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We propose a generalized virus dynamics model with distributed delays and both modes of transmission, one by virus-to-cell infection and the other by cell-to-cell transfer. In the proposed model, the distributed delays describe (i) the time needed for infected cells to produce new virions and (ii) the time necessary for the newly produced virions to become mature and infectious. In addition, the infection transmission process is modeled by general incidence functions for both modes. Furthermore, the qualitative analysis of the model is rigorously established and many known viral infection models with discrete and distributed delays are extended and improved.

Viruses are microscopic organisms that need to penetrate into a cell of their host to duplicate and multiply. Many human infections and diseases are caused by viruses such as the human immunodeficiency virus (HIV) that is responsible for acquired immunodeficiency syndrome (AIDS), Ebola that can cause an often fatal illness called Ebola hemorrhagic fever, and the hepatitis B virus (HBV) that can lead to chronic infection, cirrhosis, or liver cancer.

In viral dynamics, infection processes and virus production are not instantaneous. In reality, there are two kinds of delays: one in cell infection and the other in virus production. In the literature, these delays are modeled by discrete time delays [

On the other hand, viruses can spread by two fundamental modes, one by virus-to-cell infection through the extracellular space and the other by cell-to-cell transfer involving direct cell-to-cell contact [

As in [

Biologically, the four hypotheses are reasonable and consistent with the reality. For more details on the biological significance of these four hypotheses, we refer the reader to the works [

The probability distribution functions

The main objective of this work is to investigate the dynamical behavior of system (

For biological reasons, we suppose that the initial conditions of system (

For any initial condition

By the fundamental theory of functional differential equations [

First, we prove that

From the second and third equations of system (

Now, we prove the boundedness of the solutions. From the first equation of (

It remains to prove that

If in addition to (

When

Obviously, system (

If

If

It is clear that

Define the function

In this section, we establish the stability of the disease-free equilibrium.

The disease-free equilibrium

To study the global stability of

On the other hand, the characteristic equation at

In this section, we investigate the global stability of the chronic infection equilibrium

Assume that

We define a Lyapunov functional as follows:

In order to simplify the presentation, we will use the following notations:

In this section, we consider the following HIV infection model with distributed delays:

On the other hand, it is easy to see that the hypotheses

If

If

In this work, we have proposed a mathematical model that describes the dynamics of viral infections, such as HIV and HBV, and takes into account the two modes of transmission and the two kinds of delays, one in cell infection and the other in virus production. The transmission process for both modes is modeled by two general incidence functions that include many types of incidence rates existing in the literature. Further, the two delays are modeled by infinite distributed delays. Under some assumptions on the general incidence functions, we have proved that the global stability of the proposed model is fully determined by one threshold parameter that is the basic reproduction number

In this study, we have neglected the mobility of cells and virus. Motivated by the works in [

The authors declare that they have no conflicts of interest.