Journal of Applied Mathematics

Volume 2013 (2013), Article ID 276579, 12 pages

http://dx.doi.org/10.1155/2013/276579

Research Article

A New Approach to the Method of Lyapunov Functionals and Its Applications

Yunguo Jin¹ and Shouming Zhong1,2

¹School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

²Key Laboratory for Neuroinformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Received 18 July 2013; Revised 2 November 2013; Accepted 3 November 2013

Academic Editor: Xinyu Song

Copyright © 2013 Yunguo Jin and Shouming Zhong. 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.

Abstract.

We show some results which can replace the graph theory used to construct global Lyapunov functions in some coupled systems of differential equations. We present an example of an epidemic model with stage structure and latency spreading in a heterogeneous host population and obtain a more general threshold for the extinction and persistence of a disease. Using some results obtained by mathematical induction and suitable Lyapunov functionals, we prove the global stability of the endemic equilibrium. For some coupled systems of differential equations, by a similar approach to the discussion of the epidemic model, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible.

1. Introduction

Graph theory has developed into a substantial body of knowledge. A graph theoretic approach developed in [1, 2] is used to resolve a long-standing open problem on the uniqueness and global stability of the endemic equilibrium of a class of multigroup models in mathematical epidemiology. Using results from graph theory, a systematic approach developed in [3] allows one to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems. The graph-theoretical approach has been applied to various classes of coupled systems in engineering, ecology, and epidemiology (see, e.g., [1–13]). However, as in [14], while using the same Lyapunov function [3], sometimes graph theory can be replaced by positive operator theory. Furthermore, it seems that all authors use the graph theory under the assumption that the relative matrix is irreducible (see, e.g., [1–13]).

Motivated by the above discussion, in this paper, we show some results which can replace graph theory used to construct global Lyapunov functions in some coupled systems of differential equations (see, e.g., [1–13]) and a more general threshold without the assumption that the relative matrix is irreducible. For some coupled systems of differential equations (see, e.g., [1–13]), by a similar approach to the one discussed in this paper, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible.

Various epidemics continue to pose a public health threat to humans. One of the most important subjects in this study of epidemic models (see, e.g., [1–3, 7–13]) is to obtain a threshold that determines the persistence or extinction of a disease. In the real world, some epidemics, such as malaria, dengue, fever, gonorrhea, and bacterial infections, may have a different ability to transmit the infections in different ages. For example, measles and varicella always occur in juveniles, while it is reasonable to consider the disease transmission in adult population such as typhus and diphtheria. A heterogeneous host population can be divided into several homogeneous groups according to models of transmission, contact patterns, or geographic distributions. Since the time it takes from the moment of new infection to the moment of becoming infectious may differ from individual to individual, it is indeed a random variable. We present an example of an epidemic model with stage structure and latency spreading in a heterogeneous host population and show a general threshold to improved existing results.

Some results which can replace the graph theory used to construct global Lyapunov functions in some coupled systems of differential equations are shown in the next sections. In Section 3, a new approach to the method of Lyapunov functionals is applied to an epidemic model to obtain a more general threshold for the extinction and persistence of a disease.

2. Some Results Obtained by Mathematical Induction

In this section, we show some results which can replace graph theory used to construct global Lyapunov functions in some coupled systems of differential equations.

Assumption 1. There exist , such that

where , , .

Theorem 2. Under Assumption 1, the following results hold:

where , , are arbitrary functions. In particular, if , , then

Proof. Note that

Let

According to Assumption 1, it is easy to see that we only need to prove the following result:

In fact, if the above result holds, then we have

Using Assumption 1, we have . First, we show that . We can rewrite as

Then, using the fact that , where , , , and are arbitrary numbers, we may obtain

Using Assumption 1, we have . Next, we show that , . We can rewrite as

By a similar argument as for the discussion

in the proof of , we can obtain

Substituting (12) into (10), we have

By Assumption 1, we can get . Using and the result above, we can obtain that . This completes the proof.

Remark 3. Some results in [1–13] from graph theory can be obtained by Theorem 2. For example, in [11] and in [10].

The following result is one result of Kirchhoff’s Three Theorem (a result in graph theory). In the following, using mathematical induction, we show the result.

Theorem 4. If , , is irreducible, then Assumption 1 holds.

Proof. Obviously, the result holds for . We assume and can rewrite (1) as

Note that (14) is equivalent to the following system:

Consider the following system:

Let

We can rewrite (16) as

From (17), it is easy to see that , , is also irreducible. Note that . According to the discussion above, we can deduce that if the result (if , , is irreducible, then Assumption 1 holds) holds for , , then it holds also for . The proof is complete.

3. An Example of an Epidemic Model with Stage Structure and Latency Spreading in a Heterogeneous Host Population

In this section, we present an epidemic model with stage structure and latency spreading in a heterogeneous host population and obtain a more general threshold for the extinction and persistence of a disease. Using the results obtained by mathematical induction and suitable Lyapunov functionals, we prove the global stability of the endemic equilibrium. For some coupled systems of differential equations, by a similar approach to the discussion of the epidemic model, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible.

We formulate an epidemic model with latency spreading in a heterogeneous host population. Let , , , and denote the immature susceptible, mature susceptible, infectious, and recovered population in the th group, respectively. The disease incidence in the th group can be calculated as

where the sum takes into account cross-infections from all groups and is the transmission coefficient between compartments and . Let and represent death rates of and populations, respectively. Let be the latent period of the population. represents the individuals surviving in the latent period and becoming infective at time . Let be integrable function with . We assume that is distributed according to over the interval , where is the upper bound of the latent period. Then, we obtain the following model for a disease with latency:

where denotes influx of individuals into the immature susceptible class in the th group. is the conversion rate from immature individual to mature individual in group . , , and are the natural death rate, the disease-related death rate, and the recovery rate in the th group, respectively. All parameter values are assumed to be nonnegative and , , , , .

Let , , , and . Since the variables do not appear in the remaining three equations of (20), we can consider the following reduced system:

The initial conditions for system (21) take the form

where , the Banach space of continuous functions mapping the interval into .

We see that system (21) exits a disease-free equilibrium , where

Let

and . We assume that

(H1)there exist (, ), such that

(H2)there exist , such that

(H3)there exist , such that

Let

(i) if and only if (H1) holds;(ii) if and only if (H2) holds;(iii) if and only if (H3) holds.

Remark 5. By an approach as the one in [1–3, 7–13], we define . Let , where denotes the spectral radius. If a matrix is irreducible, then, for the eigenvalue of maximum, the associated eigenvector is positive. Note that the authors in [1–3, 7–13] discussed some coupled systems of differential equations under a definition with an approach as the one of definition of and the assumption that the relevant matrix is irreducible. In fact, if and the relevant matrix is irreducible, then ; if and the relevant matrix is irreducible, then ; if and the relevant matrix is irreducible, then . However, the reverse is not true. For example,

Furthermore, let

Obviously, holds also for arbitrary . conforms to the conditions of Theorem 8 but is not in accord with the conditions of Corollary 10. The definition of and the assumption that the relative matrix is irreducible in the results of Corollaries 9 and 10 are analogous with relative definition and assumption in [1–3, 7–13].

The equilibria of (21) are the same as those of the associated ODE system:

Let

From (30) and , we have

We derive from (32) that the region

is a forward invariant compact absorbing set with respect to (30). Let denote the interior of .

Note that is positively invariant with respect to (21). In fact, let

Then, we have

Furthermore, using the fact that , we may obtain that is positively invariant with respect to (21).

Lemma 6. If , then of system (30) is unstable in and there exists a positive equilibrium in .

Proof. Let and . Thus

If , by continuity, we obtain , in a neighborhood of in . This implies that is unstable. Using the uniform persistence result from [15] and by a similar argument to that in the proof in [1], we can show that if , the instability of implies the uniform persistence of system (30). This, together with the uniform boundedness of solutions of (30) in , implies (30) has at least a positive equilibrium in . The proof is completed.

Let

then the components of satisfy

Next, we will study the global stability of equilibria of system (21).

Theorem 7. If , , and , then of system (21) is globally asymptotically stable in .

Proof. Let . Consider a Lyapunov functional , where

Differentiating along the solution of system (21), we obtain

Differentiating along the solution of system (21), we obtain

Therefore

From (23), we know that

By (45), we obtain

We can rewrite the equation as

By the fact that is strictly decreasing function and the arithmetic-geometric mean, we have

where equality holds if and only if

Thus

If , then if and only if . If , then implies . If and , then (49) holds. If (49) holds, then, from the first two equations of (21), we may obtain , , . Therefore, if and , then we have . Therefore, if and only if and . Hence the largest invariant subset of the set where is the singleton . By LaSalle’s Invariance Principle, is globally attractive. Using the same proof as the one for Corollary 5.3.1 in [16], we can show that is locally stable. Hence, the disease-free equilibrium is globally asymptotically stable in for . This completes the proof.

Theorem 8. Under Assumption 1, of system (21) is globally asymptotically stable in , if , , and .

Proof. Set , . Consider a Lyapunov functional , where

Differentiating along the solution of system (21), we obtain

Differentiating along the solution of system (21), we obtain

Therefore

where

From (38), we know that

It follows from (39), (40), and (56) that

By Theorem 2 and the fact that is strictly decreasing function, we obtain

By the arithmetic-geometric mean, we easily see that

Let . We can rewrite as

By Theorem 2 and the fact that , where equality holds if and only if , we obtain

From (58) and (59), we see that if , then

If (62) holds, it follows from (21) that

Then, we obtain that

Thus,

This implies that

Hence, the largest invariant subset of the set where is the singleton . By LaSalle’s Invariance Principle, is globally attractive. By a similar argument to that in the proof of Theorem 7, is globally asymptotically stable in for .

Following [17], we set matrices

The next generation matrix for system (16) is

Let , where denotes the spectral radius.

By Lemma 6 and Theorems 2, 4, 7, and 8, we can obtain the following results.

Corollary 9. Assume that is irreducible and , . If , then of system (21) is globally asymptotically stable in .

Corollary 10. Assume that is irreducible and , . If , then of system (21) is globally asymptotically stable in .

Remark 11. Note that, for Theorems 7 and 8, we do not assume that is irreducible. If Theorems 7 and 8 hold, then Corollaries 9 and 10 hold too, respectively. However, it is easy to show some examples which conform to the conditions of Theorem 7 or Theorem 8 but are not in accord with the conditions of Corollary 9 or Corollary 10. It seems that all authors use the the graph theory under the assumption that the relative matrix is irreducible which is analogous with the conditions of Corollaries 9 and 10 (see, e.g., [1–13]).

Acknowledgments

This work was supported by the National Basic Research Program of China (2010CB732501), and the National Natural Science Foundation of China (61273015).

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