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Existence of a Nonautonomous SIR Epidemic Model with Age Structure
Advances in Difference Equations volume 2010, Article number: 212858 (2010)
Abstract
A nonautonomous SIR epidemic model with age structure is studied. Using integrodifferential 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 agedependence. 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 agedependent model, where birth and death rates were function of the total population. Various agestructured 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 steadystate solutions, and analyzing the global behavior of these agestructured epidemic models (see [4–7]). We may find that the epidemic models that most authors discussed mainly include SIR 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 agestructured epidemic model is like the following equations:
The nonautonomous phenomenon is so prevalent and all pervasive in the real life that modelling biological proceeding under nonautonomous environment should be more realistic than autonomous situation. The nonautonomous phenomenon is so prevalent in the real life that many epidemiological problems can be modeled by nonautonomous systems of nonlinear differential equations [8–11], 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
These works were mainly concerned with finding threshold conditions for the disease to become endemic and describing the stability of steadystate 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 nonautonomous are only concluding age structure or nonautonomous. 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 wellknown 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 nonautonomous 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 timeinterval 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 agestructured classes. Let , be the agespecific 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
where is the birth rate. We also suppose that the initial age distributions are given by , and . And the agespecific recovery rate, , is independent of the time. Then the joint dynamics of the agestructured epidemiological model for the transmission of SIR can be written as
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:
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 .
In this paper, we prove the existence and uniqueness of a nonnegative solution of the model (2.2) on any finite timeinterval. Our results are based on a process of the agedependent 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
(H_{1}) for , is a nonnegative measurable function such that the mapping belongs to for almost all . Moreover, there exists a constant such that for all ,
With the notation , , there exists another constant , such that
(H_{2}) is a nonnegative measurable function which has compact support on the variable and such that for all ,
where is another constant which depends only on . Moreover, there exists a constant such that for all
(H_{3}) has a compact support.
(H_{4}) has compact support and is a nonnegative function. We set.
(H_{5}) has a compact support and is a nonnegative function. We have.
To simplify the calculation of estimates, we perform the change
We obtain that the following system is analogous to (2.2).
where
For biological reasons, we are interested in nonnegative solutions, so we consider that
And we will look for solutions to (2.10) belonging to the following space:
endowed with the norm
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
such that
In order to prove the existence of solution of (2.10), adding in both sides of (2.16) in technical style, we have
where , , and denote the directional derivatives of and , respectively, that is,
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 zeromeasure set of ), considering
where , and integrating equalities of (2.16) along the line, we obtain the following ODS
Integrating (2.7) along , we also get for technical need
Integrating the second equation of (2.16) along , we have
Integrating the third equation of (2.16) along , we obtain
where
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
The following result provides some useful estimates.
Lemma 3.1.
Suppose ()–(), and let ,, , and . Then for ,

(i)
(3.8)

(ii)
such that
(3.9) 
(iii)
such that
(3.10) 
(iv)
such that
(3.11)
Proof.
Firstly, note that (3.8) and (3.9) are immediate. On the other hand,
We set , and then
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 ,
We use the Gronwall's inequality, and then
where and , .
Let us consider the map , where is defined by
and also can be equal to
where .
Lemma 3.3.
With the assumptions of Lemma 3.2, we have .
Proof.
In this proof we denote, for abbreviation,
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
Now, we proceed to estimate these quantities to see that . By the mean value theorem, there exists , such that
where
By the mean value theorem, there exists , such that
and by the mean value theorem, there exists , such that
We substitute , and into the formula of . Thus,
By the formula of , we have . Using (3.17) and (3.18) and substituting and into we get
By the mean value theorem, there exists , such that
By the mean value theorem, we have the following:
Thus,
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
And from Gronwall's inequality, it follows that
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 ,
Now, substituting the expression of into , we get
where and . Hence
Estimate of. By (3.10), we have
Let us now estimate . By (3.9) and (3.10), we get
Let us estimate . Thanks to (2.7), (2.8), and (3.8), we have
Second, let us estimate . Substituting the expression for into and applying (3.11), we obtain
Estimate of. By (2.6) and (3.8), , , then
Since , we have , and then
Finally, let us estimate of .
Therefore, joining all above estimates, we see that for almost all , there exist and depending only on , , , , , and , such that
Dividing both sides of this inequality by , we obtain
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 nonautonomous 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 nonautonomous 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
System (2.2) with above coefficients has a unique positive periodic solution. We can see it from Figure 1.
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.
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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).
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Yang, J., Wang, X. Existence of a Nonautonomous SIR Epidemic Model with Age Structure. Adv Differ Equ 2010, 212858 (2010). https://doi.org/10.1155/2010/212858
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Keywords
 Compact Support
 Epidemic Model
 Positive Period Solution
 Integral Differential Equation
 Nonnegative Measurable Function