Stability Analysis of a Delayed SIR Epidemic Model with Stage Structure and Nonlinear Incidence

We investigate the stability of an SIR epidemic model with stage structure and time delay. By analyzing the eigenvalues of the corresponding characteristic equation, the local stability of each feasible equilibrium of the model is established. By using comparison arguments, it is proved when the basic reproduction number is less than unity, the disease free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, sufficient conditions are derived for the global stability of an endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.


Introduction
Let S t denote the number of members of a population susceptible to a disease, I t the number of infective members, and R t the number of members who have been removed from the possibility of infection through full immunity, a standard SIR compartmental model is of the form 1 where the parameters A, β, γ, μ, ε are positive constants in which A is the recruitment rate of susceptible population, μ represents the natural death rate of the population, ε is the disease-induced death rate of the infectives, and γ is the recovery rate from the infected compartment.It is assumed further that susceptibles become infectious by contact with infectious individuals.Later they may recover and join the group of immune or dead individuals.Based on the previous idea, different types of SIR epidemic models have been investigated see, e.g., 2-6 .We note that most of the previous works assume that each species has the same contact and recovery rates ignoring the effect of stage structure.In the real world, any species has a process of growth and development, such as from immature to mature, and growth at various stages of life history showed differences in physiology.
In the recent years, there have been a fair amount of work on epidemiological models with stage structure see, e.g., 7-10 .In fact, the spread of disease is related to the species stage structure.Some diseases, such as measles, mumps, chickenpox and scarlet fever, only spread or have more opportunities to spread in children, and some other diseases, such as diphtheria, leptospirosis, a variety of sexually transmitted diseases, may spread in adult.By assuming that the mature population does not contract the disease and the immature population is susceptible to the infection in 9 , Xiao et  It is assumed that newborn individuals are the recovered population with immunity with probability γ 0 < γ < 1 and are susceptible population with probability 1−γ.β is the rate that the susceptible population become infective, and b is the rate that the infective population becomes recovered with immunity.r 1 , r 2 , r 3 are the death rates of the susceptible, infective, recovered population, respectively, and r 1 ≤ r 2 is reasonable for biological meaning.r is the death rate of the mature population.Finally, it is assumed that those immatures born at time t−τ that survive to time t exit from the immature population and enter the mature population.Xiao et al. 9 proved that if the basic reproduction number is less than unity, the disease-free equilibrium of system 1.2 is globally asymptotically stable; if the basic reproduction number is greater than unity, sufficient conditions were derived for the global stability of an endemic equilibrium.
Incidence rate plays a very important role in the research of epidemiological models; it should generally be written as βU N S/N, where N is the total population size see 1 .In classical epidemic models, bilinear incidence rate βSI and standard incidence rate βSI/N are frequently used.The bilinear incidence rate is based on the law of mass action.This contact law is more appropriate for communicable diseases such as influenza., but not for sexually transmitted diseases.For standard incidence rate, it may be a good approximation if the number of available partners is large enough and everybody could not make more contacts than is practically feasible 11 .After a study of the cholera epidemic spread in Bari in 1973, Capasso and Serio 12 introduced a saturated incidence rate g I S into epidemic models, where g I tends to a saturation level when I gets large, that is, where βI measures the force of infection of the disease, and 1/ 1 αI measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the susceptible individuals.This incidence rate seems more reasonable than the bilinear incidence rate g I S βIS, because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters 13 .Motivated by the work of Capasso and Serio 12 and Xiao et al. 9 , in this paper, we are concerned with the effect of stage structure and saturation incidence on the dynamic of an SIR epidemic model.To this end, we study the following delayed differential system

1.4
The initial conditions for system 1.4 take the form where For continuity of initial conditions, we require It is easy to show that all solutions of system 1.4 with initial conditions 1.5 and 1.7 are defined on 0, ∞ and remain positive for all t ≥ 0.
The organization of this paper is as follows.In the next section, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system 1.4 is discussed.In Section 3, we study the global stability of the diseasefree equilibrium and the endemic equilibrium of system 1.4 , respectively.Numerical simulations are carried out in Section 4 to illustrate the main theoretical results.A brief discussion is given in Section 5 to conclude this work.

Local Stability
In this section, we discuss the local stability of each of nonnegative equilibria of system 1.4 by analyzing the eigenvalues of the corresponding characteristic equations, respectively.System 1.4 always has a trivial equilibrium E 0 0, 0, 0, 0 , and a disease free equilibrium E 1 x 0 1 , 0, x 0 3 , y 0 , where The basic reproduction number is given as It is easy to prove that if R 0 > 1, system 1.4 has an endemic equilibrium

2.3
The characteristic equation of system 1.4 at the equilibrium E 0 0, 0, 0, 0 is of the form Obviously, 2.4 always has three negative real roots λ −r 1 , λ −b 2 − r 2 , and λ −r 3 .Noting that y λ and y aγe −τ λ r 3 must intersect at a positive value of λ, hence, the equation λ − aγe − λ r 3 τ 0 has a positive real root.Accordingly, E 0 is unstable.
The characteristic equation of system 1.4 at the endemic equilibrium where 2.9 Clearly, 2.8 always has a negative real root λ −r 3 .Noting that p > 0, q > 0, roots of equation λ 2 pλ q 0 have only negative real parts.In addition, from the discussion above, we see that roots of the 2.6 have only negative real parts.By the general theory on characteristic equations of delay differential equations from 14 , we see that if R 0 > 1, the endemic equilibrium E * is locally asymptotically stable.
Based on the discussions above, we have the following result.

Global Stability
In this section, we discuss the global stability of the disease-free equilibrium and the endemic equilibrium of system 1.4 , respectively.The technique of proofs is to use a comparison argument and an iteration scheme.We first consider the subsystem of 1.4

3.2
The initial conditions for system 3.2 take the form Clearly, system 3.2 has a nonnegative equilibrium A 1 z 0 , 0 , where z 0 a 2 γ 1−γ e −r 3 τ / rr 1 ; when R 0 > 1, system 3.2 has a positive equilibrium A * z * , x * , where

3.4
Moreover, from Theorem 2.1, we see that A 1 is locally asymptotically stable if R 0 < 1, and A * is locally asymptotically stable if R 0 > 1.
To study the global dynamics of system 1.4 , we need only to discuss the global behavior of solutions of system 3.2 .In the following, we investigate the global asymptotic stability of the equilibria A 1 and A * by using the comparison arguments and the iteration scheme 15 , respectively.To this end, we need the following result developed by Song and Chen in 16 .Lemma 3.1.Consider the following equation: where a, b, c, τ > 0, x t > 0 for t ∈ −τ, 0 .One has the following: Proof.Let z t , x t be any positive solution of system 3.2 with initial condition 3.3 .Let From Lemma 3.1, it is easy to show that lim t → ∞ y t y 0 y * αβe −r 3 τ r .

3.8
We derive from the first equation of system 3.2 that ż t ≤ a 1 − γ y * ε − r 1 z. 3.9 By comparison, we have lim sup Since this is true for arbitrary ε > 0 sufficiently small, it follows that Hence, for ε > 0 sufficiently small, there is a T 1 > T 0 such that, if t > T 1 , z t ≤ M z 1 ε.For ε > 0 sufficiently small, we derive from the second equation of system 3.2 that, for t > T 1 ,

3.12
Consider the following auxiliary system:

3.13
By Lemma 3.1 it follows from 3.13 that By comparison, we obtain that lim sup Since the inequality is true for arbitrary ε > 0 sufficiently small, it follows that U 2 ≤ M x 1 , where Hence, for ε > 0 sufficiently small, there is a T 2 > T 1 such that, if t > T 2 , x t ≤ M x 1 ε.For ε > 0 sufficiently small, we derive from the first equation of system 3.2 that, for 3.17 By comparison and by Lemma 3.1, we have Since the inequality is true for arbitrary ε > 0 sufficiently small, it follows that V 1 ≥ N z 1 , where

3.19
Hence, for ε > 0 sufficiently small, there is a T 3 > T 2 such that, if t > T 3 , z t ≥ N z 1 − ε.For ε > 0 sufficiently small, we derive from the second equation of system 3.2 that, for t > T 3 ,

3.20
By comparison and by Lemma 3.1, we have

3.21
Since the inequality holds for arbitrary ε > 0 sufficiently small, it follows that V 2 ≥ N x 1 , where

3.22
Therefore, for ε > 0 sufficiently small, there is a T 4 > T 3 such that if t > T 4 , x t ≥ N x 1 − ε.For ε > 0 sufficiently small, we derive from the first equation of system 3.2 that, for

3.23
By comparison and by Lemma 3.1, we have lim sup

3.24
Since the inequality holds for arbitrary ε > 0 sufficiently small, it follows that U 1 ≤ M z 2 , where Hence, for ε > 0 sufficiently small, there is a T 5 > T 4 such that, if t > T 5 , z t ≤ M z 2 ε.For ε > 0 sufficiently small, we derive from the second equation of system 3.2 that, for t > T 5 ,

3.26
By comparison and by Lemma 3.1, we have lim sup

3.27
Since the inequality holds for arbitrary ε > 0 sufficiently small, we conclude that U 2 ≤ M x 2 , where

3.28
Therefore, for ε > 0 sufficiently small, there is a T 6 > T 5 such that, if t > T 6 , x t ≤ M x 2 ε.For ε > 0 sufficiently small, we derive from the first equation of system 3.2 that, for

3.29
By comparison and by Lemma 3.1 it follows that lim inf Since this is true for arbitrary ε > 0 sufficiently small, we conclude that where

3.31
Hence, for ε > 0 sufficiently small, there is a For ε > 0 sufficiently small, we derive from the second equation of system 3.2 that, for t > T 7 ,

3.32
By comparison and by Lemma 3.1 it follows that lim inf Since this is true for arbitrary ε > 0 sufficiently small, we conclude that V 2 ≥ N x 2 , where

3.34
Hence, for ε > 0 sufficiently small, there is a

3.35
Clearly, we have

3.36
It follows from 3.36 that

3.37
Noting that M x n ≥ x * and αr 1 > β, we derive from 3.37 that

3.38
Hence, the sequence M x n is monotonically nonincreasing.Therefore, lim n → ∞ M x n exists.Taking n → ∞, it follows from 3.37 that

3.39
We therefore obtain from 3.35 and 3.39 that

3.41
We therefore have x t x * .

3.42
Noting that if R 0 > 1 and αr 1 > β hold, the positive equilibrium A * is locally asymptotically stable, we conclude that A * is globally asymptotically stable.The proof is complete.
3 , y * of system 1.4 is globally asymptotically stable; that is, the disease remains endemic.
Proof.From 3.7 , we know that lim t → ∞ y t y * y 0 aγe −r 3 τ /r.According to the results of Lemma 3.2, we prove that lim

3.43
In the following, we show the existence of lim t → ∞ x 3 t .By Lemma 3.2, it follows from 3.43 that for ε > 0 sufficiently small, there exists a T > 0, such that, if t > T, Therefore, we derive from the third equation of system 1.4 that, for t > T τ, By comparison, we have lim sup Since the inequality holds for arbitrary ε > 0 sufficiently small, we have lim sup t → ∞ x 3 t ≤ M x 3 , where Hence, for ε > 0 sufficiently small, there is a T 1 > T such that, if t > T 1 , Again, for ε > 0 sufficiently small, it follows from the third equation of system 1.4 that, for t > T 1 τ,

3.49
By comparison, we have lim inf Since the inequality holds for arbitrary ε > 0 sufficiently small, we conclude that lim inf t → ∞ x 3 t ≥ N x 3 , where

3.51
Hence, for ε > 0 sufficiently small, there is a T 2 > T 1 such that, if t > T 2 ,

3.52
It follows from 3.48 and 3.52 that lim

3.53
Noting that if R 0 > 1 and αr 1 > β hold, the endemic equilibrium E * is locally asymptotically stable, we see that E * is globally asymptotically stable.This completes the proof.

3.55
We derive from the first equation of system 3.2 that ż t ≤ a 1 − γ y 0 ε − r 1 z.

3.56
By Lemma 3.1 and by a comparison argument, we get lim sup

3.57
Since this inequality holds for arbitrary ε > 0 sufficiently small, we conclude that lim sup

3.59
For ε > 0 sufficiently small satisfying 3.54 , it follows from 3.59 and the second equation of system 3.2 that x 3 t aγ 1 − e −r 3 τ y 0 r 3 x 0 3 .

3.67
Noting that if R 0 < 1, the disease-free equilibrium E 1 is locally stable, we conclude that E 1 is globally asymptotically stable.This completes the proof.

Discussion
In this paper, we have discussed the effect of stage structure and saturation incidence rate on an SIR epidemic model with time delay.The basic reproduction number R 0 was found.The local stability of each of feasible equilibria of system 1.4 was investigated.When the basic reproduction number is greater than unity, by using the iteration scheme, we have established sufficient conditions for the global stability of the endemic equilibrium of system 1.4 .By Theorem 3.3, we see that when R 0 > 1 and αr 1 > β, the endemic equilibrium is globally stable.Biologically, these indicate that when the proportionality infection constant and /or the birth rate of the immature population is sufficiently large and the death rates of susceptible population and the mature population are sufficiently small such that R 0 > 1, then the disease remains endemic.On the other hand, by Theorem 3.4, we see that, if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable.Biologically, if the proportionality infection constant and /or the birth rate of the immature population is small enough and the death rates of susceptible population and the mature population are large enough such that R 0 < 1, then the disease fades out.We would like to point out here that Theorem 3.3 has room for improvement, we leave this for future work.
al. proposed an SIR disease transmission model with stage structure and bilinear incidence rate as follows: ẋ1 t a 1 − γ y t − r 1 x 1 t − βx 1 t x 2 t , ẋ2 t βx 1 t x 2 t − bx 2 t − r 2 x 2 t , ẋ3 t bx 2 t aγy t − aγe −r 3 τ y t − τ − r 3 x 3 t , ẏ t aγe −r 3 τ y t − τ − ry 2 t , 1.2where x 1 t , x 2 t , x 3 t denote the densities of the immature population that are susceptible, infectious population and recovered population with immunity, respectively; y t denotes the density of the mature population which does not contract the disease.The parameters a, b, β, γ, r 1 , r 2 , r 3 , r are positive constants.a is the birth rate of the immature population.