Bifurcation analysis of an age-structured alcoholism model

ABSTRACT In this paper, we investigate a new alcoholism model in which alcoholics have age structure. We rewrite the model as an abstract non-densely defined Cauchy problem and obtain the condition which guarantees the existence of the unique positive steady state. By linearizing the model at steady state and analyzing the associated characteristic transcendental equations, we study the local asymptotic stability of the steady state. Furthermore, by using Hopf bifurcation theorem in Liu et al. (Z. Angew. Math. Phys. 62 (2011) 191–222), we show that Hopf bifurcation occurs at the positive steady state when bifurcating parameter crosses some critical values. Finally, we perform some numerical simulations to illustrate our theoretical results and give a brief conclusion.


Introduction
Globally, alcohol consumption results in approximately 3.3 million deaths each year(or 5.9% of all deaths), this is greater than, for example, the proportion of deaths from HIV/AIDS (2.8%), violence (0.9%) or tuberculosis (1.7%) [28]. Alcohol consumption has been identified as a component cause for more than 200 diseases, injuries and other health conditions as described in the International Statistical Classification of Diseases and Related Health Problems (ICD) 10th Revision (WHO, 1992), more than 30 include alcohol in their name or definition. This indicates that these disease conditions do not exist at all in the absence of alcohol consumption. A strong association exists between alcohol consumption and HIV infection, sexually transmitted diseases [27]. Alcohol consumption can result in harm to other individuals, such as assault, homicide (intentional) or traffic crash, workplace accident (unintentional). Moreover, alcohol consumption results in a significant economic burden on society at large. 5.1% of the global burden of disease and injury is attributable to alcohol [28]. As discussed above, alcohol consumption has a severe effect on the health and wellbeing of individuals and populations.
Recently, it has been realized that mathematical models are important to understand the process of drinking. Mulone et al. [24] studied a two-stage model for youths with serious drinking problems and their treatment. The youths with alcohol problems were divided into two component, namely those who admitted to having the problem and those who did not admit. The stability of two steady states was analysed. Xiang et al. [32] developed a drinking model with public health educational campaigns. Mathematical analyses established that the global asymptotically stability of the equilibria were determined by the basic reproduction number R 0 . If R 0 ≤ 1, the alcohol-free equilibrium was globally asymptotically stable, and if R 0 > 1, the alcohol present equilibrium was globally asymptotically stable, and they concluded that the public health educational campaigns of drinking individuals could slow down the drinking dynamics. Huo et al. [11] introduced a novel alcoholism model which involved the impact of Twitter. Stability of all the equilibria were obtained in terms of the basic reproductive number R 0 . Backward and forward bifurcation, Hopf bifurcation were also analysed. For other alcoholism or epidemoc models, we referred to [2, 12-15, 30, 33, 34].
The age-structured models have been studied by many authors, such as Webb [31], Iannelli [16] and Cushing [7]. Age-structured epidemic models were known to be effective tools for studying epidemic dynamics [3-5, 17, 19, 35] and many authors regarded alcoholism as a social epidemic disease [11,30]. According to the WHO report, alcohol-related deaths were related to age (see Figure 1[1]). Therefore, establishment of alcoholism model should consider age factor and the model becomes more reasonable. However, so far, works on alcoholism models with age structure are very scarce.
In 1990, Thieme [26] observed that age-structured models could be regarded as non-densely defined Cauchy problems, subsequently, Magal and Ruan [22], Liu et al. [18] developed center manifold theory and Hopf bifurcation theorem for non-densely defined Cauchy problems, respectively. By using the above theory and theorem, Liu et al. [20] showed that age-structured model of consumer-resource mutualism exhibited Hopf bifurcation at the positive equilibrium under some conditions. Wang and Liu [29] considered an age-structured compartmental pest-pathogen model. Their results showed that Hopf bifurcation occurred at a positive steady state as bifurcating parameter passed values. Tang and Liu [25] studied a new predator-prey model with age structure and exhibited Hopf bifurcation at a positive steady state.
Motivated by above works, the aim of this paper is to investigate the existence of Hopf bifurcation for an alcoholism model with age-structure. Our model consists of three variables : susceptible drinkers at time t are denoted by S(t), who do not drink or drink only moderately; alcoholics at time t with alcoholism age a are denoted by A(t, a), who strongly desire to consume alcohol, having difficulties in controlling of its use, persisting in its use despite harmful consequences; and the people who recover from alcoholism after treatment are denoted by R(t). Alcoholism as a long-standing social epidemic disease, it is difficult to eliminate for a short time. So we put the alcoholism problem into the population growth model to study. So we suppose new recruits enter the population at a rate r(S(t) + +∞ 0 A(t, a) da + R(t)). The population flow among those classes is shown in the following diagram ( Figure 2).
The flow diagram leads to the following alcoholism model: where alcoholics are assumed to be age-structured, whereas susceptible drinkers and recuperator are not age-structured. r is the birth rate, μ is the natural death rate, a 1 , a 2 are the death rates of excessive drinking, respectively. δ is the transfer rate from alcoholics to recovered individuals, ρ is the relapse rate from recovered individuals to alcoholics. The coefficient α is the fraction of susceptible drinkers S(t) develop into alcoholics because of some of their own reasons, such as losses of earnings, unemployment or family problems, etc. The incidence rate at time t and alcoholism age a is S(t)β(a)A(t, a), where β(a) is the transmission rate due to pressure from alcoholics with alcoholism age a, β(a) is defined by and K 0 := +∞ 0 β(a) e −(μ+a 1 +δ)a da, i.e. β * = (μ + a 1 + δ)K 0 e (μ+a 1 +δ)τ , where τ represents the time span that a person from the initial alcoholism to a potential inviter, who invites susceptible drinkers to increase alcohol consumption. The K 0 represents the total number of secondary alcoholics produced by a primary alcoholic. For the convenience of computation, we assume that K 0 is a constant. The β * is the rate that a alcoholic with alcoholism age a (a ≥ τ ) will successfully infect a susceptible drinker.
The remainder of this paper is organized as follows. Section 2, we summarize the main results on Hopf bifurcation theorem, which obtained in [18]. In Section 3, the stability of the steady state and the existence of Hopf bifurcation are investigated. In Section 4, we perform numerical simulations to verify our analytical results. Finally, a brief conclusion is given.

Preliminaries
We recall the Hopf bifurcation theorem in Liu et al. [18] for the following non-densely defined abstract Cauchy problem: where μ ∈ R is the bifurcation parameter, A : D(A) ⊂ X → X is a linear operator on a Banach space X with D(A) not dense in X and A not necessary to be a Hille-Yosida operator, and A 0 is the part of A in X 0 , which is defined by We denote by {T A (t)} t≥0 the C 0 −semigroup of bounded linear operators on X (respectively {S A (t)} t≥0 the integrated semigroup) generated by A. The essential growth bound ω 0,ess (L) ∈ (−∞, +∞) of L by where L(X) is the space of bounded linear operators from X into X, T L (t) ess is the essential norm of T L (t) defined by T L (t) ess = κ(T L (t)B X (0, 1)), where B X (0, 1) = {x ∈ X : x X ≤ 1}, and each bounded set B ⊂ X, κ(B) = inf{ε > 0 : B can be covered by a finite number of balls of radius ≤ ε} is the Kuratovsky measure of non-compactness.
We make the following assumptions on the linear operator A and the nonlinear map F.
By Proposition 2.6 in Magal and Ruan [22], we know if Assumption 2.1 is satisfied, then A generates a unique integrated semigroup {S A (t)} t≥0 . If we assume in addition that A is a Hille-Yosida operator, then we have Next, we consider the non-homogeneous Cauchy problem where f ∈ L 1 ((0, τ ), X).

Assumption 2.3:
Let ε > 0 and F ∈ C k ((−ε, ε) × B X 0 (0, ε); X) for some k ≥ 4. Assume that the following conditions are satisfied: Base on the above discussion and assumptions, now we can state the following Hopf bifurcation theorem [18].
, which is an integrated solution of (2.3) with the parameter value equals μ(ε) and the initial value equals x ε . So for each t ≥ 0, u ε satisfies Moreover, we have the following properties (a) There exist a neighborhood N of 0 in X 0 and an open interval I in R containing 0, such that forμ ∈ I and any periodic solutionμ in N with minimal periodγ close to 2π/ω(0) of

function and we have the Taylor expansion
where ω(0) is the imaginary part of λ(0) defined in Assumption 2.3.

Stability of equilibria and existence of Hopf bifurcation
In this section, we will study stability of equilibria and existence of Hopf bifurcation for (1).

The transformation of the Cauchy problem
In system (1), let we can rewrite system (1) as the following age-structured model where Consider the Banach space (X, · ) Define the linear operator L : D(L) → X by we notice L is non-densely defined since Define the nonlinear operator F : Now we can rewrite PDEs system (5) as the following non-densely defined abstract Cauchy problem , The global existence and uniqueness of solutions of system (6) follow from the results of Magal [21] and Magal and Ruan [23].

Existence of equilibria
Hence we obtain It is easy to see the system (7) has always trivial equilibriumx 1 holds, system (7) has a unique positive equilibrium
The linearized system of system (9) at the equilibrium 0 is as follows where A = L + DF(w). Then system (9) can be written as By applying the results of Liu, Magal and Ruan [18], we obtain the following result.

Lemma 3.2:
For λ ∈ , λ ∈ ρ(L), we have the following formula It is readily checked that so L is a Hille-Yosida operator and Define the part of L in D(L) by L 0 , and we know Then, we can claim that L 0 is the infinitesimal generator of a C 0 -semigroup {T L 0 (t)} t≥0 on D(L). and for each t ≥ 0 the liear operator T L 0 (t) is defined by Now we estimate the essential growth bound of the C 0 −semigroup generated by A 0 which is the part of A in D(A). We observe that for any 0 ϕ ∈ D(L), Then DF(w) : D(L) ⊂ X → X is a compact bounded linear operator. From (10) we obtain Then we have By applying the perturbation results in Ducrot, Liu and Magal [6], we obtain Thus, by the above discussion and Theorem 3.5.5 in [1], we obtain the following proposition.

Proposition 3.1: The linear operator A is a Hille-Yoside operator, and the essential growth rate of the strongly continuous semigroup generated by A 0 is strictly negative, that is,
In order to apply Theorem 2.1, we remain to precise the spectral properties of A 0 . Setting C := DF(w), and let λ ∈ .
Consider the equation From the formula of DB(x), we know  S(λ,φ)).
From the above discussion and by using the proof of Lemma 3.5 in [29], we obtain the following lemma.

Lemma 3.3:
The following results hold: we have the following formula for the resolvent and M(λ), S(λ, ϕ) defined as above.
From the above discussion, we know that the linear operator A satisfies Assumptions 2.1-2.3(c) holds.

Stability of the trivial equilibrium
Now, we consider the stability of the trivial equilibrium E 1 (a) = (0, 0 L 1 ((0,+∞),R) , 0) T , we obtain Thus, we obtain the characteristic equation It is easy to see that If c 0 < 0 holds, then f 0 (λ) = 0 has at least one root with a real part. Hence, the equilibrium E 1 (a) is unstable. If c 0 > 0 holds, then by the Routh-Hurwitz criterion, all the roots of f 0 (λ) = 0 have negative real parts. Hence, the equilibrium E 1 (a) is stable. . α is small, then the birth rate r may be less than the mortality rate μ, the whole population is easily extinct. So in reality, the E 1 (a) is more likely to be unstable.

Stability of the positive equilibrium and Hopf bifurcation
When (H1) holds, the characteristic equation of system (1) about the positive equilibrium E * (a) can be rewritten as , we know the coefficients a, b, c, e, f , g have nothing to do with τ . It is easy to see that If τ = 0, then Since Separating the real part and the imaginary part in the above equation, we can obtain Thus, we have i.e.

Proof:
Differentiating the equation f (λ) = 0 with respect to λ, we have Separating the real part and the imaginary part in the above equation, we can obtain Now defining According to Equations (12), we know With the help of Equations (18) and (19) we deduce that G (ω 0 ) = 0.
This completes the proof.

Conclusion
In this paper, Hopf bifurcation of an age-structured alcoholism model is investigated. Real epidemic data indicate regular periodic fluctuations in disease incidence [8][9][10] but most models for epidemic diseases predict convergence to a unique stable endemic equilibrium, so it is important to examine under what conditions periodic fluctuations in disease incidence can occur. Because there are people who alcoholism for their own reasons, alcoholism, as a social epidemic disease, is somewhat different from common epidemic diseases. However, in our analysis, we found that regular periodic fluctuations can still arise.
By choosing τ as the bifurcation parameter and analyzing the corresponding characteristic equation, we can conclude that the local asymptotically stability of the trivial equilibrium E 1 (a) is determined by c 0 . If c 0 < 0, then the trivial equilibrium E 1 (a) is unstable for all τ ≥ 0. If c 0 > 0 the trivial equilibrium E 1 (a) is local asymptotically stable for all τ ≥ 0. For the positive equilibrium, if (H1) holds, we establish conditions to ensure the local stability. The parameter τ does not affect the stability of the trivial equilibrium, but can change the stability of the positive equilibrium. By using the center manifold theory [22] and Hopf bifurcation theorem [18], which developed for non-densely defined Cauchy problems, the existence of Hopf bifurcation at the positive equilibrium is obtained. In particular, a non-trivial periodic solution bifurcates from the positive equilibrium when bifurcation parameter τ passes through the critical values τ k (k = 0, 1, 2, . . .). Our analytical results indicate that introduction of parameter τ can affect the dynamic behavior of the system (1).
Moreover, due to alcoholism is often associated with some of the alcoholics' own reasons, such as losses of earnings, unemployment or family problems, etc. In this sense, as long as the population is not extinct (according to the analysis of Remark 3.1, the E 1 (a) is more likely to be unstable), alcoholics will always exist. Therefore, we hope that alcoholattributable socioeconomic burden is minimal. This optimal control problem, we will study in future work.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work is supported by the NNSF of China (11861044 and 11661050), the NSF of Gansu Province (148RJZA024), and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology.