Research Article | Open Access

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

**Academic Editor:**Xinyu Song

#### 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., [1â€“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., [1â€“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., [1â€“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., [1â€“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., [1â€“3, 7â€“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 [1â€“13] from graph theory can be obtained by Theorem 2. For example, in [11] and in [10].

The following result is one result of Kirchhoffâ€™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 [1â€“3, 7â€“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 [1â€“3, 7â€“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 [1â€“3, 7â€“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 LaSalleâ€™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