Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse.

In this paper, an SEIR epidemic model for an imperfect treatment disease with age-dependent latency and relapse is proposed. The model is well-suited to model tuberculosis. The basic reproduction number R0 is calculated. We obtain the global behavior of the model in terms of R0. If R0< 1, the disease-free equilibrium is globally asymptotically stable, whereas if R0>1, a Lyapunov functional is used to show that the endemic equilibrium is globally stable amongst solutions for which the disease is present.


1.
Introduction. Mathematical modeling is a very important tool in analyzing the propagation and controlling of infectious diseases. Age structure is an important characteristic in the modeling of some infectious diseases. The first formulation of a partial differential equation(PDE) for the age distribution of a population was due to McKendrick [21]. Since the seminar papers by Kermack and McKendrick [13]- [15], age structure models have been used extensively to study the transmission dynamics of infectious diseases, we refer to the monographs by Hoppensteadt [11], Iannalli [12] and Webb [30] on this topic.
As an ancient disease, TB peaked and declined by 1940's before it became curable, while the downtrend stopped in the middle 1980's and 1990's. As one of the top 3 deadly infectious diseases, TB would cause a higher death rate if not treated, while the disease would be latent in an individual body for months, years or even decades before it outbreaks. McCluskey [20] pointed out that the risk of activation can be modeled as a function of duration age, and this form can be used to describe more general latent period via introducing the duration age in the latent class as a variable.
On the other hand, for the infectious tuberculosis, the removed individuals often have a higher relapse rate. Actually, the recurrence as an important feature of some animal and human diseases has been studied extensively, see [4], [23]. For instance, van den Driessche and coauthors in [4], [5] established two models with a constant relapse period and a general relapse distribution respectively, which showed the threshold property of the basic reproduction number. It is interest to investigate the model with age-dependent relapse rate and to determine whether the threshold property can be preserved or not.
Recently, Wang et al. [24]- [27] considered the global stability of nonlinear agestructured models, Liu et al. [17] introduced age-dependent latency and relapse into an SEIR epidemic model and the local stability and global stability of equilibria are obtained by analyzing the corresponding characteristic equations and constructing the proper Volterra-type Lyapunov functionals, respectively. Wang et al. [28] proposed an SV EIR epidemic model with media impact, age-dependent vaccination and latency, and discussed the global dynamics of the age-structure model.
However, most of the models assumed that TB would show neither its clinical symptoms nor its infectivity during its latent period, while in fact, TB has many early clinical symptoms such as fever, fatigability, night sweat, chest pain, hemoptysis and so on. Here we formulate and analyze an SEIR epidemic model with continuous age dependent latency and relapse. We assumed, as the development of the disease, TB is infectious during its latent period with less infectivity and incomplete treatment comparing with outbreak period. Although epidemic models with age-dependent have been studied extensively, there have been still inadequate results on the full global stability. In this paper, we employ the method developed by Webb [30] for age-dependent models, namely integrating solutions along the characteristics to obtain an equivalent integral equation. We obtain the basic reproduction number in virtue of the method in [7]. Moreover, we study the asymptotic smoothness of the semi-flow generated by the system and the existence of a global attractor [3], [19]. Finally, we show the global stability of equilibria via constructing the proper Volterra-type Lyapunov functionals. For more details concerning the current Lyapunov functionals approach, we refer the reader to recent work [2]. This paper will be organized as follows: In Section 2 , we formulate our general SEIR tuberculosis model with latent age and relapse age which is described by a coupled system of ODEs and PDEs. In Section 3, we investigate the existence of equilibria and obtain the expression of the basic reproduction number R 0 . In Section 4, the local asymptotic stability of the equilibria will be derived. In Section 5, we present the results about uniform persistence. In section 6, we deal with the global stability of equilibria. Finally, some numerical simulations and useful discussions are made in the last section.
2. Model formulation. The total population is decomposed into four disjoint subclasses, susceptible class S, latent class E, infectious class I, and removed class R. More precisely, let S(t) denote the number of susceptible individuals at time t. Susceptible individuals would become new infected ones after they contact with infectious individuals at a rate β, while they enter a stage when they are infected with the disease but have little infectivity. This stage is often called latent stage, which maybe enter into the stage of removed class R by receiving treatment at a rate µ. The density of individuals in the latent class is denoted by e(t, a) where t is the duration time spenting in this class and a is called the latent-stage progression age, denoting E(t) = +∞ 0 e(t, a)da the total density of latent individuals. The number of individuals in the class I at time t is I(t). The removal rate from latent class E to infectious class I is given by the function σ(a). Thus, the total rate at which individuals progress into the infectious class alive is +∞ 0 σ(a)e(t, a)da. Infectious individuals come into the removed class after recovery due to complete treatment. Let r 1 be the recovery rate from the infectious class. The density of individuals in removed class is denoted by r(t, c) , where c represents the relapse age, denoting R(t) = ∞ 0 r(t, c)dc the total density of removed individuals. In fact, infectious individuals might come into the latent class E due to incomplete treatment at the rate r 2 . Due to the relapse of the disease, the age-dependent relapse rate in the removed class is given by the function k(c). The total rate at which individuals relapse into the infectious class alive is given by the quantity +∞ 0 k(c)r(t, c)dc. We also denote Λ , δ e , δ i as the density of the recruitment into the susceptible class, the additional death rates induced by the latent disease and infectious disease. The parameter b is the natural death rate of all individuals. All recruitment of the population enters the susceptible class and occurs with constant flux Λ. Further, all parameters are assumed to be positive. FIGURE 1 shows the schematic flow diagram of our model which can be described by a system of ordinary and partial differential equations with boundary conditions and initial conditions S(0) = S 0 , e(0, a) = e 0 (a), where S 0 , I 0 ∈ R + , and e 0 (a), r 0 (c) ∈ L 1 + (0, ∞) which is the nonnegative and Lebesgue integrable space of functions on [0, +∞).
In order to simplify the later derivation, we make the following hypotheses about the parameters of the system (1) According to Webb [30], by solving the PDE parts of (1) along the characteristic lines t − a = const and t − c = const respectively, we obtain Define the space of functions X as The norm has the biological interpretation of giving the total population size. The initial conditions (3) for system (1) can be rewritten as x 0 = (S 0 , e(0, ·), I 0 , r(0, ·)) ∈ X. Using standard methods we can verify the existence and uniqueness of solutions to model (1) with the boundary conditions (2) and initial conditions (3) (see Iannelli [12] and Webb [30]). Meanwhile, we can claim that any solution of system (1) with nonnegative initial conditions remains nonnegative. The nonnegativity of e(t, a) and r(t, c) follows from (4) and (5). Next, we shall show that S(t) > 0 for t ≥ 0 and I(t) > 0 for t ≥ 0. Otherwise, assume that S(t) would lose its positivity for the first time at t 1 > 0, i.e., S(t 1 ) = 0. However, from the first equation of (1) we obtain Similarly, assume that I(t) would lose its positivity for the first time at t 2 > 0, i.e., I(t 2 ) = 0. However, from the third equation of (1) we obtain r(t, c)dc, which is the total population at time t. We can easily see that the time derivative of N along solutions of model (1) is Similarly, by using Hence, we have It follows from the variation of constants formula that N (t) ≤ Λ b , for any t ≥ 0, which implies that Ω = (S(t), e(t, ·), I(t), r(t, ·)) ∈ R+ × L 1 is positively invariant absorbing set for system (1).
, then the following statements hold for t ≥ 0, For convenience, we rewrite (1) as follows where then we can get Thus (6) can be rewritten as where Let L : D(L) ⊂ X → X be the linear operator defined by Therefore (7) can be rewritten as an abstract Cauchy problem By using the results in Magal [19] and Chen et al. [3], there exists a uniquely determined semiflow {U (t)} t≥0 on X 0+ such that, for each , V (0) , there exists a unique continuous map U ∈ C(R + , X 0+ ), which is an integrated solution of the Cauchy problem (8), that is, for And Ω is the positively invariant absorbing set under the semi-flow U can be verified, is called asymptotically smooth if each forward invariant bounded closed set is attracted by a nonempty compact set [8], [22]. In order to obtain global properties of the dynamics of the semi-flow U (t), it is important to prove the asymptotically smooth of semi-flow U (t). First we give the following useful lemma.
In order to prove Lemma 2.2, we first decompose U : In order to verify condition (i) of Lemma 2.2, we need to prove the following proposition.
If I * = 0, then we have e * (a) = 0 and r * (c) = 0 respectively from (17) and (18). Furthermore, it is easy to know that S 0 = Λ b from the first equation of (15). Thus, system (15) has a disease-free equilibrium E 0 , and In order to find any endemic equilibrium, we introduce the basic reproduction number R 0 , which is the average number of new infections generated by a single newly infectious individual during the full infectious period [7]. It is given by the following expression .

4.
Local asymptotic stability of the equilibria. In this section, sufficient conditions for the local asymptotic stability of the equilibria will be derived. Proof. First, we introduce the change of variables as follows x 1 (t) = S(t) − S 0 , x 2 (t, a) = e(t, a), x 3 (t) = I(t), x 4 (t, c) = r(t, c).
Linearizing the system (1) about disease-free equilibrium E 0 , we obtain the following system Set where x 0 1 , x 0 2 (a), x 0 3 , x 0 4 (c) will be determined. Substituting (22) into (21), we get Integrating the first equation of (24) from 0 to a yields Similarly, we have from (26) that Substituting (27) and (28) into (25) and solving (25) gives which is the characteristic equation. Let Obviously, F (λ) is a continuously differential function and satisfies Thus, we know (29) has a unique real root λ * . Obviously, we have λ * < 0, if R 0 < 1, and λ * > 0, if R 0 > 1. Let λ = x + yi be an arbitrary complex root to (29), then which means that λ * > x. Thus, all the roots of (29) have negative real part if and only if R 0 ≤ 1 and have at least one eigenvalue with positive real part if R 0 > 1. Therefore we have that the disease-free equilibrium E 0 is locally asymptotically stable if R 0 ≤ 1 and unstable if R 0 > 1.
Theorem 4.2. The unique endemic equilibrium E * is locally asymptotically stable if R 0 > 1.
Theorem 5.2. Assume R 0 > 1, the semiflow {U (t)} t≥0 generated by system (1) is uniformly persistent with respect to the pair (∂M 0 , M 0 ), that is there exists ε > 0, such that for each y ∈ M 0 , Furthermore, there exists a compact subset A 0 ⊂ M 0 which is a global attractor for Proof. Since the infection-free equilibrium E 0 = (S 0 , 0, 0, 0, 0 L 1 , 0 L 1 ) is globally asymptotically stable in ∂M 0 , applying Theorem 4.2 in Hale and Waltman [10], we only need to investigate the behavior of the solutions starting in M 0 in some neighborhood of E 0 . Then, we will show that W s ( Since y ∈ W s ({E 0 }), we have S(t) → S 0 , I(t) → 0, as t → ∞. When R 0 > 1, we know that the function Φ(t) is not decreasing for t large enough. Thus there exists t 0 > 0 such that Φ(t) ≥ Φ(t 0 ) for all t ≥ t 0 . Since Φ(t 0 ) > 0, this prevents that the function (I(t), e(t, a), r(t, c)) converges to (0, 0 L 1 , 0 L 1 ) as t → ∞. A contradiction with S(t) → S 0 .
6. Global asymptotic stability of the endemic equilibrium. Let denote g (x) = 1 − 1 x . Thus, g : R + → R + is concave up. Also, the function g has only one extremum which is a global minimum at 1, satisfying g(1) = 0 and ∀x, y ∈ R, g(xy) ≥ g(x) + g(y).
Theorem 6.1. The unique endemic equilibrium E * is globally asymptotically stable if R 0 > 1 .
Since Λ = bS * + βS * I * , then the derivative of W s along with the solutions of (1) is Calculating the derivative of W e along with the solutions of system (1)   k(c)r * (c)(g( I I * ) + g( r(t, c) r * (c) ))dc.
7. Numerical simulations. In the following, we provide some numerical simulations to illustrate the global stability of the disease-free equilibrium and the endemic equilibrium for system (1). We choose parameters Λ = 3; b = 0.065; n = 0.02; α 1 = 0.01; α 2 = 0.03; r 1 = 0.1; r 2 = 0.2; and Under the initial values  Figure 3. The time series of S(t) and I(t), and the age distributions of e(t, a) and r(t, c) when τ = 1.
In Figure 2, we choose τ = 12 , then R 0 < 1, while in Figure 3, we choose τ = 1, then R 0 > 1. The figures show the series of S(t) and I(t) which converge to their equilibrium values, and the age distribution and time series of e(t, a) and r(t, c), respectively.
8. Discussion. In this section, we briefly summarize our results. First, a PDE tuberculosis model (1) is proposed here to incorporate the latent-stage progression age of latent individuals and the relapse age of removed individuals. In addition, we assumed that infectious individuals might come into the latent class E due to incomplete treatment, and the relapse in the removed class. Under our assumptions, the expression of the basic reproduction number R 0 is given, and we proved that if R 0 < 1 the disease-free equilibrium E 0 is globally asymptotically stable, while if R 0 > 1 the unique endemic equilibrium E * is globally asymptotically stable. Figure  2 and Figure 3 further verify our results.