A With-in Host Dengue Infection Model with Immune Response and Beddington-deangelis Incidence Rate

A model of viral infection of monocytes population by dengue virus is formulated in a system of four ordinary differential equations. The model takes into account the immune response and the incidence rate of susceptible and free virus particle as Beddington-DeAngelis functional response. By constructing a block, the global stability of the uninfected steady state is investigated. This steady state always exists. If this is the only steady state, then it is globally asymptoti-cally stable. If any infected steady state exists, then uninfected steady state is unstable and one of the infected steady states is locally asymptotically stable. These different cases depend on the values of the basic reproduction ratio and the other parameters.


Introduction
Dengue is an infections mosquito-borne viral disease.It is estimated that about 50 million infections occur annually in over 100 countries [1].There is no specific treatment for curingdengue patients.Hospital treatment in general is given as supportive care which includes bed rest, antipyretics, and analgesics.Most dengue infections are asymptomatic.Few of them suffer dengue fever and dengue haemorrhagic fever, which may end up in fatality.
Dengue virus is one of the most difficult arboviruses to isolate.There are four serotypes of the dengue virus and each of the serotype has numerous virus strains.Infection with one dengue serotype may provide lifelong immunity to that serotype, but there is no cross-protective immunity to other serotype, [2].Identification of the primary target cells of dengue virus replication in infected human body has proven to be extremely difficult.It is generally believed that the target cells of dengue virus are monocytes or its differentiated cells the macrophages [3].
It is usually believed that dengue virus is quickly cleared in human body within approximately 7 days after the day of sudden onset of fever [1].Naturally this clearing process is done by the immune system which is a result of complex dynamic reactions.Following [4], in this paper we try to understand the process using a mathematical model.
Mathematical modeling of dengue disease transmis-sion in human and mosquito populations has been done since the beginning of last century.Some of the recent models could be seen in [2][3][4][5].Several studies on infection model within human body have been done for various cases [2,3] and [5][6][7][8][9][10][11].Meanwhile, mathematical modeling for with-in host dengue viral disease is quite new.
The model for with-in host dengue viral infection with Beddington-DeAngelis incidence rate and immune response is as following.
where, 1 The constant , is the rate constant characterizing infection of the cells.The constants 0 a  ,   are positive.
In the above     


. The flow from susceptible monocytes to the infected monocytes depends on the incidence rate of susceptible monocytes and free virus particle.This rate is shown by is the incidence response of susceptible monocytes to free virus particles.The period of infected monocytes is assumed constant as 1


. We assumed virus multiplication is at constant rate k and the virus clearance rate is at constant rate  .We also assumed the immune cells are produced at constant rate  and their life span is 1


. Moreover we assumed there is stimulation of immune cells production due to the increase of infected cell which is proportional to the density infected monocytes at a constant rate c as well as from the contacts with infected cells at the rate d and the immune cells will eliminate the infected monocytes at a constant rate v. Finally, the positive constants  and  have some biological meanings.
The above model is valid for only one serotype of dengue virus circulate in an infected host and dengue infects monocytes in blood stream.
For more detail the reader is referred to [4] and references therein.
The local stability of the equilibrium points of the system (1) for Lotka-Voltera functional response i.e.
, has been discussed in [4].The model ( 1) is a generalization of the self-regulating cytotoxic T lymphocytes (CTL) response model.The predator-prey like CTL response model and the linear immune response model in chapter 6 of [5].

 
In this paper, we will analyze the global of stability of the viral free equilibrium for Beddington-DeAngelis in- . In fact we will show that if this equilibrium is the only rest point of the system (1), then it is globally asymptotically stable.If there are some other equilibria, then the local stability of them depends on the values of the parameters.

Global Stability of the Uninfected Equilibrium
In this section, at first we will find the equilibrium points of the system (1) and the eigenvalues of this system at these points.This information leads us to prove the locally asymptotical stability of the equilibrium points.
At an equilibrium point of the system (1) we must have From the first equation we obtain, Substituting this value of V into the third equation yields, . From the fourth equation we obtain . Substituting these values of , V I and Z into the second equation yields, where, In the following we consider the stability property of the equilibrium point .In order to do this we check the sign of the eigenvalues of Jacobi matrix of (1) at .The Jacobi matrix is x and 3 x have negative real part.If x has negative real part, then the equilibrium 0 is locally asymptotically stable.But  .This condition equals Thus, 1 x    and 2 x    are two of the eigenvalues and the other two are the roots of to, . This number is called the basic reproduction ratio [7].Therefore we have the following theorem.Theorem 2.1.If, 0 , the equilibrium point 0 is locally asymptotically stable and if , the equilibrium is unstable.
Now we will show that if, 0 , then the equilibrium, 0 is globally asymptotically stable.In order to see this, first of all consider the following domain in the space.
It follows that the flow generated by that system (1.1) gets into on the boundary of .Let , , , : 0 , 0, 0, 0 and , , , , and * 1 : of the boundary of , we have Thus 0 is a global attractor.Thus we have proved the following theorem.
R  , then 0 ,the uninfected equilibrium is the only equilibrium of the system (1).Moreover this equilibrium is globally asymptotically stable.y Since 0 is globally asymptotically stable for 0 y 1 R  , any other equilibrium points of the system (1) cannot exist for 0 1 R  .Therefore, is the unique equilibrium point for

Stability of the Other Equilibrium Points
In this section, we consider the stability of the other rest point of the system (1).In order to this, we consider the Equation (3).First, we consider this equation for 1 0 c  and then for 1 0 c  .There are two cases for as follows. .By using the value of I and the third equation we get . Using these  K S I V Z along the orbits of the system (1), we obtain: values into the Equation (3), we obtain, Therefore, we obtain as the second rest point of the system (1).
Notice that this rest point exists if . Now, we consider the local stability of the equilibrium 1 .By using the formula (4), the value of Jacobi matrix at is where We calculate the eigenvalues of   1 J y as follow: Thus one of the roots is x    .The other roots are , q q d 0 q we see that 2 1 , q q and are po ve.Moreover, it is easy to che at, 2 0 q q q  .By the Rouths Hurwitz Criteria, all roots of th lynomial have negative real part.Therefore we have the following theorem. Theorem en from the third equation we get, first equation of the adratic equation.
this value of V into the system (1), we obtain the following qu  is another equilibrium point of the system (1) where is the positive root of the above quadratic equation.
In the following, we consider the stability property of th points.rst consider it for .Here we check the sign of genvalues of Jacobi trix of the system ( 1 We calculate the eigenvalues of   1 J y as follows: Thus one of the roots is The other roots are given by Here B

By substituting the value
A B and o V in 2 1 , q q and 0 q we will see that Mo er, it is easy to Rout Hurwitz Criteria, al c p mial have negative real part.If J y eigenvalues of the matrix a the alge qu re given by braic e ation, Then from the above equation we get By considering the value of ** Z equatio it follows that all of the coefficients of the above n are positive, then from Routh Hourwitz Criteria we see that all of th roots have negative real parts.Therefore we have the foll wing theorem.
and the Equation (3).Here stability property of this point is not shown.

Conclusions
In order to understand the main characteristic of De ry, the author in [4] assumed that this virus can be el ed by immu e system (1).using linea he existence of the endemic rom the analysis of the endemic ngue myste iminat ne response which is described by the last equation of th By r incidence rate of susceptible and free virus particle, they analyzed t virus equilibria.
In this paper, f equilibria it is found that, for Beddington DeAngelis incidence rate of susceptible and free virus particle, the same results are valid.
The reson for this correspondence is that in both models, the feature of the immune response is described by the term cI dIZ    .However, the parameter  in Beddington DeAngles makes the elimination of dengue virus by immune response in a shorter time.This fact can be seen by comparing Tables 1 and 2.  increase, the d I-components of equilibria will decrease but the virus lo increases at the initial viral infection.
For case 1 0  and large and 0 1 R  , the model has a unique endemic virus.The V and V an ad c I compone s equilibr m point decrease as a increases and the S and Z-components of it increa as d increases.
Therefore, , d a and nts of thi iu e s  ar the important parameters to capture the phenomena that dengue v us is quickly cleared in a shorter time.

Acknowledgements
The authors would like to thank the anonymous reviewers r their valuab comments and suggestions to improve the manuscript.

4
D is a global attractor.Now in D consider the following set for : In this case, the system (1) has two equilibrium points, and another one.To see this, from the first equation real of all of the e value are negative.Theref he point 1 y is locally asymptotically stable.Now we consider the stability property of the other equilibrium point, .From the formula (4) we have

For
the following numerical simulations, we use parameters of T-cells as the parameters of immune cells, those are  days.The estimated va- lue of  is obtained by assuming that the equilibrium value of the density of immune cells in the absence of us of the disease depends disease.On the contra the inc infection is 2000 cells.In this model the endemic stat on the individual response toward incoming viruses.The larger the invasion rate a , the chance is higher to catch the rease of the elimination rate ry  of infected o cell, the risk f infection is lower.For 1 and Dengue Hemorrhagic Fever," Clinical Microbiology Reviews, Vol.11, No. 3, 1998, pp.480-496.[3] E. A. Henchal e Viruses," 6, 2009, pp.1148-1155.

Table 2 . Status of equilibri m (1) in the case um points of syste 0
, so it does not depend on the other parameters for virus load of infected cell.For lager values of  and  , the infected endemic 1