VIRAL INFECTION DYNAMICS OF HBV DNA-CONTAINING CAPSIDS WITH LOGISTICS GROWTH AND SATURATED RATE

. Knowing more about the dynamics of hepatitis B viral infection with DNA containing capsids, still under research. In this paper, we propose a new mathematical model dealing with the infection of hepatocytes with capsids logistics growth functions and saturated rate. We initiate the work with the description and the well-posedness of the formulation, then we study the stability of different equilibria to get a set of results which express that the system has tree equilibrium points. For the basic reproduction number is less then one ( R 0 < 1) the disease-free equilibrium is stable and when R 0 > 1 other conditions determinate the stability of the endemic equilibrium points. By numerical simulation, we verify numerically the theoretical founding


INTRODUCTION
As it known, the hepatitis B virus (HBV) can infect liver cells and can cause acute or chronic infection of healthy hepatocytes [1,2], World Health Organization claims this disease as worldwide public health problem and report it with more than 257 millions infected persons [3,4].
Several scientific works have been developed in order to study the dynamic of HBV infection from both fields biologic and mathematics one [5][6][7][8][9][10][11][12][13][14]. In recent paper, [15] the authors describe the HBV infection stochastically with differential model and they reconize that white noise can influence the treatment of infectious. We can find also that HBV is studied and formulated with ODEs system [6,16,17] and FDEs system [18][19][20], the propagation of HBV with spatial dependence is discussed at [21] to design reaction-diffusion. Motivated by the natural existence of viral genome DNA countaining capsids with hepatocytes B, it is describe as package comprised of an assembly of proteins, many works concider capsids as a contributing factor in the dynamics of the viral infection [8,16,17,22,23]. After reading and analyzing what was done by the previous research, we propose the formulation of the HBV viral dynamics with DNA countaining capsids as the next nonlinear differential equations system: This illustrate the dynamic between the healthy hepatocytes (H), the infected hepatocytes (I), DNA containing capsids (D) and free virus (V).
The first equation express the variation over time of the healthy hepatocytes with two functions, logistic growth with r as maximum proliferation and N as population capacity. The other function is saturated rate with k as infection rate due to virus and the polulation of hepatocytes.
The second equation give tree terms to describe the pace of the infected hepatocytes, we find logistic growth with the maximum proliferation of infected cells ρ, k HV H+I for saturation and the last term to quantify the natural elimination with δ as death rate.
The third equation is for capsids, expressed by linear fonctions, with a as proliferation rate of intracellular capsids associated to infected cells, β production rate of virus due to capsids and δ death rate of capsids.
The fourth equation descrive the variation over time of HBV as linear fonctions, with production rate and elimination rate c of virus. This paper is organized by sections as follows. In section 2, we begin the analysis by showing the non negativity and boundedness of the solution and discussing locally the stability of steady states, in section 3, we present the numerical simulations by discussing the result then we conclude the analysis.

ANALYSIS OF HBV INFECTION MODEL
In this section, we try to analyze the well posedness of the HBV dynamics formulation of the model by assuring the positivity and the bordness of the solutions to keep biological identification of the infection. Indeed, for (H, I, D,V ) ∈ R 4 + , we have: Therefore, all solutions initiating are positive, which give the result.
Secondly, for the boundness, from the system (1), we have: We deduce that H, I, D and V are bounded.

Existence and local stability of the solution.
Before discussing the stability of the viral dynamic, it is necessary to calculate the basic reproduction number (BRN), the average of the new possible cases contaminated by one infected cell and mathematically it is defined by the spectral radius of the next generation matrix.
By a simple calculation we get in our case the following formulation of BRN: The equations of the model (1) are null at any equilibrium instant E * (H * , I * , D * ,V * ). (2) So, the last two equations of (2) give: For I * = 0, that give D * = 0, V * = 0 and H * = N which implies: The disease-free equilibrium E 0 (N, 0, 0, 0).
For H * = 0 and ρ ≥ δ , that give N ρ (ρ −δ ) which implies one endemic equilibrium point: The first endemic equilibrium E 1 (0, I 1 , D 1 ,V 1 ) With: From the discussion above we can start the stability analysis studies with the following set of results: Proof. The Jacobian matrix at E 0 is: The characteristic polynomial of the matrix is: From the Routh-Hurwitz stability criterion, it follows that all roots of P E 0 have negative real part, when R 0 < 1.
Proposition 3. If ρ > δ and R 0 > r ρ , the endemic equilibrium E 1 exist and it is locally asymptotically stable.
Proof. The Jacobian matrix at E 1 is: The characteristic polynomial of the matrix is: So, when ρ > δ and ρR 0 > r, we get the stability criterion of the endemic equilibrium E 1 .
So, the first two equations of (2) give: That give : δ R 0 = (r − ρ)(1 − X * N ) + δ . We can notice that the discussion will be continued with the differentiation between the proliferating rates of proliferation is essential, 2.2.1. Same proliferation rates of healthy and infected hepatocytes. At the case when the healthy and infected hepatocytes proliferate at the same rate ρ = r. We can talk over and calculat two equilibruim points, one DFE E 0 and EE E 1 and specially when R 0 = 1, we can get an infinity of positive steady states E * .
For the stability, we can discuss the next results from above as follow: Proposition 4. If R 0 < 1, the disease-free equilibrium E 0 is locally asymptotically stable.
Proposition 5. If r > δ and R 0 > 1, the endemic equilibrium E 1 exist and it is locally asymptotically stable.
Proof. Same as it is at Proposition 3 for ρ = r.
To sum up the pervious propositions when the healthy and infected hepatocytes proliferate at the same rate, we present the following bifurcation diagram showing the relation between the basic reproduction number R 0 and steady states obtained in this case, we consider R 0 is varied via δ .

Different proliferation rates of healthy and infected hepatocytes.
At the case when the infected hepatocytes proliferating at a different rate from the healthy one, we can discuss the existence of an other endemic equilibruim point.
We pose : With: We can study the stability of this endemic equilibruim and conclude the following results.
Remarque 1. From the boundness result X can get the maximum value at N. So, R D is positive, we conclude that (R 0 > 1 and r > ρ) or (R 0 < 1 and r < ρ).
For meaningful interpretation, we consider R 0 > 1 and r > ρ, the existence of E 2 is also related to the condition giving as follow:  ) and H i < 1, ∀i ∈ {1, 2, 3}, the endemic equilibrium E 2 is locally asymptotically stable.
Proof. The Jacobian matrix at E 2 is: The characteristic polynomial of the matrix is: With: . Again, using Routh-Hurwitz stability criterion, the eigenvalues of the above matrix have negative real parts when A > 0, D > 0, AB −C > 0 and A(BC − AD) −C 2 > 0.

NUMERICAL SIMULATIONS AND DISCUSSION
In the present section, several numerical simulations are carried out to exhibit the theoretical results. Using Euler's explicit method, we program numerically the solutions of the model (1) under Matlab to illustrate this result.
With the following parameter values [14,16,24]: r = 1, ρ = 1, N = 2 × 10 11 , k = 0.0014, δ = 0.0693, a = 30, β = 0.87 and c = 0.693. In Fig. 4, we talk over the disease-free equilibrium, we discuss the case when we have the same proliferation rates of healthy and infected hepatocytes (ρ = r), we simulate the solutions and the behavior infection converge to the stable state E 0 = (2 × 10 11 , 0, 0, 0). In this case the basic reproduction number is less than unit (R 0 = 0.8100 < 1) which is confirmed with our theoretical result of the stability of DFEs.
Next, Fig. 5 shows the results when the proliferation rates of healthy and infected hepatocytes are different (ρ = r), we choose ρ = 0.4 and do not change the other parameter values, the disease dies out and go the same point E 0 . Moreover, the stability in this case is different in terme of taking more time to converge and the infected cells, capsids and virus expand more then the first case of the same proliferation rates. Fig. 6 argues the stability of the endemic equilibrium, we change the paremeter a = 100 and let the others as they are, we get the basic reproduction number is greater than unity (R 0 = 2.7 > 1).
At the case of the same proliferation rates of healthy and infected hepatocytes (ρ = r), the curves converge to E 1 = (0, 0.0186 × 10 11 , 1.9817 × 10 11 , 2.4878 × 10 11 ), we observe the persistence of the capcids and the virus. Fig. 7 shows, when the proliferation rates of healthy and infected hepatocytes are different (ρ = r), that the behavior infection converge to E 1 = (0, 0.0165×10 11 , 1.7604×10 11 , 2.2100×10 11 ), however, the persistente of the virus in this case is less than the case of the same proliferation rates.

CONCLUSION
In this work, we investigated the hepatocytes B infection with DNA-containing capsids by considering the proliferating of its dynamics following logistic growth funtions and satureted rates under ordinary differential equation (ODE) model. In accordance with the biological concept, the correct pose of the model is shown to confirm identification of the infection matematically, then we establish the analysis of the problem to prove the existence and the stability of various steady states. At the present study, we give the local stability conditions of differents equilibruim points and to better understand the progression of hepatic B diseases two scenarios possible were discuss, when the healthy and infected hepatocytes proliferate at the same, we conclude that at the disease free equilibrium the virus expand more then the case of different rates, however, at the endemic equillibruim the persistente result of the virus in this case of different rates is less than the same. Moreover, our numerical summulation confirm the theoretical results concerning the stability of the steady states.