Global Dynamics and Implications of an HBV Model with Proliferating Infected Hepatocytes

Abstract

Chronic hepatitis B (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined the within-host dynamics of the disease. Most previous models assumed that infected hepatocytes do not proliferate; however, the effect of HBV infection on hepatocyte proliferation is controversial, with conflicting data showing both induction and inhibition of proliferation. With a family of ordinary differential equation (ODE) models, we explored the dynamical impact of proliferation among HBV-infected hepatocytes. Here, we show that infected hepatocyte proliferation in this class of models generates a threshold that divides the dynamics into two categories. Sufficiently compromised proliferation in infected cells produces complex dynamics characterized by oscillating viral loads, whereas higher proliferation generates straightforward dynamics that always results in chronic infection, sometimes with liver failure. A global stability result of the liver failure state was included as it is unique to this class of models. Finally, the model analysis motivated a testable biological hypothesis: Healthy hepatocytes are present in chronic HBV infection if and only if the proliferation of infected hepatocytes is severely impaired.

Phenomenology of an In-host Model of Hepatitis C

“…Proof In order to prove that a Hopf bifurcation occurs it remains to show that the eigenvalues which lie on the imaginary axis at the bifurcation point cross the axis with non-zero velocity as a suitable parameter is varied [7]. Assuming the existence of the bifurcation point this crossing condition has been verified for the model with η = 0 in [6]. We give a new proof that this condition is satisfied for the model with η = 0.…”

“…Now we prove that in the case η = 0 there are parameters for which the condition δ = σ is satisfied. We note that no proof of this fact was given in [5] or in the later more general paper [6]. Consider fixed positive values of the parameters δ, r T and c. Then R 1 is also fixed.…”

“…Note that it is essential for this result that s > 0. It was shown in [5] and [6] that for some models related to the system (1)-( 5) with s = 0 there are solutions which tend to the origin as t → ∞. In fact it is also true, and simpler to prove, that there are solutions of this type in the model of Guedj and Neumann [4].…”

“…The bound of T + I by the quantity p 0 introduced in [11] is replaced by a bound in terms of the quantity obtained from p 0 by replacing d by d 1 and r T by r 1 . The definition of p 0 is recalled in (6). Once the solution is known to be bounded and remain away from the singular set it is straightforward to show that all concentrations remain positive as long as the solution exists.…”

“…In [6] the authors considered a system which corresponds to that obtained from (1.3a)-(1.3c) of [11] by setting d = s = η = 0, system ( 14)-( 16) of [6]. Lemma 2 can be used as above to prove that that system has Hopf bifurcations for small values of the parameter ρ in that system, which corresponds to r I in our notation.…”