Threshold Dynamics of an SIR Model with Nonlinear Incidence Rate and Age-Dependent Susceptibility

Yang, Junyuan; Wang, Xiaoyan

doi:https://doi.org/10.1155/2018/9613807

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Abstract Introduction Numerical Simulation Discussion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 9613807 | https://doi.org/10.1155/2018/9613807

Threshold Dynamics of an SIR Model with Nonlinear Incidence Rate and Age-Dependent Susceptibility

Junyuan Yang^1,2and Xiaoyan Wang³

Academic Editor: Diego R. Amancio

Received14 May 2018

Accepted17 Jul 2018

Published14 Oct 2018

Abstract

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.

1. Introduction

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 [5] 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 [8] 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 [9]. 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 [15]

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 this subsection, we employ the approach proposed by Thieme [18] to formulate system (5) as an abstract Cauchy problem. Now, we define the state space as follows: with the norm form given by

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.

Next, we will show the existence and uniqueness of system (11) by the results in [19]. First, we will give the following two lemmas.

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 .

Lemmas 1.1 and 1.2 imply that Assumption 3.1 of [19] holds. It follows from Lemma 3.1 in [19] that the existence and uniqueness of system (11) are given by the following theorem.

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 [19], we define the set as

Assumption 1.2. For any , the following conditions hold.

Obviously, . Based on Assumptions 1.1 and 1.2, the nonlinear operator on is differentiable with respect to , and the differentiation of at is given by

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 [20]. 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 [21]. 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.

Next, we prove that condition (2) of Theorem 2.1 is also satisfied. To achieve such goal, we employ the Fréchet-Kolmogorov theorem for the compactness of sets in [21].

Theorem 2.3 (Fréchet-Kolmogorov theorem, [21]).
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 [21]. 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).

With the help of Lemmas 2.2 and 2.4, together with , we conclude that has compact closure in for any . Theorem 2.1 ensures that the semiflow is asymptotically smooth.

Proposition 2.5. Let Assumption 1.1 hold, and then, the solution semiflow defined by (19) on is asymptotically smooth.

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 [22] and then developed by Inaba in [23]. Moreover, Yang and Xu in [24] 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

Moreover, we replace in (42) by (40) and obtain

Combining (40) and (43), we have

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 .

Secondly, we study the local stability of the steady states obtained in Theorem 3.1. In order to do this, we linearize system (5) at the endemic steady state and obtain

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 [21]. Therefore, we can define as

Let

Obviously, if , then as .

Definition 4.1 ([21], page 61).
If there exists an , independent of the initial conditions, such that then (5) is called to be uniformly weakly -persistent (resp., uniformly strongly -persistent), for .

To achieve the persistence of the system (5), we firstly prove the weekly -persistence and then give the global compactness of the orbit.

Proposition 4.2. If , then (5) is uniformly weakly -persistent.

Proof 6. Suppose , there exists an such that Arguing directly, there exists with Then, there exists such that Without loss of generality, we can assume that since we can achieve it by replacing the initial condition with . Then, for with a notice and (A3) of Assumption 1.1 for the function with respect to variables and , we have and this implies that Therefore, Trying to consider the third equation of (5), we have where we follow the mean value theorem that for . Taking Laplace transform on both side of (68) yields By the arbitrary of and , we take and which are small enough. Then, (62) contradicts .

A total trajectory of the semiflow is a function such that for all and all . For the definition of a total trajectory, the solution satisfies the following property for and .

Proposition 4.3. Let be a total trajectory in . Then, is positive and is positive or identically zero for all .

Proof 7. First, we show that is strictly positive for all and . From equation of system (5), we obtain that Integrating this equality with the boundary equation yields Second, we claim that is identically zero if there exists some such that . From the third equation of (5), we follow that Solving the above inequality, we have By the nonnegativity of , we conclude that is identically zero for all . Similarly, we repeat this process and conclude that for all . Consequently, our claim is true.
Finally, we show that is strictly positive if there does not exist some such that . By way of contradiction, we redo the second process and arrive at the strict positivity of for all .

By Propositions 4.2, 2.5, and 4.3, together with Theorem 3.2 in [25], the following theorem holds.

Theorem 4.4. System (5) is uniformly strongly -persistent if .

Corollary 4.1. Suppose and . Let be a total trajectory in . Then, there exists an such that and for all .

Proof 8. First of all, we claim that for all . Otherwise, suppose that there exists By the assumption of , we conclude that for all , , and . Then by the virtue of the function with respect to , we have which is a contradiction with . We repeat this process and obtain for all . In review of Theorem 4.4, we immediately have that there exists some positive constant such that . On the other hand, for such , we recall the process of 4.3 to obtain for and . Hence, is the value that we are looking for and

From Theorem 5.7 in [21], we have the following result.

Theorem 4.5. Suppose Assumptions 1.1 and 1.2 hold. If the basic reproduction number , then there exists a compact attractor that attracts all solutions with initial condition belonging to .

For constructing the Lyapunov functionals to get the global stability of the steady states, we define as

It is well known that attains a global minimum only at 1 with and for . In order to guarantee a well definition of , we give the following proposition.

Proposition 4.6. For all , the following estimation holds.for all and .

Proof 9. This is a direct result from Corollary 4.1. We omit the details.

Theorem 4.7. Suppose and (A3) of Assumption 1.1 hold, then the disease-free steady state is globally asymptotically stable.

Proof 10. Define a Lyapunov function as where , and . Deviating along the solution of system (5) yields On the other hand, We substitute (82) into (81) and obtain Deviating with respect to time yields Substituting them into the deviation of gives where we use by (A3) of Assumption 1.1. Therefore, if , then . Since is bounded on , the alpha limit set of must be contained in , the largest invariant subset of .

In order to establish the global stability of the endemic steady state, we need the additional assumption for nonlinear function f.

Assumption 4.1. For ,

Now, we are in the position to prove the following result.

Theorem 4.8. Suppose that and Assumption 4.1 hold, then the endemic steady state is globally asymptotically stable in .

Proof 11. By Theorem 3.2 and Theorem 4.5, it suffices to show . Let be a total trajectory in . By Corollary 4.1, there exists , for any and , such that for and . Define where and . Then is bounded and well defined based on Proposition 4.6.
Next, we show that the upper right derivative along the solution is nonpositive. We firstly have Furthermore, it follows from Then, Differentiating with respect to , we obtain Summing up and together with the expressions of and arrives at In the following, we show that This, together with Assumption 4.1, implies that and then . So that is nonincreasing. Since is bounded on , the -limit set of must be contained in , the largest invariant subset of . It follows from that and . Immediately, we have for and . This, together with the first equation of system (5), yields for all and . By the virtue of stated in Assumption (A1), we have that for all . Therefore, .
The above analysis indicates that the -limit set of consists of just the endemic steady state and hence for all . Thus, .

5. Numerical Simulation

In this section, we carry out numerical simulations to verify our theoretical results. To show the threshold property for system (1), we pick up the parameters as follows: and where is varied as our demands. The incidence rate takes in the form of where is the maturate rate and we fix . The initial condition is chosen as

First, we pick up the transmission rate value as . Then, we directly estimate the basic reproduction number . Theorem 4.7 indicates the global stability of disease-free equilibrium . Figure 1(a) shows that the disease-free steady state decreasingly converges to zero. Second, we take and calculate . Theorem 4.8 and Figure 1(b) show that the endemic steady state is asymptotically stable.

(a)

(b)

6. Discussion

In this paper, we proposed an SIR model with age structure in susceptibility and a general incidence rate. We gave the existence, uniqueness, and regularity of the solution of system (5). We calculated the basic reproduction number, which is the spectral radius of the next-generation operator. Based on the assumptions on the incidence function and the parameters, we obtain the asymptotical smoothness of the system and existence, uniqueness, local stability, and global stability of the steady states. The interesting thing is that the global behavior of system (5) is rigorously proved by constructing Lyapunov functional method.

Incidence rate plays an important role in investigating the disease transmission pattern. As mentioned in Introduction, bilinear type and nonlinear type had been studied in many literatures. We used a general function to extend the two classical types and so it can cover some existing results [8, 9]. Some complex dynamics (Hopf bifurcation [26], backward bifurcation [27], homoclinics [28], and so on) inducing by complex incidence rate had been investigated. While in this paper, we just focus on the global dynamics based on some rigorous limitations for the nonlinear incidence rate . Oscillation is a nature phenomenon in realistic life, and exploring the intrinsic mechanism has been become a hot topic in epidemic modelling processes. In the future, we will turn our focus on this aspect to discover some substantial mechanisms to result in disease oscillations.

Data Availability

Our data is artificial selection to support our theoretical results.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

Research is partially supported by the National Natural Science Foundation of China (no. 61573016 and no. 61203228), the Shanxi Scholarship Council of China (2015-094), the Shanxi Scientific Data Sharing Platform for Animal Diseases, and the Startup foundation for High-level Personal of Shanxi University.

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Copyright © 2018 Junyuan Yang and Xiaoyan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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