GLOBAL STABILITY OF A MULTI-GROUP MODEL WITH GENERALIZED NONLINEAR INCIDENCE AND VACCINATION AGE

. A multi-group epidemic model with general nonlinear incidence and vaccination age structure has been formulated and studied. Mathematical analysis shows that the global stability of disease-free equilibrium and endemic equilibrium of the model are determined by the basic reproduction number R 0 : the disease-free equilibrium is globally asymptotically stable if R 0 < 1, the endemic equilibrium is globally asymptotically stable if R 0 > 1. The Lyapunov functionals for the global dynamics of the multi-group model are constructed by applying the theory of non-negative matrices and a novel grouping technique in estimating the derivative.


1.
Introduction. Vaccination is one of the commonly used control measures to prevent and reduce the transmission of infectious diseases. The eradication of smallpox, which was last seen in a natural case in 1977, has been considered as the most spectacular success of vaccination. Some vaccines can offer lifelong immunity with only one dose, while others require boosters in order to maintain immunity since the acquired immunity varies with different vaccines and vaccination strategies. For example, the immunization period of hepatitis B vaccine is about five years.
It is natural to consider the vaccination and the waning immunity in modeling a disease dynamics. Li et al. [1] has investigated the global dynamics of an epidemic model with vaccination for newborns and susceptibles. Blower and McLean [2] have argued that a mass vaccination campaign may increase the severity of disease, if the vaccination is applied to only 50% of the population and the vaccine efficacy is 60%. Xiao et al. [3] assumed that the vaccinated individuals can be infected at a reduced rate compare to the susceptibles. Other mathematical models on vaccination have been studied in [4,5,6].
Although waning immunity has been included in several models [1,3,5], it was assumed that the rate of the immunity loss is a constant. A better assumption on the waning immunity is that the protection immunity depends on the vaccination 978 JINHU XU AND YICANG ZHOU age of an individual (the time from the vaccination). The epidemiological models with vaccination age structure can be a suitable choice to describe the dynamics of an infectious diseases with waning immunity after vaccination.
The mathematical models with the chronological age, disease age, and vaccination age have been widely used to describe the impact of the age on the disease evolution [7,8,9,10,11]. Iannelli et al. [9] have studied an epidemic model with vaccination age by assuming the immunity decreases with the time after vaccination. Li et al. [10] have proposed an epidemic model with vaccination age and treatment to show that backward bifurcation occurs due to a piecewise treatment function. Duan et al. [11] has simplified the model [10] by assuming no treatment and have obtained the global stabilities of the disease-free equilibrium and the endemic equilibrium. Motivated by [10,11], we formulate and study a class multi-group epidemic models with the latent class and vaccination age. The total population is classified into five epidemiological compartments, the susceptible compartment, the latent compartment, the infected compartment, the removed compartment and the vaccinated compartment.
Let S(t), E(t), I(t), and R(t) be the numbers of the individuals in the susceptible, latent, infected and removed compartments at time t, respectively. Let v(θ, t) be the vaccination age density of the individuals in the vaccinated compartment at time t, i.e., +∞ 0 v(θ, t)dθ is the number of the individuals in the vaccinated compartment at time t. In our model, it is assumed that the incidence rate is of the form βSf (I), where β is the transmission rate, and f (I) is the disease incidence function satisfying It is easy to see that the incidence rate, βSf (I), includes the bilinear form (βSI) and the saturation form ( βSI 1+mI ). Following the compartment modeling approach, we construct the SVEIR model with vaccination age to describe the disease dynamics.
The vaccinated compartment is structured by the vaccination age θ, and it is assumed that the newly vaccinated individuals enter the vaccinated class v(θ, t) with vaccination age zero. α(θ) is the vaccine wane rate, and it is a nonnegative, bounded and continuous function of θ. For two given vaccination ages θ 1 , θ 2 , (0 ≤ θ 1 ≤ θ 2 ≤ +∞), the number of the vaccinated individuals with age of vaccination θ between θ 1 and θ 2 at time t is θ2 θ1 v(θ, t)dθ. The immunity lose rate (the number of individuals moving from the vaccinated class into the susceptible class due to the waning immunity) at time t is +∞ 0 The parameters Λ, β, µ, ξ and δ are the recruitment rate of the susceptible class, the force of infection per contact per unit time, the mortality of individuals, the rate of vaccination of the susceptible individuals and the rate at which exposed individuals become infectious, respectively. (1 − p)γ and pγ (0 ≤ p < 1) are the per capital recovery rate and per capital death rate due to the disease, respectively. p = 0 implies that there is no disease induced death.
The main purpose of this paper is to study the global dynamics of the basic model (2) and the extended multi-group model (24). Organization of the paper is as follows: Preliminary results for the single group model (2) is presented in Section 2. The global stability of the equilibria for the single group model is proved in Section 3. The global stability of equilibria for the extended multi-group model is established in Section 4. A brief summary is given in the last Section.

2.
Preliminaries. Let us define the state space The initial condition for system (2) can be represented as where Integrating the second equation in (2) along the characteristic line t − θ = constant(see [12]) leads to the following formula where Γ 0 (θ) = e − θ 0 (µ+α(τ ))dτ . In order to investigate the global stability of system (2). We first define the continuous semi-flow associated with the system. Using standard methods, similar to [13,14,15], we can verify the existence and uniqueness of solutions to the system (2). Moreover, we can show that all solutions with nonnegative initial conditions X 0 ∈ X + will remain nonnegative for all t ≥ 0. Thus, we can obtain a continuous semi-flow Φ : R + × X + → X + defined by system (2) such that Thus Therefore, from equations in (2), we know that The comparison principle implies that the following inequality holds
Thus, Φ(t, X 0 ) is point dissipative and Ω attracts all points in X + . Now, we are in a position to show the positive invariance of Ω.
Lemma 2.1. Let Φ and Ω be defined by (7) and (8), respectively. Ω is positive invariant for Φ; that is, As we are now concerned with the infinite dimensional Banach space X including L 1 (0, ∞), the issue one first faces is to verify the relative compactness of the orbit {Φ(t, X 0 ) : t ≥ 0} in X in order to make use of the invariance principle. To this end, we first decompose Φ : R + × X + → X + into the following two operators Θ, Ψ : R + × X + → X + : wherẽ It is easy to see that Φ(t, X 0 ) = Θ(t, X 0 ) + Ψ(t, X 0 ), ∀t ≥ 0, and from Proposition 3.13 in [12] and Lemma 2.1, we arrive at the following lemma.
To show that conditions (i) and (ii) in Lemma 2.2 hold, we first prove the following lemma.
Then, for X 0 ∈ Ω satisfying X 0 X ≤ r, we have which completes the proof.
Proof. It follows from Lemma 2.1 that S(t), E(t), I(t) and R(t) remain in the compact set [0, Λ µ ]. Thus, we are in a position to show thatṽ(θ, t) remains in a precompact subset of L 1 + (0, ∞), which is independent of X 0 ∈ Ω. To this end, it suffices to verify the following conditions (see e.g., Theorem B.2 in [16]).
In summary, from Lemmas 2.2-2.4, we have proved the following result on the relative compactness of the orbit {Φ(t, X 0 ) : t ≥ 0}.
Note that the equation of R(t) in (2) can be dropped due to R(t) does not appear explicitly in the first four equations. Thus, the dynamical behavior of model (2) is equivalent to that of the following model For the reduced system (17), from [14], we consider the weighted space of continuous functions which is a Banach space endowed with the norm ||φ|| η := sup θ≤0 e ηθ ||φ(θ)|| < +∞, According to the similar arguments as Lemma 2.1 in [17], it is not difficult to prove that if (S(t), E(t), I(t)) be the unique solution to model (17) with initial condition defined above and that S(t), E(t), I(t) ar positive for all t ≥ 0. Since the right hand side of system (17) is continuous differentiable and the system is positive invariant. Thus, the compactness of the orbit of system (17) holds. Furthermore, we can obtain the invariant region of the reduced system (17) In the rest of this Section we will focus on the dynamics of the reduced system (17). In order to make the calculation simpler, we use the notations Γ 0 and Γ, where The biological interpretation of Γ 0 is the average time that an individual spends in the vaccinated compartment. Death and immunity loss are two ways for an individual to leave the vaccinated compartment. µΓ 0 or 1 − µΓ is the proportion that an individual moves out the vaccinated compartment because of death or the immunity loss, respectively. ξ(1 − µΓ 0 ) is the per capita rate at which individuals enter the susceptible compartment from the vaccinated compartment due to the immunity loss.
Obviously, model (17) always has a disease-free equilibrium P 0 = Λ µ(1+ξΓ0) , 0, 0 . Any positive equilibrium (S * , E * , I * ) of model (17) satisfies the system of equations The further calculation yields where, I * is the positive root of the following equation The derivative of g(I) is From the conditions in (1) we have f (I) − If (I) ≥ 0, which leads to g (I) < 0 for all I > 0. From equation (19) we have .
has the disease-free equilibrium P 0 and an endemic equilibrium P * if R 0 > 1.
3. Stability analysis of equilibria. In this section, we investigate the stability of the disease-free equilibrium P 0 and the endemic equilibrium P * of model (17).
Proof. The characteristic equation of the linearization of system (17) at P 0 is where It is easy to see that H(λ) = 0 has at least one positive root if R 0 > 1, and the disease-free equilibrium P 0 of model (17) is unstable. If R 0 < 1, then any root of H(λ) = 0 has a negative real parts, and the disease-free equilibrium P 0 of model (17) is locally stable provided that all the roots of equation (23) have negative real parts, where If λ is a root of (23) with Reλ ≥ 0, then we have which leads to a contradiction. We can conclude that all the roots of characteristic equation (21) have negative real parts, and the disease-free equilibrium P 0 of model (17) is stable.
Theorem 3.2. If R 0 < 1, then the disease-free equilibrium P 0 of model (17) is globally asymptotically stable with non-negative initial data.
Proof. Let (S(t), E(t), I(t)) be any solution of system (17) with non-negative initial data. We consider Since the function h(x) = x − 1 − ln x (x ∈ R + ), has the global minimum at x = 1 and h(1) = 0, we know that and L 1 (t) is nonnegative defined with respect to the disease-free equilibrium P 0 , which is a global minimum. Let By using the following equality we can get the the derivative of L along the solution of model (17) dθ.
When R 0 > 1 we can obtain the global stability of the endemic equilibrium. The result is given in the following theorem. Proof. Let (S(t), E(t), I(t)) be any solution of system (17) with non-negative initial data. Define the following Lyapunov functional By using the property of function h(x) = x − 1 − ln x (x ∈ R + ), we find that the function L 2 (t) is nonnegative with its global minimum at P 0 . The similar argument as in the proof of Theorem 3.2 gives the the derivative of L 2 along the solution of model (17) The equality Λ = βS * f (I * ) + (µ + ξ)S * − ∞ 0 Γ(θ)S * dθ leads to From the equations we have Since the arithmetic mean is greater than or equal to the geometric mean, we have Those two inequalities lead to with the equality holding true if and only if I = I * .
From the basic calculus we know that 1 + ln x − x ≤ 0 for x > 0, which leads we obtain that the equilibrium P * of system (17) is globally asymptotically stable. This completes the proof of Theorem 3.3.

4.
Multi-group model. One essential assumption in classical compartmental epidemic models is that the individuals are homogeneously mixed, and each individual has the same chance to get infected. More realistic models divide the host population into groups to consider the disease transmission in heterogeneous cases. The host population are classified into different groups according to their education levels, ethnic backgrounds, gender, age, professions, communities or geographic distributions for their diversities in disease transmission. The vital epidemic parameters varies among different population groups. For more information on multi-group epidemic models one can refer [5,18,19,20,21,22,23,24,25] and reference therein.
In this section, we formulate a multi-group epidemic model and study its dynamics. We assume that the disease can transmit within the same group and among different groups. The multi-group model has the following form, In model (24), S k , E k , I k , V k and R k (k = 1, 2, ..., n) denote the numbers of susceptible, latent, infectious, vaccinated and recovered individuals at time t in the k-th group, respectively. v k (θ, t) is the age density of vaccinated individuals at time t in the k-th group. The non-negative constant β kj is the transmission rate due to the contact of susceptible individuals in the k-th group with infectious individuals in the j-th group. The other non-negative constant parameters have the same meaning as those in model (2). The function f k (I k ) satisfies Similar to Section 2, we integrating the second equation in (24) along the char- where Γ 0k (θ) = e − θ 0 (µ k +α k (τ ))dτ . Then, similar to Section 2, we have where Γ k (θ) = ξ k α k (θ)Γ 0k (θ). We replace the first equation in (24) by (27) and drop the R k (t) equation since the variable R k (t) does not appear explicitly in the first four equations in (24) . The qualitative behavior of model (24) is equivalent to the following model Once the solution of model (28) is determined, we can obtain v k (θ, t) from (26) . The stability of the equilibrium of model (24) is the same as that of model (28) . We focus on the dynamical analysis of the reduced model (28). For the reduced system (28), we consider the weighted space of continuous functions C k = {φ ∈ C((−∞, 0], R) : sup θ≤0 e η k θ ||φ(θ)|| < +∞} which is a Banach space endowed with the norm ||φ|| k := sup θ≤0 e η k θ ||φ(θ)|| < +∞, and let φ t ∈ C k be such that φ t (s) = φ(t + s), s ∈ (−∞, 0] and η k is a positive constant. Thus, consider the reduce system (28) in the phase space The relationship between Γ k and Γ 0k is Γ k = ∞ 0 Γ k (θ)dθ = ξ k (1−µ k Γ 0k ). The biological interpretation of those quantities is the same as that given in Section 3. P 0 = (S 0 1 , 0, 0, ..., S 0 n , 0, 0) is the disease-free equilibrium of model (28) , where 1, 2, ..., n). We can also define the basic reproduction number Here R 0 is the spectrum radius of the matrix FV −1 . From the biological interpretation of the basic reproduction number we have the following result.
The local stability of the disease-free equilibrium of model (28) comes from the relationship between the eigenvalues of the linearized matrix and R 0 . The detailed process is omitted here. Theorem 4.2. Assume that B = (β kj ) n×n is irreducible. If R 0 < 1, then the disease-free equilibrium P 0 of system (28) is globally asymptotically stable. If R 0 > 1, then system (28) is uniformly persistent.
It follows that dL 1 dt > 0 in a small enough neighborhood of P 0 . This implies that P 0 is unstable. With a uniform persistence result from [28] and a similar argument as in the proof of Proposition 3.3 of [29] we can show that the instability of P 0 of the system (28) implies the uniform persistence of system (28) when R 0 > 1. This completes the proof of Theorem 4.2.
The uniform persistence of system (28), together with the uniform boundedness of solutions, implies the existence of an endemic equilibrium (see Theorem 2.8.6 in [30] or Theorem D.3 in [31]). Let P * = (S * 1 , E * 1 , I * 1 , ..., S * n , E * n , I * n ) be the endemic equilibrium of model (28). S * k , E * k , and I * k are positive and satisfy equations We can prove that the endemic equilibrium P * is globally asymptotically stable if it exists. The method is based on the graph-theoretical approach and Lyapunov functionals by Guo et al. [22,23] and Li and Shuai [24].
Theorem 4.3. Assume that B = (β kj ) n×n is irreducible. If R 0 > 1, then the endemic equilibrium P * of (28) is globally asymptotically stable and thus is the unique endemic equilibrium.
Proof. For convenience of notations, define B is also irreducible. By Lemma 2.1 in [22], the solution space of the linear system Bv = 0 has dimension 1 with a base (v 1 , ..., v n ) = (c 11 , ..., c nn ), where, c kk > 0 is the co-factor of the k-th diagonal entry of B. Let (S k (t), E k (t), I k (t)) be any solution of system (28) with non-negative initial data. We construct the following Lyapunov function: Computing the derivative of L 2 along the solution of model (28), we obtain that Using the equilibrium equations (29), we have .
From the properties of function f k (I k ), it is easy to obtain the following inequalities Those two inequalities imply that with equality holding if and only if I k = I * k . From S k (t − θ) > 0 and S k (t) > 0 it follows that 1 + ln with equality holding if and only if S k (t−θ) = S k (t). It is obvious that 0 for all positive S * and S k , with equality holding if and only if S * k = S k (t). The equality Bv = 0 yields From those expressions we have and the equality holds if and only if A direct result of the inequality between the arithmetical mean and the geometrical mean gives that H 1 ≤ 0, and H 1 = 0 if and only if S 1 = S * 1 , E 1 = E * 1 , I 1 = I * 1 .

JINHU XU AND YICANG ZHOU
We are going to show that H n ≤ 0 for all n ≥ 2. Let G denote the directed graph associated with matrix (β kj ). G has vertices 1, 2, .., n with a directed arc (k, j) from k to j if and only if β kj = 0. E(G) denotes the set of all directed arcs of G. Using Kirchhoff's Matrix-tree Theorem in graph theory, we know that v k = c kk can be interpreted as a sum of weights of all directed spanning subtrees T of G that are rooted at vertex k. Consequently, each term in v k β kj is the weight of a unicyclic subgraph Q of G, obtained from such a tree T by adding a directed arc (k, j) from vertex k to vertex j. Note that the arc (k, j) is part of the unique cycle CQ of Q, and that the same unicyclic graph Q can be formed when each arc of CQ is added to a corresponding rooted tree T . Therefore, the double sum in H n can be reorganized as a sum over all unicyclic subgraphs Q containing vertices {1, 2, .., n}. LaSalle's invariance principle [27], the unique endemic equilibrium P * of system (28) is globally asymptotically stable if R 0 > 1. This completes the proof of Theorem 4.3.

5.
Conclusions. One group and multi-group SVEIR epidemic models with general nonlinear incidence rate are proposed to describe heterogeneities in disease transmission. The global stability of those two models is established by using Lyapunov functions. The dynamical behavior is completely determined by the magnitude of the the basic reproduction number R 0 .
We define R 0 = ρ β kj S 0 k f j (0) µ k +γ k n×n = lim δ k →∞ R 0 to investigate the influence of the latent period on R 0 . It is obvious that R 0 > R 0 and ∂R0 ∂δ k > 0, which shows that latent period has a positive role in disease control: a long latent period may lead to the extinction of the disease. Similarly, the fact that R * 0 = ρ Λ k δ k β kj f j (0) µ k (µ k +δ k )(µ k +γ k ) n×n = R 0 | ξ k =0 > R 0 and ∂R0 ∂ξ k = ρ −Λ k δ k Γ 0k β kj f j (0) µ k (µ k +δ k )(µ k +γ k )(1+ξ k Γ 0k ) 2 n×n < 0 imply that the vaccination is helpful to eradicate the disease. If R * 0 > 1, then there exists a unique Θ such that R 0 < 1 for ξ k > Θ since ∂R0 ∂ξ k < 0 and lim ξ k →∞ R 0 = 0. The immunity of a vaccine may not be permanent, a long immunity period of vaccines is still expected for diseases prevention. Our model with vaccination age may help to track the period of vaccination and the immunity wane. The vaccination age is a prominent feature of our model, and the threshold result is the novel result of our paper. Of course, other factors, such as the discrete or continuous distributed delay for the latency, the population migration, can be integrated into the model to make it more realistic. Moreover, much attention should be paid to improve the results in this paper and make the results more completed.