# Mathematical analysis of an extended cellular model of the Hepatitis C Virus infection with non-cytolytic process

## TL;DR

This paper provides a mathematical analysis of an extended hepatitis C virus infection model, establishing conditions for stability of infection states and confirming results through numerical simulations.

## Contribution

It introduces a comprehensive stability analysis of an extended HCV model with cellular proliferation and spontaneous cure, including global stability conditions.

## Key findings

- Uninfected equilibrium is globally stable when R0 < 1 - q/(dI + q).
- Infected equilibrium is globally stable when R0 > 1.
- Numerical simulations confirm theoretical stability results.

## Abstract

The aims of this work is to analyse of the global stability of the extended model of hepatitis C virus(HCV) infection with cellular proliferation, spontaneous cure and hepatocyte homeostasis. We first give general information about hepatitis C. Secondly, We prove the existence of local, maximal and global solutions of the model and establish some properties of this solution as positivity and asymptotic behaviour. Thirdly we show, by the construction of an appropriate Lyapunov function, that the uninfected equilibrium and the unique infected equilibrium of the model of HCV are globally asymptotically stable respectively when the threshold number $\mathcal{R}_{0}<1-\frac{q}{d_{I}+q}$ and when $\mathcal{R}_{0}>1$. Finally, some numerical simulations are carried out using Maple software confirm these theoretical results.