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Infection and Atherosclerosis: Is There anAssociation?

Mojdeh Mohtashemi

MITRE Corporation,202 Burlington Road,Bedford, MA 01730-1420, USAandMIT Department of Computer Science,77 Massachusetts Avenue,Cambridge, MA 02139-4307, USA

Brandon W. Higgs

MITRE Corporation,7515 Colshire Drive,McLean, VA 22102-7508, USA

Richard Levins

Harvard School of Public Health,677 Huntington Avenue,Boston, MA 02115, USA

The role of infectious agents in the etiology of atherosclerosis has longbeen implicated. More recently, however, a few epidemiological studieshave provided data to dispute the positive association between infectionand atherosclerosis. We present a complex system approach using themethod of loop analysis to examine the association between infectionand atherosclerosis under varied assumptions. We find that both positiveand negative associations between infection and atherosclerosis can arise,depending on the study design, the etiological assumptions, and sourcesand directions of change into the system under study. Finally, we demon-strate a set of conditions under which data can be falsely interpretedas lack of any association between infection and atherosclerosis. Themethod of loop analysis can be used as an effective guide for designingepidemiological studies before such investigations are undertaken.

1. Introduction

Atherosclerotic cardiovascular disease is the leading cause of mortalityin the Western societies. The etiology of atherosclerosis is complex andinvolves both intrinsic (e.g., cholesterol balance, blood pressure) andenvironmental factors (e.g., diet, nicotine). Atherosclerosis has longbeen characterized by inflammation and swelling of the arterial wall.

Complex Systems,16(2006) 259276; 2006 Complex Systems Publications, Inc.

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260 M. Mohtashemi, B. W. Higgs, and R. Levins

As early as the 1800s, pathologists had already observed the presence ofwhite blood cells in atherosclerotic lesions suggesting atherosclerosis asan inflammatory disorder. Today, this notion is strongly supported bymounting evidence [1]. Therefore, it was quite reassuring when in 1988

Saikku et al. provided serologic evidence that Chlamydia pneumoniae(C. pneumoniae) may be a potential etiologic agent in atheroscleroticcardiovascular disease due to the inflammatory response of a workinginnate immune defense to infection [2]. Since then other infectiousagents [34], and more recently influenza [58], have been implicatedin atherosclerosis. However, C. pneumoniae is the most extensivelystudied infectious agent for its potential role in atherosclerosis. Today,despite mounting evidence in support of a positive association betweeninfection and atherosclerosis [316], there are conflicting data, some of

which report in favor of no association [1720], while others imply anegative association [21] between atherosclerosis and infection.

Recently, Daneshet al. conducted a nested case-control prospectivecohort study to determine an association between C. pneumoniae IgGtitres and coronary heart disease. The study design included male sub-jects, randomly selected from general practice registers in 24 Britishtowns during the years of 19781980. The subjects were entered intothe British Regional Heart Study [17] and followed for 16 years. Ofthe 5,661 men who provided blood samples, 496 were selected as cases

(defined as those who had either fatal coronary events or nonfatal my-ocardial infarction (MI) by 1996 and had baseline measurements ofIgG serum antibodies toC. pneumoniae). A total of 989 subjects whowere frequency matched to the cases (by towns of residence and age),were randomly selected as controls (defined as those who had survivedto the end of the study without a MI and had baseline C. pneumo-niae measurements). After the 16 year time course, the authors foundno strong association between C. pneumoniae IgG titres and coronaryheart disease.

In another study, Coles et al. investigated the association betweenatherosclerosis and C. pneumoniae with a cross-sectional populationstudy [18]. A sample of 1,034 subjects (near equal split in gender) withsera available for testing C. pneumoniae IgG and IgA antibodies wereselected from 2,000 randomly selected participants in the 1989 Aus-tralian National Heart Foundation Perth Risk Factor Prevalence survey.The authors reported a lack of any association between seropositiv-ity toC. pneumoniaeand carotid atherosclerosis (quantized by carotidultrasound analysis).

In a recent study, Jackson etal. [20] examined the association betweeninfluenza vaccination and a reduction in the risk of recurrent coronaryevents by studying a cohort of survivors from a first MI during an 11-month period. The authors reported that vaccination against influenzadid not reduce the risk of recurrent coronary events in the study cohort,

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Infection and Atherosclerosis: Is There an Association? 261

providing another study conferring no association between atheroscle-rosis and infection.

More notably, however, is a recent prospective study by Kiechl etal. in Bruneck, Italy. The authors set out to test the hypothesis that

genetic variants of toll-like receptor 4 (TLR4) are related to the devel-opment of atherosclerosis [21]. TLR4 is known to confer differencesin the inflammatory response to bacterial lipopolysaccharide (LPS). Atthe 1990 baseline evaluation, they recruited a random sample of theentire population of Bruneck, stratified by gender and age. A total of810 subjects were screened for the TLR4 polymorphisms Asp299Glyand Thr399lle. The extent of carotid atherosclerosis was assessed usinghigh-resolution duplex ultrasonography. Subjects with the Asp299GlyTLR4 allele exhibited significantly lower levels of intravascular inflam-

mation and lower risk of carotid atherosclerosis. However, they werefound to be more susceptible to several bacterial infections when com-pared to subjects with the wild type TLR4 allele. The homozygoussubjects were at an advantage (compared to heterozygotes) exhibitingless vulnerability to infection. This study presents an instance of a neg-ative association between atherosclerosis and infection, although theauthors did not specifically characterize it as such.

Incidentally, in the October 2003 issue of Science, Lee et al. [22]reported the discovery of yet another gene (PPR) that has a regulatory

effect on the inflammatory status of macrophages. The deletion of PPRfrom mice foam cells was shown to increase the availability of inflamma-tory suppressors, thereby reducing atherosclerotic lesion areas by morethan 50 percent. The authors did not, however, report on the extentof susceptibility to infection, mainly because their goal was to test thehypothesis that PPRcontrols the inflammatory status of macrophagesand thus may be a good target for treating atherosclerosis. Although itis tempting to deduce that susceptibility to infection should be height-ened in response to PPR-regulated attenuating levels of inflammation,

this conclusion remains to be tested and verified.

2. Motivation

Why are there so many conflicting studies?Epidemiological studies of patterns of association among different

components of a biological system (such as association between infec-tion and atherosclerosis) are critical to elucidating the underlying dis-ease dynamics, and vital to devising effective intervention or preventive

measures. However, statistical inferences made about patterns of asso-ciations can be misleading if the study is not designed in a manner to testsuch factors. In fact, conflicting data are prevalent in epidemiologicalstudies. When incorrect inference is made about associated patterns, if


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the data quality is to be trusted, then either the question has not beenposed correctly or the study design requires adjustment.

Suppose we ask the question: Is there an association between smokingand heart disease? The answer should invariably be: This depends on

where in the dynamics the change is initiated. If smoking is prevalentin a population, then we may also expect a higher rate of heart diseasein that population. However, if a population has a high rate of heartdisease, we may not necessarily observe a high rate of smoking, inwhich case a high incidence of heart disease may be associated withother contributing factors. Hence, the answer to the question may varydepending on the source and direction of change and how that affectsthe causal relationship in the dynamics.

When posing questions about patterns of correlation between com-

ponents of a dynamic system, the source and direction of change haveto be specified for an accurate evaluation of association. Similarly, theobserved patterns of correlation between atherosclerosis and infectiondepend on: where in the system the change is initiated, number of vari-ables included in the system, the presence or absence of links connectingthe variables, and their associated directionality. In this paper, we aimto investigate the question of whether there is an association between in-fection and atherosclerosis. Using the method of loop analysis [2330]we show that depending on the study design, the source and direction

of change, and patterns of connectivity in the underlying network, theanswer can vary. Finally, we argue that adopting such a dynamic viewis critical for understanding why there are so many conflicting epidemi-ological studies, and in particular, why data on association betweeninfection and atherosclerosis are in discord.

3. Methodology

The method of loop analysis is most appropriate for the qualitative

study of complex systems when quantitative information is either diffi-cult to obtain or simply unavailable [2330]. Such systems consist ofmany interacting components and are constantly perturbed by internalor external impacts. In the absence of numerical information, loop anal-ysis can be used to enhance understanding of the underlying dynamicsand elucidate patterns of association and dependencies between systemcomponents and the direction of change when the system is perturbedfrom equilibrium. For a thorough introduction to loop analysis we referthe reader to the appendix.

Consider the system of three variables A(atherosclerosis),N(inflam-mation), andI(infection), as depicted in Figure 1. Infection stimulatesan inflammatory response (the positive link from I to N representedby ), which in turn subsides the infection (the negative link from Nto I represented by ). Inflammation can induce atherogenecity (the


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ANI ANI ANI ANI

Figure 1. Qualitative model: a system of interactions between atherosclero-

sis (A), inflammation (N), and infection (I).

positive link from Nto A) [31, 32]. It can also be self-damping sinceit is an expression of chemicals or cells attracted to the infection sitefrom a much larger pool, and because immune components have a finite

life span (the negative loop from N to N) [31, 32]. In the absence ofintervention, microorganisms may be self-limiting due to competitiveexclusion, or crowding (the negative loop fromIto I) [33, 34]. Finallyin the absence of exogenous factors, atherosclerosis may undergo recov-ery, thus healing itself and therefore being self-damping (negative loopfromA to A). Figure 1 illustrates the INAsystem.

In order to analyze the feedback system of Figure 1, we need thenotion of feedback at levelk,Fk(see the appendix). Mathematically wehave:

Fk k

m1

(1)m1L(m, k). (1)

Intuitively, equation (1) defines feedback at level kas the net feedbackof all the subsystems consisting ofk variables in a system ofn variables(k n). HereL(m, k) is defined as the product ofm disjunct loops withk variables (m k) where disjunct loops are those loops that have novariable in common (see equation (A.5) in the appendix). For example,

the product of two disjunct loops (m 2) with three variables (k 3)would includeA A (the negative loop of size one from A to A) andthe negative loop of size two consisting of the link I N(Ipositivelyregulates N) and the link N I (N negatively regulates I). In thisexample, no variables are shared between the two loops. By definitionwe always haveF0 1.

For any loop model with nvariables there are n points of entry forparameter changes, one for each variable. A table of predictions canthen be constructed to show how the growth rate for each variable will

change (i.e., whether it increases, decreases, or remains the same), dueto the change in its own parameters, or those of other variables. Thefollowing formula computes the direction (sign) of change for every such


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Effect of parameter change

on growth rate

I N A

+??A

++-N

+++I

Ch

na

eg

in

growthrate

Table 1. Association matrix for the patterns of covariation between atheroscle-

rosis (A), inflammation (N), and infection (I). Patterns are summarized by a

positive effect (), negative effect (), and no association can be made (?).

variable

Xjc

i,k ficpjiF

compnk

Fn. (2)

Here Xj/cindicates the change in variable Xj with respect to change

in parameterc. The formula can be interpreted as follows. Take every

variable Xi that includes the parameter c, determine the direction ofchange caused by c in its own growth rate fi, that is, whether fi/cis positive, negative, or zero. Then trace each possible pathpji from

Xi to Xj, determine its sign, and multiply every such path by its com-

plementary feedback Fcompnk . Finally, sum over all such variables and

paths, and normalize by the overall feedback of the entire system ofnvariables,Fn (see equation (A.9) in the appendix for derivation). Notethat pii 1, that is, the length of the path from a variable to itself isalways the unity. This procedure results in an n ntable of patterns of

covariation among n variables (see Table 1). The table columns indi-cate the variables being affected by change while the table rows indicatethe variables through which the change is initiated. Each entry in thetable indicates the qualitative effect (or ) of change on each variablegrowth rate.

4. Results

To compute possible patterns of covariation by equation (2), first we

need to compute feedback at level three. By equation (1) we have:

F3 (1)3L(2, 3) (1)4L(3, 3)

(1)(A A)(I N)(N I)(I I)(N N)(A A)

() () ().


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Note thatL(1,3) 0 since there is no loop consisting of all three vari-ables. For the system to be stable, we must have F3 0. Then we have:

N

cI

()(I N)Fcomp1

F3

()()(1)2(A A)

() ()

A

cI

()(I N)(NA)Fcomp0

F3

()()()()

() ()

I

cI

()Fcomp2

F3

()(1)3(N N)(A A)

()

()()()()

() ().

This implies that all three variables change in the same direction. In thiscase, we get a positive association between infection and atherosclerosis(see the first row in Table 1).

Now suppose that the parameter being changed is aN of variableN (inflammation). Suppose further that fN/cN > 0. If we applyequation (2) to the three variablesA,I, andNwe get:

N

cN

()Fcomp2

F3

()(1)3(I I)(A A)

()

()()()()

() ()

A

cN

()(N A)Fcomp1

F3

()()(1)2(I I)

()

()()

() ()

I

cN

()(NI)Fcomp1

F3

()()(1)2(A A)

()

()()()()

() ().

Thus if the change in the system is initiated through a parameter ofinflammation, then inflammation and atherosclerosis change in the samedirection while inflammation and infection change in opposite directions

(see the second row in Table 1).By symmetry, iffN/cN < 0, that is, if the change in parameter aN

causes the growth rate for inflammation to decrease then we have

N

cN

()Fcomp2

F3

()(1)3(I I)(A A)

()

()()()()

() ()

A

cN

()(N A)Fcomp1

F3

()()(1)2(I I)

()

()()

() ()

IcN ()(N

I)F

comp

1F3

()()(1)2

(A

A)()

()()()()() ().

Indeed, this is the case in [14]. If the change is initiated through theexpression of the TLR4 variant Asp299Gly causing attenuated levels ofinflammation, then reduction in inflammation reduces susceptibility to


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atherosclerosis while increasing susceptibility to infection. This resultcan also be interpreted as a negative association between infection andatherosclerosis. Hence adopting a dynamic view of mechanisms ofinteraction in theINA system can result in two different yet both

valid answers to the same question.Under what condition(s) may there be no association between athero-

sclerosis and infection? There is more than one way that this can occur.Suppose the change in the system is initiated through a parameter ofatherosclerosis and thatfA/cA >0. In this case we have:

A

cA

()Fcomp2

F3

()[(1)2(N I)(I N)(1)3(I I)(N N)]

()

()[() ()]

()().

But since there is no direct link from A to either Ior N, if the changeis initiated through one of the parameters ofA, the direction of changein I and N cannot be determined under the model of Figure 1, thatis, I/cA ? and N/cA ? (see the third row in Table 1). Inother words, under the model in Figure 1, if the parameter changeis introduced through atherosclerosis, it is not clear whether there is

any association between infection and atherosclerosis. Such uncertaintycan be falsely interpreted as lack of association when in fact there isnot enough information to draw any conclusive assertion. Table 1summarizes these results.

Another condition that may incorrectly be interpreted as no associ-ation between atherosclerosis and infection is when there is more thanone point of entry for change in the system. Suppose the change in thesystem is initiated through parameters of both variablesIandN. Sup-pose further that fN/cN

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multaneous parameter changes in Iand N in the system of Figure 1.Then we have:

I

(cN, cI)

()(N I)Fcomp1 ()F

comp2

F3

() ()

()

() ()

? .

At the same time we have:

A

(cN, cI)

()(N A)Fcomp1 ()(I N)(N A)F

comp0

F3

() ()

()

() ()

().

Then what we observe is a reduction in the rate of atherosclerosis whilethe direction of change in infection cannot be determined, which can bemisinterpreted as lack of association. Indeed, this is the case with the

cited articles reporting lack of (strong) association between atheroscle-rosis and infection [1720]. These are examples of studies that are notdesigned in a manner to provide conclusive assessment of such an as-sociation because the underlying assumptions about the etiology anddirection of change have not been carefully considered. When there isinadequate information to assess the association between two compo-nents, the study often would have to be redesigned in a manner that thesource(s) of change and the interconnectivity between the componentsare well specified.

5. Discussion

We proposed a qualitative approach based on the method of loop anal-ysis to investigate why there are so many conflicting data on associationbetween infection and atherosclerosis. We then demonstrated that botha positive and negative association can arise depending on parametersof the study design which encompass the number of variables includedin the study, the patterns of interconnectivity between variables, and

sources and directions of change that affect the dynamics of etiologyin the system under study. Finally, we showed when the study designdoes not elucidate the sources of change, nor sufficiently captures theunderlying interconnectivity in the system, the observed outcomes can


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be falsely interpreted as lack of any association between infection andatherosclerosis.

Neither Daneshet al. and Coleset al. with respect toC. pneumoniae[17, 18], nor Jackson et al. with respect to influenza [20], found any

appreciable association between atherosclerosis and infection. As wasdemonstrated in our case study of Figure 1 using the method of loopanalysis, if the source(s) and direction(s) of change are not clearly speci-fied and assumptions about the etiology of the dynamics are not carefullyconsidered, then at an arbitrary point in time, it may not be possible toestablish an association between infection and atherosclerosis.

One possible scenario leading to the false conclusion of lack of anyassociation in a study design is as follows. Suppose some of the caseshave high levels of antibody to an infectious agent at baseline due to

a TLR4 polymorphism that attenuates inflammation but such a rela-tionship is not considered in the study. Suppose further that the restof the cases have wild type TLR4 so that high susceptibility to infec-tion also implies high susceptibility to atherosclerosis, yet again notconsidered in the study. Then what we observe is low incidence ofatherosclerosis in the first group since low levels of inflammation im-plies low atherogenecity, and high incidence of atherosclerosis in thesecond group. Under this situation, the association between infectionand atherosclerosis appears ambiguous and can erroneously be inter-

preted as lack of any association, when in fact more information shouldbe collected and further investigation is required. Unless the source anddirection of change are well identified, and the assumptions about theunderlying patterns of interaction between the system components arewell characterized, the study design should be considered incomplete.Consequently, any conclusions drawn about the presence or absence ofassociation between atherosclerosis and infection should be consideredat best arbitrary.

It is important to note some of the constraints as well as the utility

of the presented approach. First, the method of loop analysis does notallow one to assess degrees or strengths of association between vari-ables. As such, a strong association (positive or negative) cannot bedistinguished from a weak one, even though such a gradient might benecessary when evaluating the strengths of cause-and-effect relation-ships (e.g., for devising therapeutic regimens). However, qualitativemodels such as the method of loop analysis are most appropriate forunderstanding the underlying dynamics when numerical assessment ofthe state of variables and parameter values is difficult or impossible.

Furthermore, the method has practical implications for designing stud-ies. By considering more than one model for the study investigatorscan examine the qualitative effects of different variables and patterns ofconnectivity and choose the design that makes most sense biologicallybefore initiating the study or attempting to collect quantitative infor-


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mation about some of the biological parameters, which is often a costlyand time consuming task if not intractable.

Second, in loop analysis the underlying assumption is that the systemunder study is near or at equilibrium, and the subsequent analyses are

directed at capturing the new equilibrium values qualitatively when thesystem is perturbed from its current state of equilibrium. Clearly, suchan assumption may be at odds with the state of reality in biologicalsystems. However, as with all mathematical models, loop analysis isnot an endeavor to embody an exact presentation of the real world. Itis an attempt to simplify a real-world phenomenon in order to captureand understand those aspects of the system that are important to theinvestigator so that selected critical questions about the dynamics canbe answered.

Finally, it should be noted that the system of Figure 1 analyzed hereis quite simple. The underlying simplicity, however, is critical for illus-trating the idea. Complexity can rapidly increase as more variables andnew links are introduced into the system. As such, a well designed studyand well planned data collection scheme is evermore important to avoidcostly studies with false assertions. A complex system approach such asthe method of loop analysis can be used in conjunction with standarddesign procedures as an effective guide for designing epidemiologicalstudies before such immense investigations are undertaken.

Appendix

A. Introduction to loop analysis

The method of loop analysis is most appropriate for the qualitativestudy of complex systems when quantitative information is either diffi-cult to obtain or simply unavailable [2330]. Such systems consist ofmany interacting components and are constantly perturbed by internalor external impacts. In the absence of numerical information, loop anal-ysis can be used to enhance understanding of the underlying dynamicsand elucidate patterns of association and dependencies between systemcomponents that may not be obvious.

There is a one-to-one correspondence between loop models and sys-tems of differential equations, where a system ofn differential equationsrepresents a community ofn interacting components or variables. Thesevariables often represent component abundances or growth rates. LetX

i

be theith variable. Then we define the rate of change or the growthrate ofXi as follows:

dXidt

fi(X1, X2, . . .Xn C1, C2, . . .) (A.1)

whereChrepresents a potential parameter of the system, such as biolog-


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H PH P

11a 22a

21a

12a

H PH P

11a 22a

21a

12a

H PH P

11a 22a

21a

12a

H PH P

11a 22a

21a

12a

Figure A.1. A predatorprey system with a self-damped herbivore (H), and a

self-accelerating predator (P).

ical properties of the variables. If we assume that a system (community)is at or near equilibrium, we can then examine its local stability proper-ties whendXi/dt0, fori 1, . . . , n.

Loop models are also signed digraphs, constructed from the struc-ture of the interaction matrix, or the so-called community matrix inecology, of coefficients of theXievaluated at equilibrium [25]. Given acommunity matrixA as follows:

A

f1X1

f1X2

f1

Xn

fn

X1

fn

X2

fn

Xn

a11 a12 a1na21 a22 a2n

an1 an2 ann

where aij is the coefficient ofXj in fj, that is, the effect of change in

variable Xj on the growth rate of variable Xi, one can translate the

community matrix A uniquely into a signed digraph. The variablesof the system are the vertices of the graph, and the coefficients of thecommunity matrix are the edges or links of the graph.

The coefficients aij of the community matrix are readily taken and

translate into the effect of the variable Xj on the growth rate of the

variable Xi. If the sign ofa

ijis negative, there will be a negative link

fromXjto Xi, represented by the symbol . If the sign ofaijis positive

in the original matrix, meaningXj has a positive impact on the growth

rate of variable Xi, there will be a positive link from Xjto Xi, represented

by the symbol.In a signed digraph there are two types of loops: conjunct and dis-

junct. Conjunct loops consist of those loops that have at least onevariable in common. Disjunct loops have no variable in common. Fig-ure A.1 provides an illustration of loop models for a predatorprey

system, whereHstands for herbivore andP for predator.

A.1 Conditions for stability

The stability properties and the local behavior of systems of differentialequations in the neighborhood of their critical points have been well


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studied. Here, we relate the general conditions of stability, as under-stood in systems theory, to the notion of feedback of a system.

The characteristic polynomial p() of a community matrix A is de-fined asA I, whereIis the identity matrix. For instance, the charac-

teristic polynomial of a 3 3 matrix is:

p()

a11 a12 a13a21 a22 a23a31 a32 a33

.

Expanding the determinant we get:

p() 3 [a11a22a33]2

[(a11a22a12a21) (a11a33 a13a31) (a22a33a23a32)]

[(a11(a22a33a23a32) a12(a23a31a21a33)

a13(a21a32a22a31)] (A.2)

this can be generalized for an n-dimensional matrix as:

p() n n1

k1

(1)kDknk (A.3)

where Dk is the sum of all principal determinants of order k, corre-

sponding to subsystems ofkvariables. But every suchDk can also bewritten as a sum of products of disjunct loops as can be investigatedfrom equation (A.2). That is, we have:

Dk k

m1

(1)kmL(m, k) (A.4)

where L(m, k) is defined as the product of m disjunct loops with kvariables. Note that for an nn matrix, Dn is the determinant of thesquare matrix. We then transform this value into the measure of thefeedback of a matrix. Specifically, we define the notion of feedback atlevelk as follows:

Fk (1)k1Dk fork 1 . . .n

(1)k1k

m1

(1)kmL(m, k) by equation (A.4)

k

m1(1)(1)2kmL(m, k)

k

m1

(1)m1L(m, k) since (1)2km (1)m (1)m (A.5)

where feedback at level 0 is defined asF0 1.


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In this way, feedback at level k is the net feedback of all the subsystemsofk variables in a system ofn variables, wherek 1 . . .n.

As an example for the calculation of feedback terms, consider thepredatorprey system of Figure A.1. At level 1, F1 a11a22; and at

level 2,

F2 (1)2(a12)(a21) (1)

3(a11)(a22) a12a21a11a22.

Equipped with our definition of feedback at level k, as in equa-tion (A.5), we can rewrite equation (A.3) as:

p() n n1

k1

(1)k(1)kmL(m, k)nk

n n1k1

Fknk by equation (A.5). (A.6)

The characteristic polynomial resulting from equation (A.6) involvesthe feedback terms as coefficients. Although, one may not be able togenerally solve polynomials of higher orders, one can determine thesign of Re(i) using the coefficients of such polynomials. By the RouthHurwitz theorem, for the system to be locally stable, that is, for Re(i)

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