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

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 𝑣_𝑘>0, such that ^𝑛_𝑗=1𝛽_𝑘𝑗𝑣_𝑘−^𝑛_𝑗=1𝛽_𝑗𝑘𝑣_𝑗=0, (1) where 𝛽_𝑘𝑗≥0, 𝑘,𝑗=1,2,…,𝑛, 𝑛≥2.

Theorem 2. Under Assumption 1, the following results hold: ^𝑛_𝑘,𝑗=1𝑣_𝑘𝛽_𝑘𝑗𝐹_𝑘𝑥_𝑘−𝐹_𝑗𝑥_𝑗=0, (2) where 𝐹_𝑘(𝑥_𝑘), 𝑘=1,2,…,𝑛, are arbitrary functions. In particular, if 𝐹_𝑘(𝑥_𝑘)>0, 𝑘=1,2,…,𝑛, then ^𝑛_𝑘,𝑗=1𝑣_𝑘𝛽_𝑘𝑗𝐹1−^𝑘𝑥_𝑘𝐹_𝑗𝑥_𝑗≤0. (3)

Proof. Note that ^𝑛_𝑘,𝑗=1𝑣_𝑘𝛽_𝑘𝑗𝐹_𝑘𝑥_𝑘−𝐹_𝑗𝑥_𝑗=_{𝑛≥𝑘≥𝑗≥1}𝑣_𝑘𝛽_𝑘𝑗−𝑣_𝑗𝛽_𝑗𝑘𝐹_𝑘𝑥_𝑘−𝐹_𝑗𝑥_𝑗=∶𝑍_𝑛. (4) Let Δ_𝐿=_{𝐿≥𝑘≥𝑗≥2}𝑣_𝑘𝛽_𝑘𝑗−𝑣_𝑗𝛽_𝑗𝑘𝐹_𝑘𝑥_𝑘−𝐹_𝑗𝑥_𝑗,𝐿=2,3,…,𝑛. (5) According to Assumption 1, it is easy to see that we only need to prove the following result: 𝑍_𝑛=^𝑞+1_𝐿=2𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿+𝑣_𝐿𝛽_𝐿𝑛−𝑣_𝑛𝛽_𝑛𝐿+𝑣_𝐿𝛽_{𝐿,𝑛−1}−𝑣_𝑛−1𝛽_{𝑛−1,𝐿}+⋯+𝑣_𝐿𝛽_𝐿,𝑞+2−𝑣_𝑞+2𝛽_𝑞+2,𝐿×𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁+Δ_𝑞+1=∶𝐴_𝑞+1,𝑞=1,2,…,𝑛−2. (6) In fact, if the above result holds, then we have 𝑍_𝑛=^𝑛_𝐿=1𝑣₂𝛽_2𝐿−𝑣_𝐿𝛽_𝐿2𝐹₂𝑥₂−𝐹₁𝑥₁+Δ₂. (7) Using Assumption 1, we have 𝑍_𝑛=Δ₂=0. First, we show that 𝑍_𝑛=𝐴_𝑛−1. We can rewrite 𝑍_𝑛 as 𝑍_𝑛=^𝑛_𝐿=2𝑣_𝑛𝛽_𝑛𝐿−𝑣_𝐿𝛽_𝐿𝑛𝐹_𝑛𝑥_𝑛−𝐹_𝐿𝑥_𝐿+𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁+Δ_𝑛−1. (8) Then, using the fact that 𝑎𝑏+𝑐𝑑=𝑎(𝑏+𝑑)+𝑑(𝑐−𝑎), where 𝑎, 𝑏, 𝑐, and 𝑑 are arbitrary numbers, we may obtain 𝑍_𝑛=^𝑛_𝐿=2𝑣_𝑛𝛽_𝑛𝐿−𝑣_𝐿𝛽_𝐿𝑛𝐹_𝑛𝑥_𝑛−𝐹₁𝑥₁+𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿+𝑣_𝐿𝛽_𝐿𝑛−𝑣_𝑛𝛽_𝑛𝐿×𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁+Δ_𝑛−1=^𝑛_𝐿=2𝑣_𝑛𝛽_𝑛𝐿−𝑣_𝐿𝛽_𝐿𝑛𝐹_𝑛𝑥_𝑛−𝐹₁𝑥₁+𝑣_𝑛𝛽_𝑛1−𝑣₁𝛽_1𝑛𝐹_𝑛𝑥_𝑛−𝐹₁𝑥₁+^𝑛−1_𝐿=2𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿+𝑣_𝐿𝛽_𝐿𝑛−𝑣_𝑛𝛽_𝑛𝐿×𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁+Δ_𝑛−1=^𝑛_𝐿=1𝑣_𝑛𝛽_𝑛𝐿−𝑣_𝐿𝛽_𝐿𝑛𝐹_𝑛𝑥_𝑛−𝐹₁𝑥₁+Δ_𝑛−1+^𝑛−1_𝐿=2𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿+𝑣_𝐿𝛽_𝐿𝑛−𝑣_𝑛𝛽_𝑛𝐿×𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁. (9) Using Assumption 1, we have 𝑍_𝑛=𝐴_𝑛−1. Next, we show that 𝐴_𝑞+1=𝐴_𝑞, 𝑞=1,2,…,𝑛−2. We can rewrite 𝐴_𝑞+1 as 𝐴_𝑞+1=^𝑞+1_𝐿=2𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿+𝑣_𝐿𝛽_𝐿𝑛−𝑣_𝑛𝛽_𝑛𝐿+𝑣_𝐿𝛽_{𝐿,𝑛−1}−𝑣_𝑛−1𝛽_{𝑛−1,𝐿}+⋯+𝑣_𝐿𝛽_𝐿,𝑞+2−𝑣_𝑞+2𝛽_𝑞+2,𝐿×𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁+Δ_𝑞+^𝑞+1_𝑗=2𝑣_𝑞+1𝛽_𝑞+1,𝑗−𝑣_𝑗𝛽_𝑗,𝑞+1𝐹_𝑞+1𝑥_𝑞+1−𝐹_𝑗𝑥_𝑗. (10) By a similar argument as for the discussion ^𝑛_𝐿=2𝑣_𝑛𝛽_𝑛𝐿−𝑣_𝐿𝛽_𝐿𝑛𝐹_𝑛𝑥_𝑛−𝐹_𝐿𝑥_𝐿+𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁=^𝑛_𝐿=1𝑣_𝑛𝛽_𝑛𝐿−𝑣_𝐿𝛽_𝐿𝑛𝐹_𝑛𝑥_𝑛−𝐹₁𝑥₁+^𝑛−1_𝐿=2𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿+𝑣_𝐿𝛽_𝐿𝑛−𝑣_𝑛𝛽_𝑛𝐿×𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁, (11) in the proof of 𝑍_𝑛=𝐴_𝑛−1, we can obtain ^𝑞+1_𝐿=2𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁+^𝑞+1_𝑗=2𝑣_𝑞+1𝛽_𝑞+1,𝑗−𝑣_𝑗𝛽_𝑗,𝑞+1𝐹_𝑞+1𝑥_𝑞+1−𝐹_𝑗𝑥_𝑗=^𝑞+1_𝐿=1𝑣_𝑞+1𝛽_𝑞+1,𝐿−𝑣_𝐿𝛽_𝐿,𝑞+1𝐹_𝑞+1𝑥_𝑞+1−𝐹₁𝑥₁+^𝑞_𝐿=2𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿+𝑣_𝐿𝛽_𝐿,𝑞+1−𝑣_𝑞+1𝛽_𝑞+1,𝐿×𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁. (12) Substituting (12) into (10), we have 𝐴_𝑞+1=^𝑞+1_𝐿=2𝑣_𝐿𝛽_𝐿𝑛−𝑣_𝑛𝛽_𝑛𝐿+𝑣_𝐿𝛽_{𝐿,𝑛−1}−𝑣_𝑛−1𝛽_{𝑛−1,𝐿}+⋯+𝑣_𝐿𝛽_𝐿,𝑞+2−𝑣_𝑞+2𝛽_𝑞+2,𝐿×𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁+Δ_𝑞+^𝑞+1_𝐿=1𝑣_𝑞+1𝛽_𝑞+1,𝐿−𝑣_𝐿𝛽_𝐿,𝑞+1𝐹_𝑞+1𝑥_𝑞+1−𝐹₁𝑥₁+^𝑞_𝐿=2𝑣_𝐿𝛽_𝐿1−𝑣₁𝛽_1𝐿+𝑣_𝐿𝛽_𝐿,𝑞+1−𝑣_𝑞+1𝛽_𝑞+1,𝐿×𝐹_𝐿𝑥_𝐿−𝐹₁𝑥₁=^𝑛_𝐿=1𝑣_𝑞+1𝛽_𝑞+1,𝐿−𝑣_𝐿𝛽_𝐿,𝑞+1𝐹_𝑞+1𝑥_𝑞+1−𝐹₁𝑥₁+𝐴_𝑞. (13) By Assumption 1, we can get 𝐴_𝑞+1=𝐴_𝑞. Using 1−(𝐹_𝑘(𝑥_𝑘)/𝐹_𝑗(𝑥_𝑗))+ln(𝐹_𝑘(𝑥_𝑘)/𝐹_𝑗(𝑥_𝑗))≤0 and the result above, we can obtain that ∑^𝑛_𝑘,𝑗=1𝑣_𝑘𝛽_𝑘𝑗(1−(𝐹_𝑘(𝑥_𝑘)/𝐹_𝑗(𝑥_𝑗)))≤0. This completes the proof.

Remark 3. Some results in [1–13] from graph theory can be obtained by Theorem 2. For example, ∑^𝑛_𝑘,𝑗=1𝑣_𝑘𝛽_𝑘𝑗ln((𝐸_𝑘𝐸^∗_𝑗)/(𝐸^∗_𝑘𝐸_𝑗))=0 in [11] and ∑^𝑛_𝑘,𝑗=1𝑣_𝑘𝛽_𝑘𝑗[𝑛+2−(𝑆^∗_𝑘/𝑆_𝑘)−((𝑆_𝑘𝐼_𝑗𝑦^∗_𝑘,1)/(𝑆^∗_𝑘𝐼^∗_𝑗𝑦_𝑘,1))−∑^𝑛_𝑖=2((𝑦_{𝑘,𝑖−1}𝑦^∗_𝑘,𝑖)/(𝑦^∗_{𝑘,𝑖−1}𝑦_𝑘,𝑖))−((𝑦_𝑘,𝑛𝐼^∗_𝑘)/(𝑦^∗_𝑘,𝑛𝐼_𝑘))≤0] 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 (𝛽_𝑘𝑗)_𝑛×𝑛, 𝑛≥2, is irreducible, then Assumption 1 holds.

Proof. Obviously, the result holds for 𝑛=2. We assume 𝑛≥3 and can rewrite (1) as _𝑙≠1𝛽_1𝑙−𝛽₂₁⋯−𝛽_𝑛1−𝛽_12𝑙≠2𝛽_2𝑙⋯−𝛽_𝑛2−⋮⋮⋱⋮𝛽_1𝑛−𝛽_2𝑛⋯_𝑙≠𝑛𝛽_𝑛𝑙𝑣₁𝑣₂⋮𝑣_𝑛=00⋮0. (14) Note that (14) is equivalent to the following system: _𝑙≠1𝛽_1𝑙−𝛽₂₁⋯−𝛽_𝑛10_𝑙≠2𝛽_2𝑙−𝛽₂₁₁𝛽_2𝑙≠1𝛽_1𝑙⋯−𝛽_𝑛2−𝛽_𝑛11𝛽_2𝑙≠1𝛽_1𝑙⋮⋮⋱⋮0−𝛽_2𝑛−𝛽₂₁₁𝛽_𝑛𝑙≠1𝛽_1𝑙⋯_𝑙≠𝑛𝛽_𝑛𝑙−𝛽_𝑛11𝛽_𝑛𝑙≠1𝛽_1𝑙×𝑣₁𝑣₂⋮𝑣_𝑛=00⋮0. (15) Consider the following system:_𝑙≠2𝛽_2𝑙−𝛽₂₁₁𝛽_2𝑙≠1𝛽_1𝑙−𝛽₃₂−𝛽₃₁₁𝛽_2𝑙≠1𝛽_1𝑙⋯−𝛽_𝑛2−𝛽_𝑛11𝛽_2𝑙≠1𝛽_1𝑙−𝛽₂₃−𝛽₂₁₁𝛽_3𝑙≠1𝛽_{1𝑙𝑙≠3}𝛽_3𝑙−𝛽₃₁₁𝛽_3𝑙≠1𝛽_1𝑙⋯−𝛽_𝑛3−𝛽_𝑛11𝛽_3𝑙≠1𝛽_1𝑙−⋮⋮⋱⋮𝛽_2𝑛−𝛽₂₁₁𝛽_𝑛𝑙≠1𝛽_1𝑙−𝛽_3𝑛−𝛽₃₁₁𝛽_𝑛𝑙≠1𝛽_1𝑙⋯_𝑙≠𝑛𝛽_𝑛𝑙−𝛽_𝑛11𝛽_𝑛𝑙≠1𝛽_1𝑙×𝑣₂⋮𝑣_𝑛=0⋮0. (16)Let−𝑎₁₁−𝑎₂₁⋯−𝑎_𝑛−1,1−𝑎₁₂−𝑎₂₂⋯−𝑎_𝑛−1,2⋮⋮⋱⋮−𝑎_1,𝑛−1−𝑎_2,𝑛−1⋯−𝑎_{𝑛−1,𝑛−1}=^𝑙≠2𝛽_2𝑙−𝛽₂₁₁𝛽_2𝑙≠1𝛽_1𝑙−𝛽₃₂−𝛽₃₁₁𝛽_2𝑙≠1𝛽_1𝑙⋯−𝛽_𝑛2−𝛽_𝑛11𝛽_2𝑙≠1𝛽_1𝑙−𝛽₂₃−𝛽₂₁₁𝛽_3𝑙≠1𝛽_{1𝑙𝑙≠3}𝛽_3𝑙−𝛽₃₁₁𝛽_3𝑙≠1𝛽_1𝑙⋯−𝛽_𝑛3−𝛽_𝑛11𝛽_3𝑙≠1𝛽_1𝑙−⋮⋮⋱⋮𝛽_2𝑛−𝛽₂₁₁𝛽_𝑛𝑙≠1𝛽_1𝑙−𝛽_3𝑛−𝛽₃₁₁𝛽_𝑛𝑙≠1𝛽_1𝑙⋯_𝑙≠𝑛𝛽_𝑛𝑙−𝛽_𝑛11𝛽_𝑛𝑙≠1𝛽_1𝑙. (17) We can rewrite (16) as _𝑙≠1𝑎_1𝑙−𝑎₂₁⋯−𝑎_𝑛−1,1−𝑎_12𝑙≠2𝑎_2𝑙⋯−𝑎_𝑛−1,2⋮⋮⋱⋮−𝑎_1,𝑛−1−𝑎_2,𝑛−1⋯_{𝑙≠𝑛−1}𝑎_{𝑛−1,𝑙}𝑣₂𝑣₃⋮𝑣_𝑛=00⋮0. (18) From (17), it is easy to see that (𝑎_𝑘𝑗)_{(𝑛−1)×(𝑛−1)}, 𝑛≥3, is also irreducible. Note that ∑_𝑙≠1𝛽_1𝑙≠0. According to the discussion above, we can deduce that if the result (if (𝛽_𝑘𝑗)_𝑛×𝑛, 𝑛≥2, is irreducible, then Assumption 1 holds) holds for 𝑛=𝐿, 𝐿≥2, then it holds also for 𝑛=𝐿+1. 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 𝑆_𝑘⁽¹⁾, 𝑆_𝑘⁽²⁾, 𝐼_𝑘, and 𝑅_𝑘 denote the immature susceptible, mature susceptible, infectious, and recovered population in the 𝑘th group, respectively. The disease incidence in the 𝑘th group can be calculated as ^2𝑛_𝑖=1𝑗=1𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)𝐼_𝑗, (19) where the sum takes into account cross-infections from all groups and 𝛽^(𝑖)_𝑘𝑗 is the transmission coefficient between compartments 𝑆_𝑘^(𝑖) and 𝐼_𝑗. Let 𝑑_𝑘⁽¹⁾ and 𝑑_𝑘⁽²⁾ represent death rates of 𝑆_𝑘⁽¹⁾ and 𝑆_𝑘⁽²⁾ populations, respectively. Let 𝜏 be the latent period of the population. 𝑒^{−𝑑𝑘(𝑖)𝜏}𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏) represents the individuals surviving in the latent period 𝜏 and becoming infective at time 𝑡. Let 𝑝_𝑘^(𝑖)(𝜏)∶[0,ℎ]→ℝ₊ be integrable function with ∫^ℎ₀𝑝_𝑘^(𝑖)(𝜏)𝑑𝜏=1. We assume that 𝜏 is distributed according to 𝑝_𝑘^(𝑖)(𝜏) over the interval [0,ℎ], where ℎ is the upper bound of the latent period. Then, we obtain the following model for a disease with latency: 𝑆̇_𝑘⁽¹⁾=𝑏_𝑘−𝑑_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝑆_𝑘⁽¹⁾𝐼_𝑗−𝑎_𝑘𝑆_𝑘⁽¹⁾,𝑆̇_𝑘⁽²⁾=𝑎_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽²⁾_𝑘𝑗𝑆_𝑘⁽²⁾𝐼_𝑗−𝑑_𝑘⁽²⁾𝑆_𝑘⁽²⁾,𝐼̇_𝑘=^2𝑛_𝑖=1𝑗=1^ℎ₀𝑝_𝑘^(𝑖)(𝜏)𝑒^{−𝑑𝑘(𝑖)𝜏}𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗()−𝑑𝑡−𝜏𝑑𝜏_𝑘+𝜇_𝑘+𝛾_𝑘𝐼_𝑘,𝑅̇_𝑘=𝛾_𝑘𝐼_𝑘−𝑑_𝑘𝑅_𝑘,𝑘=1,2,…,𝑛, (20) where 𝑏_𝑘⁽¹⁾ denotes influx of individuals into the immature susceptible class in the 𝑘th group. 𝑎_𝑘 is the conversion rate from immature individual to mature individual in group 𝑘. 𝑑_𝑘, 𝜇_𝑘, and 𝛾_𝑘 are the natural death rate, the disease-related death rate, and the recovery rate in the 𝑘th group, respectively. All parameter values are assumed to be nonnegative and 𝑏_𝑘, ℎ, 𝑎_𝑘, 𝑑_𝑘^(𝑖), 𝑑_𝑘>0.

Let 𝑓_𝑘^(𝑖)(𝜏)=𝑝_𝑘^(𝑖)(𝜏)𝑒^{−𝑑𝑘(𝑖)𝜏}, 𝑐_𝑘^(𝑖)=∫^ℎ₀𝑝_𝑘^(𝑖)(𝜏)𝑒^{−𝑑𝑘(𝑖)𝜏}𝑑𝜏, 𝜑_𝑘(𝑆_𝑘⁽¹⁾)=𝑏_𝑘−𝑑_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾, and 𝑚_𝑘=𝑑_𝑘+𝜇_𝑘+𝛾_𝑘. Since the variables 𝑅_𝑘 do not appear in the remaining three equations of (20), we can consider the following reduced system: 𝑆̇_𝑘⁽¹⁾=𝜑_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝑆_𝑘⁽¹⁾𝐼_𝑗−𝑎_𝑘𝑆_𝑘⁽¹⁾,𝑆̇_𝑘⁽²⁾=𝑎_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽²⁾_𝑘𝑗𝑆_𝑘⁽²⁾𝐼_𝑗−𝑑_𝑘⁽²⁾𝑆_𝑘⁽²⁾,𝐼̇_𝑘=^2𝑛_𝑖=1𝑗=1^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑑𝜏−𝑚_𝑘𝐼_𝑘,𝑘=1,2,…,𝑛. (21) The initial conditions for system (21) take the form 𝑆_𝑘⁽¹⁾(𝜃)=𝜙_𝑘⁽¹⁾(𝜃)≥0,𝑆_𝑘⁽²⁾(𝜃)=𝜙_𝑘⁽²⁾(𝜃)𝐼≥0,_𝑘(𝜃)=𝜙_𝑘(𝜃)≥0,𝜙_𝑘^(𝑖)(0)>0,𝜙_𝑘(0)[],>0,𝑖=1,2,𝑘=1,2,…,𝑛,𝜃∈−ℎ,0 (22) where (𝜙₁⁽¹⁾(𝜃),𝜙₁⁽²⁾(𝜃),…,𝜙_𝑛(𝜃))∈𝐶([−ℎ,0],ℝ₊^3𝑛), the Banach space of continuous functions mapping the interval [−ℎ,0] into ℝ₊^3𝑛.

We see that system (21) exits a disease-free equilibrium 𝑃₀=(𝑆⁽¹⁾₁₀,𝑆⁽²⁾₁₀,…,𝑆⁽¹⁾_𝑛0,𝑆⁽²⁾_𝑛0,0,0,…,0), where 𝜑_𝑘𝑆⁽¹⁾_𝑘0=𝑑_𝑘⁽²⁾𝑆⁽²⁾_𝑘0,𝑎_𝑘𝑆⁽¹⁾_𝑘0=𝑑_𝑘⁽²⁾𝑆⁽²⁾_𝑘0,𝑘=1,2,…,𝑛. (23) Let 𝐴_𝑘𝑗=∑²_𝑖=1𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝑆^(𝑖)_𝑘0𝑚_𝑘,𝑘,𝑗=1,2,…,𝑛, (24) and 𝐐=(𝐴_𝑘𝑗)_𝑛×𝑛. We assume that(H1)there exist (𝑤₁,𝑤₂,…,𝑤_𝑛)>0 (𝑤_𝑘>0, 𝑘=1,2,…,𝑛), such that 𝑤₁,𝑤₂,…,𝑤_𝑛𝐐<𝑤₁,𝑤₂,…,𝑤_𝑛; (25)(H2)there exist (𝑤₁,𝑤₂,…,𝑤_𝑛)>0, such that 𝑤₁,𝑤₂,…,𝑤_𝑛𝐐=𝑤₁,𝑤₂,…,𝑤_𝑛; (26)(H3)there exist (𝑤₁,𝑤₂,…,𝑤_𝑛)>0, such that 𝑤₁,𝑤₂,…,𝑤_𝑛𝐐>𝑤₁,𝑤₂,…,𝑤_𝑛. (27)Let(i)𝑅₀<1 if and only if (H1) holds;(ii)𝑅₀=1 if and only if (H2) holds;(iii)𝑅₀>1 if and only if (H3) holds.

Remark 5. By an approach as the one in [1–3, 7–13], we define 𝑟₀. Let 𝑟₀=𝜌(𝐐), where 𝜌 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 𝑟₀ and the assumption that the relevant matrix is irreducible. In fact, if 𝑟₀<1 and the relevant matrix is irreducible, then 𝑅₀<1; if 𝑟₀=1 and the relevant matrix is irreducible, then 𝑅₀=1; if 𝑟₀>1 and the relevant matrix is irreducible, then 𝑅₀>1. However, the reverse is not true. For example, 𝐴_𝑘𝑗2×2=1012. (28) Furthermore, let 𝐴_𝑘𝑗4×4=1100110000110011. (29) Obviously, 𝑅₀>1 holds also for arbitrary (𝑤₁,𝑤₂,𝑤₃,𝑤₄)>0. (𝐴_𝑘𝑗)_4×4 conforms to the conditions of Theorem 8 but is not in accord with the conditions of Corollary 10. The definition of 𝑟₀ 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: 𝑆̇_𝑘⁽¹⁾=𝜑_𝑘𝑆_𝑘⁽¹⁾−𝑆_𝑘^𝑛(1)_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝐼_𝑗−𝑎_𝑘𝑆_𝑘⁽¹⁾,𝑆̇_𝑘⁽²⁾=𝑎_𝑘𝑆_𝑘⁽¹⁾−𝑆_𝑘^𝑛(2)_𝑗=1𝛽⁽²⁾_𝑘𝑗𝐼_𝑗−𝑑_𝑘⁽²⁾𝑆_𝑘⁽²⁾,𝐼̇_𝑘=²_𝑖=1𝑐_𝑘^(𝑖)𝑆_𝑘^𝑛(𝑖)_𝑗=1𝛽^(𝑖)_𝑘𝑗𝐼_𝑗−𝑚_𝑘𝐼_𝑘,𝑘=1,2,…,𝑛. (30) Let 𝑁_𝑘=𝑆_𝑘⁽¹⁾+𝑆_𝑘⁽²⁾+𝐼_𝑘,𝑑_𝑘=min𝑑_𝑘⁽¹⁾,𝑑_𝑘⁽²⁾,𝑑_𝑘,𝑘=1,2,…,𝑛. (31) From (30) and 𝑐_𝑘^(𝑖)≤1, we have 𝑁̇_𝑘≤𝜑_𝑘𝑆_𝑘⁽¹⁾+𝑑_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾−𝑑_𝑘𝑁_𝑘. (32) We derive from (32) that the region Γ=𝑆₁⁽¹⁾,𝑆₁⁽²⁾,…,𝑆_𝑛⁽¹⁾,𝑆_𝑛⁽²⁾,𝐼₁,…,𝐼_𝑛∈ℝ₊^3𝑛∶𝑆_𝑘⁽¹⁾≤𝑆⁽¹⁾_𝑘0,𝑆_𝑘⁽²⁾≤𝑆⁽²⁾_𝑘0,𝑆_𝑘⁽¹⁾+𝑆_𝑘⁽²⁾+𝐼_𝑘≤𝜑_𝑘(0)+𝑑_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0𝑑_𝑘,𝑘=1,2,…,𝑛 (33) is a forward invariant compact absorbing set with respect to (30). Let Γ^∘ denote the interior of Γ.
Note that Γ is positively invariant with respect to (21). In fact, let 𝐸̇_𝑘=^2𝑛_𝑖=1𝑗=1𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)𝐼_𝑗−^2𝑛_𝑖=1𝑗=1^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑑𝜏−𝑑_𝑘𝐸_𝑘,𝑘=1,2,…,𝑛. (34) Then, we have 𝑁̇_𝑘+𝐸̇_𝑘≤𝜑_𝑘𝑆_𝑘⁽¹⁾+𝑑_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾−𝑑_𝑘𝑁_𝑘+𝐸_𝑘. (35) Furthermore, using the fact that 𝑁_𝑘≤𝑁_𝑘+𝐸_𝑘, we may obtain that Γ is positively invariant with respect to (21).

Lemma 6. If 𝑅₀>1, then 𝑃₀ of system (30) is unstable in Γ and there exists a positive equilibrium 𝑃^∗ in Γ^∘.

Proof. Let 𝐈=(𝐼₁,𝐼₂,…,𝐼_𝑛) and 𝑀=∑^𝑛_𝑘=1((𝜔_𝑘𝐼_𝑘)/(𝑚_𝑘)). Thus 𝑀̇=^𝑛_𝑘=1𝜔_𝑘∑²_𝑖=1𝑐_𝑘^(𝑖)∑^𝑛_𝑗=1𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)𝐼_𝑗𝑚_𝑘−𝐼_𝑘≤𝜔₁,𝜔₂,…,𝜔_𝑛𝐐𝐈^𝑇−𝐈^𝑇. (36) If 𝑅₀>1, by continuity, we obtain 𝑀̇>0, in a neighborhood of 𝑃₀ in Γ^∘. This implies that 𝑃₀ 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 𝑅₀>1, the instability of 𝑃₀ implies the uniform persistence of system (30). This, together with the uniform boundedness of solutions of (30) in Γ^∘, implies (30) has at least a positive equilibrium 𝑃^∗ in Γ^∘. The proof is completed.

Let 𝑃^∗=𝑆₁^(1)∗,𝑆₁^(2)∗,…,𝑆_𝑛^(1)∗,𝑆_𝑛^(2)∗,𝐼^∗₁,𝐼^∗₂,…,𝐼^∗_𝑛; (37) then the components of 𝑃^∗ satisfy 𝜑_𝑘𝑆_𝑘^(1)∗=²_𝑖=1𝑆_𝑘^{𝑛(𝑖)∗}_𝑗=1𝛽^(𝑖)_𝑘𝑗𝐼^∗_𝑗+𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗,_2𝑖=1𝑐_𝑘^(𝑖)𝑆_𝑘^{𝑛(𝑖)∗}_𝑗=1𝛽^(𝑖)_𝑘𝑗𝐼^∗_𝑗=𝑚_𝑘𝐼^∗_𝑘,𝑎_𝑘𝑆_𝑘^(1)∗=𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗+𝑆_𝑘^𝑛(2)∗_𝑗=1𝛽⁽²⁾_𝑘𝑗𝐼^∗_𝑗,𝑘=1,2,…,𝑛. (38)(39)(40) Next, we will study the global stability of equilibria of system (21).

Theorem 7. If 𝑐_𝑘⁽¹⁾≥𝑐_𝑘⁽²⁾, 𝑘=1,2,…,𝑛, and 𝑅₀≤1, then 𝑃₀ of system (21) is globally asymptotically stable in Γ.

Proof. Let 𝐒^𝟎=(𝑆⁽¹⁾₁₀,𝑆⁽²⁾₁₀,…,𝑆⁽¹⁾_𝑛0,𝑆⁽²⁾_𝑛0). Consider a Lyapunov functional 𝐿₁+𝐿₂, where 𝐿₁=^𝑛_𝑘=1𝜔_𝑘𝑚_𝑘²_𝑖=1𝑐_𝑘^(𝑖)𝑆_𝑘^(𝑖)−𝑆^(𝑖)_𝑘0−𝑆^(𝑖)_𝑘0𝑆ln^𝑘(𝑖)𝑆^(𝑖)_𝑘0+𝐼_𝑘,𝐿₂=^𝑛_𝑘=1𝜔_𝑘𝑚_𝑘^2𝑛_𝑖=1𝑗=1𝛽^(𝑖)_𝑘𝑗^ℎ₀𝑓_𝑘^(𝑖)(𝜏)^𝑡_𝑡−𝜏𝑆_𝑘^(𝑖)(𝑥)𝐼_𝑗(𝑥)𝑑𝑥𝑑𝜏. (41) Differentiating 𝐿₁ along the solution of system (21), we obtain 𝐿̇₁=^𝑛_𝑘=1𝜔_𝑘𝑚_𝑘𝑐_𝑘⁽¹⁾𝜑_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝑆_𝑘⁽¹⁾𝐼_𝑗−𝑎_𝑘𝑆_𝑘⁽¹⁾+𝑐_𝑘⁽²⁾𝑎_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽²⁾_𝑘𝑗𝑆_𝑘⁽²⁾𝐼_𝑗−𝑑_𝑘⁽²⁾𝑆_𝑘⁽²⁾+^2𝑛_𝑖=1𝑗=1^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝛽^(𝑖)_𝑘𝑗𝑆^(𝑖)_𝑘𝑗(𝑡−𝜏)×𝐼_𝑗(𝑡−𝜏)𝑑𝜏−𝑚_𝑘𝐼_𝑘−𝑐_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽¹⁾𝜑_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝑆_𝑘⁽¹⁾𝐼_𝑗−𝑎_𝑘𝑆_𝑘⁽¹⁾−𝑐_𝑘⁽²⁾𝑆⁽²⁾_𝑘0𝑆_𝑘⁽²⁾𝑎_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽²⁾_𝑘𝑗𝑆_𝑘⁽²⁾𝐼_𝑗−𝑑_𝑘⁽²⁾𝑆_𝑘⁽²⁾. (42) Differentiating 𝐿₂ along the solution of system (21), we obtain 𝐿₂=^𝑛_𝑘=1𝜔_𝑘𝑚_𝑘^2𝑛_𝑖=1𝑗=1𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)𝐼_𝑗−^2𝑛_𝑖=1𝑗=1𝛽^(𝑖)_𝑘𝑗^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑑𝜏. (43) Therefore 𝐿̇=^𝑛_𝑘=1𝜔_𝑘𝑚_𝑘𝜑_𝑘𝑆_𝑘⁽¹⁾𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽¹⁾+𝑑_𝑘⁽²⁾𝑆⁽²⁾_𝑘0×𝑐_𝑘⁽²⁾−𝑐_𝑘⁽²⁾𝑆_𝑘⁽²⁾𝑆⁽²⁾_𝑘0+𝑎_𝑘𝑆⁽¹⁾_𝑘0×𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0+𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0−𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆⁽²⁾_𝑘0𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽²⁾+^2𝑛_𝑖=1𝑗=1𝑆^(𝑖)_𝑘0𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝐼_𝑗−𝑚_𝑘𝐼_𝑘=^𝑛_𝑘=1𝜔_𝑘𝑚_𝑘𝜑_𝑘𝑆_𝑘⁽¹⁾𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽¹⁾+^2𝑛_𝑖=1𝑗=1𝑆^(𝑖)_𝑘0𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝐼_𝑗−𝑚_𝑘𝐼_𝑘+𝑑_𝑘⁽²⁾𝑆⁽²⁾_𝑘0𝑐_𝑘⁽¹⁾+𝑐_𝑘⁽²⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0+𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0−𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆⁽²⁾_𝑘0𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽²⁾−𝑐_𝑘⁽²⁾𝑆_𝑘⁽²⁾𝑆⁽²⁾_𝑘0. (44) From (23), we know that 𝜑_𝑘𝑆⁽¹⁾_𝑘0𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽¹⁾=𝑑_𝑘⁽²⁾𝑆⁽²⁾_𝑘0𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽¹⁾,𝑘=1,2,…,𝑛. (45) By (45), we obtain 𝐿̇=^𝑛_𝑘=1𝜔_𝑘𝑚_𝑘𝜑_𝑘𝑆_𝑘⁽¹⁾−𝜑_𝑘𝑆⁽¹⁾_𝑘0𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽¹⁾+^2𝑛_𝑖=1𝑗=1𝑆^(𝑖)_𝑘0𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝐼_𝑗−𝑚_𝑘𝐼_𝑘+𝑑_𝑘⁽²⁾𝑆⁽²⁾_𝑘02𝑐_𝑘⁽¹⁾+𝑐_𝑘⁽²⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0−𝑐_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽¹⁾+𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0−𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆⁽²⁾_𝑘0𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽²⁾−𝑐_𝑘⁽²⁾𝑆_𝑘⁽²⁾𝑆⁽²⁾_𝑘0. (46) We can rewrite the equation as 𝐿̇=^𝑛_𝑘=1𝜔_𝑘𝑚_𝑘𝜑_𝑘𝑆_𝑘⁽¹⁾−𝜑_𝑘𝑆⁽¹⁾_𝑘0𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽¹⁾+^2𝑛_𝑖=1𝑗=1𝑆^(𝑖)_𝑘0𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝐼_𝑗−𝑚_𝑘𝐼_𝑘+𝑑_𝑘⁽²⁾𝑆⁽²⁾_𝑘02𝑐_𝑘⁽¹⁾+𝑐_𝑘⁽²⁾−𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽²⁾×𝑆_𝑘⁽¹⁾𝑆⁽¹⁾_𝑘0+𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽¹⁾−𝑐_𝑘⁽²⁾𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽¹⁾+𝑆_𝑘⁽¹⁾𝑆⁽²⁾_𝑘0𝑆⁽¹⁾_𝑘0𝑆_𝑘⁽²⁾+𝑆_𝑘⁽²⁾𝑆⁽²⁾_𝑘0. (47) By the fact that 𝜑_𝑘 is strictly decreasing function and the arithmetic-geometric mean, we have 𝐿̇≤^𝑛_𝑘=1𝜔_𝑘𝑚_𝑘^2𝑛_𝑖=1𝑗=1𝑆^(𝑖)_𝑘0𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝐼_𝑗−𝑚_𝑘𝐼_𝑘+𝑑_𝑘⁽²⁾𝑆⁽²⁾_𝑘0×2𝑐_𝑘⁽¹⁾+𝑐_𝑘⁽²⁾−2𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽²⁾−3𝑐_𝑘⁽²⁾=^𝑛_𝑘=1𝜔_𝑘𝑚_𝑘^2𝑛_𝑖=1𝑗=1𝑆^(𝑖)_𝑘0𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝐼_𝑗−𝑚_𝑘𝐼_𝑘=∶𝑈, (48) where equality holds if and only if 𝑆_𝑘⁽¹⁾=𝑆⁽¹⁾_𝑘0,𝑆_𝑘⁽²⁾=𝑆⁽²⁾_𝑘0,𝑘=1,2,…,𝑛. (49) Thus 𝑈=^𝑛_𝑘=1𝜔_𝑘∑²_𝑖=1𝑐_𝑘^(𝑖)∑^𝑛_𝑗=1𝛽^(𝑖)_𝑘𝑗𝑆^(𝑖)_𝑘0𝐼_𝑗𝑚_𝑘−𝐼_𝑘=𝜔₁,𝜔₂,…,𝜔_𝑛𝐐𝐈^𝑇−𝐈^𝑇. (50) If 𝑅₀<1, then 𝐿̇=0 if and only if 𝐈^𝑇=𝟎. If 𝑅₀=1, then 𝐿̇=0 implies 𝑈=0. If 𝑅₀=1 and 𝑈̇=0, then (49) holds. If (49) holds, then, from the first two equations of (21), we may obtain ∑^𝑛_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝑆_𝑘⁽¹⁾𝐼_𝑗≡0, ∑^𝑛_𝑗=1𝛽⁽²⁾_𝑘𝑗𝑆_𝑘⁽²⁾𝐼_𝑗≡0, 𝑘=1,2…,𝑛. Therefore, if 𝑅₀=1 and 𝑈̇=0, then we have 𝐼_𝑘=0. Therefore, 𝐿̇=0 if and only if 𝐈=𝟎 and 𝐒=𝐒^𝟎. Hence the largest invariant subset of the set where 𝐿̇=0 is the singleton {𝑃₀}. By LaSalle’s Invariance Principle, 𝑃₀ is globally attractive. Using the same proof as the one for Corollary 5.3.1 in [16], we can show that 𝑃₀ is locally stable. Hence, the disease-free equilibrium 𝑃₀ is globally asymptotically stable in Γ for 𝑅₀≤1. This completes the proof.

Theorem 8. Under Assumption 1, 𝑃^∗ of system (21) is globally asymptotically stable in Γ, if 𝑐_𝑘⁽¹⁾≥𝑐_𝑘⁽²⁾, 𝑘=1,2,…,𝑛, and 𝑅₀>1.

Proof. Set 𝛽_𝑘𝑗=∑²_𝑖=1𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗, 1≤𝑘,𝑗≤𝑛. Consider a Lyapunov functional 𝑉=𝑉₁+𝑉₂, where 𝑉₁=^𝑛_𝑘=1𝑣_𝑘²_𝑖=1𝑐_𝑘^(𝑖)𝑆_𝑘^(𝑖)−𝑆_𝑘^(𝑖)∗−𝑆_𝑘^(𝑖)∗𝑆ln^𝑘(𝑖)𝑆_𝑘^(𝑖)∗+𝐼_𝑘−𝐼^∗_𝑘−𝐼^∗_𝑘𝐼ln^𝑘𝐼^∗_𝑘,𝑉₂=^𝑛_𝑘=1𝑣_𝑘^2𝑛_𝑖=1𝑗=1𝛽^(𝑖)_𝑘𝑗×^ℎ₀𝑓_𝑘^(𝑖)(𝜏)^𝑡_𝑡−𝜏𝑆_𝑘^(𝑖)(𝑥)𝐼_𝑗(𝑥)−𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗−𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗𝑆×ln^𝑘(𝑖)(𝑥)𝐼_𝑗(𝑥)𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗𝑑𝑥𝑑𝜏. (51) Differentiating 𝑉₁ along the solution of system (21), we obtain 𝑉̇₁=^𝑛_𝑘=1𝑣_𝑘𝑐_𝑘⁽¹⁾𝜑_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝑆_𝑘⁽¹⁾𝐼_𝑗−𝑎_𝑘𝑆_𝑘⁽¹⁾+𝑐_𝑘⁽²⁾𝑎_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽²⁾_𝑘𝑗𝑆_𝑘⁽²⁾𝐼_𝑗−𝑑_𝑘⁽²⁾𝑆_𝑘⁽²⁾+^2𝑛_𝑖=1𝑗=1^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑑𝜏−𝑚_𝑘𝐼_𝑘−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾𝜑_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝑆_𝑘⁽¹⁾𝐼_𝑗−𝑎_𝑘𝑆_𝑘⁽¹⁾−𝑐_𝑘⁽²⁾𝑆_𝑘^(2)∗𝑆_𝑘⁽²⁾𝑎_𝑘𝑆_𝑘⁽¹⁾−^𝑛_𝑗=1𝛽⁽²⁾_𝑘𝑗𝑆_𝑘⁽²⁾𝐼_𝑗−𝑑_𝑘⁽²⁾𝑆_𝑘⁽²⁾−𝐼^∗_𝑘𝐼_𝑘^2𝑛_𝑖=1𝑗=1^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)(𝑡−𝜏)×𝐼_𝑗(𝑡−𝜏)𝑑𝜏−𝑚_𝑘𝐼_𝑘. (52) Differentiating 𝑉₂ along the solution of system (21), we obtain 𝑉̇₂=^𝑛_𝑘=1𝑣_𝑘^2𝑛_𝑖=1𝑗=1𝛽^(𝑖)_𝑘𝑗×𝑐_𝑘^(𝑖)𝑆_𝑘^(𝑖)𝐼_𝑗−^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)+𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗𝑆×ln^𝑘(𝑖)𝐼_𝑗𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗().𝑡−𝜏𝑑𝜏 (53) Therefore 𝑉̇=^𝑛_𝑘=1𝑣_𝑘𝜑_𝑘𝑆_𝑘⁽¹⁾𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾+𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗𝑐_𝑘⁽²⁾−𝑐_𝑘⁽²⁾𝑆_𝑘⁽²⁾𝑆_𝑘^(2)∗+²_𝑖=1𝑐_𝑘^𝑛(𝑖)_𝑗=1𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)∗𝐼_𝑗+𝑚_𝑘𝐼^∗_𝑘−𝑚_𝑘𝐼_𝑘+𝑎_𝑘𝑆_𝑘^(1)∗×𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗+𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗−𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(2)∗𝑆_𝑘^(1)∗𝑆_𝑘⁽²⁾−𝐼^∗_𝑘𝐼_𝑘^2𝑛_𝑖=1𝑗=1^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)×(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑑𝜏−Δ, (54) where Δ=^2𝑛_𝑖=1𝑗=1𝛽^(𝑖)_𝑘𝑗^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗𝑆ln^𝑘(𝑖)𝐼_𝑗𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑑𝜏. (55) From (38), we know that 𝜑_𝑘𝑆_𝑘^(1)∗𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾=^2𝑛_𝑖=1𝑗=1𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗+𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾. (56) It follows from (39), (40), and (56) that 𝑉̇=^𝑛_𝑘=1𝑣_𝑘𝜑_𝑘𝑆_𝑘⁽¹⁾−𝜑_𝑘𝑆_𝑘^(1)∗𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾+𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗𝑐_𝑘⁽²⁾−𝑐_𝑘⁽²⁾𝑆_𝑘⁽²⁾𝑆_𝑘^(2)∗+²_𝑖=1𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗𝐼_𝑗𝐼^∗_𝑗−𝐼_𝑘𝐼^∗_𝑘+1+𝑆_𝑘^𝑛(1)∗_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝐼^∗_𝑗𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾+𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗+𝑆_𝑘^𝑛(2)∗_𝑗=1𝛽⁽²⁾_𝑘𝑗𝐼^∗_𝑗×2𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗+𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗−𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(2)∗𝑆_𝑘^(1)∗𝑆_𝑘⁽²⁾−𝐼^∗_𝑘𝐼_𝑘×^2𝑛_𝑖=1𝑗=1^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗().𝑡−𝜏𝑑𝜏−Δ (57) By Theorem 2 and the fact that 𝜑_𝑘 is strictly decreasing function, we obtain𝑉̇≤^𝑛_𝑘=1𝑣_𝑘𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗𝑐_𝑘⁽²⁾−𝑐_𝑘⁽²⁾𝑆_𝑘⁽²⁾𝑆_𝑘^(2)∗+²_𝑖=1𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗+𝑆_𝑘^𝑛(1)∗_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝐼^∗_𝑗𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾+𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗+𝑆_𝑘^𝑛(2)∗_𝑗=1𝛽⁽²⁾_𝑘𝑗𝐼^∗_𝑗×2𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗+𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗−𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(2)∗𝑆_𝑘^(1)∗𝑆_𝑘⁽²⁾−𝐼^∗_𝑘𝐼_𝑘^2𝑛_𝑖=1𝑗=1^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)×(𝑡−𝜏)𝐼_𝑗()=𝑡−𝜏𝑑𝜏−Δ_𝑛𝑘=1𝑣_𝑘𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗𝑐_𝑘⁽²⁾+2𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗+𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗−𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(2)∗𝑆_𝑘^(1)∗𝑆_𝑘⁽²⁾−𝑐_𝑘⁽²⁾𝑆_𝑘⁽²⁾𝑆_𝑘^(2)∗+𝑆_𝑘^𝑛(2)∗_𝑗=1𝛽⁽²⁾_𝑘𝑗𝐼^∗_𝑗2𝑐_𝑘⁽¹⁾+𝑐_𝑘⁽²⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗+𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗−𝑐_𝑘⁽²⁾𝑆_𝑘⁽¹⁾𝑆_𝑘^(2)∗𝑆_𝑘^(1)∗𝑆_𝑘⁽²⁾+𝑆_𝑘^𝑛(1)∗_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝐼^∗_𝑗2𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾−𝐼^∗_𝑘𝐼_𝑘^2𝑛_𝑖=1𝑗=1^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)×(𝑡−𝜏)𝐼_𝑗()=𝑡−𝜏𝑑𝜏−Δ_𝑛𝑘=1𝑣_𝑘𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗𝑐_𝑘⁽²⁾+2𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽²⁾×𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾+𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗−𝑐_𝑘⁽²⁾𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾+𝑆_𝑘⁽¹⁾𝑆_𝑘^(2)∗𝑆_𝑘^(1)∗𝑆_𝑘⁽²⁾+𝑆_𝑘⁽²⁾𝑆_𝑘^(2)∗+𝑆_𝑘^(2)∗×^𝑛_𝑗=1𝛽⁽²⁾_𝑘𝑗𝐼^∗_𝑗2𝑐_𝑘⁽¹⁾+𝑐_𝑘⁽²⁾−𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽²⁾×𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾+𝑆_𝑘⁽¹⁾𝑆_𝑘^(1)∗−^ℎ₀𝑓_𝑘⁽²⁾(𝜏)×𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾+𝑆_𝑘⁽¹⁾𝑆_𝑘^(2)∗𝑆_𝑘^(1)∗𝑆_𝑘⁽²⁾+𝑆_𝑘⁽²⁾𝐼^∗_𝑘𝑆_𝑘⁽²⁾×(𝑡−𝜏)𝐼_𝑗()×𝑆𝑡−𝜏_𝑘(2)∗𝑆_𝑘⁽²⁾𝐼_𝑘𝐼^∗_𝑗⁻¹𝑑𝜏+𝑆_𝑘^𝑛(1)∗_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝐼^∗_𝑗×2𝑐_𝑘⁽¹⁾−^ℎ₀𝑓_𝑘⁽¹⁾(𝜏)×𝑆_𝑘^(1)∗𝑆_𝑘⁽¹⁾+𝑆_𝑘⁽¹⁾𝐼^∗_𝑘𝑆_𝑘⁽¹⁾(𝑡−𝜏)𝐼_𝑗()×𝑆𝑡−𝜏_𝑘(1)∗𝑆_𝑘⁽¹⁾𝐼_𝑘𝐼^∗_𝑗⁻¹𝑑𝜏−Δ=∶𝐵₁. (58) By the arithmetic-geometric mean, we easily see that 𝐵₁≤^𝑛_𝑘=1𝑣_𝑘𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗𝑐_𝑘⁽²⁾+2𝑐_𝑘⁽¹⁾−2𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽²⁾−3𝑐_𝑘⁽²⁾+𝑆_𝑘^𝑛(2)∗_𝑗=1𝛽⁽²⁾_𝑘𝑗𝐼^∗_𝑗2𝑐_𝑘⁽¹⁾+𝑐_𝑘⁽²⁾−2𝑐_𝑘⁽¹⁾−𝑐_𝑘⁽²⁾−3^ℎ₀𝑓_𝑘⁽²⁾(𝜏)×𝐼^∗_𝑘𝑆_𝑘⁽²⁾(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑆_𝑘⁽²⁾𝐼_𝑘𝐼^∗_𝑗^1/3𝑑𝜏+𝑆_𝑘^𝑛(1)∗_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝐼^∗_𝑗×2𝑐_𝑘⁽¹⁾−2^ℎ₀𝑓_𝑘⁽¹⁾(𝜏)×𝐼^∗_𝑘𝑆_𝑘⁽¹⁾(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑆_𝑘⁽¹⁾𝐼_𝑘𝐼^∗_𝑗^1/2=𝑑𝜏−Δ_𝑛𝑘=1𝑣_𝑘3𝑆_𝑘^𝑛(2)∗_𝑗=1𝛽⁽²⁾_𝑘𝑗𝐼^∗_𝑗×𝑐_𝑘⁽²⁾−^ℎ₀𝑓_𝑘⁽²⁾(𝜏)×𝐼^∗_𝑘𝑆_𝑘⁽²⁾(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑆_𝑘⁽²⁾𝐼_𝑘𝐼^∗_𝑗^1/3𝑑𝜏+2𝑆_𝑘^(1)∗×^𝑛_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝐼^∗_𝑗𝑐_𝑘⁽¹⁾−^ℎ₀𝑓_𝑘⁽¹⁾(𝜏)×𝐼^∗_𝑘𝑆_𝑘⁽¹⁾(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑆_𝑘⁽¹⁾𝐼_𝑘𝐼^∗_𝑗^1/2𝑑𝜏−Δ=∶𝐵₂. (59) Let 𝑌(𝑥)=1−𝑥+ln𝑥. We can rewrite 𝐵₂ as ^𝑛_𝑘=1𝑣_𝑘3𝑆_𝑘^𝑛(2)∗_𝑗=1𝛽⁽²⁾_𝑘𝑗𝐼^∗_𝑗×^ℎ₀𝑓_𝑘⁽²⁾(𝜏)𝑌×𝐼^∗_𝑘𝑆_𝑘⁽²⁾(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑆_𝑘⁽²⁾𝐼_𝑘𝐼^∗_𝑗^1/3𝑑𝜏+2𝑆_𝑘^𝑛(1)∗_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝐼^∗_𝑗×^ℎ₀𝑓_𝑘⁽¹⁾(𝜏)𝑌×𝐼^∗_𝑘𝑆_𝑘⁽¹⁾(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝑆_𝑘⁽¹⁾𝐼_𝑘𝐼^∗_𝑗^1/2−𝑑𝜏_{2𝑛𝑖=1𝑗=1}𝛽^(𝑖)_𝑘𝑗×^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗𝐼×ln^∗𝑘𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝐼_𝑘𝐼^∗_𝑗𝑆_𝑘^(𝑖)𝑑𝜏−Δ. (60) By Theorem 2 and the fact that 𝑌(𝑥)≤0, where equality holds if and only if 𝑥=1, we obtain 𝐵₂≤^𝑛_𝑘=1𝑣_𝑘−^2𝑛_𝑖=1𝑗=1𝛽^(𝑖)_𝑘𝑗^ℎ₀𝑓_𝑘^(𝑖)(𝜏)𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗𝐼×ln^∗𝑘𝑆_𝑘^(𝑖)(𝑡−𝜏)𝐼_𝑗(𝑡−𝜏)𝐼_𝑘𝐼^∗_𝑗𝑆_𝑘^(𝑖)𝑑𝜏−Δ=−^𝑛_𝑘=1𝑣_𝑘^2𝑛_𝑖=1𝑗=1𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝑆_𝑘^(𝑖)∗𝐼^∗_𝑗𝐼ln^∗𝑘𝐼_𝑗𝐼_𝑘𝐼^∗_𝑗=−^𝑛_𝑘,𝑗=1𝑣_𝑘𝛽_𝑘𝑗𝐼ln^∗𝑘𝐼_𝑗𝐼_𝑘𝐼^∗_𝑗=0. (61) From (58) and (59), we see that if 𝑉̇=0, then 𝑆_𝑘^(𝑖)=𝑆_𝑘^(𝑖)∗,𝑖=1,2,𝑘=1,2,…,𝑛. (62) If (62) holds, it follows from (21) that 0=𝜑_𝑘𝑆_𝑘^(1)∗−^𝑛_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝑆_𝑘^(1)∗𝐼_𝑗−𝑎_𝑘𝑆_𝑘^(1)∗,0=𝑎_𝑘𝑆_𝑘^(1)∗−^𝑛_𝑗=1𝛽⁽²⁾_𝑘𝑗𝑆_𝑘^(2)∗𝐼_𝑗−𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗,𝑘=1,2,…,𝑛. (63) Then, we obtain that ^𝑛_𝑗=1𝛽⁽¹⁾_𝑘𝑗𝑆_𝑘^(1)∗𝐼_𝑗=𝜑_𝑘𝑆_𝑘^(1)∗−𝑎_𝑘𝑆_𝑘^(1)∗,_𝑛𝑗=1𝛽⁽²⁾_𝑘𝑗𝑆_𝑘^(2)∗𝐼_𝑗=𝑎_𝑘𝑆_𝑘^(1)∗−𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗,𝑘=1,2,…,𝑛. (64) Thus, 𝐼̇_𝑘=𝑐_𝑘⁽¹⁾𝜑_𝑘𝑆_𝑘^(1)∗−𝑎_𝑘𝑆_𝑘^(1)∗+𝑐_𝑘⁽²⁾𝑎_𝑘𝑆_𝑘^(1)∗−𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗−𝑚_𝑘𝐼_𝑘,𝑘=1,2,…,𝑛. (65) This implies that lim_𝑡→+∞𝐼_𝑘=𝑐_𝑘⁽¹⁾𝜑_𝑘𝑆_𝑘^(1)∗−𝑎_𝑘𝑆_𝑘^(1)∗+𝑐_𝑘⁽²⁾𝑎_𝑘𝑆_𝑘^(1)∗−𝑑_𝑘⁽²⁾𝑆_𝑘^(2)∗𝑚_𝑘=𝐼^∗_𝑘. (66) Hence, the largest invariant subset of the set where 𝑉̇=0 is the singleton {𝑃^∗}. By LaSalle’s Invariance Principle, 𝑃^∗ is globally attractive. By a similar argument to that in the proof of Theorem 7, 𝑃^∗ is globally asymptotically stable in Γ for 𝑅₀>1.

Following [17], we set matrices 𝐅∶=²_𝑖=1𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝑆^(𝑖)_{𝑘0𝑛×𝑛},𝐕∶=diag𝑚₁,𝑚₂,…,𝑚_𝑛. (67) The next generation matrix for system (16) is 𝐐∶=𝐅𝐕^−𝟏=∑²_𝑖=1𝑐_𝑘^(𝑖)𝛽^(𝑖)_𝑘𝑗𝑆^(𝑖)_𝑘0𝑚_{𝑘𝑛×𝑛}. (68) Let 𝑟₀=𝜌(𝐐), where 𝜌 denotes the spectral radius.

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

Corollary 9. Assume that 𝐁=[∑²_𝑖=1𝛽^(𝑖)_𝑘𝑗] is irreducible and 𝑐_𝑘⁽¹⁾≥𝑐_𝑘⁽²⁾, 𝑘=1,2,…,𝑛. If 𝑟₀≤1, then 𝑃₀ of system (21) is globally asymptotically stable in Γ.

Corollary 10. Assume that 𝐁=[∑²_𝑖=1𝛽^(𝑖)_𝑘𝑗] is irreducible and 𝑐_𝑘⁽¹⁾≥𝑐_𝑘⁽²⁾, 𝑘=1,2,…,𝑛. If 𝑟₀>1, then 𝑃^∗ of system (21) is globally asymptotically stable in Γ.

Remark 11. Note that, for Theorems 7 and 8, we do not assume that 𝐁=[∑²_𝑖=1𝛽^(𝑖)_𝑘𝑗] 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).