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This paper studied a structured model by age of tuberculosis. A population divided into two parts was considered for the study. Each subpopulation is submitted to a program of vaccination. It was allowed the migration of vaccinated people only between the two patches. After the determination of and , the local and global stability of the disease-free equilibrium was studied. It showed the existence of three endemic equilibrium points. The theoretical results were illustrated by a numeric simulation.

Tuberculosis (TB) (short for tubercle bacillus) is a widespread, infectious disease caused by various strains of mycobacteria, usually Mycobacterium tuberculosis (MTB). Tuberculosis typically attacks the lungs, but can also affect other parts of the body [

It is well known that factors such as the emergence of drug resistance against tuberculosis, the growth of the incidence of HIV in recent years, as well as other diseases favor the development of Koch bacillus in the body call for improved strategies to control this deadly disease [

Two-patch age structured model of tuberculosis was considered. The model is to split the population into two subpopulations. The recruitment is only possible in the class of susceptible and the vaccinated individuals were able to migrate between the two subpopulations. Each subpopulation is divided into five classes based on their epidemiological status: susceptible, vaccinated, latent, infectious or treated. We denote these subgroups

The age-structured model for the transmission of TB (see

with initial and boundary conditions:

and

assume that assume that

(see Greenhalgh, 1988 [

By summing equations of system (1) and (2), we obtain the following equations for the total population

where

Let

The system (1) can be normalized as the following system:

with boundary conditions

with

In this section we will prove that the system (5) has a unique positive solution, and to achieve this we will write the system (5) in compact form (abstract Cauchy problem).

Consider the Banach space X defined by

where

The state space of system (5), where

To determine the components

After replacing

With

where

And

where

Let

thus, we can rewrite the system (5) as an abstract Cauchy problem:

where

According to these results we have the following results (see [

Lemma 1. The operator F is continuously Fréchet differentiable on X.

Lemma 2. The operator A generates a

Theorem 1. For each

Proof. The proof of this theorem can be found in [

A steady state

with initial value conditions

Therefore, we obtain the disease-free steady state

To study the stability of the disease-free steady state, we denote the perturbations of system by

The perturbations satisfy the following equations:

with boundary conditions:

we consider the exponential solutions of system (16) of the form:

The system (16) becomes:

with boundary conditions:

Let

From Equation (18), we obtain:

Hence, by Equations ((20) and (21)) after changing order of integration, we obtain:

Injecting (22) in the expression of

Denote the right-hand side of Equation (23) by

We define the net reproductive number as

We can obtain an expression for

Let

Theorem 2. The infection-free steady-state (5) is locally asymptotically stable (l.a.s.) if

Proof. Noticing that

We know that Equation (23) has a unique negative real solution

indicating that

In this corollary, we have the three cases of the unstability of the disease free equilibrium.

Corollary 1. 1) whenever

2) whenever

3) whenever

Since

Corollary 2. Assume that

Theorem 3. The disease-free equilibrium of system (5) is globally asymptotically stable if

Proof. The proof consist to show that

Integrating system (5) along characteristic lines we get

Injecting (27) in (28), and changing order of integration, we obtain:

Injecting (29) in

By using corollary 2, inequality (*) and Fatou’s lemma, we have

Since

Corollary 3. The disease-free equilibrium is globally asymptotically in:

1) the first sub-population if

2) the second sub-population if

For this disease can disappear without any form of intervention, according to these results we must ensure that there is no new infected and the infectious rate does not reach a certain spread.

There exists three endemic steady state of system (5) whenever

Theorem 4. A boundary endemic equilibrium of the form

Proof. The method commonly used to find an endemic steady state for age-structure models consists of obtaining explicit expressions for a time independent solution of system (5)

with the initial conditions:

Let

Integrating system (31), we obtain:

By injecting (37) in (34), we obtain:

Injecting (40) in the expression of

Let

Since

i.e.

We now see that an endemic steady state exists if Equation (41) has a positive solution.

Since

Since

In particular, for

Theorem 5. A boundary endemic equilibrium of the form

Proof. (Ideas of proof)

with the initial conditions:

Let

Integrating system (51), we obtain:

Hence, by the similar method using in theorem 4, we obtain the result. W

Theorem 6. An interior endemic equilibrium of the form

whenever

Proof.

with the initial conditions:

Let

By injecting (58) in (59), we obtain:

By injecting (63) in the expression of

Let

Since

We now see that an endemic steady state exists if Equation (64) has a positive solution. Since

Since

In particular, for

In this section, when

When

In this paper, an age structured model of two-patch for tuberculosis was analyzed and discussed. Each sub-population is subjected to a vaccination program. Apart from age; the vaccinated compartment, we introduced as a class of treated in the model proposed by Tewa J. Jules in [

We thank the Editor and the referee for their comments. We would like to thank Numerical Analysis student group for their valuable comments and the authors whose works have been used in this article. We also thank the ministry of Higher Education of Research an Innovation who kindly supported the costs of the publication.

Wahid, B.K.A. and Bisso, S. (2016) Mathematical Analysis and Simulation of an Age-Structured Model of Two-Patch for Tuberculosis (TB). Applied Mathematics, 7, 1882-1902. http://dx.doi.org/10.4236/am.2016.715155