Stochastic Dynamics of Hepatitis B Virus Infection: Analysis, Stability, and Numerical Simulation

TL;DR

This work develops a rigorous stochastic within-host HBV model with multiplicative environmental noise, establishing global existence and positivity of solutions, along with conditions for almost sure extinction of infected compartments. It shows that stochastic perturbations can stabilize the infection-free state and induce ergodic long-term behavior, even when deterministic analysis predicts persistence. The study combines Lyapunov methods, ergodicity criteria, and Euler–Maruyama simulations to connect analytic results with numerical evidence, illustrating noise-driven reductions in viral load and stationary distributions centered near the healthy hepatocyte equilibrium. These findings highlight the crucial role of randomness in shaping HBV outcomes and suggest potential stochastic-informed approaches to therapy and disease management.

Abstract

This study develops and analyzes a stochastic differential equation (SDE) model for the dynamics of hepatitis B virus (HBV) infection. While deterministic frameworks have yielded important insights into viral behavior, they cannot adequately describe the intrinsic randomness and fluctuations present in biological processes. To address this limitation, we construct a stochastic model incorporating multiplicative environmental noise to account for variability in infection rates, cellular mortality, and viral replication. We establish a rigorous theoretical foundation by proving the existence, uniqueness, and global positivity of solutions for all biologically relevant initial conditions. Stability properties are investigated in detail, including stability in probability and almost sure exponential stability, with particular emphasis on conditions under which random perturbations stabilize the infection-free state. Furthermore, we demonstrate the existence of a unique ergodic stationary distribution and derive convergence properties of the uninfected hepatocyte population. Numerical simulations, performed via the Euler-Maruyama method with sufficiently small time steps to ensure positivity and accuracy, validate the analytical results and illustrate the impact of stochastic fluctuations on system dynamics. The simulations confirm that environmental noise can induce viral extinction even in parameter regimes where deterministic analysis predicts persistence. These findings enhance the mathematical understanding of HBV infection dynamics and underscore the significant role of stochastic effects in shaping long-term disease outcomes.

Figures (7)

• Figure 1: Disease–free and endemic scenarios. Left to right: time traces of $x(t)$, $\log y(t)$, and $\log z(t)$. Bottom: phase portraits $(x,y)$, $(y,z)$, and the full 3D trajectory $(x,y,z)$. Blue curves: high treatment efficacy ($\eta=0.9$), which yields rapid decay of $y$ and $z$ in agreement with Theorem \ref{['Syz']}; red dashed curves: low efficacy ($\eta=0.1$), which sustains an endemic state with persistent $y$ and $z$ over the simulated window. All paths remain nonnegative, consistent with Theorem \ref{['Existence']}.
• Figure 2: Noise intensity sweep. Rows show $x(t)$, $\log y(t)$, $\log z(t)$, and the $(x,y)$ phase portrait for common noise magnitude $\sigma_1=\sigma_2=\sigma_3\in\{0,0.01,0.05,0.1,0.2\}$. Increasing stochasticity suppresses the long–time level of infection and can force the extinction of $y$ and $z$, even when the deterministic configuration is near–endemic. This illustrates noise–induced stabilization in line with Theorem \ref{['Syz']}.
• Figure 3: Deterministic path, stochastic mean, and ensemble. Left: $x(t)$. Middle: $\log y(t)$. Right: $\log z(t)$. Light lines are $10$ independent stochastic realizations, the red line is the deterministic trajectory obtained by suppressing the noise terms, and the blue line is the sample mean of the stochastic ensemble. The ensemble average of $x(t)$ aligns with the drift–based evolution and concentrates near $\Lambda/\mu_1$ in agreement with Lemma \ref{['lem4']} and Theorem \ref{['x-x1']}. The infected compartments exhibit faster decay in the stochastic runs than in the deterministic run, consistent with noise–induced stabilization.
• Figure 4: Stability diagnostics. Top left: infected load versus time for increasing treatment efficacy $\eta$ in the drift–only setting, showing the classical transition across $R_0=1$. Top right: infected load versus time for increasing standard noise level $\sigma$, which progressively drives the infection toward extinction. Bottom left: $R_0(\eta)$ with a horizontal threshold at $1$. Bottom right: a stochastic surrogate $R_0^{\mathrm{stoch}}(\sigma)$ illustrating the effective reduction of the reproduction number as noise grows.
• Figure 5: Long–time behavior and empirical stationary distributions. Top row: long simulations of $x(t)$, $\log y(t)$, and $\log z(t)$ over $T=500$ days with $\Delta t=10^{-2}$. Bottom row: empirical probability densities of the terminal window for $x$, $\log_{10}y$, and $\log_{10}z$. The $x$-histogram centers near $\Lambda/\mu_1$ as predicted by Lemma \ref{['lem4']}, while the infected compartments concentrate near zero under the stable configuration of Theorem \ref{['Syz']}. The presence of tight, time–invariant histograms is consistent with the ergodic conclusion of Theorem \ref{['thm:ergodic']}.

Theorems & Definitions (17)

• Definition 3.1: Strong solution of an SDE
• Lemma 3.1
• proof
• Theorem 4.1: Global Existence, Uniqueness, and Positivity
• proof
• Definition 5.1: Stochastic Stability gardiner2004handbookRef1
• Theorem 5.1: Exponential Extinction of $y(t)$ and $z(t)$
• proof
• Remark 5.1
• Lemma 5.1: Analysis of Linear Reference Process