Open Access

Existence of a Nonautonomous SIR Epidemic Model with Age Structure

Advances in Difference Equations20102010:212858

DOI: 10.1155/2010/212858

Received: 15 December 2009

Accepted: 1 February 2010

Published: 23 February 2010

Abstract

A nonautonomous SIR epidemic model with age structure is studied. Using integro-differential equation and a fixed point theorem, we prove the existence and uniqueness of a positive solution to this model. We conclude our results and discuss some problems to this model in the future. We simulate our analyzed results.

1. Introduction

Age structure of a population affects the dynamics of disease transmission. Traditional transmission dynamics of certain diseases cannot be correctly described by the traditional epidemic models with no age-dependence. A simplemodel was first proposed by Lotka and Von Foerster [1, 2], where the birth and the death processes were independent of the total population size and so the limitation of the resources was not taken into account. To overcome this deficiency, Gurtin and MacCamy [3], in their pioneering work considered a nonlinear age-dependent model, where birth and death rates were function of the total population. Various age-structured epidemic models have been investigated by many authors, and a number of papers have been published on finding the threshold conditions for the disease to become endemic, describing the stability of steady-state solutions, and analyzing the global behavior of these age-structured epidemic models (see [47]). We may find that the epidemic models that most authors discussed mainly include S-I-R that is, the total population of a country or a district was subdivided into two or three compartments containing susceptibles, infectives, or immunes; it was assumed that there is no latent class, so a person who catches the disease becomes infectious instantaneously. The basic SIR age-structured epidemic model is like the following equations:

(1.1)

The non-autonomous phenomenon is so prevalent and all pervasive in the real life that modelling biological proceeding under non-autonomous environment should be more realistic than autonomous situation. The non-autonomous phenomenon is so prevalent in the real life that many epidemiological problems can be modeled by non-autonomous systems of nonlinear differential equations [811], which should be more realistic than autonomous differential equations. In one case, the incidence of many infectious diseases fluctuates over time and often exhibits periodic behavior. The basic SIR model is formulated by

(1.2)

These works were mainly concerned with finding threshold conditions for the disease to become endemic and describing the stability of steady-state solutions, often under the assumption that the population has reached its steady state and the diseases do not affect the death rate of the population.

However, all of the models which are not mixed age structure and non-autonomous are only concluding age structure or non-autonomous. Birth rate or input function is dependent on age or dependent on time in these models cited therein. In fact, birth rate or input function is dependent not only on age and time but also on the total population . We know the resource is limited. As recognized by authors, there was only one paper [3, 12] related them. In [3, 12], their model are two dimensions about epidemic dynamics. The population is increasing year after year. The birth rate is a decrease function until the population attend certain level such as Logistic growth rate. At the same time, the death rate should be dependent on the total population . We can consider now more realistic and complex models in which the epidemic acts in a different way on infected, susceptible and recovered (immune). We consider a well-known expression for the force of infection which is justified in the literature. We choose as the natural space for the solution because the total population is finite.

This paper is organized as follows: Section 2 introduces a non-autonomous SIR model with age structure. In Section 3, existence and uniqueness of a solution for an epidemic model with different mortality rates on any finite time-interval is obtained. In Section 4, we conclude our results and discuss the defect of our model.

2. The Model Formulation

This section describes the basic model we are going to analyze in this paper. The population is divided into three subclasses: susceptible, infected, and recovered. Where denote the associated density functions with these respective epidemiological age-structured classes. Let , be the age-specific mortality of the susceptible, the infective and the recovered individuals at time , respectively. We assume that the disease affects the death rate, so we have , and . We assume that all new born are susceptible whose birth process is described by

(2.1)

where is the birth rate. We also suppose that the initial age distributions are given by , and . And the age-specific recovery rate, , is independent of the time. Then the joint dynamics of the age-structured epidemiological model for the transmission of SIR can be written as

(2.2)

We supposes and belong to . So, and , as . It is logical to satisfy the biological meaning. The horizontal transmission of the disease occurs according to the following law:

(2.3)

where is the rate at which an infective individual of age comes into a disease transmitting contact with a susceptible individual of age . Summing the equations of (2.2), we obtain the following problem for the population density .

(2.4)

In this paper, we prove the existence and uniqueness of a nonnegative solution of the model (2.2) on any finite time-interval. Our results are based on a process of the age-dependent problem for the susceptible the infected and the removed, and then a fixed point method. To study existence and uniqueness of a solution for an epidemic model with different mortality rates, we need the following hypotheses. Given , we denote and we suppose that

(H1) for , is a nonnegative measurable function such that the mapping belongs to for almost all . Moreover, there exists a constant such that for all ,
(2.5)
With the notation , , there exists another constant , such that
(2.6)
(H2) is a nonnegative measurable function which has compact support on the variable and such that for all ,
(2.7)
where is another constant which depends only on . Moreover, there exists a constant such that for all
(2.8)

(H3) has a compact support.

(H4) has compact support and is a nonnegative function. We set .

(H5) has a compact support and is a nonnegative function. We have .

To simplify the calculation of estimates, we perform the change

(2.9)

We obtain that the following system is analogous to (2.2).

(2.10)

where

(2.11)

For biological reasons, we are interested in nonnegative solutions, so we consider that

(2.12)

And we will look for solutions to (2.10) belonging to the following space:

(2.13)

endowed with the norm

(2.14)

where is a positive constant which will be chosen later and denotes the usual norm in that is,

Namely, by a solution to (2.10), we mean a function

(2.15)

such that

(2.16)

In order to prove the existence of solution of (2.10), adding in both sides of (2.16) in technical style, we have

(2.17)

where , , and denote the directional derivatives of and , respectively, that is,

(2.18)

Generally, will not be differentiable everywhere; of course,when this occurs, , and .

3. Existence of a Solution to the System

If we assume that is smooth along the characteristics (except perhaps for a zero-measure set of ), considering

(3.1)

where , and integrating equalities of (2.16) along the line, we obtain the following ODS

(3.2)

Integrating (2.7) along , we also get for technical need

(3.3)

Integrating the second equation of (2.16) along , we have

(3.4)

Integrating the third equation of (2.16) along , we obtain

(3.5)

where

(3.6)

We can easily see that solving (2.16) is equivalent to finding a solution to (3.2), (3.4) and (3.5) or (3.3), (3.4), and (3.5) (see [3]). So, in the sequel, we restrict our attention to these integral equations.

Let us consider with , and fixed. Consider the set

(3.7)

The following result provides some useful estimates.

Lemma 3.1.

Suppose ( )–( ), and let , , , and . Then for ,
  1. (i)
    (3.8)
     
  2. (ii)
    such that
    (3.9)
     
  3. (iii)
    such that
    (3.10)
     
  4. (iv)
    such that
    (3.11)
     

Proof.

Firstly, note that (3.8) and (3.9) are immediate. On the other hand,
(3.12)
We set , and then
(3.13)

Lemma 3.2.

Suppose ( )–( ), if satisfies (3.2), (3.4), and (3.5), or (3.3), (3.4), and (3.5), then there exists a constant , depending only on and , such that with defined in (3.7).

Proof.

Suppose that satisfies the above assumptions. Considering (3.2), (3.4) and (3.5), or (3.3), (3.4), and (3.5), thanks to (3.7) and an obvious change of variables in the integrals, we have for all ,
(3.14)
We use the Gronwall's inequality, and then
(3.15)

where and , .

Let us consider the map , where is defined by

(3.16)
and also can be equal to
(3.17)
(3.18)
(3.19)

where .

Lemma 3.3.

With the assumptions of Lemma 3.2, we have .

Proof.

In this proof we denote, for abbreviation,
(3.20)

If , then . Then , , , by (2.5) and (2.7). Hence, is clearly measurable in and essentially bounded on .

By (3.18), , a.e. . So, we only need to show that , , a.e. , , and , a.e. . We assume that (the discussion for is similar). Using (3.11) and (3.19) and substituting and into we get

(3.21)
Now, we proceed to estimate these quantities to see that . By the mean value theorem, there exists , such that
(3.22)
where
(3.23)
By the mean value theorem, there exists , such that
(3.24)
and by the mean value theorem, there exists , such that
(3.25)
We substitute , and into the formula of . Thus,
(3.26)

By the formula of , we have . Using (3.17) and (3.18) and substituting and into we get

(3.27)
By the mean value theorem, there exists , such that
(3.28)
By the mean value theorem, we have the following:
(3.29)
Thus,
(3.30)

So that , , a.e. , and we can conclude that for each , .

Theorem 3.4.

Suppose ( )–( ), for each and for each , with , there exists a unique satisfying (3.2), (3.4), and (3.5), or (3.3), (3.4), and (3.5). And so, is the unique solution to problem (2.10).

Proof.

In order to prove the theorem, it remains to be shown that (defined by (3.11) and (3.18)) has a unique point fixed in .

Let be defined by (3.7); then for being large enough maps into . Indeed, by estimate of , we get, for almost all

(3.31)
And from Gronwall's inequality, it follows that
(3.32)

for depending on , , and . Hence, we have proved that maps into .

Let us assume that is fixed such that remains in for in . Clearly, is closed in and to prove that has a unique fixed point in , it suffices to prove that is a strict contraction, for instance for the norm defined in definition of with suitable. For convenience in the following we denote a certain (which may change) but which is independent of and . For , , let us estimate .

First, for almost all ,

(3.33)

Now, substituting the expression of into , we get

(3.34)
where and . Hence
(3.35)
Estimate of . By (3.10), we have
(3.36)
Let us now estimate . By (3.9) and (3.10), we get
(3.37)

Let us estimate . Thanks to (2.7), (2.8), and (3.8), we have

(3.38)
Second, let us estimate . Substituting the expression for into and applying (3.11), we obtain
(3.39)
Estimate of . By (2.6) and (3.8), , , then
(3.40)
Since , we have , and then
(3.41)
Finally, let us estimate of .
(3.42)
Therefore, joining all above estimates, we see that for almost all , there exist and depending only on , , , , , and , such that
(3.43)
Dividing both sides of this inequality by , we obtain
(3.44)

And thus for great enough is a strict contraction with a unique fixed point in , and so in . This concludes the proof.

4. Discussion

In this paper, existence of positive period solution of a non-autonomous SIR epidemic model with age structure is studied. We obtained existence and uniqueness of this model using integral differential equation and a fixed theorem. The model is different from the classical age structure epidemic model and non-autonomous epidemic model. The initial condition is nonlocal and dependent on total population. In addition, incidence law is not Lipschitzianity. The classical methods are not valid. We construct a new norm and prove the existence of our model under definition of the new norm. We can illustrate this through two simulates examples. We set
(4.1)
System (2.2) with above coefficients has a unique positive periodic solution. We can see it from Figure 1.
Figure 1

The temporal solution found by numerical integration of problem with initial values s0(a) = 10, i0(a) = 10, r0(a) = 5. They show that system (2.2) has a unique positive periodic solution.

In the future, there are some problems that will be solved. The existence of steady state and stability of the steady state are still discussed. If birth rate is impulsive, what results will occur. The simulation of the age structure still to be resolved. Furthermore, what effect will occurs, if we introduce the delay in our model.

Declarations

Acknowledgments

This work is Supported by the National Sciences Foundation of China (10971178), the Sciences Foundation of Shanxi (20090110053), and the Sciences Exploited Foundation of Shanxi (20081045).

Authors’ Affiliations

(1)
Department of Applied Mathematics, Yuncheng University
(2)
Beijing Institute of Information and Control

References

  1. Lotka AJ: Elements of Mathematical Biology. Dover, New York, NY, USA; 1956.MATHGoogle Scholar
  2. Von Foerster H: Some remarks on changing populations. In The Kinetics of Cellular Proliferation. Grune & Stratton, New York, NY, USA; 1959:382-407.Google Scholar
  3. Gurtin ME, MacCamy RC: Non-linear age-dependent population dynamics. Archive for Rational Mechanics and Analysis 1974, 54: 281-300. 10.1007/BF00250793MathSciNetView ArticleMATHGoogle Scholar
  4. Busenberg SN, Iannelli M, Thieme HR: Global behavior of an age-structured epidemic model. SIAM Journal on Mathematical Analysis 1991,22(4):1065-1080. 10.1137/0522069MathSciNetView ArticleMATHGoogle Scholar
  5. Iannelli M, Milner FA, Pugliese A: Analytical and numerical results for the age-structured S-I-S epidemic model with mixed inter-intracohort transmission. SIAM Journal on Mathematical Analysis 1992,23(3):662-688. 10.1137/0523034MathSciNetView ArticleMATHGoogle Scholar
  6. El-Doma M: Analysis for an SIR age-structured epidemic model with vertical transmission and vaccination. International Journal of Ecology and Development 2005, 3: 1-38.Google Scholar
  7. Webb GF: Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics. Volume 89. Marcel Dekker, New York, NY, USA; 1985:vi+294.Google Scholar
  8. Thieme HR: Uniform weak implies uniform strong persistence for non-autonomous semiflows. Proceedings of the American Mathematical Society 1999,127(8):2395-2403. 10.1090/S0002-9939-99-05034-0MathSciNetView ArticleMATHGoogle Scholar
  9. Wang W, Zhao X-Q: Threshold dynamics for compartmental epidemic models in periodic environments. Journal of Dynamics and Differential Equations 2008,20(3):699-717. 10.1007/s10884-008-9111-8MathSciNetView ArticleMATHGoogle Scholar
  10. Zhang T, Teng Z: Permanence and extinction for a nonautonomous SIRS epidemic model with time delay. Applied Mathematical Modelling 2009,33(2):1058-1071. 10.1016/j.apm.2007.12.020MathSciNetView ArticleMATHGoogle Scholar
  11. Teng Z, Liu Y, Zhang L: Persistence and extinction of disease in non-autonomous SIRS epidemic models with disease-induced mortality. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2599-2614. 10.1016/j.na.2007.08.036MathSciNetView ArticleMATHGoogle Scholar
  12. Delgado M, Molina-Becerra M, Suárez A: Analysis of an age-structured predator-prey model with disease in the prey. Nonlinear Analysis: Real World Applications 2006,7(4):853-871. 10.1016/j.nonrwa.2005.03.031MathSciNetView ArticleMATHGoogle Scholar

Copyright

© J. Yang and X. Wang. 2010

This article is published under license to BioMed Central Ltd. 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.