Nonlinear Analysis: Real World Applications 14 (2013) 414–422

Contents lists available at SciVerse ScienceDirect

Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

A computer virus model with graded cure ratesLu-Xing Yang a,b,∗, Xiaofan Yang a,∗, Qingyi Zhu a, Luosheng Wen b

a College of Computer Science, Chongqing University, Chongqing, 400044, Chinab College of Mathematics and Statistics, Chongqing University, Chongqing, 400044, China

a r t i c l e i n f o

Article history:Received 31 January 2012Accepted 1 July 2012

Keywords:Computer virusDynamical modelEquilibriumGlobal stabilityLyapunov functionControl strategy

a b s t r a c t

A dynamical model characterizing the spread of computer viruses over the Internet isestablished, in which two assumptions are imposed: (1) a computer possesses infectivityonce it is infected, and (2) latent computers have a lower cure rate than seizing computers.The qualitative properties of this model are fully studied. First, the basic reproductionnumber, R0, for this model is determined. Second, by introducing appropriate Lyapunovfunctions, it is proved that the virus-free equilibrium is globally asymptotically stable ifR0 ≤ 1, whereas the viral equilibrium is globally asymptotically stable if 1 < R0 ≤ 4. Next,the sensitivity analysis of R0 to three system parameters is conducted, and the dependenceof R0 on the remaining system parameters is investigated. On this basis, a set of policies isrecommended for eradicating viruses spreading across the Internet effectively.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Computer viruses are malicious codes or programs which can replicate themselves and spread via wired or wirelessnetworks. With the ever increasing number of Internet applications, computer viruses have come to be a great threat to ourwork and daily life. With the advent of the Internet of Things, this threat would become increasingly serious. Consequently,it is urgent to understand how computer viruses spread over the Internet and to propose effective measures to cope withthis issue. To achieve this goal, and in view of the fact that the spread of virus among computers resembles that of biologicalvirus among a population, it is suitable to establish dynamicalmodels describing the propagation of computer viruses acrossthe Internet by appropriately modifying epidemic models [1].

One common feature shared by a computer virus and a biological virus is infectivity [2]. Based on this fact, some classicepidemicmodels, such as the SIRS model [3–7], SEIRmodel [8,9], SEIRS model [10], SEIQV model [11] and SEIQRS model [12],were usually borrowed to depict the spread of a computer virus in such a way that (1) susceptible individuals correspondto uninfected computers, (2) latent patients correspond to infected computers in which all viruses are in latency, and(3) infecting patients correspond to infected computers in which at least one virus is breaking out. In biological background,it is well known that an infected individual who is in latency cannot infect other individuals. In computer background,however, an infected computer which is in latency can infect other computers through, say, file copying or file downloading.Unfortunately, all the previous computer virus models failed to consider this passive infectivity. Indeed, a reasonablecomputer virus model should assume that both seizing and latent computers have infectivity.

On one hand, a computer user will clear viruses within his computer by running antivirus software immediately when heclearly perceives the existence of viruses (he feels that his computer is suffering from significant performance degradation,say). On the other hand, a computer user might also try to clear viruses spontaneously even if he is not sure that viruses arestaying in his computer possibly because

∗ Corresponding authors.E-mail addresses: ylx910920@gmail.com (L.-X. Yang), xf_yang1964@yahoo.com (X. Yang).

1468-1218/$ – see front matter© 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2012.07.005

L.-X. Yang et al. / Nonlinear Analysis: Real World Applications 14 (2013) 414–422 415

(1) he is accustomed to running antivirus program regularly, or(2) he is informed that viruses are spreading over the Internet.

As a consequence, latent computers would be cured at a lower (but positive) rate than seizing computers. When attemptingto model a computer virus, this feature of graded cure rates should be taken into consideration.

In this paper, a new computer virus propagation model, which incorporates the two features mentioned above, isproposed. One major difficulty in studying the qualitative properties of this model lies in the construction of suitableLyapunov functions. We choose to use linear combinations of quadratic functions in independent variables as the candidateLyapunov functions. Equippedwith this tool, it is proved that the dynamic behavior of themodel is determinedby a thresholdR0. Specifically, the virus-free equilibrium is globally asymptotically stable if R0 ≤ 1, whereas the viral equilibrium is locallyasymptotically stable if R0 > 1 and, furthermore, is globally asymptotically stable if R0 ≤ 4. Based on these results andfurther analysis, some effective strategies for eradicating computer viruses are recommended.

The subsequent materials of this paper is organized as follows: Section 2 formulates the new model and determines itsbasic reproduction number. Section 3 proves the global stability of the virus-free equilibrium. Section 4 examines the localand global stabilities of the viral equilibrium. In Section 5, the dependence of R0 on the system parameters is analyzed, andsome policies for controlling the spread of computer virus are posed. Finally, Section 6 summarizes this work.

2. Mathematical model

A computer is internal or external depending on whether it is currently connected to the Internet or not. For ourpurpose, only internal computers are concerned, and all internal computers are categorized into three classes: uninfectedcomputers (i.e., virus-free computers), infected computers that are currently latent (latent computers, for short), andinfected computers that are currently breaking out (seizing computers, for short). Due to the fact that in the future, thetotal amount of computers in the world would tend to saturation, it is reasonable to suppose that this total number isconstant. Let S(t), L(t) and B(t) denote, at time t , the percentages of uninfected, latent and seizing computers in all internalcomputers, respectively. Then S(t) + L(t) + B(t) ≡ 1. Unless otherwise stated, let S, L and B stand for S(t), L(t) and B(t),respectively.

By carefully considering the features of a computer virus, the following hypotheses are made:(H1) External computers are connected to the Internet at positive constant rate δ, and internal computers are disconnected

from the Internet also at this rate.(H2) All newly connected computers are virus-free.(H3) The percentage of internal computers infected at time t increases by βS(L + B), where β is a positive constant. This

hypothesis says that both seizing and latent computers have infectivity. In contrast, all traditional models assumedthat only seizing individuals have infectivity, i.e., at time t the force of infection can be described as βSf (B) [13].

(H4) Latent computers break out at positive constant rate α.(H5) Latent computers are cured at positive constant rate γ1, while breaking-out computers are cured at positive constant

rate γ2. For the graded cure rates, we have γ2 > γ1 > 0.

Based on the previous assumptions, one can derive the following computer virus propagation model:S = δ − βS(L + B) + γ1L + γ2B − δS,L = βS(L + B) − γ1L − αL − δL,B = αL − γ2B − δB.

(1)

Usually, the basic reproduction number for a virus propagation model is defined as the average number of previouslyvirus-free computers that are infected by a single viral computer during its life cycle. By the physical meanings of the systemparameters in model (1), the following results are obtained:(a) The average lifetime of a latent computer is T1 =

1α+γ1+δ

.(b) A latent computer converts an uninfected computer to a latent one at rate υ1 = β .(c) The average lifetime of a seizing computer is T2 =

1γ2+δ

.(d) A seizing computer converts an uninfected computer to a seizing one at rate υ2 =

βα

α+γ1+δ.

Thus, the basic reproduction number is obtained as

R0 = T1υ1 + T2υ2 =β(α + γ2 + δ)

(α + γ1 + δ)(γ2 + δ). (2)

Because S(t) + L(t) + B(t) ≡ 1, it is sufficient to consider the following two-dimensional subsystem:L = β(1 − L − B)(L + B) − γ1L − αL − δL,B = αL − γ2B − δB

(3)

with initial conditions L(0) ≥ 0 and B(0) ≥ 0. The feasible region for system (3) is Ω = (L, B) : L ≥ 0, B ≥ 0, L + B ≤ 1,which is positively invariant.

An equilibrium of system (3) is virus-free or viral depending on whether L + B = 0 or not.

416 L.-X. Yang et al. / Nonlinear Analysis: Real World Applications 14 (2013) 414–422

Fig. 1. Evolutions of L, B and L + B in the case α = 0.6, β = 0.3, δ = 0.1, γ1 = 0.1, γ2 = 0.3, L(0) = 0.1 and B(0) = 0.

3. The virus-free equilibrium and its stability

Clearly, system (3) always has a virus-free equilibrium E0(0, 0). Now, let us check the stability of this equilibrium withrespect to the feasible region Ω .

Theorem 1. The virus-free equilibrium E0 is globally asymptotically stable with respect to Ω if R0 ≤ 1.

Proof. By use of the Lyapunov Direct Method with undetermined coefficients (see [14]). Consider the candidate function

V (L, B) =12

× (L2 + aB2), (4)

where a is a positive constant to be determined. Clearly, V is positive definite. The time derivative of V along an orbit ofsystem (3) is

dVdt

(3)

= βSL(L + B) − (α + γ1 + δ)L2 + aαLB − a(γ2 + δ)B2

=

βS −

(γ2 + δ)(α + γ1 + δ)

α + δ + γ2

L2 +

βS −

(γ2 + δ)(α + γ1 + δ)

α + δ + γ2

LB

−(α + γ1 + δ)α

α + δ + γ2L2 +

(γ2 + δ)(α + γ1 + δ)

α + δ + γ2+ aα

LB − a(γ2 + δ)B2.

Let a =(α+γ1+δ)(γ2+δ)

α(α+δ+γ2), then

dVdt

(3)

=

βS −

(γ2 + δ)(α + γ1 + δ)

α + δ + γ2

L2 +

βS −

(γ2 + δ)(α + γ1 + δ)

α + δ + γ2

LB

−(α + γ1 + δ)α

α + δ + γ2

L −

γ2 + δ

αB2

= β

S −

1R0

L2 + β

S −

1R0

LB −

(α + γ1 + δ)α

α + δ + γ2

L −

γ2 + δ

αB2

.

Because R0 ≤ 1, it can be seen that dVdt |(3) ≤ 0 holds for all (L, B) ∈ Ω . Furthermore, it is easily verified that dV

dt |(3) = 0 ifand only if (L, B) = (0, 0). Besides, V (L, B) → ∞ as L → ∞ or B → ∞. It follows from the LaSalle Invariance Principle [15]that E0 is globally asymptotically stable with respect to Ω if R0 ≤ 1.

Example 1. Consider system (3) with α = 0.6, β = 0.3, δ = 0.1, γ1 = 0.1, γ2 = 0.3, L(0) = 0.1 and B(0) = 0. By formula(2), we have R0 =

1516 . It follows by Theorem 1 that the virus-free equilibrium is globally asymptotically stable. Fig. 1 displays

the time plot of L(t), B(t) and L(t) + B(t).

L.-X. Yang et al. / Nonlinear Analysis: Real World Applications 14 (2013) 414–422 417

4. The viral equilibrium and its stability

When R0 > 1, system (3) has a unique viral equilibrium E∗(L∗, B∗), where

L∗ =

(γ2 + δ)1 −

1R0

α + δ + γ2

, B∗ =

α1 −

1R0

α + δ + γ2

.

In this section, we study the stability of the viral equilibrium. First, we have

Theorem 2. The viral equilibrium E∗ is locally asymptotically stable if R0 > 1.

Proof. For the linearized system of system (3) at E∗, the corresponding characteristic equation is

λ2+ a1λ + a2 = 0, (5)

where

a1 = 2β(L∗ + B∗) + α + 2δ + γ1 + γ2 − β,

a2 = [2β(L∗ + B∗) + α + γ1 + δ − β](γ + δ) + [2β(L∗ + B∗) − β]α.

Straightforward calculations yield

a1 = β −2(α + γ1 + δ)(γ2 + δ)

α + γ2 + δ+ α + 2δ + γ1 + γ2

= (R0 − 1)(α + γ1 + δ)(γ2 + δ)

α + γ2 + δ+ α + 2δ + γ1 + γ2 −

(α + γ1 + δ)(γ2 + δ)

α + γ2 + δ> 0

and

a2 = [2β(1 − S∗) + α + γ1 + δ − β](γ2 + δ) + [2β(1 − S∗) − β]α

= 2β1 −

1R0

(γ2 + δ) + (α + γ1 + δ)(γ2 + δ) + 2αβ

1 −

1R0

− β(α + γ2 + δ)

= β(α + γ2 + δ)

1 −

1R0

> 0.

It follows from the Hurwitz criterion [15] that the two roots of Eq. (5) have negative real parts. Hence, E∗ is locally asymp-totically stable.

Now, it is the turn to study the global stability of E∗. Let Ω ′= Ω − E0, then we have

Theorem 3. The viral equilibrium E∗ is globally asymptotically stable with respect to Ω ′ if 1 < R0 ≤ 4.

Proof. Consider the candidate function

V (L, B) =12[(L − L∗) + (B − B∗)]

2+

12a(L − L∗)

2, (6)

where a is a positive constant to be determined. It can be seen that V is positive definite with respect to (L∗, B∗). LetS∗ = 1 − L∗ − B∗. The time derivative of V along an orbit of system (3) is

dVdt

(3)

= (S − S∗)dSdt

+ a(L − L∗)dLdt

= (S − S∗)[−β(S − S∗)(1 − S − S∗) + (γ1 − γ2)(L − L∗) − (γ2 + δ)(S − S∗)]

+ a(L − L∗)[β(S − S∗)(1 − S − S∗) − (α + γ1 + δ)(L − L∗)]

= [β(S + S∗ − 1) − (γ2 + δ)](S − S∗)2+ [γ1 − γ2 + aβ(1 − S − S∗)](L − L∗)(S − S∗)

−a(α + γ1 + δ)(L − L∗)2

= β(S − 1)(S − S∗)2+

(γ1 − γ2)(γ2 + δ)

α + γ2 + δ(S − S∗)

2

+ [γ1 − γ2 + aβ(1 − S − S∗)](L − L∗)(S − S∗) − a(α + γ1 + δ)(L − L∗)2.

418 L.-X. Yang et al. / Nonlinear Analysis: Real World Applications 14 (2013) 414–422

Fig. 2. Evolutions of L, B and L + B in the case α = 0.3, β = 0.4, δ = 0.1, γ1 = 0.1, γ2 = 0.3, L(0) = 0.1 and B(0) = 0.2.

By the Cauchy–Schwartz inequality, we have

[γ1 − γ2 + aβ(1 − S − S∗)](L − L∗)(S − S∗) ≤(γ2 − γ1)(γ2 + δ)

α + γ2 + δ(S − S∗)

2

+[γ1 − γ2 + aβ(1 − S − S∗)]

2

4×

α + γ2 + δ

(γ2 − γ1)(γ2 + δ)(L − L∗)

2.

So,

dVdt

(3)

≤ β(S − 1)(S − S∗)2

+

[γ1 − γ2 + aβ(1 − S − S∗)]

2

4×

α + γ2 + δ

(γ2 − γ1)(γ2 + δ)− a(α + γ1 + δ)

(L − L∗)

2.

Let a =γ2−γ1βS∗

. In view of R0 ≤ 4, the coefficient of (L − L∗)2 becomes

14[γ2 − γ1 + aβ(S + S∗ − 1)]2

α + γ2 + δ

(γ2 − γ1)(γ2 + δ)− a(α + γ1 + δ)

≤ maxS∈[0,1]

14[γ2 − γ1 + aβ(S + S∗ − 1)]2

α + γ2 + δ

(γ2 − γ1)(γ2 + δ)− a(α + γ1 + δ)

=

14(γ2 − γ1 + aβS∗)

2 α + γ2 + δ

(γ2 − γ1)(γ2 + δ)− a(α + γ1 + δ) = 0.

This implies that dVdt |(3) ≤ 0 holds for (L, B) ∈ Ω ′. It is easily verified that dV

dt |(3) = 0 if and only if (L, B) = (L∗, B∗).Therefore, the largest compact invariant set in (L, B) ∈ Ω ′

: V ′= 0 is the singleton E∗. Besides, V (L, B) → ∞ as L → ∞

or B → ∞. Thus, it follows from the LaSalle Invariance Principle that E∗ is globally asymptotically stable with respect to Ω ′

if 1 < R0 ≤ 4.

Example 2. Consider system (3) with α = 0.3, β = 0.4, δ = 0.1, γ1 = 0.1, γ2 = 0.3, L(0) = 0.1 and B(0) = 0.2. ThenR0 = 1.4 < 4, E∗ = ( 8

49 ,649 ). By Theorem 3, E∗ is globally asymptotically stable. Fig. 2 exhibits the time plot of L(t), B(t) and

L(t) + B(t).

Example 3. Consider system (3) with α = 0.05, β = 0.8, δ = 0.1, γ1 = 0.05, γ2 = 0.1, L(0) = 0.1 and B(0) = 0.2. ThenR0 = 5 > 4, E∗ = (0.64, 0.16). By Theorem 2, E∗ is locally asymptotically stable. Fig. 3 demonstrates the time plot of L(t),B(t) and L(t) + B(t), from which one can see that, in this case, E∗ is globally asymptotically stable.

Based on plenty of numerical experiments, we put forward the followingConjecture E∗ is globally asymptotically stable if R0 > 4.In realworld applications, ourmain objective is to control the percentage of viral computers by taking effectivemeasures.

Theorems 1 and 3 tell us that, to eradicate viruses from the Internet, one should take actions to control the systemparametersso that R0 is well below one. Therefore, the global stability of the viral equilibrium in the case R0 > 4, although being aninteresting mathematical problem, is of no practical importance.

L.-X. Yang et al. / Nonlinear Analysis: Real World Applications 14 (2013) 414–422 419

Fig. 3. Evolutions of L, B and L + B in the case α = 0.05, β = 0.8, δ = 0.1, γ1 = 0.05, γ2 = 0.1, L(0) = 0.1 and B(0) = 0.2.

Fig. 4. Evolution of L + B for different values of β in the case α = 0.3, δ = 0.1, γ1 = 0.3, γ2 = 0.2, L(0) = 0.1 and B(0) = 0.1.

5. Discussions

As was indicated in the previous section, it is critical to take various actions to control the system parameters so that R0is remarkably below one. This section is intended to propose some effective measures for achieving this goal.

Clearly, β denotes the infection rate of uninfected computers, whereas γ1 and γ2 denote the cure rates of infectedcomputers. For our purpose, it is instructive to examine the sensitivities of R0 to these three systemparameters, respectively.Following Arriola and Hyman [16], the normalized forward sensitivity indices with respect to β , γ1 and γ2 are calculated,respectively, as follows:

∂R0R0∂β

β

=β

R0

∂R0

∂β= 1 > 0,

∂R0R0∂γ1γ1

=γ1

R0

∂R0

∂γ1=

−γ1

α + γ1 + δ< 0,

∂R0R0∂γ2γ2

=γ2

R0

∂R0

∂γ2=

−αγ2

(α + γ2 + δ)(γ2 + δ)< 0.

It can be seen that, among these three parameters, R0 is most sensitive to the change in β . Indeed, an increase in β wouldyield an increase of the same proportion in R0 (equivalently, a decrease in β would lead to an equal decrease in R0; they

420 L.-X. Yang et al. / Nonlinear Analysis: Real World Applications 14 (2013) 414–422

Fig. 5. Evolution of L + B for different values of γ1 in the case α = 0.3, δ = 0.2, γ2 = 0.3, β = 0.4, L(0) = 0.1 and B(0) = 0.1.

Fig. 6. Evolution of L + B for different values of γ2 in the case α = 0.3, δ = 0.2, γ1 = 0.1, β = 0.4, L(0) = 0.1 and B(0) = 0.1.

are directly proportional) (Fig. 4). As opposed to this, γi(i = 1, 2) have an inversely proportional relationship with R0; anincrease in γi(i = 1, 2) will bring about a decrease in R0 (Figs. 5 and 6), with a proportionally smaller size of decrease.This sensitivity analysis informs us that the old adage ‘‘an ounce of prevention is worth a pound of cure’’ is certainly truein the context of controlling the spread of virus over the Internet. Therefore, it is strongly recommended that one shouldperiodically acquire and run antivirus software of the newest version. Besides, filtering and blocking suspicious messageswith a firewall is also suggested.

As for the two remaining parameters, direct calculations demonstrate that ∂R0∂α

and ∂R0∂δ

are less than zero. So R0 isdecreasing with α (Fig. 7) and with δ (Fig. 8), respectively. This shows that higher disconnecting rate from the Internetis beneficial to control the spread of computer virus. Hence, it is highly recommended that one disconnect his computerfrom the Internet whenever this connection is unnecessary.

6. Conclusions

A new computer virus propagation model has been introduced, where it is assumed that all infected computerspossess infectivity, and latent computers have a lower cure rate than seizing computers. Because this model is remarkablydifferent from all previously proposedmodels, its study needs techniques different from those used for traditional epidemicmodels [17,18]. By taking linear combinations of quadratic functions of independent variables as the candidate Lyapunovfunctions, we have successfully studied the global stability of the proposedmodel. Moreover, we have proposed a collectionof effective measures for controlling the spread of a computer virus over the Internet.

In our opinion, the proposed model collects some features of computer virus better than previous models. Toward thisdirection, our next work is to develop new models which can characterize more features of a computer virus (to studycomputer virus models with delay or impulse).

L.-X. Yang et al. / Nonlinear Analysis: Real World Applications 14 (2013) 414–422 421

Fig. 7. Evolution of L + B for different values of α in the case δ = 0.2, γ1 = 0.1, γ2 = 0.3, β = 0.4, L(0) = 0.1 and B(0) = 0.1.

Fig. 8. Evolution of L + B for different values of δ in the case α = 0.3, γ1 = 0.1, γ2 = 0.3, β = 0.4, L(0) = 0.1 and B(0) = 0.1.

Acknowledgments

The authors are grateful to the anonymous reviewers for their valuable suggestions. This work is supported by NaturalScience Foundation of China (#10771227) and Doctorate Foundation of Educational Ministry of China (#20110191110022).

References

[1] J.O. Kephart, S.R. White, D.M. Chess, Computers and epidemiology, IEEE Spectr. (1993) 20–26.[2] F. Cohen, Computer virus: theory and experiments, Comput. Secur. 6 (1987) 22–35.[3] X. Han, Q.L. Tan, Dynamical behavior of computer virus on internet, Appl. Math. Comput. 217 (2010) 2520–2526.[4] B.K. Mishra, N. Jha, Fixed period of temporary immunity after run of anti-malicious software on computer nodes, Appl. Math. Comput. 190 (2007)

1207–1212.[5] B.K. Mishra, S.K. Pandey, Fuzzy epidemic model for the transmission of worms in computer network, Nonlinear Anal. Real World Appl. 11 (2010)

4335–4341.[6] J.G. Ren, X.F. Yang, L.X. Yang, Y.H. Xu, F.Z. Yang, A delayed computer virus propagation model and its dynamics, Chaos Solitons Fractals 45 (2012)

74–79.[7] J.G. Ren, X.F. Yang, Q.Y. Zhu, L.X. Yang, C.M. Zhang, A novel computer virus model and its dynamics, Nonlinear Anal. Real World Appl. 13 (2012)

376–384.[8] B.K. Mishra, S.K. Pandey, Dynamic model of worms with vertical transmission in computer network, Appl. Math. Comput. 217 (2011) 8438–8445.[9] H. Yuan, G.Q. Chen, Network virus-epidemic model with the point-to-group information propagation, Appl. Math. Comput. 206 (2008) 357–367.

[10] B.K. Mishra, D.K. Saini, SEIRS epidemic model with delay for transmission of malicious objects in computer network, Appl. Math. Comput. 188 (2007)1476–1482.

[11] F.W.Wang, Y.K. Zhang, C.G.Wang, J.F. Ma, S.J. Moon, Stability analysis of a SEIQV epidemicmodel for rapid spreadingworms, Comput. Secur. 29 (2010)410–418.

[12] B.K. Mishra, N. Jha, SEIQRS model for the transmission of malicious objects in computer network, Appl. Math. Model. 34 (2010) 710–715.[13] N.F. Britten, Essential Mathematical Biology, Springer-Verlag, 2003.[14] J.Q. Li, Y.L. Yang, Y.C. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal. Real World Appl. 12 (2011)

2163–2173.

422 L.-X. Yang et al. / Nonlinear Analysis: Real World Applications 14 (2013) 414–422

[15] R.C. Robinson, An Introduction to Dynamical Systems: Continuous and Discrete, Prentice Hall, 2004.[16] L. Arriola, J. Hyman, Forward and adjoint sensitivity analysis with applications in dynamical systems, Lecture Notes in Linear Algebra andOptimization

(2005).[17] H. Hethcote, Z. Ma, S. Liao, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci. 180 (2002) 60–141.[18] J. Mena-Lorca, H. Hethcote, Dynamic models for infectious diseases as regulators of population sizes, J. Math. Biol. 30 (1992) 693–716.