Stability analysis in a delayed SIR epidemic model with a saturated incidence rate

Abstract. We formulate a delayed SIR epidemic model by introducing a la tent period into susceptible, and infectious individuals in incidence rate . This new reformulation provides a reasonable role of incubation period on the dynamics of SIR epidemic model. We show that if the basic reproduction number, denoted, R0, is less than unity, the diseasefree equilibrium is locally asymptotically stable. Moreov er, we prove that ifR0 > 1, the endemic equilibrium is locally asymptotically stable. In the end some numerical simulations are given to compare our model with existing mod el.


Introduction and mathematical models
Epidemic models have been studied by many authors.Most of them are interesting in the formulation of the incidence rate, i.e., the infection rate of susceptible individuals through their contacts with infectious (see, for example, [1][2][3][4][5]).In order to model this disease transmission process several authors employ the following incidence functions: The first one is the bilinear incidence rate βSI, where S and I are respectively the number of susceptible and infective individuals in the population, and β is a positive constant [6][7][8][9][10].The second one is the saturated incidence rate of the form βSI 1+α1S , where α 1 is a positive constant.The effect of saturation factor (refer to α 1 ) stems from epidemic control (tacking appropriate preventive measures) [11][12][13][14].The third one is the saturated incidence rate of the form βSI 1+α2I , where α 2 is a positive constant.In the last one, the number of effective contacts between infective and susceptible individuals may saturate at high infective levels due to crowding of infective individuals or due to the protection measures by the susceptible individuals [7,15,16].
In a recent paper [17], we considered a delayed SIR epidemic model with a saturated incidence rate as follows: where S is the number of susceptible individuals, I is the number of infectious individuals, R is the number of recovered individuals, A is the recruitment rate of the population, µ is the natural death of the population, α is the death rate due to disease, β is the transmission rate, α 1 and α 2 are the parameter that measure the inhibitory effect, γ is the recovery rate of the infective individuals, and τ is the incubation period [14,[16][17][18].
In the SIR model ( 1), the number of the new infective cases produced in the period (t − τ, t] is neglected in the evolution of the susceptible class, and is taken into consideration in the evolution of the infectious class.However, it may be more realistic to tack this period into consideration in the evolution of susceptible class, and not in the evolution of infectious class, because susceptible individuals infected at time t − τ is able to spread the disease at time t.In this paper, incubation period is introduced into the SIR epidemic model (1) to formulate a new delayed SIR model as follow: , ( The first two equations in system (2) do not depend on the third equation, and therefore this equation can be omitted without loss of generality.Hence, system (2) can be rewritten as This model provides a reasonable role of incubation period on the dynamics of SIR epidemic model.We show that if the basic reproduction number, denoted, R 0 , is less than unity, the disease-free equilibrium is locally asymptotically stable, and disease always dies out.Moreover, we prove that if R 0 > 1, the endemic equilibrium is locally asymptotically stable, so that the disease, if initially present, will persist at the unique endemic equilibrium.In the end some numerical simulations are given to compare our model with existing model.

Steady state and local stability analysis
In this section, we discuss the local stability of an endemic equilibrium and a diseasefree equilibrium of system (3) by analyzing the corresponding characteristic equations, respectively [19].System (3) always has a disease-free equilibrium E 1 = ( A µ , 0).Further, if admits a unique endemic equilibrium E * = (S * , I * ), where where .

Remark 1. (i)
The basic reproduction number, R 0 representing how many secondary infectious result from the introduction of one infected individual into a population of susceptible [20].
Now let us start to discuss the local behavior of the system (3) of the equilibrium points E 1 = ( A µ , 0), and E * = (S * , I * ).At the equilibrium E 1 , characteristic equation is We have the following result Proposition 1.If R 0 < 1, then the disease free equilibrium E 1 is locally asymptotically stable.And if R 0 > 1, then the equilibrium point E 1 is unstable.

µ+α1A
. Hence, if R c < 1, then the disease free equilibrium E 1 is locally asymptotically stable for τ = 0.By Corollary 2.4 in Ruan and Wei [21, p. 867], it follows that if instability occurs for a particular value of the delay τ , a characteristic root of (4) must intersect the imaginary axis.Suppose that (4) has a purely imaginary root iω, with ω > 0.Then, by separating real and imaginary parts in (4), we have Hence, For R c < 1, equation ( 7) has no positive solution.Thus from Remark 1, if R 0 < 1, then the disease free equilibrium E 1 is locally asymptotically stable for all τ ≥ 0.
If R c > 1, then the disease free equilibrium E 1 is unstable for τ = 0.By Kuang's theorem [22, p. 77], it follows that E * is unstable for all τ ≥ 0. This concludes the proof.
Let x = S − S * and y = I − I * .Then by linearizing system (3) around E * , we have The characteristic equation associated to system (8) is where The local stability of the steady state E * is a result of the localization of the roots of the characteristic equation (9).In order to investigate the local stability of the steady state, we begin by considering the case without delay τ = 0.In this case the characteristic equation ( 9) reads as where hence, according to the Hurwitz criterion and Remark 1, we have the following proposition.
Proposition 2. For τ = 0, the equilibrium E * is locally asymptotically stable if and only if R 0 > 1.
We now return to the study of equation ( 9) with τ > 0.
Theorem 1.If R 0 > 1, then the steady state E * is locally asymptotically stable for all τ ≥ 0.

Concluding remarks and future research
In this paper, we considered a delayed SIR model with a modified saturated incidence rate βSI 1+α1S+α2I , and a constant recruitment, A. We showed that the local stability of the endemic equilibrium point, E * , depend on the basic reproduction number, R 0 , and doesn't change with respect to time delay, τ , (the incubation period); If R 0 < 1, the diseasefree equilibrium, E 1 , is locally asymptotically stable so the disease dies out.Moreover, we prove that if R 0 > 1, the disease-free equilibrium, E 1 , is unstable and the endemic equilibrium, E * , is locally asymptotically stable for all τ ≥ 0 (the disease approaches the endemic value E * ).In the end some numerical simulations are given to illustrate the theoretical analysis and to compare our model with existing model in [17].
For the future research, we consider a delayed SIR model with a saturated incidence rate of the form βSI 1+α1S+α2I and a logistic growth.In this case we show that the local stability of the endemic equilibrium point, E * , depend on time delay, τ .