Qualitative Analysis of a Generalized Virus Dynamics Model with Both Modes of Transmission and Distributed Delays

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 proposedmodel, 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 becomemature 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.


Introduction
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.
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 [16][17][18][19].For these reasons, we propose the following generalized virus dynamics model with both modes of transmission and distributed delays: where (), (), and V() are the concentrations of uninfected cells, infected cells, and free virus particles at time , respectively.The uninfected cells are produced at rate , die at rate , and become infected either by free virus at rate (, , V)V or by direct contact with an infected cell at rate (, ).Hence, the term (, , V)V + (, ) denotes International Journal of Differential Equations the total infection rate of uninfected cells.The parameters  and  are, respectively, the death rates of infected cells and free virus. is the production rate of free virus by an infected cell.In this proposed model, we assume that the virus or infected cell contacts an uninfected target cell at time  − , and the cell becomes infected at time , where  is a random variable taken from a probability distribution ℎ 1 ().The term  − 1  represents the probability of surviving from time  −  to time , where  1 is the death rate for infected but not yet virus-producing cells.Similarly, we assume that the time necessary for the newly produced virions to become mature and infectious is a random variable with a probability distribution ℎ 2 ().The term  − 2  denotes the probability of surviving the immature virions during the delay period, where 1/ 2 is the average life time of an immature virus.Therefore, the integral ∫ ∞ 0 ℎ 2 () − 2  ( − ) describes the mature viral particles produced at time .
( 3 ) (, , V) is a monotone decreasing function with respect to  and V.
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 [20][21][22].Further, the general incidence functions (, , V) and (, ) include various types of incidence rates existing in the literature.
The main objective of this work is to investigate the dynamical behavior of system (1).For this end, we start with the existence, the positivity, and boundedness of solutions, which implies that our model is well posed.After that, we determine the basic reproduction number and steady states of the model.The global stability of the disease-free equilibrium and the chronic infection equilibrium is established in Sections 3 and 4 by constructing appropriate Lyapunov functionals.An application of our results is presented in Section 5. Finally, the conclusion is summarized in Section 6.

Well-Posedness and Equilibria
For biological reasons, we suppose that the initial conditions of system (1) satisfy Define the Banach space for fading memory type as follows: where  is a positive constant and Theorem 1.For any initial condition  = ( 1 ,  2 ,  3 ) ∈   satisfying (2), system (1) has a unique solution on [0, +∞).Furthermore, this solution is nonnegative and bounded for all  ≥ 0.
Proof.By the fundamental theory of functional differential equations [24][25][26], system (1) with initial condition  ∈   has a unique local solution on (0,  max ), where  max is the maximal existence time for solution of system (1).
From the second and third equations of system (1), we get which implies that () and V() are nonnegative for all  ∈ (0,  max ).
International Journal of Differential Equations 3 Now, we prove the boundedness of the solutions.From the first equation of (1), we have ẋ () ≤  − () which implies that lim sup Then () is bounded.Let Since () is bounded and ∫ ∞ 0 ℎ 1 () = 1, the integral in () is well defined and differentiable with respect to .Hence, where  = min{, } and Thus, () ≤  fl max{(0),  1 /}, which implies that () is bounded.It remains to prove that V() is bounded.By third equation of system (1) and the boundedness of (), we deduce that Then V() ≤ max{V(0),  2 /}.Therefore, V() is also bounded.We have proved that all variables of system (1) are bounded which implies that  max = +∞ and the solution exists globally.
), all solution of (1) with initial condition  is positive for all  ≥ 0.

Stability of the Disease-Free Equilibrium
In this section, we establish the stability of the disease-free equilibrium.
Theorem 4. The disease-free equilibrium   is globally asymptotically stable when  0 ≤ 1 and becomes unstable when  0 > 1.

Stability of the Chronic Infection Equilibrium
In this section, we investigate the global stability of the chronic infection equilibrium  * by assuming that  0 > 1 and the functions  and  satisfy, for all , , V > 0, the following hypothesis: Therefore, we get the following result.

Application
In this section, we consider the following HIV infection model with distributed delays: where  1 and  2 are nonnegative constants that measure the saturation effect.The parameters  1 and  2 are the virusto-cell infection rate and the cell-to-cell transmission rate, respectively.The other parameters have the same biological meanings as in model ( 1).Further, system ( 26) is a special case of (1) with (, , V) =  1 /(1 +  1 V) and (, ) =  2 /(1 +  2 ).Notice that the HIV infection model presented by Lai and Zou [23] is a particular case of our model (26), it suffices to take  1 =  2 = 0 and ℎ 2 () = ().In addition, system (26) always has a disease-free equilibrium   (/, 0, 0) and a unique chronic infection equilibrium  * ( * ,  * , V * ) when On the other hand, it is easy to see that the hypotheses ( 0 )-( 3 ) are satisfied.Furthermore, we have ) (  (,  * , V * )  (, , V) Consequently, the hypothesis ( 4 ) is satisfied.By applying Theorems 4 and 5, we get the following result.
(ii) If  0 > 1, then the disease-free equilibrium   becomes unstable and the chronic infection equilibrium  * of ( 26) is globally asymptotically stable.

Conclusion
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  0 .In addition, the viral infection models with infinite distributed delays and the corresponding results presented in several previous studies are extended and generalized.
In this study, we have neglected the mobility of cells and virus.Motivated by the works in [28][29][30][31][32], we will consider this mobility in our future project in order to improve our present model.