Research Article | Open Access
Threshold Dynamics of an SIR Model with Nonlinear Incidence Rate and Age-Dependent Susceptibility
We propose an SIR epidemic model with different susceptibilities and nonlinear incidence rate. First, we obtain the existence and uniqueness of the system and the regularity of the solution semiflow based on some assumptions for the parameters. Then, we calculate the basic reproduction number, which is the spectral radius of the next-generation operator. Second, we investigate the existence and local stability of the steady states. Finally, we construct suitable Lyapunov functionals to strictly prove the global stability of the system, which are determined by the basic reproduction number and some assumptions for the incidence rate.
Mathematical modelling has been paid attention on investigating the intrinsic mechanisms of the disease prevalence. Many factors have been illustrated to affect dynamics of diseases spread. One of them is the susceptibility, which describes the fragility of individuals attacked during their lifetime. The susceptible ability is determined by individuals’ behaviors and the immunity of individuals. As Brucellosis, humans associated with animals are more easy to be transmitted compared with ones apart from these kinds of jobs. There are few literatures [1, 2] studying this factor to affect the dynamics of systems. In this paper, we introduce susceptible age to evaluate the susceptibility for the modelling process.
Besides, incidence rates play an important role in the disease spread pattern. Compartment modelling method was firstly proposed by Kermack and McKendrick [3, 4]. The baseline models are -type and -type, respectively, in which denotes the susceptible individuals, represents the infectious individuals, and denotes the recovered individuals ( is the total population). From mathematical points of view, there are two classical forms: one is bilinear denoted and the other is standard incidence denoted by and , where represents the transmission rate. As literatures indicated, the two incidence rates have distinct limitations, that is, the bilinear one is suitable for an initial phase of some disease spread, while the other one is applicable for the final phase. In order to overcome these defaults, Capasso and Serio in  introduced a saturated incidence rate to connect the above two cases. There is no doubt that it is more realistic and is more meaningful than the old ones. Thereafter, a general case given by was introduced into the models (see, e.g., [6, 7]). Furthermore, a general incidence rate written as in  had been used to investigate the global dynamics of and epidemic models.
To our best knowledge, Magal et al. are the pioneers to investigate the global stability of the steady states for age-structured model by constructing Lyapunov functional in . Since then, many researchers [10–13] have extensively studied the global behavior for age-structured epidemic models and bilinear incidence rates and the references therein. Since the complex structure for age-structured models, there are few references to establish global dynamics of epidemic models combining age structure and nonlinear incidence [14–17]. This issue has been becoming a challenging task for lacking well-established tools.
As we know, the recovered individuals can gain permanent disease-induced immunity for some diseases such as chickenpox, mumps, and measles. Motivated by what has been discussed, we integrate the susceptible age with a general incidence rate into an model given as follows: with initial condition where is the set of integrable functions from into . The total population is classified into three classes: susceptible , infected , and recovered . denotes the natural death rate, denotes the death rate due to disease, and is the recovery rate. is the transmission rate with respect to susceptible age ; represents the transmission incidence and it concludes many common forms such as , , and . All the parameters satisfy the following assumptions:
Assumption 1.1. (A1) and are positive; and are nonnegative.
(A2) The function where is the set of all bounded and uniformly continuous functions from to . Denote (A3) For , , ; , for ; is an increasing function for , while for ; ; is a decreasing function with respect to for .
The function satisfies the local Lipschitz continuous in and , which coincides the assumption in 
Equation is separated with other equations of system (1), and it can be omitted. Hence, system (1) can lower as a two-dimensional system: where . The goal of this paper is to establish global dynamics for (5). This paper supplements or generalizes many existing results [10, 14] if we take some special incidences or some specific parameters.
Note that the total population satisfies the following equation: which implies that .
1.1. Integrated Semigroup Formula
In order to formulate system (5) as an abstract Cauchy problem, we define the linear operator on as follows: where and denote the Sobolev space. The nonlinear operator on is given.
Note that and . Denote and , we can rewrite system (5) as an abstract Cauchy problem.
Lemma 1.1. There exists a real constant and such that , and where denotes the resolvent set of and denotes the identity operator. Furthermore, for all , .
Proof 1. For any and , it follows from the definitions of the linear operator that which implies that We directly calculate and obtain So that we have the following estimation. Hence, and are the values that we are looking for in the presentation of the theorem.
Define . Under Assumption 1.1, we have the following lemma for the nonlinear operator on .
Lemma 1.2. Suppose Assumption 1.1 holds. For all , there exists such that for any and .
Proof 2. For any , , we have where .
Theorem 1.3. There exists a unique semiflow for system (11). For all and for any ,
In what follows, we will give the regularity of the semiflow . By the proof process of Theorem 6.3 in , we define the set as
Assumption 1.2. For any , the following conditions hold.
From what has been discussed, we obtain the regularity of the semiflow on .
Theorem 1.4. For any , then the semiflow is continuously differentiable with respect to , and is the global classical solution of the abstract Cauchy problem (11).
Based on Assumption 1.1, we have the following priori estimations.
Lemma 1.5. For and , the following estimations hold: and Furthermore, and
Before ending this section, we define a set
Then, it follows from Lemma 1.5 that is positively invariant. This indicates that , , and . In what follows, for convenience, we denote as .
The paper is devoted to study the global stability of the steady states of the system by employing the Lyapunov functionals. The rest of this paper is organized as follows: Section 2 gives the asymptotic smoothness of the solution semiflow . In Section 3, we calculate the basic reproduction number by renewal equation and discuss the existence and local stability of steady states of system (5). In Section 4, the global dynamics of system (5) is established by considering the persistence of system (5) and constructing suitable Lyapunov functionals.
2. Asymptotic Smoothness
In view of system (5), we note that it has one integrate term. Hence, it is necessary to investigate the existence of a compact attractor of all bounded sets of . In order to establish this goal, we need the results of Theorem 2.33 and Theorem 2.46 in . To apply them, we decompose the solution semiflow , where where and
Theorem 2.1. The semiflow is asymptotically smooth if there are two maps and such that , and for any bounded set , the following statements hold: (1)For any , there exists a function such that for any with , then .(2)There exists such that has a compact closure for each .
Lemma 2.2. For any , there exists a function such that for any , and
Proof 3. This lemma follows the process as Theorem 2.46 in . Let be a bounded subset of . It is easy to see that , and , are nonnegative. Using (30) leads to where . Hence, the assumption in (1) of Theorem 2.1 holds.
Theorem 2.3 (Fréchet-Kolmogorov theorem, ).
Let be a subset of . Then, has compact closure if and only if the following conditions hold: (i)(ii) uniformly in (iii) uniformly in (iv) uniformly in
Lemma 2.4. maps any bounded subsets of into sets with compact closure in .
Proof 4. For any fixed and any bounded set , the set is precompact. With a notice, is bounded by the virtue of and Lemma 1.5. This implies that is precompact. It is enough to show that is precompact. This can be achieved by applying Fréchet-Kolmogorov Theorem 2.3 in . First, and indicate that is bounded. This implies that condition (i) of Fréchet-Kolmogorov theorem holds. Second, it is an immediate result that approaches zero as goes to infinity. By the boundedness of , condition (iv) of Fréchet-Kolmogorov theorem is well established. Finally, condition (iii) of the Fréchet-Kolmogorov theorem should be verified. In order to prove it, we should show that is uniformly continuous on the bounded set or Equation (34) holds obviously when since . So, we need to show that the result holds for . Since we are concerned with the limit as tends to , we assume that . Then, since , for , , and . This estimate immediately yields (34).
3. The Basic Reproduction Number, Existence, and Local Stability of Steady States
In this section, we give the general approach for calculating the basic reproduction number, which is the spectral radius of the next-generation operator. The basic reproduction number is the most important value for characterizing the renewal process in structured population. An epoch-making method was proposed by Diekmann et al. in  and then developed by Inaba in . Moreover, Yang and Xu in  extend this method to calculate the basic reproduction of disease models on complex networks. Let be a solution of system (5), and then, it satisfies the following equations:
Obviously, system (36) has a unique disease-free steady state . Now, we are in position to define the basic reproduction number by analyzing the renewal equation. The linear equation in disease-free invasion phase is given by
Solving (37) yields
Hence, the basic reproduction number is calculated as
In epidemiology, the basic reproduction number gives the average number of cases that a typical infectious individual generates, if introduced into a susceptible population, over the whole infectious period.
Next, we look for the solution . It follows from the third equation of (36) that
Solving the first equation of system (36), we have
Then, we integrate (41) to obtain
Borrowing the first equation of system (44), we obtain the following equation consisting of one variable
With a notice,
Then if . Besides, we note that and . Therefore, there exists at least one positive solution of (45) for .
Next, we will show the uniqueness of the endemic steady state of system (44). By way of contradiction, we assume that there exist two positive solutions of , denoted by and , respectively. They satisfy that . It follows from the relations between and that , that is, with . By the second equation of (44) and the property of with respect to , we obtain
This is a contradiction with assumption . From what has been discussed, we have the following result concerning the existence of steady states.
Theorem 3.1. Consider system (5) with defined in (39). If , then there is a unique steady state, which is the disease-free steady state ; while if , then there are two steady states, the disease-free steady state and the endemic steady state .
Now, we focus on system (48) and obtain the local stability of the steady states.
Theorem 3.2. Let define in (39). If , then the disease-free steady state is locally asymptotically stable and if , the unique endemic steady state is locally asymptotically stable.
Proof 5. Let and be a solution of (48). Then, system (48) can be formally written as
Solving the first equation yields
Then substituting (50) into the equation , we have
After variable change and integration by parts, we obtain
Plugging (53) into the second equation of (49), we admit
Equation (53) is the characteristic equation of (5). Then, the steady state is locally (asymptotically) stable if all eigenvalues of the characteristic equation have negative real parts and it is unstable if at least one eigenvalue has a positive real part.
Firstly, we note that the characteristic equation at is Integrating the inner formula yields We claim that all roots of (55) have negative real parts. Otherwise, let be a root of (55) with . Then, the module of the left hand side of (55) is larger than , while the module of the right hand side of (55) is . If , this leads to a contradiction. Therefore, all the eigenvalues of the characteristic (54) have negative real part and then disease-free steady state is locally asymptotically stable if .
Secondly, by the virtue of , we readily obtain Next, we show that the characteristic equation (53) has no eigenvalues with nonnegative real parts. Arguing directly, assume that there is one eigenvalue with . Then On the other hand, by the right hand side of (53), we have Consequently, the endemic steady state is locally asymptotically stable.
4. Global Stability Analysis
In order to consider the compactness of the orbit , we use the invariance principle which is referred in Theorem 4.2 of Chapter IV . Therefore, we can define as