A generalized virus dynamics model with cell-to-cell transmission and cure rate

Khalid Hattaf,Noura Yousfi

doi:10.1186/s13662-016-0906-3

Khalid Hattaf, Noura Yousfi

Open Access

https://doi.org/10.1186/s13662-016-0906-3

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Abstract

Viruses can be spread and transmitted through two fundamental modes, one by virus-to-cell infection and the other by direct cell-to-cell transmission. In this paper, we propose a new generalized virus dynamics model, which incorporates both modes and takes into account the cure of infected cells. We first show mathematically and biologically the well-posedness of our model. Further, an explicit formula for the basic reproduction number $R_{0}$ of the model is determined. By analyzing the characteristic equations we establish the local stability of the disease-free equilibrium and the chronic infection equilibrium in terms of $R_{0}$ . The global behavior of the model is investigated by constructing an appropriate Lyapunov functional for disease-free equilibrium and by applying geometrical approach to chronic infection equilibrium. Moreover, mathematical virus models and results presented in many previous studies are generalized and improved.

Highlights

Many viruses infect humans and cause different infectious diseases such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), Ebola virus, and more recently Zika virus. They are often transmitted in body by two fundamentally distinct modes, either by virus-to-cell infection through the extracellular space or by cell-to-cell transmission involving direct cell-to-cell contact [ – ]
A part of infected cells returns to the uninfected state by loss of all covalently closed circular DNA from their nucleus [ – ]
Motivated by the mentioned biological and mathematical considerations, we propose the following generalized virus dynamics model with two transmission modes and cure rate:

Summary

Introduction

Many viruses infect humans and cause different infectious diseases such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), Ebola virus, and more recently Zika virus. Motivated by the mentioned biological and mathematical considerations, we propose the following generalized virus dynamics model with two transmission modes and cure rate:. The incidence function g(x, y) for direct cell-to-cell transmission mode is assumed to be continuously differentiable in the interior of R + and satisfy the following property:. The first assumption on the function g(x, y) means that the incidence rate by cell-to-cell transmission is equal to zero if there are no susceptible cells. The last assumption on the function g(x, y) means that the greater the amount of infected cells, the lower the average number of cells infected by direct contact in the unit time.

We have ψ

Proof Construct the Lyapunov functional

It follows then that