Van Den Driessche Research Articles

Inherited LU-factorizations of matrices

Various types of LU-factorizations for nonsingular matrices, where L is a lower triangular matrix and U is an upper triangular matrix, are defined and characterized. These types of LU-factorizations are extended to the general m × n case. The more general conditions are considered in the light of the structures of [C.R. Johnson, D.D. Olesky, P. Van den Driessche, Inherited matrix entries: LU factorizations, SIAM J. Matrix Anal. Appl. 10 (1989) 99–104]. Applications to graphs and adjacency matrices are investigated. Conditions for the product of a lower and an upper triangular matrix to be the zero matrix are also obtained.

Spectrally arbitrary patterns: Reducibility and the 2 n conjecture for n = 5

A sign pattern Z (a matrix whose entries are elements of {+, −, 0}) is spectrally arbitrary if for any self-conjugate spectrum there is a real matrix with sign pattern Z having the given spectrum. Spectrally arbitrary sign patterns were introduced in [J.H. Drew, C.R. Johnson, D.D. Olesky, P. van den Driessche, Spectrally arbitrary patterns, Linear Algebra Appl. 308 (2000) 121–137], where it was (incorrectly) stated that if a sign pattern Z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern, then Z is a spectrally arbitrary sign pattern, and it was conjectured that the converse is true as well; we present counterexamples to both of these statements. In [T. Britz, J.J. McDonald, D.D. Olesky, P. van den Driessche, Minimal spectrally arbitrary patterns, SIAM J. Matrix Anal. Appl. 26 (2004) 257–271] it was conjectured that any n × n spectrally arbitrary sign pattern must have at least 2 n nonzero entries; we establish that this conjecture is true for 5 × 5 sign patterns. We also establish analogous results for nonzero patterns.

A max version of the generalized spectral radius theorem

Let Ψ be a bounded set of n × n nonnegative matrices in max algebra. In this paper we propose the notions of the max algebra version of the generalized spectral radius μ( Ψ) of Ψ, and the max algebra version of the joint spectral radius η( Ψ) of Ψ. The max algebra version of the generalized spectral radius theorem μ( Ψ) = η( Ψ) is established. We propose the relationship between the generalized spectral radius ρ( Ψ) of Ψ (in the sense of Daubechies and Lagarias) and its max algebra version μ( Ψ). Moreover, a generalization of Elsner and van den Driessche’s lemma is presented as well.

Basic Reproduction Number for HIV Model Incorporating Commercial Sex and Behavior Change

The basic reproduction number is obtained for an HIV epidemic model incorporating direct and indirect commercial sex as well as behavior change by the female commercial sex workers (CSWs) and their male customers in response to the proliferation of the disease in the community. A recent result by van den Driessche P., and Watmough J. (Math. Biosci. 180:29-48, 2002) is utilized to compute the threshold parameters for the local asymptotic stability of the Disease-Free Equilibrium (DFE), by considering the transfers in and out of the infective classes. Numerical examples are used to describe the uniqueness and global properties of the endemic equilibrium when DFE is unstable. Biological interpretation of the results obtained in this work is discussed, as are the implications of our results for the design of public health policies such as targeting strategy to target intervention and control measures toward specific high-risk population groups in order to reduce infections. We show that targeting any one sector of the commercial sex alone for prevention will be difficult to have a decided effect on eradicating the epidemic. However, if the aim of the targeted intervention policy is not eradication of the epidemic but decrease in HIV incidence of a particular high-risk group, then concentrated targeting strategy could be sufficient, if properly implemented. This work also demonstrates the usefulness of the theorem of van den Driessche and Watmough (Math. Biosci. 180:29-48, 2002) in obtaining threshold parameters for complicated infectious diseases models.

Multiple bifurcations in a predator–prey system with monotonic functional response

In this paper, we study the dynamics of a predator-prey system with Holling type III functional response of the formx^'=rx1-xK-x^2x^2+ay,y^'=y-d+@wx^2x^2+a-h.When h=0, Chen and Zhang (J.P. Chen, H.D. Zhang, The qualitative analysis of two species predator-prey model with Holling's type III functional response, Appl. Math. Mech. 7 (1) (1986) 73-80) and Kazarinoff and Driessche (N.D. Kazarinoff, P. Van Den Driessche, A model predator-prey system with functional response, Math. Biosci. 39 (1-2) (1978) 125-134) gave qualitative analysis of the above system and obtained some conditions for the existence and stability of positive equilibria and limit cycles. The dynamical phenomena they concluded are normal, and there is no degenerate singularities. But when the constant harvesting is present, we will show that the system may have more complex and more rich dynamics. We carry out bifurcation analysis by choosing the death rate and the harvesting rate of the predator as the bifurcation parameters and show that the system can undergo the Bogdanov-Takens bifurcation. Also the figures of all degenerate structures are given.

On determining minimal spectrally arbitrary patterns

A new family of minimal spectrally arbitrary patterns is presented which allow forarbitrary spectrum by using the Nilpotent-Jacobian method introduced in [J.H. Drew, C.R. Johnson,D.D. Olesky, and P. van den Driessche. Spectrally arbitrary patterns.Lin. Alg. and Appl. 308:121-137, 2000]. The novel approach here is the use of the Intermediate Value Theorem to avoid findingan explicit nilpotent realization of the new minimal spectrally arbitrary patterns.

Comment on ‘How plausible are high-frequency sediment supply-driven cycles in the stratigraphic record?’ by S. Castelltort and J. Van Den Driessche (2003, Sedimentary Geology, 157, 3–13)

Comment on ‘How plausible are high-frequency sediment supply-driven cycles in the stratigraphic record?’ by S. Castelltort and J. Van Den Driessche (2003, Sedimentary Geology, 157, 3–13)

Reply to comment on “How plausible are high-frequency sediment supply-driven cycles in the stratigraphic record?” by Jasper Knight

The comment by Jasper Knight on ‘‘How plausible are high-frequency sediment supply-driven cycles in the stratigraphic record?’’ (Castelltort and Van Den Driessche, 2003) mostly tries to defend that rivers and their stratigraphic record show a complexity across many spatial and temporal scales that we would have overlooked in our paper. This would have made our work, based on the sedimentary system concept, ‘‘narrow in approach and outlook’’, and lead us to ‘‘conceptual weaknesses’’. Jasper Knight then claims proposing ‘‘some alternative ways at looking at the operation of river sediment systems’’. In fact, Jasper Knight only enumerates some geological truisms about fluvial systems, erosion and sedimentation, whereas in our paper we made it clear that the concept of macroscale sedimentary systems on geological timescales ‘‘is aimed at distilling the first-order characteristics and dynamics of real systems from their natural complexity’’. This leads him,

Call for Papers: Special Issue in honor of Pauline van den Driessche

Call for Papers: Special Issue in honor of Pauline van den Driessche

Reply to Measurement of Oro-Cecal Transit Time in Young Children Using Lactose [13C] Ureide Requires Further Validation

Reply to Measurement of Oro-Cecal Transit Time in Young Children Using Lactose [13C] Ureide Requires Further Validation To the Editors: In reply to the letter to the editor by Christian et al. concerning our article entitled “The Lactose-[13C]Ureide Breath Test: A New, Non-Invasive Technique to Determine Orocecal Transit Time in Children,” we have the following comments. First of all, we are delighted by their shared interest and meaningful input. In our article, we demonstrated by incubating stools that infants before complete weaning do not have the appropriate bacterial enzymatic activity (Clostridium innocuum) to hydrolyze lactose ureide. Results of 11 LU breath tests on 10 children performed by Christian et al. showed that although these children were weaned, they did not all produce a clear 13CO2 rise above baseline. This result can be due to the difference in pre-dose and dose of testmeal. They gave 1×100 mg pre-dose, whereas we administered 3×500 mg the day before the test. Their testmeal contained 100 mg 13C LU; our testmeal 250 mg 13C LU. Since the literature in adults and our own experience showed that pre-dosing is important (no predose corresponds with no or a low response), the low dosages might be the reason for a lack of response. In vitro induction with unlabeled LU one hour before the 13CO2 collection showed no difference with no induced stools. We fully agree with the comment of Christian that in addition to weaning, the type of weaning foods may be important and that further investigation is required. M. Van Den Driessche G. Veereman-Wauters

A Lotka-Volterra competition model and its global convergence to a definite axial equilibrium.

We consider a four-species model based on competition and show that the whole four-species system collapses to a definite single species equilibrium at its carrying capacity. To do so, we use the results of Hirsch, Van Den Driessche and Zeeman, Hofbauer and Sigmund, and the product theorem of the Conley connection matrix theory by Mischaikow and Reineck.

Modifying the power method in max algebra

In the max algebra system, the eigenequation for an n×n irreducible nonnegative matrix A=[a ij] is A⊗x=μ(A)x. Here (A⊗x) i= max ja ijx j and μ(A) is the maximum circuit geometric mean. The complexity of the power method given in [L. Elsner, P. van den Driessche, Linear Algebra Appl., to appear] to compute μ(A) and x is considered. Under some assumptions on the critical matrix, it is shown that the algorithm may have time complexity O(n 4) . A modified power method, based on Karp's formula, is presented. For this new algorithm, with no assumptions on the critical matrix, μ(A) and x can be computed in O(n 3) time. Furthermore, this algorithm can be used to compute all linearly independent eigenvectors corresponding to μ(A).

University and university in Belgium

In Belgium, the model of the university museum as an outgrowth of and responsive to a closely knit academic community is gradually giving way to that of an institution which aims to serve a larger public. But this evolution has been uneven and is far from universal, as explained by Bernard van den Driessche, administrator of the Museum of Louvain‐la‐Neuve (Catholic University of Louvain), which he helped set up in 1979. Vice‐chairman from 1992 to 1995 of the ICOM Belgian National Committee, he was co‐ordinator of several issues of La vie des musées (Museum Life) published by the French‐speaking Association of Belgian Museums. He is currently president of Museums and Society in Wallonia, an association formed in 1998.

When Nomads Settle: The Effects of Commoditization, Nutritional Change, and Formal Education on Ariaal and Rendille Pastoralists

When Nomads Settle: The Effects of Commoditization, Nutritional Change, and Formal Education on Ariaal and Rendille Pastoralists

Stability with Respect to the Delay: On a Paper of K. L. Cooke and P. van den Driessche

A proposal for the correction of the main result of the title's paper is made. Some accompanying comments on other points are given. So, a simpler central result is derived. An explicit representation of the set of stable delays is achieved.

Gene therapy in the treatment of disease

Gene therapy or transfer is increasingly used as a therapeutic modality in diseases that span the medical spectrum. The American Academy of Allergy and Immunology and the National Institute of Allergy and Infectious Diseases jointly sponsored the Symposium, "Gene Therapy in the Treatment of Disease," which was held December 6, 1993 at the National Institutes of Health (NIH). The program committee consisted of Lawrence Lichtenstein, MD, PhD, (Baltimore) Chairman; Philip Fireman, MD, (Pittsburgh); Howard B. Dickler, MD, (Bethesda); William Paul, MD, (Bethesda); and Raif Geha, MD, (Boston). Eight outstanding presentations on current and potential uses of gene therapy were provided by speakers who are leading researchers in this area of broad interest and unusually promising clinical application. This report summarizes the highlights of those presentations.

Superinfection and the evolution of parasite virulence

Earlier ideas that parasites evolve toward becoming harmless to their hosts have, in recent years, given way to more analytic studies, focused on the 'basic reproductive rate', R0, of individual parasites. In general, the biology of the parasite life cycle will lead to constraining relations between virulence (parasite-associated host death or reduction in fertility) and transmissibility: the maximum R0 may then be attained by virulence being high, or low, or at some intermediate level, depending on the details of the constraining relations. Such studies have not generally included superinfection (where an already-infected host is infected by another parasite). Here we propose a general, but simple, model of superinfection, which is amenable to analytical treatment. In such models selection does not simply act to maximize R0; superinfection leads to selection for higher levels of virulence, highly polymorphic parasite populations and very complicated dynamics. We calculate the equilibrium distribution of parasite strains and the maximum level of virulence that can be maintained by superinfection. We also note the equivalence between our 'superinfection model' and recent approaches to the study of the meta-population dynamics of multi-species interactions.

Syn-extension rotations in the Permian St Affrique basin (Massif Central, France): paleomagnetic constraints

We present the results of a paleomagnetic study of the Permian sedimentary basin of St Affrique (southern Massif Central, France). This study was devised to test for tectonic rotations in the Montagne Noire area during the Stephano-Permian extension in the Hecynian chain and to complete a preliminary study [1] in which we suggested an counterclockwise rotation of St Affrique basin with respect to the neighbouring Lodève basin. This rotation was attributed to the domino-like rotation of the basement of St Affrique (i.e., the Paleozoic Monts de Lacaune imbricate units) as expected from the tectonic interpretation of the Montagne Noire area made by Van Den Driessche and Brun [2,3]. However, the rotation of the basin during Permian should induce declination differences between the basal and the upper part of the stratigraphic sequence. We therefore conducted vertical sampling of the basin throughout the Permian sequence. Two sites of Stephanian age gave no results, but twelve sites from the lower to the upper Permian have been sampled in the St Affrique basin and did provide results. Paleomagnetic analysis using thermal procedures confirms our previous analysis and allowed us to demonstrate three magnetization components. The low temperature A component is unblocked by 300°C and conforms to the present-day dipole field direction. An intermediate temperature component (B) has also been isolated, with unblocking temperatures of 300° to 550°–600°C, and this component was possibly carried by magnetite. Between 600° and 680°C a third component (C) may be suspected at some sites, but it has not been isolated with confidence. At a site showing a tilted-block geometry produced by syn-sedimentary normal faulting (site p7), it is shown that the B component is an early magnetization acquired during faulting of the succession, and we are therefore able to give it a stratigraphic age. The mean directions of the B component, which are computed at the stratigraphic stage level, show the following counterclockwise deviations from the reference magnetic field direction for stable Europe: 21° ± 4.5° for the upper Autunian, 10° ± 7° for the Saxonian, 10° ± 9° for the lower Thuringian and 4° ± 6° and 3.5° ± 4.5° for two successive stages of middle/upper Thuringian age. We thus consider that (1) the comparison of the lowest units of St Affrique and Lodève basins [1], and (2) the evolution in paleomagnetic declination during the Permian within the St Affrique basin itself provide evidence, in space and time, of a rotation of the basin during its formation. We attribute this rotation to the domino-like rotation of its basement (the Monts de Lacaune imbricate units) during the Stephano-Permian extension in the Hercynian chain [2,3].

Partly zero eigenvectors and matrices that satisfyAu=b

For given vectors u,b ∈ F n (where F is a field with at least 3 elements), we establish criteria for deciding whether a digraph allows in its pattern class a matrix A which satisfies Au = b. As corollaries to this we give necessary and sufficient conditions for a pattern class to allow a matrix which has an eigenvector with a particular zero/nonzero pattern. Moreover we establish whether or not that eigenvector can correspond to a zero or nonzero eigenvalue. We use these results to establish the analogous results for loopfree digraphs, and thus we obtain results additional to work already done by Maybee, Olesky, and Van Den Driessche in this area. We then use bipartite graphs to generalize our criteria for solutions to Au = b to the rectangular case.

Paleomagnetic evidence for rotation during Stephano-Permian extension in southern Massif Central (France)

We present results of a preliminary paleomagnetic study devised as a test of a tectonic model proposed by Van Den Driessche and Brun [1] to interpret the main structural features of the Montagne Noire (South Massif Central, France). This model suggests that crustal extension during Stephanian to Permian times is responsible for a major flat-lying detachment. The inferred geometrical relationships between fault azimuth and extension direction is believed to induce a counter-clockwise rotation of the Paleozoic cover during its northward motion. In order to test this model, we have sampled Autunian (Lower Permian) grey and red sandstone beds within two basins opened during the extension, the Lodève and the St-Affrique basins. The paleomagnetic study of six sites (three in each basin) using thermal demagnetization procedures, reveals the presence of 3 magnetization components. A first one (A), which is unblocked below 400°C, is interpreted as a recent overprint due to weathering of the sandstones. The B component is unblocked between 400° and 575°C. This B component is assumed to be primary, although the gentle dip of the beds does not allow a fold test. It fits the Lower Permian field direction in the Lodève basin ( in situ: D m = 194° , I m = 6° , k = 50, tilt corrected: D m = 194° , I m = −10° , k = 53). In contrast, it shows a systematic difference in declination in the St-Affrique basin ( in situ: D m = 173° , I m = 2.5° , k = 101, tilt corrected: D m = 172° , I m = 6° , k = 37). This is interpreted as a post-Autunian counter-clockwise rotation of at least 20° of the St-Affrique basin, thus supporting one of the main implications of the tectonic model. Finally, red sediments from two sites of the Lodève basin yield a third component (C) with unblocking temperatures up to 670°C and carried by hematite. Its direction is also consistent with the Permian field direction, showing that hematite crystallisation occurred early in the diagenesis history of the sediments.