Understanding the Complex Patterns Observed during Hepatitis B Virus Therapy

Data from human clinical trials have shown that the hepatitis B virus (HBV) follows complex profiles, such as bi-phasic, tri-phasic, stepwise decay and rebound. We utilized a deterministic model of HBV kinetics following antiviral therapy to uncover the mechanistic interactions behind HBV dynamics. Analytical investigation of the model was used to separate the parameter space describing virus decay and rebound. Monte Carlo sampling of the parameter space was used to determine the virological, pharmacological and immunological factors that separate the bi-phasic and tri-phasic virus profiles. We found that the level of liver infection at the start of therapy best separates the decay patterns. Moreover, drug efficacy, ratio between division of uninfected and infected cells, and the strength of cytotoxic immune response are important in assessing the amount of liver damage experienced over time and in quantifying the duration of therapy leading to virus resolution in each of the observed profiles.

Proof. Note that (1) is locally Lipschitz at t = 0. Therefore, a solution exists and is unique on [0, b) for some b > 0. Assume that there exists t 1 ∈ (0, b) such that V (t 1 ) = 0 and all variables are and so Proposition 2. If max{r T , r I } > min{d T , δ}, then any solution (T (t), I(t)) of (1) subject to (2) remains bounded on [0, b) for some b > 0.
Proof. Let F = T + I, r max = max{r T , r I } and d min = min{d T , δ}. Then and, since r max > d min , Note that X − Y > 0, and so F (t) is bounded. Therefore T (t) and I(t) are bounded.
Proof. We first show positivity for F . Assume that there exists t 1 ∈ (0, b) such that F (t 1 ) = 0 and all variables are positive on [0, t 1 ). Assume also that T and I are bounded on [0, t 1 ), i.e., there exist M 1 and M 2 such that T (t) ≤ M 1 and and so Since we assume all the variables positive on [0, t 1 ), this implies that both T (t) and I(t) are positive for all t ∈ [0, t 1 ].
Proof. If I(t) is bounded on [0, b), then there exists a number M > 0 such that Then and so Stability analysis. We study the local asymptotic stability of system (1)'s equilibria for = η = 0. The system has four equilibria: a liver death equilibria E * = (0, 0, 0), a disease-free equilibrium a chronic infection equilibrium with total liver infection and a chronic equilibrium with partial liver infection is the basic reproduction number, representing the number of secondary infections induced by an infected cell in a naive population.
Proposition 5. The liver death equilibrium is unstable.
Proof. The Jacobian matrix for the system is When evaluated at E * , J becomes: whose eigenvalues λ 1 = r T > 0 and λ 2 = r I − δ > 0. Therefore E * is unstable.
Proposition 6. The free disease equilibrium is locally asymptotically stable if R 0 < 1.
Proof. The Jacobian matrix for the system evaluated at E 0 becomes: whose eigenvalues are negative when R 0 < 1.
Proposition 7. The equilibrium E tot.liv exists when r I > δ, is locally asymptotically stable when and is unstable otherwise.
Proof. E tot.liv exists when r I > δ. It can be shown that the characteristic equation for E tot.liv is given by Since the other two eigenvalues are always negative, this condition is enough to ensure local asymptotic stability of equilibrium E tot.liv .
Proposition 8. The equilibrium E is locally asymptotically stable if r I > δ and 1 < R 0 and R 0 r I − δ r T < 1.
The proof is messy and it will not be presented here. When the treatment is initiated, we assume that the chronic equilibrium E is stable, i.e. 1 < R 0 < r T r I −δ . A successful combination drug therapy 0 < ≤ 1 and 0 < η ≤ 1 will lead to virus clearance if the clearance equilibrium in the presence of therapy, E d 0 = (K, 0, 0) (same as E 0 in the absence of therapy), becomes the locally asymptotically stable steady state. This occurs when Net liver gain 0.8 Figure S3: Density of bi-phasic (blue) and tri-phasic (pink) V (t) samples versus: (A) Liver turnover; (B) Net liver gain, for = 0.99, r T /r I = 2.5, 0.01 ≤ δ ≤ 0.1 d −1 and τ = 100 days.