+a_1({\mu }_C+d_C+{\displaystyle \frac{{\sigma }_2^2}{2}})-a_2{\displaystyle \frac{N_s{r}_c{I}_C}{V_C}}+a_2({\sigma }_C+{\displaystyle \frac{{\sigma }_3^2}{2}}),\hfill \end{array}$$

$$\begin{array}{cc}\hfill =& -{\displaystyle \frac{{\Lambda }_C}{S_C}}-a_1{\displaystyle \frac{{\beta }_C{S}_C{V}_C}{I_C}}-a_2{\displaystyle \frac{N_s{r}_c{I}_C}{V_C}}+{\beta }_C{\Lambda }_C+({\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}})+a_1({\mu }_C+d_C+{\displaystyle \frac{{\sigma }_2^2}{2}})+a_2({\sigma }_C+{\displaystyle \frac{{\sigma }_3^2}{2}}).\hfill \end{array}$$

The first three terms can be rewritten by applying the fact that ${\displaystyle a+b\ge 2\sqrt{ab}}$ for all ${\displaystyle a,b>0}$.

$$\begin{array}{c}\le -3{\left(a_1{a}_2{\beta }_C{\Lambda }_C{N}_s{r}_c\right)}^{{\displaystyle \frac{1}{3}}}+({\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}})+a_1({\mu }_C+d_C+{\displaystyle \frac{{\sigma }_2^2}{2}})+a_2({\sigma }_C+{\displaystyle \frac{{\sigma }_3^2}{2}})+{\beta }_C{\Lambda }_C.\end{array}$$

This condition was chosen such that

$$a_1({\mu }_C+d_C+{\displaystyle \frac{{\sigma }_2^2}{2}})=a_2({\sigma }_C+{\displaystyle \frac{{\sigma }_3^2}{2}})={\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}},$$

and then ${\displaystyle a_1=\frac{{\mu }_C+\frac{{\sigma }_1^2}{2}}{({\mu }_C+d_C+\frac{{\sigma }_2^2}{2})}}$ was obtained, and ${\displaystyle a_2=\frac{{\mu }_C+\frac{{\sigma }_1^2}{2}}{({\sigma }_C+\frac{{\sigma }_3^2}{2})}}$. Hence,

$$\begin{array}{c}\hfill \mathcal{L}{V}_1 \le -3{\left[{\displaystyle \frac{{({\mu }_C+\frac{{\sigma }_1^2}{2})}^3{\beta }_C{\Lambda }_C{N}_s{r}_c}{({\mu }_C+\frac{{\sigma }_1^2}{2})({\mu }_C+d_C+\frac{{\sigma }_2^2}{2})({\sigma }_C+\frac{{\sigma }_3^2}{2})}}\right]}^{{\displaystyle \frac{1}{3}}}+3({\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}})+{\beta }_C{\Lambda }_C,\end{array}$$

$$\begin{array}{c}\le -3({\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}})\left[{\left({\displaystyle \frac{{\beta }_C{\Lambda }_C{N}_s{r}_c}{({\mu }_C+\frac{{\sigma }_1^2}{2})({\mu }_C+d_C+\frac{{\sigma }_2^2}{2})({\sigma }_C+\frac{{\sigma }_3^2}{2})}}\right)}^{{\displaystyle \frac{1}{3}}}-1\right]+{\beta }_C{\Lambda }_C,\end{array}$$

$$\begin{array}{c}\le -3({\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}})[{\left(R_0^s\right)}^{{\displaystyle \frac{1}{3}}}-1]+{\beta }_C{\Lambda }_C,\end{array}$$

$$\begin{array}{cc}\hfill \mathcal{L}{V}_1:=& -\Psi +{\beta }_C{\Lambda }_C.\hfill \end{array}$$

(13)

where

$$\Psi =3({\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}})[{\left(R_0^s\right)}^{{\displaystyle \frac{1}{3}}}-1].$$

Next, we arrived at the following:

$$\begin{array}{cc}\hfill \mathcal{L}{V}_2=& -{\displaystyle \frac{{\Lambda }_C}{S_C}}+{\beta }_C{\Lambda }_C+{\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}}\hfill \end{array}$$

$$\begin{array}{c}\mathcal{L}{V}_2 \le -{\displaystyle \frac{{\Lambda }_C}{S_C}}+{\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}},\end{array}$$

(14)

$$\begin{array}{cc}\hfill \mathcal{L}{V}_3=& -{\displaystyle \frac{{\beta }_C{S}_C{V}_C}{I_C}}+{\mu }_C+d_C+{\displaystyle \frac{{\sigma }_2^2}{2}},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \mathcal{L}{V}_4=& -{\displaystyle \frac{N_s{r}_c{I}_C}{V_C}}+{\sigma }_C+{\displaystyle \frac{{\sigma }_3^2}{2}},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \mathcal{L}{V}_5=& {(S_C+I_C+V_C)}^{\theta }[{\Lambda }_C-{\mu }_C(S_C+I_C)-{\sigma }_C{V}_C-(d_C+N_s{r}_c)I_C]+\hfill \\ & {(S_C+I_C+V_C)}^{\theta -1}\times {\displaystyle \frac{{\sigma }_1^2{S}_C^2+{\sigma }_2^2{I}_C^2+{\sigma }_3^2{V}_C^2}{2}},\hfill \end{array}$$

$$\begin{array}{c}\le {(S_C+I_C+V_C)}^{\theta }[{\Lambda }_C-{\mu }_C(S_C+I_C+V_C)]+{\displaystyle \frac{\theta }{2}}{(S_C+I_C+V_C)}^{\theta +1}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2),\end{array}$$

$$\begin{array}{cc}\hfill =& {\Lambda }_C{(S_C+I_C+V_C)}^{\theta }-\left[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)\right]\times {(S_C+I_C+V_C)}^{\theta +1},\hfill \end{array}$$

$$\begin{array}{cc}\hfill =& {\Lambda }_C{(S_C+I_C+V_C)}^{\theta }-{\displaystyle \frac{1}{2}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)]{(S_C+I_C+V_C)}^{\theta +1}-\hfill \\ & {\displaystyle \frac{1}{2}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)]\times {(S_C+I_C+V_C)}^{\theta +1},\hfill \end{array}$$

$$\begin{array}{c}\mathcal{L}{V}_5 \le A-{\displaystyle \frac{1}{2}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1}).\end{array}$$

(17)

where

$$A=\underset{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3}{sup}\left\{{\Lambda }_C{(S_C+I_C+V_C)}^{\theta }-{\displaystyle \frac{1}{2}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)]{(S_C+I_C+V_C)}^{\theta +1}\right\}<\infty .$$

Therefore, applying (13)–(17), we have

$$\mathcal{L}\overline{V}=\mathcal{L}{V}_1+\mathcal{L}{V}_2+\mathcal{L}{V}_3+\mathcal{L}{V}_4+\mathcal{L}{V}_5,$$

$$\begin{array}{cc}\hfill \mathcal{L}\overline{V}=& -M\Psi +M{\beta }_C{V}_C-{\displaystyle \frac{{\Lambda }_C}{S_C}}+{\mu }_C+{\displaystyle \frac{{\sigma }_1^2}{2}}-{\displaystyle \frac{{\beta }_C{S}_C{V}_C}{I_C}}+{\mu }_C+d_C+{\displaystyle \frac{{\sigma }_2^2}{2}}-{\displaystyle \frac{N_s{r}_c{I}_C}{V_C}}+{\sigma }_C\hfill \\ & +{\displaystyle \frac{{\sigma }_3^2}{2}}+A-{\displaystyle \frac{1}{2}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1}),\hfill \end{array}$$

$$\begin{array}{c}\le -M\Psi +M{\beta }_C{V}_C-{\displaystyle \frac{{\Lambda }_C}{S_C}}-{\displaystyle \frac{{\beta }_C{S}_C{V}_C}{I_C}}-{\displaystyle \frac{N_s{r}_c{I}_C}{V_C}}-{\displaystyle \frac{1}{2}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1})+F,\end{array}$$

(18)

where ${\displaystyle F=2{\mu }_C+{\sigma }_C+d_C+\frac{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2}{2}+A}$

$$\begin{array}{c}\mathcal{L}\overline{V} \le -M\Psi +M{\beta }_C{V}_C-{\displaystyle \frac{{\Lambda }_C}{S_C}}-{\displaystyle \frac{{\beta }_C{S}_C{V}_C}{I_C}}-{\displaystyle \frac{N_s{r}_c{I}_C}{V_C}}+B.\end{array}$$

where ${\displaystyle B}$ is defined as

$$B=\underset{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3}{sup}\left\{-{\displaystyle \frac{1}{2}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1})+F\right\}.$$

Recall that, as stated in Equation (12), the matrix ${\displaystyle M}$ must satisfy the condition ${\displaystyle -M\Psi +B \le -2}$. Now, we construct a compact subset ${\displaystyle D}$ such that Lemma 1 holds for ${\displaystyle A_2}$. The following was how a bounded closed set was defined:

$$D=\left\{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3:ϵ \le S_C \le {\displaystyle \frac{1}{ϵ}},ϵ \le I_C \le {\displaystyle \frac{1}{ϵ}},{ϵ}^3 \le V_C \le {\displaystyle \frac{1}{{ϵ}^3}}\right\}.$$

where ${\displaystyle 0<ϵ<1}$ is a sufficiently small number. Then, in ${\displaystyle {\mathrm{R}}_{+}^3\setminus D}$, this was chosen such that

$$-{\displaystyle \frac{{\Lambda }_C}{ϵ}}+C \le -1,$$

(19)

$$M{\beta }_Cϵ+C \le -1,$$

(20)

$$-{\displaystyle \frac{{\beta }_C}{ϵ}}+C \le -1,$$

(21)

$$-{\displaystyle \frac{N_s{r}_c}{ϵ}}+C \le -1,$$

(22)

$$-{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2)]{\displaystyle \frac{1}{{ϵ}^{\theta +1}}}+C \le -1,$$

(23)

$$-{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2)]{\displaystyle \frac{1}{{ϵ}^{3(\theta +1)}}}+C \le -1.$$

(24)

where ${\displaystyle C}$ is a constant that needs to be determined below. To fulfill the requirement of condition ${\displaystyle A_2}$ of Lemma 1, ${\displaystyle {\mathrm{R}}_{+}^3\setminus D}$ can be divided into seven (7) domains as follows:

$$D_1=\{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3:0<S_C<ϵ\},$$

$$D_2=\{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3:0<V_C<{ϵ}^3\},$$

$$D_3=\{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3:0<I_C<ϵ,S_C\ge ϵ,V_C\ge {ϵ}^3\},$$

$$D_4=\{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3:0<V_C<{ϵ}^3,I_C\ge ϵ\},$$

$$D_5=\{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3:S_C>{\displaystyle \frac{1}{ϵ}}\},$$

$$D_6=\{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3:I_C>{\displaystyle \frac{1}{ϵ}}\},$$

$$D_7=\{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3:V_C>{\displaystyle \frac{1}{{ϵ}^3}}\}.$$

Clearly,

$$D^c={\mathrm{R}}_{+}^3\setminus D=D_1\cup D_2\cup D_3\cup D_4\cup D_5\cup D_6\cup D_7.$$

The next step is to demonstrate that, for each ${\displaystyle (S_C, I_C, V_C)\in D^c}$, ${\displaystyle \mathbb{L}\overline{V}(S_C, I_C, V_C) \le -1}$. This is equal to demonstrating it on the aforementioned seven (7) domains, respectively.

• Case 1: For any ${\displaystyle (S_C,I_C,V_C)\in D^c}$, by (18), we obtain

  $$\mathcal{L}\overline{V} \le -{\displaystyle \frac{{\Lambda }_C}{S_C}}+M{\beta }_C{V}_C-{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1})+F,$$

  $$\le -{\displaystyle \frac{{\Lambda }_C}{S_C}}+C \le -{\displaystyle \frac{{\Lambda }_C}{ϵ}}+C \le -1.$$

  (25)

  which follows from (19) and

  $$C=\underset{(S_C,I_C,V_C)\in {\mathrm{R}}_{+}^3}{sup}\left\{M{\beta }_C{V}_C-{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1})+F\right\}.$$

  (26)

• Case 2: For any ${\displaystyle (S_C,I_C,V_C)\in D^c}$, by (18), this was obtained as follows:

  $$\mathcal{L}\overline{V} \le -M\Psi +M{\beta }_C{V}_C+B,$$

  $$\le -M\Psi +B+M{\beta }_Cϵ+C,$$

  $$\le -2+1 \le -1,$$

  (27)

  which follows from (12) and (20).

Case 3: For any ${\displaystyle (S_C,I_C,V_C)\in D^c}$, by (18), this result was obtained as follows:

$$\begin{array}{cc}\hfill \mathcal{L}\overline{V} \le & -{\displaystyle \frac{{\beta }_C{S}_C{V}_C}{I_C}}-{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1})+F,\hfill \end{array}$$

$$\le -{\displaystyle \frac{{\beta }_C}{ϵ}}+C \le -1.$$

(28)

which follows from (21) and (26).

Case 4: For any ${\displaystyle (S_C,I_C,V_C)\in D^c}$, by (18), this was obtained as follows:

$$\begin{array}{cc}\hfill \mathcal{L}\overline{V} \le & -{\displaystyle \frac{N_s{r}_c{I}_C}{V_C}}-{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1})+F,\hfill \end{array}$$

$$\le -{\displaystyle \frac{N_s{r}_c}{ϵ}}+C \le -1.$$

(29)

which follows from (22) and (26).

Case 5: For any ${\displaystyle (S_C,I_C,V_C)\in D^c}$, by (18), this result was obtained as follows:

$$\begin{array}{cc}\hfill \mathcal{L}\overline{V} \le & -{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1})+C,\hfill \end{array}$$

$$\le -{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2)]{\displaystyle \frac{1}{{ϵ}^{\theta +1}}}+C \le -1.$$

(30)

which follows from (23) and (26).

Case 6: For any ${\displaystyle (S_C,I_C,V_C)\in D^c}$, by (18), the result obtained was

$$\begin{array}{cc}\hfill \mathcal{L}\overline{V} \le & -{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1})+C,\hfill \end{array}$$

$$\le -{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2)]{\displaystyle \frac{1}{{ϵ}^{\theta +1}}}+C \le -1.$$

(31)

which follows from (23) and (26).

Case 7: For any ${\displaystyle (S_C,I_C,V_C)\in D^c}$, by (18), it was obtained that

$$\begin{array}{c}\mathcal{L}\overline{V} \le -{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2)](S_C^{\theta +1}+I_C^{\theta +1}+V_C^{\theta +1})+C,\end{array}$$

$$\le -{\displaystyle \frac{1}{4}}[{\mu }_C-{\displaystyle \frac{\theta }{2}}({\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2)]{\displaystyle \frac{1}{{ϵ}^{3(\theta +1)}}}+C \le -1.$$

(32)

which follows from (24) and (26).

Evidently, for every ${\displaystyle (S_C,I_C,V_C)\in D^c}$, it has been demonstrated in (25) and (27)–(32) that, for sufficiently small ${\displaystyle ϵ}$,

$$\mathcal{L}\overline{V}(S_C, I_C, V_C) \le -1.$$

Hence, condition ${\displaystyle A_2}$ in Lemma 1 is satisfied.

By invoking this lemma, the study confirms the existence of a unique stationary distribution for the multiscale system (2). When ${\displaystyle R_0^s>1}$ and the system possesses the ergodic property characterized by the invariant measure ${\displaystyle \pi (·)}$, it indicates that the virus persists under stochastic influences. Importantly, the stochastic reproduction number ${\displaystyle R_0^s}$ matches the control reproduction number of the deterministic model (Equation (3.5) in [9]), denoted ${\displaystyle R_0}$, when white noise effects are absent, i.e., ${\displaystyle {\sigma }_i^2=0}$ for ${\displaystyle i=1,2,3}$, at the between-cell scale. These findings not only extend the conclusions of the deterministic multiscale model (system (2.8) in [9]) but also enhance and solidify the theoretical understanding of stationary distribution existence in the stochastic nested multiscale framework. Specifically, when ${\displaystyle R_0>1}$, the deterministic model (system (2.8)) admits a unique endemic equilibrium, thereby ensuring virus persistence. Furthermore, the expression for the stochastic version of the within-cell viral load, ${\displaystyle N_s}$ (derived in Appendix A), coincides with its deterministic counterpart in [9] when white noise is excluded, that is, when ${\displaystyle {\sigma }_i^2=0}$ for ${\displaystyle i=4,5,6,7,8}$. Consequently, the stochastic reproduction number of the within-cell model, ${\displaystyle R{e}_0^s}$, also coincides with the reproduction number of the deterministic within-cell model. Additionally, the intensities of the white noise directly modulate all threshold parameters in the between-cell scale models, thereby influencing the system’s dynamics under stochastic perturbations.

5. Virus Eradication or Extinction

The primary motivation for formulating mathematical models of infectious disease systems is to understand and regulate disease dynamics, ultimately enabling effective control and eventual eradication over time. Meanwhile, this study has reformulated the nested multiscale model of hepatitis B virus (HBV) into a stochastic version by incorporating white noise at both the within-cell and between-cell levels. In this section, we focus on establishing sufficient conditions for the eradication or extinction of the virus. Specifically, we identify the disease-free state of the stochastic system (2) and derive the conditions under which viral extinction occurs. The proof of the main result relies on the findings presented in Lemmas 2 and 3, which are used to establish the key theorem in this section.

Theorem 3.

Let ${\displaystyle (S_C\left(t\right),I_C\left(t\right),V_C\left(t\right))}$ be the solution to the model (2) with value ${\displaystyle (S_C\left(0\right),I_C\left(0\right),V_C\left(0\right))\in R_{+}^3}$. If ${\displaystyle {\mu }_C>\frac{{\sigma }_1^2\vee {\sigma }_2^2\vee {\sigma }_3^2}{2}}$ and ${\displaystyle \widehat{R_0^s}=\frac{4({\beta }_C{\Lambda }_C-{\mu }_C^2)}{{\mu }_C({\sigma }_2^2\wedge {\sigma }_3^2)}}$; then, the virus will die out exponentially with probability one, that is, ${\displaystyle \underset{t⟶\infty }{lim}{I}_C\left(t\right)=0}$, ${\displaystyle \underset{t⟶\infty }{lim}{V}_C\left(t\right)=0}$${\displaystyle a.s.}$ with ${\displaystyle \underset{t⟶\infty }{lim}{S}_C\left(t\right)=\frac{{\Lambda }_C}{{\mu }_C}}$${\displaystyle a.s.}$

Proof. 

By the first equation of system (2), this was obtained as follows:

$$d{S}_C=[{\Lambda }_C-{\beta }_C{S}_C{V}_C-{\mu }_C{S}_C]dt+{\sigma }_1{S}_{C}d{B}_1\left(t\right).$$

Evaluating the rate of change of ${\displaystyle S_C\left(t\right)}$ in (2),

$${\int }_0^{t}d{S}_C={\int }_0^t{\Lambda }_{C}dt-{\beta }_C{\int }_0^t{S}_C{V}_{C}ds-{\mu }_C{\int }_0^t{S}_{C}ds+{\sigma }_1{\int }_0^t{S}_{C}d{B}_1\left(t\right),$$

$${\displaystyle \frac{S_C\left(t\right)-S_C\left(0\right)}{t}}={\Lambda }_C-{\beta }_C{\displaystyle \frac{{\int }_0^t{S}_C{V}_{C}ds}{t}}-{\displaystyle \frac{{\mu }_C}{t}}{\int }_0^t{S}_{C}ds+{\displaystyle \frac{{\sigma }_1}{t}}{\int }_0^t{S}_{C}d{B}_1\left(t\right).$$

By ergodic theorem [16], the result was

$${\displaystyle \frac{S_C\left(t\right)-S_C\left(0\right)}{t}} = {\Lambda }_C-{\beta }_C<S_C{V}_C>-{\mu }_C<S_C>+{\displaystyle \frac{{\sigma }_1}{t}}{\int }_0^t{S}_{C}d{B}_1\left(t\right),$$

$${\mu }_C<S_C> = {\Lambda }_C-{\displaystyle \frac{S_C\left(t\right)-S_C\left(0\right)}{t}}-{\beta }_C<S_C{V}_C>+{\displaystyle \frac{{\sigma }_1}{t}}{\int }_0^t{S}_{C}d{B}_1\left(t\right),$$

$$<S_C> = {\displaystyle \frac{1}{{\mu }_C}}\left[{\Lambda }_C-{\displaystyle \frac{S_C\left(t\right)-S_C\left(0\right)}{t}}-{\beta }_C<S_C{V}_C>+{\displaystyle \frac{{\sigma }_1}{t}}{\int }_0^t{S}_{C}d{B}_1\left(t\right)\right],$$

$$\underset{t⟶\infty }{lim}<S_C> \le \underset{t⟶\infty }{lim}{\displaystyle \frac{1}{{\mu }_C}}\left[{\Lambda }_C+{\displaystyle \frac{{\sigma }_1}{t}}{\int }_0^t{S}_{C}d{B}_1\left(t\right)\right].$$

It follows from Lemmas 2 and 3 that ${\displaystyle \underset{t⟶\infty }{lim}<S_C> \le \underset{t⟶\infty }{lim}\frac{{\Lambda }_C}{{\mu }_C}}$${\displaystyle a.s.}$

• Next, this was defined as follows:

$$K\left(t\right)=I_C\left(t\right)+V_C\left(t\right),$$

$$K\left(t\right)=[{\beta }_C{S}_C{V}_C-{\mu }_C(I_C\left(t\right)+V_C\left(t\right))-(d_C-N_s{r}_c)]dt+{\sigma }_2{I}_{C}d{B}_2\left(t\right)+{\sigma }_3{I}_{C}d{B}_3\left(t\right),$$

$$dlnK={\displaystyle \frac{1}{I_C\left(t\right)+V_C\left(t\right)}}\left\{[{\beta }_C{S}_C{V}_C-{\mu }_C(I_C\left(t\right)+V_C\left(t\right))-(d_C-N_s{r}_c)]dt+{\sigma }_2{I}_{C}d{B}_2\left(t\right)+{\sigma }_3{I}_{C}d{B}_3\left(t\right)\right\},$$

$$\begin{array}{cc}\hfill dlnK \le & \left\{{\beta }_C{S}_C-{\mu }_C-{\displaystyle \frac{1}{2{(I_C\left(t\right)+V_C\left(t\right))}^2}}({\sigma }_2^2{I}_C^2+{\sigma }_3^2{V}_C^2)\right\}dt+{\displaystyle \frac{{\sigma }_2{I}_C}{(I_C\left(t\right)+V_C\left(t\right))}}d{B}_2\left(t\right)+\hfill \\ & {\displaystyle \frac{{\sigma }_3{V}_C}{(I_C\left(t\right)+V_C\left(t\right))}}d{B}_3\left(t\right),\hfill \end{array}$$

$$\le \left\{{\beta }_C{S}_C-{\mu }_C-{\displaystyle \frac{{\sigma }_2^2\wedge {\sigma }_3^2}{4(I_C^2+V_C^2)}}(I_C^2+V_C^2)\right\}dt+{\displaystyle \frac{{\sigma }_2{I}_C}{(I_C\left(t\right)+V_C\left(t\right))}}d{B}_2\left(t\right)+{\displaystyle \frac{{\sigma }_3{V}_C}{(I_C\left(t\right)+V_C\left(t\right))}}d{B}_3\left(t\right),$$

$$\le \left\{{\beta }_C{S}_C-{\mu }_C-{\displaystyle \frac{{\sigma }_2^2\wedge {\sigma }_3^2}{4}}\right\}dt+{\displaystyle \frac{{\sigma }_2{I}_C}{(I_C\left(t\right)+V_C\left(t\right))}}d{B}_2\left(t\right)+{\displaystyle \frac{{\sigma }_3{V}_C}{(I_C\left(t\right)+V_C\left(t\right))}}d{B}_3\left(t\right).$$

Integrating from ${\displaystyle 0}$ to ${\displaystyle t}$ and dividing by ${\displaystyle t}$ yields the following result:

$$\begin{array}{cc}\hfill {\displaystyle \frac{lnK\left(t\right)-lnK\left(0\right)}{t}} \le & {\displaystyle \frac{{\beta }_C}{t}}{\int }_0^t{S}_C\left(s\right)ds-{\mu }_C-{\displaystyle \frac{{\sigma }_2^2\wedge {\sigma }_3^2}{4}}+{\displaystyle \frac{{\sigma }_2}{t}}{\int }_0^t{\displaystyle \frac{I_C\left(s\right)}{(I_C\left(s\right)+V_C\left(s\right))}}d{B}_2\left(s\right)+\hfill \\ & {\displaystyle \frac{{\sigma }_3}{t}}{\int }_0^t{\displaystyle \frac{V_C\left(s\right)}{(I_C\left(s\right)+V_C\left(s\right))}}d{B}_3\left(s\right).\hfill \end{array}$$

Applying the superior limits and invoking Lemma 3, it follows that

$$\underset{t⟶\infty }{lim\; sup}{\displaystyle \frac{lnK\left(t\right)}{t}} \le {\displaystyle \frac{{\beta }_C{\Lambda }_C}{{\mu }_C}}-{\mu }_C-{\displaystyle \frac{1}{4}}({\sigma }_2^2\wedge {\sigma }_3^2),$$

$$={\displaystyle \frac{1}{4}}({\sigma }_2^2\wedge {\sigma }_3^2)\left[{\displaystyle \frac{4({\beta }_C{\Lambda }_C-{\mu }_C^2)}{{\mu }_C({\sigma }_2^2\wedge {\sigma }_3^2)}}-1\right],$$

$$={\displaystyle \frac{1}{4}}({\sigma }_2^2\wedge {\sigma }_3^2)(\widehat{R_0^s}-1)<0a.s.$$

This implies that ${\displaystyle \underset{t⟶\infty }{lim}{I}_C\left(t\right)=0}$ and ${\displaystyle \underset{t⟶\infty }{lim}{V}_C\left(t\right)=0}$ ${\displaystyle a.s.}$ ${\displaystyle \square }$

Theorem 3 establishes that the stochastic nested multiscale model represented by Equation (2) predicts the possibility of virus extinction or eradication under certain conditions. Central to this result is the threshold parameter ${\displaystyle \widehat{R_0^s}}$, which serves as a critical indicator for determining whether the hepatitis B virus will persist or be eliminated from the host population. Notably, this threshold does not depend on the within-cell viral load denoted by ${\displaystyle N_s}$. The parameter ${\displaystyle N_s}$ is a composite term that encapsulates the within-cell viral dynamics and serves as a bridge between the microscopic (within-cell) and macroscopic (between-cell) scales. Specifically, it quantifies the effective viral output from infected cells to the extracellular environment, thereby contributing to the community-level viral load through shedding or excretion processes. Despite its biological relevance in the overall disease transmission dynamics, ${\displaystyle N_s}$ does not directly influence the condition for eradication as defined by ${\displaystyle \widehat{R_0^s}}$. This implies that the stochastic extinction of the virus, as dictated by Theorem 3, is governed by factors that are independent of the intracellular viral replication and instead are determined by parameters at the population- or between-cell level. In essence, as long as the condition ${\displaystyle \widehat{R_0^s}<1}$ is satisfied, the model predicts that the hepatitis B virus will be eradicated from the system regardless of the magnitude of the within-cell viral load ${\displaystyle N_s}$. For an in-depth explanation of the model structure, parameter definitions, and multiscale modeling framework used to analyze hepatitis B virus infection, readers are referred to [9]. This source provides comprehensive insights into how within-cell processes are integrated into the between-cell transmission dynamics through composite parameters like ${\displaystyle N_s}$, and how these contribute to the overall understanding of disease progression and control strategies.

6. Conclusions

This study advances the stochastic multiscale to investigate the dynamic behavior of hepatitis B viral infection. By implication, this study constructs a suitable stochastic Lyapunov function, which provides sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solution to model system (2). Furthermore, the study establishes sufficient conditions for the extinction of the virus at the cellular level of biological organization. It should be noted that threshold parameters ${\displaystyle R_0^s}$, which determines the existence of an ergodic stationary distribution, and ${\displaystyle \widehat{R_0^s}}$, which characterizes the condition for virus eradication, are not reproduction numbers in the classical sense. Specifically, they do not correspond to the basic reproduction number typically derived in deterministic nonlinear ordinary differential equation (ODE) models used in the nested multiscale model presented in [9]. Instead, these thresholds arise from the stochastic analysis of system (2) and serve distinct roles in describing the system’s long-term behavior under random perturbations. The results show that a distinct ergodic stationary distribution (indicative of viral persistence) can be guaranteed under lower levels of white noise. Conversely, higher levels of white noise may lead to the extinction of the virus. The analytical findings reveal that, while the condition for disease persistence depends on the composite parameter ${\displaystyle N_s}$, the condition for disease eradication does not. This indicates that ${\displaystyle N_s}$ plays a significant role in shaping the overall dynamics of the model system. This study focused on developing an analytical solution for a stochastic nested multiscale model describing hepatitis B viral infections, capturing the complex interactions between within-cell viral replication dynamics and between-cell transmission processes under random environmental fluctuations. The model framework provides a rigorous mathematical basis for understanding how stochastic effects influence key epidemiological thresholds, such as the reproduction number, and the long-term behavior of the infection, including persistence or eradication scenarios. In future research, the scope will be broadened to integrate the effects of medical interventions into the model. These will include treatment strategies such as Direct-Acting Antivirals (DAAs), which target specific stages of the viral life cycle to suppress replication, as well as preventive measures like vaccination and enhanced public health initiatives aimed at reducing transmission rates. The extended model will also account for varying levels of treatment adherence, drug resistance development, and the potential synergistic effects of combining therapies with preventive approaches. To complement the analytical results, numerical simulations will be performed to assess the robustness of the theoretical predictions under a wide range of parameter values and stochastic intensities. These simulations will serve as a validation tool, allowing for comparison between predicted and simulated infection dynamics, and will provide deeper insights into optimal intervention strategies for controlling and ultimately eradicating hepatitis B in both deterministic and stochastic settings.