A Model for Hepatitis B with Chronological and Infection Ages

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:

Studies like [9][10][11][12][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:

p a s t a da e t t p a s t a da
Here 0 Λ > is some constant entering in flux, μ>0 is the natural death rate, γ i 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 Finally this model is supplemented together with some initial data 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 R 0 is greater than 1 and study the global asymptotically stability of the disease free equilibrium if 1>R 0 . Finally Section 6 presents discussion.

Mathematical assumptions
We assume that: a.e. and not identically 0 and 1. ), as well as the non-densely defined linear operator : as well as the nonlinear map 0 F : X X → defined by 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 together with the initial data We also consider the positive cones 3 1 3 0 0 (0, ) , is a Hille-Yosida operator. More precisely we have ( ) X and be given. Then the equation rewrites as the following system On the other hand one has 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 X 0+ 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

t x x A U s xds F U s x ds for each t
Moreover we have for each Proof: Let us first notice that for each M>0 there exists λ>0 such that With B X (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 X 0+ 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, ( ,.), ( ,.), ( ,.)) =

T U t x s t i t e t
Consider the quantity the total population at time t. Then it satisfies the differential inequality

Stationary States
The disease free equilibrium The disease free equilibrium corresponds to a stationary (that is time independent solution) . As a consequence we have the following lemma where we have set

Endemic equilibrium
We look for stationary solutions (s; i; e) such that (i; e) not identically zero satisfying We are looking for endemic stationary state, that is λ>0, so that

Dynamical Properties Assumption 4.1
Assume that the maps a→β i (a) and a→β e (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  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
. This means that for each bounded set where δ is defined as Moreover A is locally asymptotically stable. Next one considers the following quantities As a consequence we obtain for each ∈  s , t>0 and a>0 that We will use the lemma above in the proof of the following theorem. Note that

R p a s a da p a s a da
Then one gets (by density):

Simulations
We first simplify the model by assuming that β e and β i are We will use data in Tables 1-3 for the case of Cameroon.

Discussion
Simulations 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 R 0 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 R 0 ). Simulations conducted follow our results and suggest the fact that the endemic equilibrium is asymptotically stable if R 0 >1 (Figures 1-9).

Age
prevalence p prevalence q Ref.