Global Stability of an SEIS Epidemic Model with General Saturation Incidence

We present an SEIS epidemic model with infective force in both latent period and infected period, which has different general saturation incidence rates. It is shown that the global dynamics are completely determined by the basic reproductive number R 0 . If R 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable in T by LaSalle’s Invariance Principle, and the disease dies out. Moreover, using the method of autonomous convergence theorem, we obtain that the unique epidemic equilibrium is globally asymptotically stable in T, and the disease spreads to be endemic.


Introduction
Epidemiology is the study of hot spots of the spread of infectious disease, with the objective to trace factors that contribute to their occurrence.Mathematical epidemiology models describing the population dynamics of infectious diseases have been playing an important role in better understanding of epidemiological patterns and disease control for a long time.Epidemiological models are now widely used as more epidemiologists realize the role that modeling can play in basic understanding and policy development.In recent years, many epidemiological models of ordinary differential equations have been studies by a number of authors [1][2][3][4].
The most general form of an epidemiological model is an SEIRS model consisting of four population subclasses: susceptible, -exposed, -infected, and -recovered.All other models are limiting cases of the SEIRS model under various parameter restrictions.If there is no immunity and hence no R class, the SEIS model is obtained, which can be regarded when the average period of immunity tends to zero.
Many epidemic models with the infectious force in the latent period have been performed.Guihua and Zhen [5,6] studied global stability of an SEI model with general incidence or standard incidence.Mukhopadhyay and Bhattacharyya [7] discussed global stability of an SEIS model with standard incidence.Global dynamics of an SEI model with acute and chronic stages were given by Yuan and Yang [8].
Incidence rate plays a very important role in the research of epidemiological models.Comparing with bilinear and standard incidence rate, saturating incidence rate may be more suitable for our real word, which should generally be written as ()/, where  is the total population size.Michaelis and Menten combined the two previous approaches by assuming that if the number of available partners  is low, the number of actual per capita partners () is proportional to  whereas if the number of available partners is large, there is a saturation effect which makes the number of actual partners constant.Specifically, it has the form (Michaelis-Menten contact rate): Obviously, incidence with above form suggests that the number of new cases per unit time is saturated with the total population.Using a mechanistic argument, Heesterbeek and Metz [9] derived the expression for the saturating contact rate of individual contacts in a population that mixes randomly; that is, Furthermore, () is nondecreasing and ()/ is nonincreasing.

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The above discussion reveals the importance of incidence functions in epidemic models.Different nonlinear forms of incidence can exhibit very dynamics and hence are able to unearth some otherwise unknown features of disease dynamics.Though the aspect of nonlinearity in incidence has found a significant importance in the existing literature, the fact that population subclasses with different infection statuses should have different incidence rates has received little attention among mathematical epidemiologists.Thus in an SEIS epidemic model, since there is a difference in relative measure of infectiousness between the exposed and the infected populations, the incidence rate between the susceptible fraction  and the infected fraction  should be different from that between  with the exposed fraction .
The present analysis aims to explore the impact of this distinct incidence for exposed and infected populations under the influence of spatial heterogeneity.As a model system, We have divided the population in researched area into three classes: -susceptible, -exposed with the infectious force, and -infected.
In the next section, we establish the model discussed in this paper and determine the basic reproductive number.In Section 3, we analyze the global stability of the disease-free equilibrium.In Section 4, we resolve the unique existence and global stability of the epidemic equilibrium.In Section 5, we present some numerical simulation of examples which validate these theoretical results.The paper ends with a brief discussion in Section 6.

The Model and the Basic Reproductive Number
The model, we consider, has the following population subclasses: (i) -the susceptible, (ii) -the exposed, and (iii) -the infected.The total population size, denoted by , is () = () + () + ().The transfer mechanism from the class  to the class  is guided by the function where  1 and  2 are average numbers of adequate contacts of an exposed individual and an infectious individual per unit time, respectively, and   () ( = 1, 2) are relevant saturation contact rate, which satisfy the following assumptions, for  > 0, The assumptions (i) and (ii) are biologically motivated.As the total population  increases, the probability of a contact with a susceptible individual decreases, and thus the force of the exposed or the infected is expected to be a decreasing function of .And the assumption (iii) implies that the contact rate   () is saturated.
The population transfer among compartments is schematically depicted in the transfer diagram in Figure 1.The transfer diagram leads to the following SEIS epidemic model of ordinary differential equations: where Λ is the recruitment rate of the population,  is the natural death rate, and  is the death rate for the infected. individuals move to the class  at the rate  and  individuals recover at the rate , which are assumed to join the susceptible class.The above parameters are positive.Summing up the three equations in system (4), then the time derivative of () along a solution of system (4) is Therefore,   ≤ Λ−, equivalently,   + ≤ Λ. Applying a theorem on differential inequalities [10], we get 0 ≤  ≤ Λ/ for  → +∞.Thus, the three-dimensional simplex is positively invariant with respect to system (4), where R 3 + denotes the nonnegative cone of R 3 including its lower dimensional faces.
By using  = −− and (5), we get the following system: The dynamical behavior of system (4) in  is equivalent to that of system (7).Thus, in the rest of the paper, we will study the system (7) in the feasible region which can be shown to be a positive invariant set for system (7).Now, we derive the basic reproductive number of system (4) by the method of next-generation matrix formulated in [11].
We call the next generation matrix for system (9).According to [11,Theorem 2], the basic reproductive number of system (4), which is the number of secondary infectious cases produced by an exposed individual and an infectious individual during their effective infectious period when introduced in a population of susceptible, is where () denotes the spectral radius of matrix .

Stability Analysis of the Disease-Free Equilibrium
In this section, we discuss the global stability of the diseasefree equilibrium.It is obvious that system (7) always has the unique disease-free equilibrium  0 = (0, 0, Λ/) in .About  0 , we have the following main results.
Proof.The Jacobian matrix of system (7) at  0 = (0, 0, Λ/) goes as follows: which has a eigenvalue  1 = − < 0, obviously.The other two eigenvalues  2 and  3 are determined by the following equation: If  0 > 1, we can have easily Therefore,  2 and  3 are two opposite-sign real roots.Thus,  0 is unstable.

Existence and Stability of the Endemic Equilibrium
In this section, we first discuss the existence and uniqueness of the endemic equilibrium  * of system (7) when  0 > 1.
Whereafter, we focus on investigating the local stability of  * .We have to prove that the Jacobian matrix ( * ) is stable; namely, all its eigenvalues have negative real parts.This is routinely done by verifying the Routh-Hurwitz conditions.Finally, we study the global stability of the endemic equilibrium  * of system (4) with the method of autonomous convergence theorem of Li and Muldowney in [13].
The coordinates of the endemic equilibrium (positive equilibrium) of system (7) are the positive solutions of equations in   .Let  =  +  + , by the direct calculation, we can get the following equation of  easily as Because   () ( = 1, 2) satisfy conditions (i), (ii), and (iii), thus () is an increasing continuous function, and Proof.The Jacobian matrix of system (7) at where thereinto Therefore, the characteristic equation of ( * ) is where By calculation, we have By Routh-Hurwitz stability theorem [10], all the three eigenvalues of ( * ) have negative real parts.Thus, the endemic equilibrium  * is locally asymptotically stable in   , when  0 > 1.
Denote the boundary and the interior of  by  and   , we also obtain for system (4).Now, we briefly outline the autonomous convergence theorem in [13] for proving global stability of the endemic equilibrium  * .
Let  ⊂ R  be an open set, and let  → () ∈ R  be a  1 function defined in .We consider the autonomous system in R  : Let  be an equilibrium of (29); that is, () = 0. We recall that  is said to be globally stable in  if it is locally stable and all trajectories in  converge to .Assume that the following hypothesis hold: (H1)  is simply connected; (H2) there exists a compact absorbing set Γ ⊂ ; (H3)  is the only equilibrium of (29) and is locally stable in .
The basic job is to find conditions under which the global stability of  with respect to  is implied by its local stability.The difficulty associated with this problem is largely due to the lack of practical tools.A new approach to the global stability problem has emerged from a series of papers on higher-dimensional generalizations of the criteria of Bendixson and Dulac for planar systems and on so-called autonomous convergence theorems.First, we now introduce a definition, which will appear in the following context.Definition 6 (see [13]).Suppose system (29) has a periodic solution  = () with least period  > 0 and orbit  = {() : 0 ≤  ≤ }.This orbit is orbitally stable if for each  > 0, there exists a  > 0 such that any solution (), for which the distance of (0) from  is less than , remains at a distance less than  from  for all  ≥ 0. It is asymptotically orbitally stable if the distance of () from  also tends to zero as  → ∞.This orbit  is asymptotically orbitally stable with asymptotic phase if it is asymptotically orbitally stable and there is a  > 0 such that any solution (), for which the distance of (0) from  is less than , satisfies |() − ( − )| → 0 as  → ∞ for some  which may depend on (0).Theorem 7 (see [14]).A sufficient condition for a period orbit  = {() : 0 ≤  ≤ } of (29) is asymptotically orbitally stable with asymptotic phase such that the linear system   () = (  [2]   ( ()))  () (30) is asymptotically stable.
Remark 8. Equation ( 30) is called the second compound equation of (29) and  [2] / is the second compound matrix of the Jacobian matrix / of .
It is also demonstrated that Theorem 7 generalizes a class of Poincare for the orbital stability of periodic solutions to planar autonomous systems.

Example and Numerical Simulation
In this paper, we considered an SEIS model with saturation incidence.Now, we give the number simulations for system (4) (see Figures 2 and 3).

Discussion
In this paper, we present a complete mathematical analysis for the global stability problem at the equilibria of an SEIS epidemic model with saturation incidence.The basic reproductive number  0 is obtained as a sharp threshold parameter, which represents the average number of secondary infections from a single exposed host and infectious host.If  0 ≤ 1, the disease-free equilibrium  0 is globally asymptotically stable in the feasible region  by Lyapunov function, and thus the disease always dies out.If  0 > 1, the unique disease equilibrium  * is globally asymptotically stable in   , so that the disease, if initially present, will persist at the unique endemic equilibrium level.The global stability of  * in model is proved using a geometrical approach in [13].We expect that these approaches can be applied to solve global stability problems in many other epidemic models.

Appendix Compound Matrices
Let  be a linear operator on R  and also denote its matrix representation with respect to the standard basis of R  .Let ∧ 2 R  denote the exterior product of R  . induces canonically a linear operator  [2] on ∧ 2 R  ; for  1 ,  2 ∈ R  , define  [2] ( 1 ∧  2 ) :=  ( 1 ) ∧  2 +  1 ∧  ( 2 ) (A.1) and extend the definition over ∧ 2 R  by linearity.The matrix representation of  [2] with respect to the canonical basis in ∧ 2 R  is called the second additive compound matrix of .This is an (  2 )×(  2 ) matrix and satisfies the property (+) [2] =  [2] + [2] .The entries in  [2] are linear relations of those in .Let  = (  ).For any integer  = 1, 2 . . ., (  2 ), let () = ( 1 ,  2 ) be the th member in the lexicographic ordering integer 10 ISRN Applied Mathematics pairs such that 1 ≤  1 <  2 ≤ .Then, the entry in the th column of  =  [2] is For any integer 1 ≤  ≤ , the th additive compound matrix   of  is defined canonically.For detailed discussions of compound matrices and their properties, we refer the reader to [20].A comprehensive survey on compound matrices and their relations to differential equations is given in [20].For  = 2, 3, and 4, the second additive compound matrix  [2] of an  ×  matrix  = (  ) is, respectively,

Figure 2 :Figure 3 :
Figure 2: Movement paths of , , and  as functions of time .For (a), we have  0 = 0.85 and  0 is globally stable.The disease is extinct.For (b), we have  0 = 1.4 and * is globally stable.The disease spreads to be endemic.