ABSTRACT

A SEI model for hepatitis B is constructed where the susceptibility and other crucial transmission probabilities depend on the chronological age and the basic reproduction rate is derived. Under suitable (biological and mathematical) assumptions in a closed population, results of Houpa D. D. E. et al. [1] are extended from constant case of and to age-dependent case: the disease-free equilibrium is globally asymptotically stable (GAS) if. On the other hand, induces that endemic equilibrium is GAS and the system is uniformly persistent.

Keywords: Hepatitis B; PDE Model; Global Stability; Lyapunov-LaSalle Functionals

1. Introduction

This paper studies a system of equations modelling the dynamic of hepatitis B with age-dependent susceptibility in a closed population. Its manifestations in human body are shown by hepatitis B antigens (small spherical particles, tubular forms and a large shelled spherical particles) because of their association with a high risk of hepatitis [2] . Hepatitis B caused acute hepatitis and severe chronic liver disease. Hepatitis is endemic in Africa [3] [4] . According to Pasquini et al. [5] (with a computer model), Bonzi et al. [6] (with an EDOs model), Inaba et al. [7] (theoretically with a PDE) or D. J. Nokes et al. [8] (with statistics tools) and L. Zou et al. [9] (with PDE by fitting model to data), age factor is important in epidemiology of disease like hepatitis and reveals most of time useful informations on the dynamics of the epidemic.

A SEI model for hepatitis B is constructed where the susceptibility and other crucial transmission probabilities depend on the chronological age and the basic reproduction rate is derived. Under suitable (biological and mathematical) assumptions in a closed population, it is proved that the disease-free equilibrium is globally asymptotically stable (GAS) if and induces that endemic equilibrium is GAS and the system is uniformly persistent.

The work is organized as follows. After the presentation of the mathematical model with its main results, Section 2 studies the well posedness of the PDE and derives preliminary results useful to study the long-term behaviour of the model. Moreover, it deals with the wellposedness of the model and proves the global asymptotic stability of the disease-free equilibrium when the basic reproduction number and stability of the endemic equilibrium (EE) with the carriers (E) transmission rate small enough to be considered as zero. These results are verified through numerical simulations extended by a discussion and conclusions in Section 3.

2. Mathematical Model

2.1. Presentation

In this study we will consider the following (chronological) age-dependent susceptibility model:

(1.1)

posed for and. Here denotes the age-specific density of susceptible, and denotes respectively the age-specific densities of acute infected (that can be symptomatic or asymptomatic) and chronic carriers. In addition is a given function such that while. Function represents the age-specific probability to become a chronic carrier when becoming infected. Function denotes the probability to develop an acute infection when getting the infection at age ([8] studied the age-dependence susceptibility to the infection). We conditionally extend in some sens results of Houpa D. D. E. et al. [1] who analyzed the case where and are constant. Parameter denotes the natural death rate at age, and denotes the exit rates associated to each infected class. Clearly at each age,. is the transition rate from to. Obviously,. In some studies (like Kouakep et al. [10] ) authors set. The term corresponds to the age-specific force of infection and follows the usual law of mass-action, that reads as

This problem (1.1) is supplemented together with the boundary conditions:

(1.2)

and initial data

(1.3)

This model (1.1) is suggested by Melnik et al. [11] for the age-dependent susceptibility concept supplemented with Kouakep et al. [10] introducing and.

We recall that according to WHO [4] , Bonzi et al. [6] and Fall et al. [3] , asymptomatic carriers has a low infectious rate. As a consequence in most part of this work one will assume that

(1.4)

Then

(1.5)

In the above model (1.1)-(1.5), we do not take into account possible vertical transmission and we do not consider any control strategy such as vaccination campaign. It seems to be relevant together the assumption of WHO [4] wich considers that vertical transmission of the disease exists in sub-Saharan Africa. But its influence on the dynamics of the disease is rather small because the proportion of chronic infections acquired perinatally is low [12] .

Using the data, Nokes et al. in [8] constructed the prototype (useful for us) in the simulations:

(1.6)

We do not focus on chronological age in the infective classes.

2.2. Main Results and Simulations

The basic reproduction rate is defined by

(1.7)

The DFE is defined by

(1.8)

For endemic equilibrium, we obtain only in the case,

(1.9)

linked to

That means 

Assumption 1. Assume that the maps is bounded and uniformly continuous from into itself.

Let. The function has only one extremum which is a global minimum 0 at 1, satisfying (see [13] ). We make these assumptions for the endemic equilibrium when:

Assumption 2.

1. has a constant sign on.

2. On the attractor (an invariant compact attractor of all bounded sets following the Proposition 2 therein), the following inequality holds true:

We make also this assumption for the disease free equilibrium when:

Assumption 3. has a constant sign on.

The global stability of the steady states is resumed in the following Theorem 1.

Theorem 1. Assume Assumptions 1, 2 and 3. Then:

Ÿ  If, then the DFE, the disease free equilibrium, is globally asymptotically stable.

Ÿ  If, then there exists an endemic equilibrium that is globally asymptotically stable for all, and. Moreover, in that case the system is uniformly persistent.

Remark 1. We will see that disease free equilibrium (DFE) exists whenever or. But endemic equilibrium exists only when.

We denote in Tables 1 and 2: “p” for people(s), “yr” for year and “nbb” for “new born babies”. We made simulations with the values in Tables 1 and 2 and denoted by the constant birth rate at any positive time (with year unit). We consider the following parameters for DFE case (related to Figures 1-3).

For endemic case (related to Figures 4-10), we consider the values in the Table 2.

We have tested our Assumption 2-2 on the Figures 9 and 10 with and: it is verified in our simulations up to some time (considered as origin by time shifting or rescalling, wich is not very important in our case for long term dynamics in our simulations or calculations) with the global asymptotic stability of the endemic case. One could see that Assumption 2 and 3 could be relaxed by proving them for with an arbitrary positive real constant (or number). In all the cases, we observe a period of stability after a severe outbreak of the disease.

2.3. Technical Materials

Let us introduce the Banach space and endowed with the usual product norm as well as its positive cone defined (with) by:

with.

We consider also the linear operator defined by

with the non densily domain in.

Finally let us introduce the nonlinear and Frechet differentiable map defined by:

Identifying and, one obtains that System (1.1)-(1.5) rewrites as the following non-densely defined Cauchy problem (1.10):

(1.10)

We first derive that the above abstract Cauchy problem (1.10)-(1.11) generates a unique globally defined and positive semiflow. Moreover satisfies the Hille-Yosida property. Then standard methodologies apply to provide the existence and uniqueness of mild solution for system (1.10)-(1.11) (see for instance [10] [14] -[17] ):

Proposition 1. Let Mathematical Assumption 1 be satisfied.

Then there exists a continuous semiflow that is bounded dissipative on into itself such that for each, the map is the unique integrated solution of (1.10)-(1.11) with initial data, namely satisfies

(i)(ii) for each.

Remark 2. One can prove the proposition 1 by using ideas of corollaries 1 and 2 in Melnik et al. [11] .

By using results in Sell and You [18] , one can prove that is asymptotically smooth. Then using results of Hale [19] [20] , Hale et al. [21] , one obtains the following proposition.

Proposition 2. Let Mathematical Assumption 1 be satisfied. Then there exists a compact set such that

(i) is invariant under the semiflow.

(ii) attracts the bounded sets of under. This means that for each bounded set we have

where is defined as

Moreover is locally asymptotically stable.

We will widely adapt ideas of Magal et al. [13] and Melnik et al. [11] here with Lyapunov functionals on for the global stability of DFE and EE.

1) Stability of the DFE:

Let us introduce the positive map defined on:

is positive defin ite at the DFE. We evaluate as

with, and. The equations of the system 1.1 help us to get for:

We would like to prove that

Three cases occur by Assumption 3:

1. If, one obtains:

And by integrating from 0 to, one gets:

Then

2. If, one gets:

3. If, one gets:

And by integrating from 0 to, one gets:

but and,

that implies

Hence

Finally 

Hence by recalling that,

Finally by global stability Lyapunov-LaSalle theorem [11] [13] [22] , the DFE = is globally asymptotically stable because the largest invariant set of orbits verifying is reduced for all positive and, to, and corresponding to the disease free steady state (DFE) seen as.

2) Stability of the endemic equilibrium:

Any solution of system (1.1) with positive initial condition remains positive indefinitely: then the system (1.1) is uniformly persistent (the tools are similar to Melnik [11] ).

Let. The function has only one extremum which is a global minimum 0 at 1, satisfying (see [13] ). Then, we will analyse the Lyapunov functional

We notice that and is positive definite at EE = that provides the minimum of. Moreover is defined for all, , and

With

we obtain:

We set By assumption,

Then

Three cases appear by Assumption 2-1:

1. If, then

By integrating from 0 to, one gets:

Then

And finally

2. If we get:. But

1) For and, one has:

then

2) For one obtains:

To get, it is enough to show that:

We set:

We want to prove that:

By definition of we have

then

It enough to verify this sufficient condition

Recall that

Then

if and only if

that means (see Figures 9 and 10 in the simulations of subsection 1.2.2)

and,

By Assumption 2-2,

we obtain,

and

3. If then:

By integrating from 0 to, one gets:

Then,

but. So

by using results in case, one has:.

Then by global stability Lyapunov-LaSalle theorem [11] [13] [22] , the endemic equilibrium (EE) is globally asymptotically stable because the largest invariant set of orbits verifying is reduced for all positive and, to, and corresponding to the endemic steady state.

3. Conclusions

We observe that our computations for stability of DFE and EE are confirmed by simulations. It is also established that increasing the transmission coefficient, increases the basic reproduction rate. In a forthcoming work, we will introduce vertical transmission (because of the contreversal article Sall et al. [23] on WHOs [12] neglection of vertical transmission in sub-Saharan Africa), studies of (optimal) vaccination strategies and immigration by other ways than birth. The results of this work extend those of Melnik et al. [11] and Kouakep et al. [10] on a more realistic case applied to hepatitis B situation. One can study the stability of the endemic equilibrium (EE) with small enough (like Ducrot et al. [10] ) using perturbation arguments of Magal [24] . For the case (avoid here) where and the map is bounded and uniformly continuous from into itself, Ducrot et al. [25] deal with global stability of the disease free equilibrium with (constant) functions and by considering a particular case of the Lyapunov functional (similar to Magal et al. [13] and Kouakep et al. [10] ) which is non-increasing along the complete orbits with and such that

and

Note that for,

Ducrot et al. [25] used also arguments like those in (Demasse et al. [17] , proposition 4.1 and its proof). For global stability of endemic equilibrium in the case, Ducrot et al. [25] used the following Lyapunov functional (under special assumptions on) with a well-chosen positive constant:

(1.11)

The model (1.1)-(1.5) is formally equivalent (with) to the following model:

(1.12)

supplemented together with the boundary conditions:

(1.13)

and initial data

(1.14)

By replacing (chronological age) by (infection age) in the infectives classes and, the model (1.1)-(1.5) is equivalent (with) to the following model (see Kouakep et al. [10] ):

(1.15)

supplemented together with the boundary conditions:

(1.16)

it remains to model, the force of infection, and those general form can be written in the form

(1.17)

where is the chronological age and is the time since the infective(s) are contaminated. Another similar problem (with) is:

(1.18)

supplemented together with the boundary conditions:

(1.19)

We strongly believe that Assumptions 2 and 3 could be relaxed if we use usual tools of functional analysis by splitting functions and in the form as a difference of two well-chosen positive functions and. Then one can use the constant-sign cases on and.

Acknowledgements

Authors are grateful to Reviewers, Pr Bekolle, Dr. A. Ducrot, Dr. Damakoa Irepran, Dr. Kamgang J. C. and GDM-MIAP group for helpful remarks or comments on the manuscript.

References

NOTES

^*Corresponding author.