- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 925648, 17 pages

http://dx.doi.org/10.1155/2012/925648

## An SLBRS Model with Vertical Transmission of Computer Virus over the Internet

School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received 5 July 2012; Revised 20 August 2012; Accepted 23 August 2012

Academic Editor: Yanbing Liu

Copyright © 2012 Maobin Yang et al. 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

By incorporating an additional recovery compartment in the SLBS model, a new model, known as the SLBRS model, is proposed in this paper. The qualitative properties of this model are investigated. The result shows that the dynamic behavior of the model is determined by a threshold . Specially, virus-free equilibrium is globally asymptotically stable if , whereas the viral equilibrium is globally asymptotically stable if . Next, the sensitivity analysis of to four system parameters is also analyzed. On this basis, a collection of strategies are advised for eradicating viruses spreading across the Internet effectively.

#### 1. Introduction

Malicious computer virus programs form a great threat to information society by acquiring information and hard drives illegally, damaging motherboard, congesting network traffic, making devices out of control, and the like [1–5]. The popularization of the Internet further strengthens this threat; computer viruses can propagate through downloading files, opening email attachments, or even instant messaging [3, 4]. Although a considerable amount of work has been done to protect against the propagation of virus, the human effort to struggle against virus is still in its infancy [1–8].

Because of the development and unpredictability of computer virus, the creation of antivirus software, which aims to analyze the structures of viral codes, always lags behind the creation of virus programs. As a result, antivirus techniques are incompetent to effectively forecast the trend of evolution of virus [3, 4, 7, 8]. For the purpose of understanding the long-term behavior of viruses and posing effective strategies of controlling their spread across the Internet, and in view of that the propagation of computer virus across the Internet resembles that of biological virus across a population, it is appropriate to macroscopically establish and study computer virus propagation models by properly modifying their biological counterparts. Indeed, some classical epidemiological models were simply borrowed to establish computer virus models so as to characterize the way virus propagates [9–16], which share a common assumption that an infected computer in which the virus resides is in latency cannot infect other computers. Very recently, Yang et al. [17–20] proposed a new model, the SLBS model, by taking into account the fact that a computer immediately possesses infection ability once it is infected. This model, however, does not consider that those recovered computers can gain temporary immunity.

In this paper, we propose a new computer virus model, the SLBRS model, which incorporates an additional recovery compartment. In SLBRS model, all the computers connected to the Internet are partitioned into four compartments: uninfected computers having no immunity (*S* computers), infected computers that are latent (*L* computers), infected computers that are breaking out (*B* computers), and uninfected computers having temporary immunity (*R* computers) from which virus is removed after its outbreak. Because the incidence rate of SLBRS model is , not conventional [11, 12], it is a challenge to research this new model. Specially, the global stability of the viral equilibrium for SLBRS model is mostly difficult to resolve by means of constructive Lyapunov function.

To establish our results, we apply a so-called geometric approach to global stability to obtain sufficient conditions for global stability of viral equilibrium, namely, equilibrium with all positive components. It is a generalization of the Poincare-Bendixson criterion for systems of ordinary differential equations, which first appeared in Smith [21] and was further developed in Li and Muldowney [22, 23] and Li [24, 25], the majority of applications refer to epidemic models, as SIR, SEIR, SEIS, SEIRS models (see, e.g., [26–31]), as well as other population dynamics context [32, 33]. It is proved that the dynamical behavior of the SLBRS model is fully determined by a threshold : the virus-free equilibrium is globally asymptotically stable if , whereas the viral equilibrium is globally asymptotically stable if . The theoretical results obtained imply some practical means of eradicating computer viruses distributed over the Internet.

The subsequent materials are organized in this pattern: Sections 2 formulates the SLBRS model and gives its basic reproduction number as well as virus-free and viral equilibria, respectively. Sections 3 examine the global stabilities of the virus-free and viral equilibria, respectively. By analyzing the dependence of on the system parameters, Section 4 poses a set of effective strategies for controlling the spread of computer virus across the Internet. Finally, Section 5 makes a brief summary of this work.

#### 2. SLBRS Model

In this paper, all computers connected to the Internet are partitioned into four compartments: uninfected computers having no immunity (*S* computers), infected computers that are latent (*L* computers), infected computers that are breaking out (*B* computers), and uninfected computers having temporary immunity (*R* computers). Let , , , and denote their corresponding percentages of all computers at time , respectively. The basic assumptions in concern with our model are presented below. All newly connected computers are all virus free. External computers are connected to the Internet at positive constant rate . Also, internal computers are disconnected from the Internet at the same rate . Each virus-free computer gets contact with an infected computer at a bilinear incidence rate , where is positive constant. Latent computers break out at nonnegative constant rate . Breaking-out computers are cured at nonnegative constant rate . Recovered computers become susceptibly virus-free again at nonnegative constant rate . According to the above assumptions, the transfer diagram is depicted in Figure 1.

Therefor, the corresponding computer virus model is of the form:

Because , system (2.1) simplifies to the following three-dimensional subsystem: on the closed, positively invariant set , with initial conditions , , .

Hence, we consider only solutions with initial conditions inside the region , in which the usual existence, uniqueness of solutions, and continuation results hold.

Clearly, the system (2.2) always has the virus-free equilibrium:

As one of the most useful threshold parameters mathematically characterizing the spread of virus, the basic reproduction number, , is defined as the expected number of secondary cases produced by a single (typical) infection in a population of susceptible computers [24]. Let , it follows from system (2.1) that where Let be the Jacobian matrix of at , and the Jacobian matrix of at , respectively, then we get The spectral radius of the matrix is exactly the basic reproduction number of the model, that is, . Hence

Further, system (2.2) also has an interior equilibrium called viral equilibrium given by where and . . It is clear that the viral equilibrium exists if and only if .

In the following sections, we will study the dynamical behavior of the system (2.2).

#### 3. Global Stability

In this section, we study the global stabilities of virus-free and viral equilibria, respectively. First, we have the following.

Theorem 3.1. *
(a) If , then is the only equilibrium and it is globally stable in . (b) If , then is unstable and there exists a unique vrial equilibrium . Furthermore, all solutions starting in and sufficiently close to move away from if . *

*Proof. *Consider a Lyapunov function:
Clearly, is positively definite. By calculation, we get
Because , that is, . It can be seen that holds for . Furthermore, if and only if . Besides, as or . It follows from LaSalle invariance principle [34] that is globally asymptotically stable with respect to if . This proves claim (a).

We linearize the system (2.2) at , giving the Jacobian matrix:
The corresponding eigenvalues of are
Obviously and if which implies . By the stability theory [35], is unstable and there exists a unique vrial equilibrium if . This proves claim (b).

Secondly, we study the global stability of viral equilibrium, . So we have the following.

Lemma 3.2. *The viral equilibrium is locally asymptotically stable with respect to if . *

*Proof. *The Jacobian matrix of the linearized system of system (2.2) evaluated at is
The characteristic equation of is , where
We have
Clearly, , , , and if . By Routh-Hurwitz criterion, is locally asymptotically stable.

It is obvious that is the maximum invariant set on the boundary of . By Theorem 3.1, all orbits started from the interior of will not get away from if . Furthermore, the stable set of , , is equal to and on the boundary of . Then, we can derive the following proposition.

Proposition 3.3. *If , the system (2.2) is uniformly persistent. *

System (2.2) is said to be uniformly persistent in , or rather there exists constant such that provided [9, 36–38]. Here, is independent of initial conditions.

The uniform persistence of (2.2) in the bounded set is equivalent to the existence of a compact that is absorbing for (2.2), namely, each compact set satisfies for sufficiently large , where denotes the solution of (2.2) such that [38]. We state our main result in the following theorem. Appendix A outlines a general mathematical framework for providing global stability, which will be used in the following to prove the Theorem 3.4.

Theorem 3.4. * is globally stable in if . *

*Proof. *From the Appendix A, we know that system (2.2) satisfies the following assumptions. There exists a compact absorbing set . System (2.2) has a unique equilibrium . Let and denote the vector field of system (2.2), the Jacobian matrix associated with a general solution of system (2.2) is
and its second additive compound Jacobian matrix [39, 40] is
We consider the matrix-valued function as
where
and . Here, is the uniform persistence constant in (3.8). Then,
and the matrix in (A.4) can be written in block form:
where
where donates .

Let denote the vectors in , we select a norm in as
and let denote the Lozinskii measure with respect to this norm.

As described in [41], we have the estimate:
Here, , are matrix norms with respect to the vector norm, and denotes the Lozinskii measure with respect to the norm.

We thus obtain
Rewriting (2.2), we have
The uniform persistence constant in (3.8) can be adjusted so that there exists independent of the compact absorbing set, such that
Substituting (3.21) into (3.19) and (3.22) into (3.20) and using (3.23) and our choice of , we obtain, for ,
Therefore, for by (3.18) and (3.24). Along each solution ) to (2.2) such that and for , we thus have
which implies that
from (3.18), proving Theorem 3.4.

Consider system (2.2) with parameter values shown in Table 1.

For , by Theorem 3.1, the virus-free equilibrium of system (2.2) is globally asymptotically stable. Figures 2(a)-2(b) demonstrate how , , , and evolve with the elapse of time in the case of , and 0.9944 in Table 1, respectively.

For , by Theorem 3.4, the viral equilibrium of system (2.2) is globally asymptotically stable. Figures 3(a)–3(d) demonstrate how , , and evolve with the elapse of time in the case of , 4.2273, 9.9440 and 109.0909 in Table 1, respectively.

#### 4. Discussions

As was indicated in the previous section, it is critical to take various actions to control the system parameters so that is remarkably below one. This section is intended to propose some effective measures for achieving this goal.

For our purpose, it is instructive to examine the sensitivities of to four system parameters: , , , and , respectively. Following Arriola and Hyman [42], the normalized forward sensitivity indices with respect to , , , and are calculated, respectively, as follows: It can be seen that, among these four parameters, , in proportion with , is the most sensitive to the change in . As opposed to this, the other three parameters , , and have an inversely proportional relationship with , an increase in or , or will bring about a decrease in , with a proportionally smaller size of decrease.

Below, let us explain how these properties of model (2.2) can be utilized to control the spread of computer virus.(1)Filtering and blocking suspicious messages with firewall located at the gateway of a domain, the parameter can be kept low and, hence, the chance that a virus-free computer within the domain is infected by a viral computer outside the domain can be significantly decreased, yielding a lower threshold value .(2)Timely updating and running antivirus software of the newest version on computers, the breaking out rate of latent computers, , and the cure rate of breaking out computers, , can be remarkably enhanced, leading to a low .(3)Timely disconnecting computers from the Internet when the connections are unnecessary, the disconnecting rate of computers, , can be made high, bringing about an ideal .

In practice, all of these measures are strongly recommended to achieve a threshold value well below unity, so that viruses within the Internet approach extinction.

#### 5. Conclusion

In nearly all previous computer virus propagation models with latent compartment, to our knowledge, latent computers are assumed not to infect other computers, which does not accord with the real situations. To overcome this defect, the SLBRS model proposed in this paper assumes that all latent computers have infectivity. The dynamics of this model has been fully studied. The results concerning this model include the following. two equilibria, the virus-free equilibrium and the viral equilibrium , as well as the basic reproduction ratio are obtained. The dynamical behavior is determined completely by the value of : implies the global stability of , whereas implies the global stability of . By conducting a sensitive analysis of with respect to various model parameters and on the condition that , a series of measures of strategies is proposed for controlling the spread of virus through the Internet effectively.

Our proof of the global stability of viral equilibrium when utilizes a general approach established in [43], and relies on the construction of a new Lyapunov function for the second compound system.

#### Appendices

#### A. The General Mathematical Framework of Geometric Approach to Global Stability

The presentation here follows that in [22]. Let be a function for in an open set . Consider the differential equation:
Denote by the solution to (A.1) such that . We make the following two assumptions. There exists a compact absorbing set . Equation (A.1) has a unique equilibrium in *D*.

The equilibrium is said to be *globally stable* in if it is locally stable and all trajectories in *D converge to *. The following global-stability problem is formulated in [22].

*Global-stability problem.* Under the assumptions (H1) and (H2), find conditions on the vector field such that the local stability of implies its global stability in .

The assumptions (H1) and (H2) are satisfied if is globally stable in . For , a *Bendixson criterion * is a condition satisfied by which precludes the existence of nonconstant periodic solutions of (A.1). A Bendixson criterion is said to be robust under local perturbations of at if, for sufficiently small and neighborhood of , it is also satisfied by such that the support and , where
Such will be called local -perturbations of at . It is easy to see that the classical Bendixson's condition for is robust under local perturbations of at each . Bendixson criteria for higher dimensional systems that are robust are discussed in [21, 22, 44].

A point is wandering for (A.1) if there exists a neighborhood of and such that is empty for all . Thus, for example, all equilibria and limit points are nonwandering. The following is a version of the local closing lemma of Pugh [45, 46] as stated in [43].

Lemma A.1. *Let . Suppose that is a nonwandering point of (A.1) and that . Also assume that the positive semiorbit of has compact closure. Then, for each neighborhood of and , there exists a local -perturbation of at such that *(1)* and*(2)*the perturbed system has a nonconstant periodic solution whose trajectory passes through .*

The following general global-stability principle is established in [43].

Theorem A.2. *Suppose that assumptions (H1) and (H2) hold. Assume that (A.1) satisfies a Bendixson criterion that is robust under local perturbations of at all nonequilibrium nonwandering points for (A.1). Then, is globally stable in provided it is stable. *

The main idea of the proof in [43] for Theorem A.2 is as follows. Suppose that system (A.1) satisfies a Bendixson criterion. Then it does not have any nonconstant periodic solutions. Moreover, the robustness assumption on the Bendixson criterion implies that all nearby differential equations have no nonconstant periodic solutions. Thus, by Lemma A.1, all nonwandering points of (A.1) in must be equilibria. In particular, each omega limit point in must be an equilibrium. Therefore, for all since is the only equilibrium in .

A method of deriving a Bendixson criterion in is developed in [35]. The idea is to show that the second compound equation, with respect to a solution to (A.1), is uniformly asymptotically stable, and the exponential decay rate of all solutions to (A.3) is uniform for in each compact subset of . Here, is the second additive compound matrix of the Jacobian matrix ; see Appendix B. It is an matrix, and thus (A.3) is a linear system of dimension . If is simply connected, the above-mentioned stability property of (A.3) implies the exponential decay of the surface area of any compact two-dimensional surface in , which in turn precludes the existence of any invariant simple closed rectifiable curve in , including periodic orbits. The required uniform asymptotic stability of the linear system (A.3) can be proved by constructing a suitable Lyapunov function.

Let be an matrix-valued function that is for . Assume that exists and is continuous for , the compact absorbing set.

Set
where the matrix is obtained by replacing each entry of by its derivative in the direction of , . Let be a suitable vector norm for , , and let be the *Lozinskii* measure of with respect to the induced matrix norm in , defined by
Define a quantity as
It is shown in [22] that if is simply connected and , the function is a Lyapunov function for (A.3), and hence (A.1) has no orbit that gives rise to an invariant simple closed rectifiable curve, such as periodic orbits, homoclinic orbits, and heteroclinic cycles. Hence, is a Bendixson criterion for (A.1) in . Moreover, it is robust under local perturbations of near any nonequilibrium point that is nonwandering. In particular, the following global-stability result is proved in Theorem 3.5 of [22].

Theorem A.3. *Assume that is simply connected and that the assumptions (H1), (H2) hold. Then, the unique equilibrium of (A.1) is globally stable in if . *

We remark that, under the assumptions of Theorem A.3, the condition also implies the local stability of , since, assuming the contrary, is both the alpha and the omega limit point of a homoclinic orbit that is ruled out by the condition .

#### B. The Second Additive Compound Matrix

Let be a linear operator on and also denote its matrix representation with respect to the standard basis of . Let denote the exterior product of . A induces canonically a linear operator on : for , define
and extend the definition over by linearity. The matrix representation of with respect to the canonical basis in is called *the second additive compound* matrix of . This is an matrix and satisfies the property . In the special case, when , we have . In general, each entry of is a linear expression of those of . For instance, when , the second additive compound matrix of is
For detailed discussions of compound matrices and their properties, we refer the reader to [39, 40]. A comprehensive survey on compound matrices and their relations to differential equations is given in [40].

#### Acknowledgment

This work is partially supported by the National Natural Science Foundation of China (Grant no. 61071195) and Chongqing Municipal Natural Science Foundation (Grants no. CSTC2009BA2090, CSTCjjA40014 and CSTC, 2010bb2417).

#### References

- K. Ashton, “That ‘Internet of Things’ Thing,” In: RFID Journal, Abgerufen, 2009, Abgerufen, 2011.
- T. Chen and J. -M. Robert, “The evolution of viruses and worms,” in
*Statistical Methods in Computer Security*, 2004. - F. Cohen,
*A Short Course on Computer Viruses*, John Wiley & Sons, New York, NY, USA, 2nd edition, 1994. - F. Cohen, “Computer virus: theory and experiments,”
*Computers & Security*, vol. 6, no. 1, pp. 22–35, 1987. View at Publisher · View at Google Scholar - “An architectural approach towards the future internet of things,” in
*Architecting the Internet of Things*, U. Dieter, H. Mark, and F. Michahelles, Eds., p. 8, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar - A. Solomon, “A brief history of PC viruses,”
*Computer Fraud & Security Bulletin*, vol. 1993, no. 12, pp. 9–19, 1993. - P. Szor,
*The Art of Computer Virus Research and Defense*, Addison-Wesley Professional, 2005. - J. von Neumann and A. Burks,
*Theory of Self-Reproducing Automata*, U. of Illinois Press, Urbana, Ill, USA, 1966. - X. Han and Q. Tan, “Dynamical behavior of computer virus on Internet,”
*Applied Mathematics and Computation*, vol. 217, no. 6, pp. 2520–2526, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. K. Mishra and N. Jha, “Fixed period of temporary immunity after run of anti-malicious software on computer nodes,”
*Applied Mathematics and Computation*, vol. 190, pp. 1207–1212, 2007. - B. K. Mishra and N. Jha, “SEIQRS model for the transmission of malicious objects in computer network,”
*Applied Mathematical Modelling*, vol. 34, no. 3, pp. 710–715, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. K. Mishra and S. K. Pandey, “Dynamic model of worms with vertical transmission in computer network,”
*Applied Mathematics and Computation*, vol. 217, no. 21, pp. 8438–8446, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. K. Mishra and S. K. Pandey, “Fuzzy epidemic model for the transmission of worms in computer network,”
*Nonlinear Analysis*, vol. 11, pp. 4335–4341, 2010. - J. R. C. Piqueira and V. O. Araujo, “A modified epidemiological model for computer viruses,”
*Applied Mathematics and Computation*, vol. 213, no. 2, pp. 355–360, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Ren, X. Yang, L.-X. Yang, Y. Xu, and F. Yang, “A delayed computer virus propagation model and its dynamics,”
*Chaos, Solitons & Fractals*, vol. 45, no. 1, pp. 74–79, 2012. View at Publisher · View at Google Scholar - J. Ren, X. Yang, Q. Zhu, L.-X. Yang, and C. Zhang, “A novel computer virus model and its dynamics,”
*Nonlinear Analysis*, vol. 13, no. 1, pp. 376–384, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L.-X. Yang, X. Yang, L. Wen, and J. Liu, “A novel computer virus propagation model and its dynamics,”
*International Journal of Computer Mathematics*. In press. View at Publisher · View at Google Scholar - L.-X. Yang, X. Yang, L. Wen, and J. Liu, “Propagation behavior of virus codes in the situation that infected computers are connected to the Internet with
positive probability,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 693695, 13 pages, 2012. View at Publisher · View at Google Scholar - L.-X. Yang, X. Yang, Q. Zhu, and L. Wen, “A computer virus model with graded cure rates,”
*Nonlinear Analysis: Real World Applications*, vol. 14, no. 1, pp. 414–422, 2013. View at Publisher · View at Google Scholar - X. Yang and L.-X. Yang, “Towards the epidemiological modeling of computer viruses,”
*Discrete Dynamics in Nature and Society*, In press. - R. A. Smith, “Some applications of Hausdorff dimension inequalities for ordinary differential equations,”
*Proceedings of the Royal Society of Edinburgh A*, vol. 104, no. 3-4, pp. 235–259, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Y. Li and J. S. Muldowney, “A geometric approach to global-stability problems,”
*SIAM Journal on Mathematical Analysis*, vol. 27, no. 4, pp. 1070–1083, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Y. Li and J. S. Muldowney, “On R. A. Smith's autonomous convergence theorem,”
*The Rocky Mountain Journal of Mathematics*, vol. 25, no. 1, pp. 365–379, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Y. Li, “Dulac criteria for autonomous systems having an invariant affine manifold,”
*Journal of Mathematical Analysis and Applications*, vol. 199, no. 2, pp. 374–390, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Y. Li, “Periodic solutions to a class of autonomous systems,” in
*Differential Equations and Applications to Biology and to Industry*, M. Martelli, K. Cooke, E. Cumberbatch, B. Tang, and H. Thieme, Eds., pp. 299–308, World Scientific, Singapore, 1996. View at Zentralblatt MATH - J. Arino, C. C. McCluskey, and P. van den Driessche, “Global results for an epidemic model with vaccination that exhibits backward bifurcation,”
*SIAM Journal on Applied Mathematics*, vol. 64, no. 1, pp. 260–276, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Buonomo, A. d'Onofrio, and D. Lacitignola, “Global stability of an SIR epidemic model with information dependent vaccination,”
*Mathematical Biosciences*, vol. 216, no. 1, pp. 9–16, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Buonomo and D. Lacitignola, “On the dynamics of an SEIR epidemic model with a convex incidence rate,”
*Ricerche di Matematica*, vol. 57, no. 2, pp. 261–281, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Y. Li and J. S. Muldowney, “Global stability for the SEIR model in epidemiology,”
*Mathematical Biosciences*, vol. 125, no. 2, pp. 155–164, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Y. Li, H. L. Smith, and L. Wang, “Global dynamics an SEIR epidemic model with vertical transmission,”
*SIAM Journal on Applied Mathematics*, vol. 62, no. 1, pp. 58–69, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Wang and M. Y. Li, “Mathematical analysis of the global dynamics of a model for HIV infection of $\mathrm{CD}{4}^{+}$ T cells,”
*Mathematical Biosciences*, vol. 200, no. 1, pp. 44–57, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. Beretta, F. Solimano, and Y. Takeuchi, “Negative criteria for the existence of periodic solutions in a class of delay-differential equations,”
*Nonlinear Analysis*, vol. 50, no. 7, pp. 941–966, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Buonomo and D. Lacitignola, “General conditions for global stability in a single species population-toxicant model,”
*Nonlinear Analysis*, vol. 5, no. 4, pp. 749–762, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. P. LaSalle,
*The Stability of Dynamical Systems*, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1976. - J. Hale,
*Theory of Functional Differential Equations*, Springer, New York, NY, USA, 1977. - G. J. Butler and P. Waltman, “Persistence in dynamical systems,”
*Proceedings of the American Mathematical Society*, vol. 96, pp. 425–430, 1986. - H. I. Freedman, S. G. Ruan, and M. X. Tang, “Uniform persistence and flows near a closed positively invariant set,”
*Journal of Dynamics and Differential Equations*, vol. 6, no. 4, pp. 583–600, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Waltman, “A brief survey of persistence in dynamical systems,” in
*Delay Differential Equations and Dynamical Systems*, vol. 1475, pp. 31–40, Springer, Berlin, Germany, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Fiedler, “Additive compound matrices and an inequality for eigenvalues of symmetric stochastic matrices,”
*Czechoslovak Mathematical Journal*, vol. 24, no. 99, pp. 392–402, 1974. View at Zentralblatt MATH - J. S. Muldowney, “Compound matrices and ordinary differential equations,”
*The Rocky Mountain Journal of Mathematics*, vol. 20, no. 4, pp. 857–872, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. H. Martin, Jr., “Logarithmic norms and projections applied to linear differential systems,”
*Journal of Mathematical Analysis and Applications*, vol. 45, pp. 432–454, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Arriola and J. Hyman, “Forward and adjoint sensitivity analysis with applications in dynamical systems,” Lecture Notes in Linear Algebra and Optimization, 2005.
- M. W. Hirsch, “Systems of differential equations that are competitive or cooperative. VI. A local ${C}^{r}$ closing lemma for 3-dimensional systems,”
*Ergodic Theory and Dynamical Systems*, vol. 11, no. 3, pp. 443–454, 1991. View at Publisher · View at Google Scholar - M. Y. LI and J. S. Muldowney, “On R.A. Smith’s autonomous convergence theorem,”
*Rocky Mountain Journal of Mathematics*, vol. 25, pp. 365–379, 1995. View at Zentralblatt MATH - C. C. Pugh, “The closing lemma,”
*American Journal of Mathematics*, vol. 89, pp. 956–1009, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. C. Pugh and C. Robinson, “The ${C}^{1}$ closing lemma, including Hamiltonians,”
*Ergodic Theory and Dynamical Systems*, vol. 3, no. 2, pp. 261–313, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH