Stability Results for an Age-Structured SIS Epidemic Model with Vector Population

Abstract

We formulate an age-structured SIS epidemic model with periodic parameters, which includes host population and vector population. The host population is described by two partial differential equations, and the vector population is described by a single ordinary differential equation. The existence problem for endemic periodic solutions is reduced to a fixed point problem of a nonlinear integral operator acting on locally integrable periodic functions. We obtain that if the spectral radius of the Frechet derivative of the fixed point operator at zero is greater than one, there exists a unique endemic periodic solution, and we investigate the global attractiveness of disease-free steady state of the normalized system.

He-Long Liu 1 and Jing-Yuan Yu 2 and Guang-Tian Zhu 3

Academic Editor:Zhidong Teng

1, College of Mathematics and Information Science, Xinyang Normal University, Henan 464000, China
2, Beijing Institute of Information and Control, Beijing 100037, China
3, Academy of Mathematics and System Science, C.A.S., Beijing 100080, China

Received 15 November 2014; Revised 13 January 2015; Accepted 15 January 2015; 24 February 2015

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.

1. Introduction

Some infectious diseases such as malaria, Chagas disease, and dengue fever are transmitted via vector. The seasonality of vector population decides that the transmission of infectious diseases is periodic. So, in order to model the seasonal spread of infectious diseases many authors have studied differential equations systems with periodic parameters [1-6].

In order to reflect the effect of demographic behavior of individuals, scholars have recognized that age-structured epidemic models are more realistic, since any disease prevention policy depends on the age structure of host population, and instantaneous death and infection rates depend on the age. Since the pioneer work of McKendrick [3], many authors have studied various age-structured epidemic models [2, 7-12]. However, due to their relatively complex form, the analyses of mathematical properties are difficult especially in the local and global stability of steady states.

The model we study in this paper is an age-structured SIS epidemic model with vector population, host population is divided into two compartments, susceptibles and infectives, and we assume that recovered individuals cannot obtain immunity and directly go back to the susceptibles. The vector population is divided into two groups, susceptibles and infectives. Generally speaking, the number of susceptible population is much greater than the number of infective population in a vector population. So, we may assume that the change of infectives' size does not affect the total number of susceptible vector population.

The age-structured SIS epidemic models have been studied in [8, 9, 12-14]. The general form of these models to a periodic system was given in [6]; the paper analyzed age structured SIS models with seasonal periodicities and vertical transmission and studied the global stability of a nontrivial endemic periodic solution.

The age-structured epidemic models with vector population have been studied in [15-18]. In [15], authors proved that the population dynamics of malaria, formulated vector-host model for malaria, and used the system of ordinary differential equations to describe the model. Paper [16] discussed a vector-host model for the spread of Chagas disease with infection-age. In [17], a deterministic model showed that the age-structured model underwent the phenomenon of backward bifurcation at [figure omitted; refer to PDF] under certain conditions and that the backward bifurcation feature was caused by malaria-induced mortality in humans. In [18], authors studied the existence and uniqueness of endemic periodic solution of an age-structured SIS epidemic model with periodic parameters.

In our model, host population is described by two partial differential equations, and infective vector population is described by a single ordinary differential equation. Integrating the partial differential equations along the characteristic lines, we can normalize them to a partial differential equation of fraction of infected population and get an expression of the infective host population. Integrating ordinary differential equation, we obtain an expression of the infected vector population. Using these expressions, we can obtain an integral equation, which is a fixed point equation in locally integrable time-periodic functions. From the fixed point theory, we obtain that there exists a unique endemic periodic solution under certain conditions and investigate the global attractiveness of disease-free steady state of the normalized system.

This paper is organized as follows: Section 2 introduces an age-structured SIS epidemic model with vector population. In Section 3, we show the well-posedness of the time evolution problem. In Section 4, we prove the existence of endemic periodic solution of the system in case that threshold value is greater than one. In Section 5, we get that the nontrivial solution is unique if threshold value is greater than one. In Section 6, we study the global attractiveness of infection-free state [figure omitted; refer to PDF] . Section 7 contains some discussions of the results.

2. The Model

In this section, we formulate an age-structured SIS model, which includes host population and vector population. The host population is divided into two classes: susceptible and infective. Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be the age-densities of, respectively, the susceptible and infective host population at time [figure omitted; refer to PDF] . In the vector population, we assume the number of susceptible individuals is far greater than the number of infective individuals, and let [figure omitted; refer to PDF] (known [figure omitted; refer to PDF] -periodic function) be the number of susceptible population and [figure omitted; refer to PDF] the number of infective population. Let [figure omitted; refer to PDF] be the density with respect to age of the total number of the host population and satisfy [figure omitted; refer to PDF] where the constant [figure omitted; refer to PDF] is the total size of the host population, [figure omitted; refer to PDF] is the crude death rate of the host population, and [figure omitted; refer to PDF] denotes the instantaneous death rate at age [figure omitted; refer to PDF] of the host population, is nonnegative, locally integrable on [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] denotes the maximum attainable age), and satisfies [figure omitted; refer to PDF] The crude death rate of the host population is determined such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the survival function. We have the relation [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] be the number of bites per vector per unit time and [figure omitted; refer to PDF] be the proportion of infected bites that give rise to infection. Then the force of infection for the host population is defined by [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] be the proportion of bites to infected hosts that give rise to infection in vector. Then the number of new infection of vectors per unit time from infected hosts is given by [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] be the age-specific recovery rate in the host population and [figure omitted; refer to PDF] the per capita death rate of vectors. Biologically speaking, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are periodic in time [figure omitted; refer to PDF] , maybe different in period. But for theoretical analysis, we assume that their periods are the same. Moreover we assume that the death rate of the host population is not affected by the presence of the disease.

With these assumptions, we obtain the following system of equations which describes the dynamics of the vector-host model: [figure omitted; refer to PDF] with boundary and initial conditions: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] To simplify the model, let [figure omitted; refer to PDF] The system (7) and (8) can be rewritten as [figure omitted; refer to PDF] with boundary and initial conditions: [figure omitted; refer to PDF] The following results will be used in Section 3.

From the third equation of the system (11), we have [figure omitted; refer to PDF] Integrating differential inequality (13), we have [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -periodic functions, we obtain [figure omitted; refer to PDF]

3. Existence and Uniqueness of Solution

From the first equation and the second equation of the system (11) and (12), we obtain that [figure omitted; refer to PDF] . So, the system (11) and (12) can be reduced to two equations for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] as [figure omitted; refer to PDF] with boundary and initial conditions: [figure omitted; refer to PDF]

We consider the initial-boundary value problem of the system (16) and (17) as an abstract Cauchy problem: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] endowed with the norm [figure omitted; refer to PDF] Suppose [figure omitted; refer to PDF] ; we define [figure omitted; refer to PDF] The system (18) is a semilinear nonautonomous Cauchy problem, we easily obtain that the operator [figure omitted; refer to PDF] is the infinitesimal generator of [figure omitted; refer to PDF] -semigroup [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is continuously Frechet differentiable on [figure omitted; refer to PDF] . Then for each [figure omitted; refer to PDF] , there exists a maximal interval of existence [figure omitted; refer to PDF] and a unique mild solution [figure omitted; refer to PDF] (see [8, 14, 19]), which satisfies (18), where either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , in the case [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] , and from (15) [figure omitted; refer to PDF] , we easily obtain [figure omitted; refer to PDF] . So, we have the following theorem.

Theorem 1.

The initial-boundary value problem (18), that is, the system (16) and (17), has a unique nonnegative mild solution [figure omitted; refer to PDF] .

4. Existence of Endemic Periodic Solution

In this section, we discuss existence of endemic periodic solution of the system (16) and (17). Let [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the set of locally integrable [figure omitted; refer to PDF] -periodic [figure omitted; refer to PDF] -valued functions with norm [figure omitted; refer to PDF] [figure omitted; refer to PDF] is the set of locally integrable [figure omitted; refer to PDF] -periodic functions with norm [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are their positive cone, respectively. We give the state space of the system (16) and (17) as follows: [figure omitted; refer to PDF] If [figure omitted; refer to PDF] is an endemic [figure omitted; refer to PDF] -periodic solution of the system (16) and (17), then it satisfies [figure omitted; refer to PDF] Integrating the first equation of (25) along the characteristic lines, we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

From the second equation of (25), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Substituting (26) into (27), we have [figure omitted; refer to PDF] According to expression of [figure omitted; refer to PDF] , we define a nonlinear positive operator [figure omitted; refer to PDF] on [figure omitted; refer to PDF] .

If [figure omitted; refer to PDF] has a nontrivial fixed point [figure omitted; refer to PDF] , then from (26) there is a [figure omitted; refer to PDF] -periodic solution [figure omitted; refer to PDF] . So, the system (16) and (17) has an endemic [figure omitted; refer to PDF] -periodic solution in a weak sense.

In the following, we investigate such a fixed point [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . First, we define a positive bounded linear operator [figure omitted; refer to PDF] as [figure omitted; refer to PDF] which is the Frechet derivative of operator [figure omitted; refer to PDF] at [figure omitted; refer to PDF] , and it is a majorant of [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] .

Next we prove the following lemmas.

Lemma 2.

The operator [figure omitted; refer to PDF] is monotone nondecreasing and uniformly bounded for [figure omitted; refer to PDF] .

Proof.

For [figure omitted; refer to PDF] , from (29) we obtain [figure omitted; refer to PDF] From (31), we have that [figure omitted; refer to PDF] is monotone nondecreasing and obtain [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , similar to (15), we have [figure omitted; refer to PDF] From (33), we obtain that [figure omitted; refer to PDF] is uniformly bounded.

Lemma 3.

Let [figure omitted; refer to PDF] be the spectral radius of operator [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then it is a positive eigenvalue of [figure omitted; refer to PDF] associated with a positive eigenvector [figure omitted; refer to PDF] .

Proof.

It is easy to get that [figure omitted; refer to PDF] is a linear map from [figure omitted; refer to PDF] into itself and leaves the cone invariant.

Let [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] [figure omitted; refer to PDF] can be rewritten as [figure omitted; refer to PDF] We extend the domain of [figure omitted; refer to PDF] and define [figure omitted; refer to PDF] for [figure omitted; refer to PDF] or [figure omitted; refer to PDF] ; then we have [figure omitted; refer to PDF] From (35) and (36), we have [figure omitted; refer to PDF] Combining (35) and (37), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF] is well defined and is uniformly bounded; the sum in the expression of [figure omitted; refer to PDF] is a finite sum due to [figure omitted; refer to PDF] , [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .

Hence, [figure omitted; refer to PDF] can be regarded as an operator on [figure omitted; refer to PDF] . From the well-known compactness criteria in [figure omitted; refer to PDF] [20], we obtain that [figure omitted; refer to PDF] is compact.

Similar to [figure omitted; refer to PDF] , we get the expression of [figure omitted; refer to PDF] as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF] is well defined and is uniformly bounded, and [figure omitted; refer to PDF] is compact in [figure omitted; refer to PDF] .

Since [figure omitted; refer to PDF] , we obtain that [figure omitted; refer to PDF] is positive, linear, and compact. If [figure omitted; refer to PDF] , from the Krein-Rutman theorem [21], there is [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] holds.

Using the above lemmas, we have the following theorem.

Theorem 4.

If [figure omitted; refer to PDF] holds, then operator [figure omitted; refer to PDF] has at least one nontrivial fixed point [figure omitted; refer to PDF] : [figure omitted; refer to PDF] that is, the system (16) and (17) has at least an endemic [figure omitted; refer to PDF] -periodic solution.

Proof.

From Lemma 3, we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] are uniformly bounded, let [figure omitted; refer to PDF] We have [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] whose positivity follows from [figure omitted; refer to PDF] . From (29), (44), and (46), we have [figure omitted; refer to PDF] From Lemma 2, we obtain that operator [figure omitted; refer to PDF] is monotone nondecreasing. So, we define a monotone sequence [figure omitted; refer to PDF] From Lemma 2, we have that sequence [figure omitted; refer to PDF] is bounded above. From B. Levi's theorem, there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

5. Uniqueness of Endemic Periodic Solution

If [figure omitted; refer to PDF] are the endemic [figure omitted; refer to PDF] -periodic solutions of the system (16) and (17), respectively. Let [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -periodic functions, then [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , are periodic functions. From (16), we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] / [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Considering that [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -periodic functions, we obtain [figure omitted; refer to PDF] For a number [figure omitted; refer to PDF] and the nonlinear positive operator [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] From (52), we have [figure omitted; refer to PDF] Combine (53) and (54) to get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Now we use (52) and (55) to investigate the uniqueness of endemic periodic solution.

From (52) we have [figure omitted; refer to PDF] If [figure omitted; refer to PDF] , we take [figure omitted; refer to PDF] . From (55) we have [figure omitted; refer to PDF] Noting [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , from Lemma 2 and (52) we obtain [figure omitted; refer to PDF] If [figure omitted; refer to PDF] , we take [figure omitted; refer to PDF] and repeat the above process to have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , then there exists a number [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] So, if [figure omitted; refer to PDF] from (57) and if [figure omitted; refer to PDF] from (61), we always have [figure omitted; refer to PDF] Exchanging the role of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , similar to the above arguments, we can obtain [figure omitted; refer to PDF] Equations (62) and (64) imply that [figure omitted; refer to PDF] . From (26), we have [figure omitted; refer to PDF] .

From Theorem 4 and the above arguments, we obtain the theorem.

Theorem 5.

If [figure omitted; refer to PDF] holds, the system (16) and (17) has a unique endemic [figure omitted; refer to PDF] -periodic solution in a weak sense.

6. Stability of the Disease-Free Steady State

The system (16) always has the infection-free steady state: [figure omitted; refer to PDF] In this section, we will prove that the disease-free steady state is global attractiveness. From the first equation of (16), we have [figure omitted; refer to PDF] From the second equation of (16), we have [figure omitted; refer to PDF]

Substituting (66) into (67), we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] For [figure omitted; refer to PDF] , by variable substitution, [figure omitted; refer to PDF] can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Noting [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] From (15) we obtain that [figure omitted; refer to PDF] is bounded above. Moreover, from (72) we have [figure omitted; refer to PDF] Before proving the global attractiveness of the disease-free steady state, we first prove a lemma.

Lemma 6.

Consider [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Proof.

For any given [figure omitted; refer to PDF] , we choose [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; we have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -periodic functions, we obtain [figure omitted; refer to PDF] From (75), (76), we take the limit supreme on both sides of (75) when [figure omitted; refer to PDF] [figure omitted; refer to PDF] and have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] is arbitrary, the lemma follows.

Let [figure omitted; refer to PDF] Using Lemma 6, we have the following theorem.

Theorem 7.

If [figure omitted; refer to PDF] holds, then the infection-free steady state [figure omitted; refer to PDF] is global attractiveness; that is, [figure omitted; refer to PDF]

Proof.

From (73) and Lemma 6, we take the limit supreme on both sides of (68) when [figure omitted; refer to PDF] [figure omitted; refer to PDF] and have [figure omitted; refer to PDF] If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] , which implies [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] , from (66) we have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , similar to the proof of Lemma 6, for any given [figure omitted; refer to PDF] , we choose [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] . From (82) to get [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] is arbitrary, inequality (83) implies [figure omitted; refer to PDF] This completes our proof.

7. Discussion

In this paper, in order to reflect the dependence of vector-borne diseases progress on seasonality of vector population and chronological age of the host population, our SIS epidemic model has periodic parameters, and the host population is structured by the chronological age. So, the SIS epidemic model has a relatively complex form. Due to the difficulty in the analysis, to our knowledge, we do not obtain threshold value for the existence and uniqueness of nontrivial endemic periodic solution of the age-structured SIS epidemic model and do not prove the infection-free state is globally stable and only get the condition under which the infection-free state is global attractiveness. In addition, for our age-dependent case, the stability of the endemic periodic solution [figure omitted; refer to PDF] for [figure omitted; refer to PDF] has been left as an open problem.

Acknowledgment

The authors are grateful to two reviewers for their careful reading of the original manuscript and their many valuable comments and suggestions.

Conflict of Interests

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

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