GLOBAL ASYMPTOTIC STABILITY OF A GENERALIZED SIRS EPIDEMIC MODEL WITH TRANSFER FROM INFECTIOUS TO SUSCEPTIBLE∗

In this paper, we propose a generalized SIRS epidemic model with varying total population size caused by the death rate due to the disease and transfer from infectious to susceptible, where the incidence rate employed includs a wide range of monotonic and concave incidence rates. Applying the geometric approach developed by Smith, Li and Muldowey, we prove that the endemic equilibrium is globally asymptotically stable provided that the rate of loss of immuity δ is in a critical interval [η, δ̄) when the basic reproduction number R0 is greater than unity.


Introduction
Mathematical models of infectious diseases have been proven to be very important in better understanding of epidemiological patterns and disease control in human populations.The classical SIR model is the Kermack and McKendrick model, which the total population is divided into three compartmentsSusceptible(S), Infected(I), and Recovery(R).In most communicable diseases such as cholera, pertussis, influenza and malaria, it has been observed that recovered individuals can return to the susceptible period (temporary immunity).This observation has been described by SIRS and SEIRS models (see [9-11, 13, 15, 19-21] and the references therein).
Many researchers were devoted to generalizing the nonlinear incidence rates or considering different factors to probe into the preserved dynamics in epidemic models.By direct Lyapunov method, Korobeinikov [9] studied the global stability for the SIR and SIRS models with the more general incidence rates f (S, I).Moreover, the author observed that the same Lyapunov function could be applied to examine the global property for the SIR with the varying total population size caused by the death rate due to the disease.Similar conclusions for an SEIR model were reached in work [10].Most works (see [9,10,20]) assumed that the incidence rates f (S, I) are monotonic and concave with respect to I to achieve threshold dynamics of epidemic models.Recently, Li etc [13] proposed an SIRS model with the incidence Sf (I) (also see [11]) and transfer from infectious to susceptible.Tang etc [19] studied the SIRS model with the incidence βf (S)g(I) and vaccination in susceptible.Singh etc [17] developed an effective SEIRS model by taking into consideration the human immunity.Chen etc [2] considered the stability and attraction for the endemic equilibrium of an SEIQ epidemic model with transfer from infectious to susceptible.
Motivated by the above discussions, we propose the following SIRS epidemic model with varying total population size and transfer from infectious to susceptible with the initial conditions where S is the number of susceptible individuals, I is the number of infectious individuals, R is the number of recovered individuals, Λ is the recruitment rate of susceptible individuals (we assume that all recruitment are susceptible), µ is the natural death rate, γ 1 is the transfer rate from the infected class to the susceptible class, γ 2 is the transfer rate from the infected class to the recovered class, α is the disease-induced death rate, δ is the immunity loss rate.Asumme that all the parameters Λ, µ, α, γ 1 , γ 2 and δ are positive constants with Λ > µ.
In our proposition, f (S, I) is a continuously differentiable function on R 2 + satisfying f (0, I) = f (S, 0) = 0 for S, I ≥ 0 and the following hypotheses [5]: (H 1 ) f is a strictly monotonically increasing function of S ≥ 0, for any fixed I > 0, and f is a strictly monotonically increasing function of I > 0, for any fixed S ≥ 0; (H 2 ) Φ(S, I) = f (S,I) I is a bounded and monotonically decreasing function of I > 0, for any fixed S ≥ 0, and ϕ(S) = lim I→0 + Φ(S, I) is continuous on S ≥ 0.
It is easy to see that the function Φ(S, I) = f (S,I) I is a monotonically decreasing function of I > 0 for any fixed S ≥ 0 iff f (S,I) I − ∂f (S,I) ∂I ≥ 0. As for the function f (S, I), in some cases where is monotonically increases with respect to S, I and concave with respect to I (i.e., ∂ 2 f (S,I) ∂I 2 ≤ 0), the hypotheses (H 1 )-(H 2 ) naturally hold.Therefore, the function f (S, I) employed in this paper includes the following forms: the bilinear incidence βSI introduced in [2,17], the standard incidence βSI S+I+R advanced in [15,21] and the saturated incidence βSI 1+α1S+α2I proposed in [8] where α 1 , α 2 are positive constants.
The global stability of equilibria for SIRS models or other differential equations has been extensively studied by applying Lyapunov functions(see [1, 9-11, 13, 19, 21] and the references therein).In this paper, we investigate the global stability of the endemic equilibrium only for a reasonably small δ applying the geometric approach developed by Smith [18], Li and Muldowney [12].We will prove that the endemic equilibrium is globally asymptotically stable provided that the rate of loss of immuity δ is less than a critical value δ when the basic reproduction number R 0 is greater than unity.
The rest of the paper is organized as follows.In "Basic properties of the generalized SIRS Model" section, the basic properties of the generalized SIRS model (1.1) are investigated.We calculate the basic reproduction number and give the existence and uniqueness of endemic equilibrium.Furthermore, the local stability of equilibria and the uniform persistence of model (1.1) are carefully addressed.In "Global asymptotic stability of the endemic equilibrium" section, the globle asymptotic stability of the endemic equilibrium is discussed.A brief discussion is presented to conclude this paper.

Basic properties of the generalized SIRS model
In this section, we state the basic proprieties of model (1.1) with disease-induce mortality.The varying total population size N satisfies the equation N = S +I +R.From model (1.1), we have which implies that the biologically feasible region for model (1.1) is bounded and positively invariant.Obviously, every solution of model (1.1) with the initial condition (1.2) exists globally and is nonnegative.
In the following, we calculate the basic reproduction number of model (1.1) using the method of the next generation matrix developed by van den Driessche and Watmough [4].
Set X = (R, I) T .Then it follows from model (1.1) , where The Jacobian matrices of F 1 (X) and V 1 (X) at the point (0, 0) with S = Λ µ are, respectively, thus the next generation matrix is We obtain the basic reproduction number of model (1.1) as follows: The Jacobian matrices of F 2 (X) and V 2 (X) at the point (0, 0) with S = Λ µ are, respectively, thus the next generation matrix is We obtain the basic reproduction number of model (1.1) as follows: We note that the basic reproduction number In this paper, we use the basic reproduction number R 0 .It is easy to verify that model (1.1) always has a disease-free equilibrium E 0 ( Λ µ , 0, 0).Next, we show the existence and uniqueness of the endemic equilibrium.Proof.If (S, I, R) is an equilibrium of model (1.1), then we have the following system We consider the following function By (H 1 ) and (H 2 ), we know that h is a continuous and strictly monotone decreasing function of I > 0, and for R 0 > 1, and Hence, if R 0 > 1, there exists a unique endemic equilibrium E * (S * , I * , R * ).
We will obtain the local stability of E 0 and E * by the Routh-Hurwitz criterion.
Theorem 2.2.If R 0 < 1, then the disease-free equilibrium E 0 is locally asymptotically stable; If R 0 > 1, then the disease-free equilibrium E 0 is unstable.
Proof.The Jacobian matrix of model (1.1) at E 0 is The characteristic equation of J 0 is given by where If R 0 < 1, all the roots of above equation have negative real parts which implies that E 0 is locally asymptotically stable.If R 0 > 1, the above equation has a positive root which implies that E 0 is unstable.
Proof.The Jacobian matrix of model (1.1) at E * is where L = ∂f (S * ,I * )
In order to obtain sufficient conditions that E * is globally asymptotically stable for R 0 > 1, we prove the uniform persistence of model (1.1).Let Ω 1 denote the interior of Ω and ∂Ω denote the boundary of Ω.We can prove the following results.
Proof.From Theorem 2.1 we get for R 0 > 1, there exists a unique endemic equilibrium E * .From Theorem 2.2 we know that R 0 > 1 implies that the diseasefree equilibrium E 0 is unstable.By Theorem 4.3 in [6], the instability of E 0 , together with E 0 ∈ ∂Ω, imply the uniform persistence of the state variables of model (1.1).Therefore, there exists a positive constant C such that every solution (S, I, R) of model (1.1) with the initial data (S(0), where C is independent of initial data in Ω 1 .
The uniform persistence, together with boundedness of Ω, are equivalent to the existence of a compact set in the interior of Ω which is absorbing for model (1.1)(see [7]).So, we have Theorem 2.5.If R 0 > 1, then there exists a compact absorbing set K ⊂ Ω 1 .

Global asymptotic stability of the endemic equilibrium
From Theorem 2.1 and Theorem 2.3, we find that for R 0 > 1, the unique endemic equilibrium E * exists and is locally asymptotically stable.In this section we discuss the possible global asymptotical stability of E * using the geometric approach developed by Smith [18], Li and Muldowney [12].We rewrite model (1.1) as the following autonomous dynamical system ẋ = g(x), ( where g : Ω → R 3 + and g ∈ C 1 (Ω), x = (S, I, R) T .From Section 2, we have the following conditions hold: Let A be a matrix-valued function that is C 1 on Ω 1 and set B = A g A −1 + AJ [2] A −1 , where the matrix A g is (a ij (x)) g = (▽a ij (x) • g(x)), and J [2] is the second additive compound matrix of J = ∂g ∂x (see [16]).The Lozinskiȋ measure μ(B) of B with respect to the norm | • | in R 3 is defined as (see [3]) Now, we present an important lemma and prove the main result as follows.
Theorem 3.1.Assume that R 0 > 1.Then there exist η and δ > 0 such that the unique endemic equilibrium E * is globally asymptotically stable in Ω 1 when δ is in the critical interval [η, δ).
Proof.The Jacobian matrix associated with a solution x(t, x 0 ) of (3.1) is given by where m = µ + γ 1 + γ 2 + α.The second additive compound matrix J [2] of J is given by

Discussion
In this paper, we propose a generalized SIRS epidemic model.The incidence function f (S, I) employed in this paper can be applied generally for a wide class of incidence functions such as the bilinear incidence, the standard incidence and the saturated incidence.We calculate the basic reproduction number R 0 by using the method of the next generation matrix and give some basic properties of the generalized SIRS model.We investigate the global stability of the endemic equilibrium by applying the geometric approach.When R 0 is greater than unity, there exists a unique endemic equilibrium, which is globally asymptotically stable provided that the rate of loss of immuity δ is in a critical interval [η, δ).
and |B 12 |, |B 21 | are matrix norms with respect to the l 1 vector norm and μ1 (B 22 ) is the Lozinskiȋ measure of the matrix B 22 with respect to the l 1 norm in R 2 .