Volume 3 • Issue 1 • 1000148J Appl Computat MathISSN: 2168-9679 JACM, an open access journal

Open AccessResearch Article

Kouakep et al., J Appl Computat Math 2013, 3:1 DOI: 10.4172/2168-9679.1000148

Keywords: Nonlinear dynamical systems; Global stability; Lyapunov methods; Differential infectivity models; HBV; Cameroon

A Simplified Model without Vertical Transmission and With Control Measure

Hepatitis B is endemic in Africa [1-3] (see also the references cited therein) and some models have been constructed in order to understand it’s dynamic with ODE deterministic (with delay or not) or stochastic processes [2,4-8] or partial differential equations [9,10].

Studies like [9-13] recognized the importance of the age factor in the dynamics of infectious diseases like hepatitis B [10]. Moreover some studies like [4,14] obtained results with ODEs with discrete age(s) that could be generalized with continuous age assumption more realistic and relevant.

We introduce then a model with differential infectivity and chronological or infection ages. We denote by s (t,a) the density of susceptible at time t with chronological aged a. We denote by i (t,τ) the density of infective that will develop acute disease at time t contaminated since a time and e (t,τ) the density of infective that will not develop acute disease (asymptomatic carrier) at time t contaminated since a time τ. The model we shall consider reads as follows:

( ) ( , ) ( , ) ( ) ( , ), 0, 0,( ,0) ,

( ) ( , ) ( ) ( , ), 0, 0,( ) ( , ) ( ) ( , ), 0, 0,

τ

τ

µ λ

τ µ γ τ ττ µ γ τ τ

∂ + ∂ = − − > >= Λ

∂ + ∂ = − + > >∂ + ∂ = − + > >

t a

t i

t e

s t a s t a t s t a t as t

i t i t te t e t t

(1.1)

00

( ,0) ( ) ( ) ( , ) , ( ,0) ( ) (1 ( )) ( , )λ λ∞

∞= = −∫ ∫i t t p a s t a da e t t p a s t a da

Here 0Λ > is some constant entering in flux, μ>0 is the natural death rate, γi is the additional death rate due to the disease, a→p(a) ∈ [0,1] is the proportion of individuals going to the acute infective class while 1-p(a) is the proportion to not develop the acute disease when infection occurs. Finally, it remains to model λ (t,a), the force of infection, those general form can be written in the form

0( ) ( ( ) ( , ) ( ) ( , ))λ β τ τ β τ τ τ

∞= +∫ i et i t e t d (1.2)

Finally this model is supplemented together with some initial data1

01 2

0 0 0 0

(0,.) (.) (0, )

(0,.) (.), ( ,.) (.) ( , ) (0, )+

+

= ∈ ∞

= = ∈ ∞

s s Li i e o e with i e L

The above model takes into account the chronological age of susceptible. This parameter has strong implication in the dynamics of

infection. Indeed, depending on the age at which susceptible enters the infective's classes, the disease will develop indifferent way. For hepatitis B virus (HBV), young infections lead to chronic infection while older infection leads to acute disease.

In the above model, we do not take into account possible vertical transmission and we do not consider any control strategy such vaccination campaign. It seems to be relevant together the assumption of WHO [3] that consider that vertical transmission of the disease does occur in sub-saharian Africa, but its influence of the dynamics of the disease is rather small because the proportion of chronic infections acquired prenatally is low. Under the above assumption, we assume that the chronological age for the infective classes do not play an important role. But the time since infection is a relevant biological variable because of the possibility to have a long latent period (especially for the asymptomatic carrier class, until several years).

The work is organized as follows. In Section 2, we prove the wellposedness of the PDE (1.1-1.2), derive preliminary results useful to study the long term behavior of the model. Sections 3, 4, and 5 is devoted to the uniqueness of endemic equilibrium when the biological basic reproduction rate R0 is greater than 1 and study the global asymptotically stability of the disease free equilibrium if 1>R0. Finally Section 6 presents discussion.

Abstract Cauchy Problem ReformulationMathematical assumptions

We assume that:

a) ( ) 0, ( ) 0γ τ γ γ τ≡ > ≡i i e

b) The function (0, )∞∈ ∞p L with ( ) [0,1]∈p a a.e. and notidentically 0 and 1.

c) Function 0, (0, )β β ∞∈ ∞i e L .

*Corresponding author: T Y Kouakep, Ngaoundere University, PO Box454 Ndang, Ngaoundere, Cameroon, Tel: 237 22 25 40 40; E-mail: kouakep@aims-senegal.org

Received July 26, 2013; Accepted December 04, 2013; Published December 23, 2013

Citation: Kouakep TY, Ducroty A, Houpa DDE (2013) A Model for Hepatitis B with Chronological and Infection Ages. J Appl Computat Math 3: 148. doi:10.4172/2168-9679.1000148

Copyright: © 2013 Kouakep TY, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

AbstractWe construct and study a differential infectivity model with chronological and infection age. The application

is done on hepatitis B in Cameroon. We prove the global stability of the disease free equilibrium when the basic reproduction ratio R0 is less than one and the existence and uniqueness of an endemic equilibrium when R0>1 .

A Model for Hepatitis B with Chronological and Infection AgesT Y Kouakep1*, A Ducroty2 and Houpa D D E3

1Ngaoundere University, PO Box 454 Ndang, Ngaoundere, Cameroon2Universite Bordeaux Segalen, UFR Sciences et Modelisation, 3 ter place de la Victoire, Bat. D, 2eme etage, 33076 Bordeaux Cedex, France3Ngaoundere University, PO Box 454 Ndang, Ngaoundere, Cameroon

Journal of Applied & Computational Mathematics

ISSN: 2168-9679

Jour

nal o

f App

lied & Computational M

athematics

Citation: Kouakep TY, Ducroty A, Houpa DDE (2013) A Model for Hepatitis B with Chronological and Infection Ages. J Appl Computat Math 3: 148. doi:10.4172/2168-9679.1000148

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Volume 3 • Issue 1 • 1000148J Appl Computat MathISSN: 2168-9679 JACM, an open access journal

Abstract Cauchy problem

The aim of this section is to deal with (1.1). To do so, we consider the Banach spaces

33 1 3 1 3

0(0, ) , {0 } (0, ) ,= × ∞ = × ∞

X L X L

endowed the norm

1

2

33 1

1 (0, , )11

2

3

( ),

ααα

α ψψψψ

∞=

= +

∑i

i

X

L

as well as the non-densely defined linear operator : ( ) ⊂ →A D A X X defined by

{ }3 1,1 3( ) ( , )= × ∞

D A O W O

together with

1

2

3'1 11'

2 2 2'

3 3 3

(0)0(0)0(0)0

,( ) ( )

( ) ( ) ( )

( ) ( ) ( )

ϕϕϕ

ϕ µϕϕϕ ϕ τ µ γ ϕ τϕ ϕ τ µ γ ϕ τ

− − − = − − − − + − − +

i

e

A a a

as well as the nonlinear map 0F : X X→ defined by

1 2 30 0

1 1 2 30 0

22 30

3

00

( ) ( ) ( ( ) ( ) ( ) ( ))0,

(1 ( )) ( ) ( ( ) ( ) ( ) ( ))

( ( ) ( ) ( ) ( ))

ϕ β τ ϕ τ β τ ϕ τ τ

ϕ ϕ β τ ϕ τ β τ ϕ τ τϕ

β τ ϕ τ β τ ϕ τ τϕ

∞ ∞

∞ ∞

∞

Λ + = − + − +

∫ ∫∫ ∫

∫

i e

i e

i e

p a a da dF

p a a da d

d

Let us notice that 0( ) =D A X

Now by identifying (s (t, .), I (t, .), e(t, .)) in (1.1) together with u(t) = (0, 0, 0, s(t, .), i(t, .), e(t, ,))T , one obtains that u(t) satisfies the

following abstract Cauchy problem ( ) ( ) ( ( )), 0,= + >du t Au t F u t t

dt (2.3)

together with the initial data 0 0 0 0(0) (0,0,0, , , )= = ∈Tu x s i e X .

We also consider the positive cones 3 1 30 0(0, ) ,+

+ + + += × ∞ = ∩X L X X X

Lemma 2.1: (Hille-Yosida property) Operator A: ( ) ⊂ →D A X X is a Hille-Yosida operator. More precisely we have ( ), ( )µ ρ− ∞ ⊂ A the resolvent set of A and for each λ>-μ, each 1 2 3 1 2 3( , , , , , )α α α ψ ψ ψ ∈T X we have

3

1

2

311

12

3 2

3

0

( )

αααϕ

λψϕ

ϕ ψψ

−

− =

A

( ) ( )( )1 1 10

( ) ( )( )2 2 20

( ) ( )( )3 3 30

( ) ( )

( ) ( )

( ) ( )

λ µ λ µ

λ µ γ λ µ γ

λ µ γ λ µ γ

ϕ α ψ

ϕ α ψ

ϕ α ψ

− + − + −

− + + − + + −

− + + − + + −

⇔

= +

= + = +

∫∫∫

i i

e e

aa a s

aa a s

aa a s

a e e s ds

a e e s ds

a e e s ds

Moreover we have for each λ>-μ1

0( )λ −+ +− ⊂A X X (2.4)

Proof: Let 1 2 3 1 2 3( , , , , , )α α α ψ ψ ψ λ µ= ∈ > −Tx X and be given. Then the equation

3

1

2

3

0

( ) ,ϕ

λϕϕ

− =

A x

rewrites as the following system

1 1 1

2 2 2

3 3 3

1 1 2 2 3 3

( ) ' ( ) ( ) ( )( ) ' ( ) ( ) ( )

( ) ' ( ) ( ) ( ) 0(0) , (0) , ( ) ,

ϕ λ µ ϕ ψϕ λ µ γ ϕ ψ

ϕ λ µ γ ϕ ψϕ α ϕ α ϕ α

= − + + = − + + + = − + + + ∀ ≥

= = =

i

e

a a aa a a

a a a aa

that is

( ) ( )( )1 1 10

( ) ( )( )2 2 20

( ) ( )( )3 3 30

( ) ( )

( ) ( )

( ) ( )

λ µ λ µ

λ µ γ λ µ γ

λ µ γ λ µ γ

ϕ α ψ

ϕ α ψ

ϕ α ψ

− + − + −

+ + + + −− −

+ + + + −− −

= +

= + = +

∫∫∫

i i

e e

aa a s

aa a s

aa a s

a e e s ds

a e e s ds

a e e s ds

On the other hand one has 1

1 1 (0, )11

12 2 (0, )1

2

13 3 (0, )1

3

(0, ) ,

(0, ) ,

(0, )

α ϕϕ

λ µ

α ϕϕ

λ µ

α ϕϕ

λ µ

∞

∞

∞

+ ∞ ≤

+

+∞ ≤ +

+ ∞ ≤ +

L

L

L

L

L

L

As a consequence,

( , ) ( )µ− ∞ ⊂ p A and 1 1( ) ( ) ,λ λ µλ µ

−− ≤ ∀ > −+

A L X

This completes the proof of the Hille-Yosida property. Finally the explicit formula of the resolvent operator implies that (2.4) holds true.

Theorem 2.2: There exists a continuous semi flow {U(t)}t ≥ 0 on X0+ into itself such that for each 0+∈x X , the map t→U(t)x is the unique integrated solution of (2.3) with initial data x, namely t→U(t)x satisfies

(i) 0( ) ( ), 0∈ ∀ ≥∫

tU s xds D A t

(ii) 0 0

( ) ( ) ( ( ) ) 0= + + ≥∫ ∫t t

U t x x A U s xds F U s x ds for each t

Citation: Kouakep TY, Ducroty A, Houpa DDE (2013) A Model for Hepatitis B with Chronological and Infection Ages. J Appl Computat Math 3: 148. doi:10.4172/2168-9679.1000148

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Volume 3 • Issue 1 • 1000148J Appl Computat MathISSN: 2168-9679 JACM, an open access journal

Moreover we have for each 0+∈x X :

liminf ( ) limsup ( )µ γ γ µ→∞ →∞

Λ Λ≤ ≤ ≤

+ + t ti e

U t x x U t x x

Proof: Let us first notice that for each M>0 there exists λ>0 such that

0( ) , (0, )λ + ++ ∈ ∀ ∈ ∩xF x x X x B M X

With BX(0,M) denote the ball of radius M centered at 0. One obtains the existence of a maximal positive semi flow for (2.3) on X0+ into itself. It remains to prove that this semi flow is globally defined. To do so, let 0+∈x X be given and recall that ( ) (0,0,0, ( ,.), ( ,.), ( ,.))= TU t x s t i t e t

Consider the quantity

0 0 0( ) ( ) ( , ) ( , ) ( , ) ,τ τ τ τ

∞ ∞ ∞= = + +∫ ∫ ∫P t U t x x s t a da i t d e t d

the total population at time t. Then it satisfies the differential inequality

( ) ( )µ≤∧−dP t P t

dt

Thus the map t→P(t) cannot blow up in finite and the global existence result follows.

Let us, in addition, notice that, from this inequality one gets (by density)

0limsup ( ) ,µ +

→∞

Λ≤ ∀ ∈

tU t x x x X

One the other hand one has

( )0 0

( ) ( ) ( , ) ( , )µ γ τ τ γ τ τ∞ ∞

= Λ− − +∫ ∫i edP t P t i t d e t d

dt ( ) ( ),µ γ γ≥ ∧− + +i e P t

so that 0liminf ( ) ,µ γ γ +

→∞

Λ≥ ∀ ∈

+ +x i e

U t x x x X

This completes the proof of the result.

Stationary StatesThe disease free equilibrium

The disease free equilibrium corresponds to a stationary (that is time independent solution)

( ( ), 0, 0),≡ ≡ ≡F F F Fs s a i eof (1.1)-(1.2). As a consequence we have the following lemma

Lemma 3.1: The dynamical system provided by (1.1)-(1.2) has a unique disease free equilibrium where sF is given by ( ),≡ ∧s a e

From now one we set

1 20 0( ) ( ) , ( ) ( ) ,

∞ ∞= =∫ ∫F FS p a s a da S p a s a da

and we consider the biological basic reproduction rate

0 0( ( ) ( ) )τ τ

τβ τ β τ∞ − −= Λ +∫ i ev v

i i e eR P e P e d (3.5)

where we have set

Endemic equilibrium

We look for stationary solutions (s; i; e) such that (i; e) not identically zero satisfying

0 0

0

'( ) ( , ) ( ), 0,(0) ,'( ) ( ) ( ), 0,'( ) ( ) ( ), 0,

(0) ( ) ( ) , (0) (1 ( )) ( )

( ( ) ( ) ( ) ( )) ,

µ λ

τ µ γ τ ττ µ γ τ τ

λ λ

λ β τ τ β τ τ τ

∞ ∞

∞

= − − >= Λ= − + >= − + >

= = −

+

∫ ∫∫

i

e

i e

s a s t a s a asi ie e

i p a s a da e p a s a da

i e d

Theorem 3.2: Recalling definition (3.5), if R0>1, system (1.1)-(1.2) has a unique endemic equilibrium point denoted by (sE; iE; eE).

Proof: we have( )

( ) ( )

0

( ) ( )

0

( ) ,

( ) ( )

( ) (1 ( ))

µ λ

µ λ τ µ λ

µ λ τ µ λ

τ λ

τ λ

− +

∞− + − +

∞− + − +

= Λ

= Λ

= Λ −

∫∫

i

e

a

a

s a e

i e p a e da

e e p a e da

Thus

( ) ( )

0 0

( ) ( )

0 0

( ) ( )

( ) (1 ( ))

µ γ τ µ λ

µ γ τ µ λ

λ β τ τλ

β τ τλ

∞ ∞− + − +

∞ ∞− + − +

= Λ

+ Λ −

∫ ∫∫ ∫

i

e

ai

ae

e d p a e da

e d p a e da

We are looking for endemic stationary state, that is λ>0, so that

( ) ( )

0 0

( ) ( )

0 0

1 ( ) ( )

( ) (1 ( ))

µ γ τ µ λ

τ µ γ τ µ λ

β τ τλ

β τ τλ

∞ ∞− + − +

∞− + − +

= Λ

+ Λ −

∫ ∫∫ ∫

i

e

ai

ae

e d p a e da

e d p a e da

Now the map λ→f(λ) is non increasing with f defined by

( ) ( )

0 0

( ) ( )

0 0

( ) ( ) ( )

( ) (1 ( ))

µ γ τ µ λ

τ µ γ τ µ λ

λ β τ τλ

β τ τλ

∞ ∞− + − +

∞− + − +

= Λ

+ Λ −

∫ ∫∫ ∫

i

e

ai

ae

f e d p a e da

e d p a e da

and f(λ)→0 when λ→∞. As a consequence, since R0=f(0)>1, there exists a unique λE>0 such that

f λE=1.

Finally, the functions( )

( ) ( )

0

( )( )

0

( ) ,

( ) ( )

( ) (1 ( )

µ λ

µ γ τ µ λ

µ λµ γ τ

τ λ

τ λ

− +

∞− + − +

∞ − +− +

= Λ

= Λ

= Λ −

∫∫

E

i E

E

aE

aE E

aeE E

s a e

i e p a e da

e e p a e da

provides the unique endemic stationary state of system (1.1)-(1.2).

Dynamical PropertiesAssumption 4.1

Assume that the maps a→βi (a) and a→βe (a) are bounded and uniformly continuous from [0,∞) into itself.

Volterra Integral Formulation:

The solutions of (1.1)-(1.2) can be reformulated as follows

0

( )( ,0) , (0,.) ,

µ λ∂ + ∂ = − −= Λ =

t as s s t ss t s sWith

Citation: Kouakep TY, Ducroty A, Houpa DDE (2013) A Model for Hepatitis B with Chronological and Infection Ages. J Appl Computat Math 3: 148. doi:10.4172/2168-9679.1000148

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Volume 3 • Issue 1 • 1000148J Appl Computat MathISSN: 2168-9679 JACM, an open access journal

0( ) ( ) ( , ) ( ) ( , ) ,

τλ β τ τ β τ τ τ= +∫ i et i t e t d

and( ) ( )

00( )( )

( ) 0,( ) 0,( , ) , ( , )

( )( ) 0

µ γ µ γ

µ γµ γτ τ+ +

++

− − ≥− − ≥= =

−− − <

i

ei

t e t

aaei

e e a t if a te i a t if a ti t e t

e b t ae b t a if a t

where t→bi(t)=i(t,0) and t→be(t)=e(t;0) are the unique continuous functions satisfying for k=i.e:

0( ) ( ,0) [ ( ,.)] ( ( ) ( ) ( ) ( ))τ τβ τ τ β τ τ τ= = − + −∫ i e

t v vk k i i e eb t k t p s t e b t e b t d

0 0[ ( ,.)] ( ( ) ( ( ) ( ))β τ τ β τ τ τ∞

+ − + −∫ i ev t v tk i et

p s t e i t e e t d

where we have set

0 0[ ] ( ) ( ) , [ ] (1 ( )) ( )ϕ ϕ ϕ ϕ

∞ ∞= = −∫ ∫i ep p a a da p p a a da .

By using results in Sell and You [15], one can find suitable conditions to prove that {U(t)}t ≥ 0 is asymptotically smooth and derived the following proposition.

Proposition 4.2: Let Assumption 4.1 be satisfied. Then there exists a compact set 0+⊂A X such that

(i) A is invariant under the semi flow { } 0( )

≥tU t

(ii) A attracts the bounded sets of X0+ under { } 0( )

≥tU t . This means

that for each bounded set 0+⊂B X we have

lim ( ( ) , ) 0δ→∞

=t

U t B A ,

where δ is defined as

( , ) sup infδ∈∈

= −y Bx A

A B x y

Moreover A is locally asymptotically stable. Next one considers the following quantities sup{ 0 : ( ) 0}, { , }τ τ β τ= ≥ > ∈k k k i e and the following set:

{ }1 2

0 0( , ) (0, ) : ( ) ( ) 0

τ τϕ ψ ϕ τ τ ψ τ τ+= ∈ ∞ + >∫ ∫

i iM L d d

We set also

{ }

31

0 0(0, ) ,+ += × ∞ × ⊂

M O L M X

0 0 0\+∂ =M X M

From this Volterra integral formulation one obtains the following lemma:

Lemma 4.3: The sets M0 and ∂M0 are positively invariant under the semi flow 0{ ( )} ≥tU t . Moreover if 0∈∂x M then

lim ( ) 0→∞

− =FtU t x x

with ( )( )3.0 , . ,0,0µ−= = Λ

T

F Fx s e

Global Stability of the Disease free Equilibrium when R0<1 and SimulationsStability with a Lyapunov like function

Lemma 5.1: Let t € R and a>0 be given. For a globally in time solution s, we get the following inequality:

( , ) ( )≤ Fs t a s a

Proof: At first we want to prove that: ( , ) ( )≤ Fs t a s a , 0∀ >a . We have on characteristics:

( )0

( )00

0( , ) ( )

0

µ λ

µ λ

− +∫

− +∫

− ≥= − − <Λ

t s ds

t s ds

e if a ts t a s a t

if a te (5.6)

As a consequence we obtain for each ∈s , t>0 and a>0 that

( )0

( ) ( )0

( , ) 0( , )

0

µ λ

µ λ

− + +∫

− − +∫

− − ≥+ = ∧ − <

tt s ds

ta t t ds

e s s a t if a ts t s a

e if a t

If t=a then for each ∈s :

0( )

( , ) ( ,0) ( )µ λ− −∫+ = ≤

aa s ds

Fs s a a e s s S a

Let ∈t and a>0 be given. Choose ∈s such that s=t-a. The

above equality re-writes as

( , ) ( )≤ Fs t a s a ,

and the result follows.

We will use the lemma above in the proof of the following theorem.

Theorem 5.2: For the model (1.1)-(1.2), if R0<1, then the disease free equilibrium is globally asymptotically stable in ∂M0.

Proof: Recall that

( )00 0

( )

0 0

( ) ( )

( ) (1 ( )) ,

µ γ τ µ

τ µ γ τ µ

β τ τ

β τ τ

∞ ∞− + −

∞− + −

=∧

+ −

∫ ∫

∫ ∫

i

e

ai

ae

R e d p a e da

e d p a e da

Choose and such that

( )

0

' ( ) ( ) ( ) ( )

(0) ( ) ,µ γ

τ µ γ τ β τ

β∞ − +

Γ = + Γ −Γ = ∫ i

i i i i

si ie s ds

And

( )

0

' ( ) ( ) ( ) ( )

(0) ( ) ,µ γ

τ µ γ τ β τ

β∞ − +

Γ = + Γ −

Γ = ∫ e

e e e e

se ee s ds

Note that

0 0 0(0) ( ) ( ) (0) (1 ( )) ( )

∞ ∞= Γ +Γ −∫ ∫i F e FR p a s a da p a s a da

Then one gets (by density):

0 0 0( ) ( , ) ( ) ( , ) ( ) ( ) ( , )ττ τ τ τ τ τ µ γ τ τ τ

∞ ∞ ∞Γ = − Γ − + Γ∫ ∫ ∫i i i i

d i t d i t d i t ddt

0 0( ) (0) ( ) ( , ) [ ' ( ) ( ) ( )] ( , )λ τ γ τ τ τ

∞ ∞= Γ + Γ − Γ∫ ∫i i i it p a s t a da i t d

0 0 0( ) ( , ) ( ) ( , ) ( ) ( ) ( , )ττ τ τ τ τ τ µ γ τ τ τ

∞ ∞ ∞Γ = − Γ − + Γ∫ ∫ ∫e e e e

d e t d e t d e t ddt

0 0( ) (0) (1 ( )) ( , ) [ ' ( ) ( ) ( )] ( , )λ τ µ γ τ τ τ

∞ ∞= Γ − + Γ − + Γ∫ ∫e e e et p a s t a da e t d

so that

0[ ( ) ( , ) ( ) ( , )]τ τ τ τ τ

∞Γ +Γ∫ i e

d i t e t ddt

Citation: Kouakep TY, Ducroty A, Houpa DDE (2013) A Model for Hepatitis B with Chronological and Infection Ages. J Appl Computat Math 3: 148. doi:10.4172/2168-9679.1000148

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Volume 3 • Issue 1 • 1000148J Appl Computat MathISSN: 2168-9679 JACM, an open access journal

0 0

0

(0) ( ) ( , ) (0) (1 ( )) ( )

( ) ( , ) ( ) ( , )β τ τ β τ τ τ

∞ ∞

∞

= Γ +Γ − × +

∫ ∫

∫

i e F

i e

p a s t a da p a s a da

i t e t d

0( ) ( , ) ( ) ( , )β τ τ β τ τ τ

∞ − + ∫ i ei t e t d

0[ ( ) ( , ) ( ) ( , ) ]τ τ τ τ τ

∞Γ +Γ∫ i e

d i t e t ddt

so that

0

[ ( ) ( , ) ( ) ( , ) ]τ τ τ τ τ∞Γ +Γ∫ i e

d i t e t ddt

0 0(0) ( ) ( , ) (0) (1 ( )) ( )

∞ ∞ ≤ Γ +Γ − × ∫ ∫i e Fp a s t a da p a s a da

0( ) ( , ) ( ) ( , )β τ τ β τ τ τ

∞ + ∫ i ei t e t d

Since 0 1≤R then,

0[ ( ) ( , ) ( ) ( , ) ]τ τ τ τ τ

∞Γ +Γ∫ i e

d i t e t ddt

0 0(0) ( ) ( , ) (0) (1 ( )) ( )

∞ ∞ ≤ Γ +Γ − × ∫ ∫i e Fp a s t a da p a s a da

0( ) ( , ) ( ) ( , )β τ τ β τ τ τ

∞ + ∫ i ei t e t d

0 0(0) ( ) ( , ) ( ) (0) (1 ( ))[ ( , ) ( )]

∞ ∞ ≤ Γ − +Γ − − ∫ ∫i F e Fp a s t a s a da p a s t a s a da

0( ) ( , ) ( ) ( , )β τ τ β τ τ τ

∞ × + ∫ i ei t e t d

On the attractor A we check by lemma 5.1 that

( , ) ( ), , 0,≤ ∀ ∈ >Fs t a s a t aso that the functional

0[ , ]( ) : [ ( ) ( , ) ( ) ( , )] ,τ τ τ τ τ

∞= Γ +Γ∫ i eV i e t i t e t d

is non-increasing along the complete orbits.

V [i,e] is a strict Lyapunov function for DFE on 0⊂ ∂A M and global stability of DFE when R0<1 follows.

Simulations

We first simplify the model by assuming that βe and βi are both constant parameters. Then introducing

0( ) ( , )τ τ

∞= ∫I t i t d and

0( ) ( , )τ τ

∞= ∫E t e t d .

We will use data in Tables 1-3 for the case of Cameroon. DiscussionSimulations illustrate the asymptotic stability of DFE in section 5.

The model described by equations (1.1-1.2) exhibit a rich dynamic. We observe that the biological basic reproduction rate R0 is fundamental for the study of the basic dynamical properties. Applied to hepatitis B, the model suggests that infection rates play a great role in the description of the disease (see expression of R0). Simulations conducted follow our results and suggest the fact that the endemic equilibrium is asymptotically stable if R0>1 (Figures 1-9).

Age prevalence p prevalence q Ref.0 to 1 month (excluded) 0.1 or 10% 0.9 or 90% [11,5]1 to 6 months (included) 0.2 or 20% 0.8 or 80% [5]7 to 12 months (included) 0.45 or 45% 0.55 or 55% [5]1 to 5 years (included) 0.5-0.25 or 50-25% 0.25-0.5 or 25-50% [11,5]>5 years 0.94-0.9 or 94-90% 0.06-0.1 or 6-10% [11,5]

Table 1: "Data-(C)", some other data collected on Cameroon.

Age (years) p(a) βI ΒE VI VE μ[0; A=60 yrs] 0.811.e(-288*a) 10-6 10-20 0.036 0.02 15.10-5

Table 2: Values for R0<1.

Age (years) p(a) βI ΒE VI VE μ[0; A=60 yrs] 0.811.e(-288*a) 0.00006 10-20 0.09 0.05 15.10-5

Table 3: Values for R>1.

Figure 1: Construct function q(a)=0.811*e(-0.288*a) from data coming from "Data-(C)" in Table 1.

Figure 2: Function 0

( , )→ ∫A

t s t a da from Data-"C", R0<1.

Citation: Kouakep TY, Ducroty A, Houpa DDE (2013) A Model for Hepatitis B with Chronological and Infection Ages. J Appl Computat Math 3: 148. doi:10.4172/2168-9679.1000148

Page 6 of 7

Volume 3 • Issue 1 • 1000148J Appl Computat MathISSN: 2168-9679 JACM, an open access journal

Figure 3: Function E(t) from Data-"C", R0<1.Figure 6: Function E(t) from Data-"C", R0>1.

Figure 7: Function I(t) from data of Data-"C", R0>1.

Figure 8: Prevalence of asymptomatic carriers E(t) from Data-"C", R0>1.

Figure 4: Function I(t) from data of Data-"C", R0<1.

Figure 5: Function 0

( , )→ ∫A

t s t a da from Data-"C", R0>1.

Citation: Kouakep TY, Ducroty A, Houpa DDE (2013) A Model for Hepatitis B with Chronological and Infection Ages. J Appl Computat Math 3: 148. doi:10.4172/2168-9679.1000148

Page 7 of 7

Volume 3 • Issue 1 • 1000148J Appl Computat MathISSN: 2168-9679 JACM, an open access journal

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