Nonnegative Matrices in the Mathematical Sciences

Berman, Abraham; Plemmons, Robert J.

doi:10.1137/1.9781611971262

Cited by 2,693 publications

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“…A particularly convenient definition of irreducibility of a non-negative matrix is in graphical terms, as it relates to a specific connectedness property of an associated graph [7]. Suppose that Z is an n × n non-negative matrix.…”

Section: First What Does Irreducibility Of P Q and Qp Mean?mentioning

confidence: 99%

“…For a proof of this fact, see [7]. Below we use the notation x min = min i (x i ) and x max = max i (x i ) for a given vector x in R n .…”

Section: Bounds For R(s) and Implications For Controlmentioning

confidence: 99%

“…For instance 0 < x means that x is a nonzero, non-negative vector. We collect some relevant properties of quasi-monotone matrices, which are consequences of the celebrated Perron-Frobenius Theorem, see for example [7]: Theorem 6. Let A be a quasi-monotone and irreducible matrix.…”

Section: Appendicesmentioning

confidence: 99%

“…It is well-known that a non-negative matrix X is irreducible if and only if there exists an integer k > 0 such that X + X 2 + · · · + X k >> 0, see [7]. Note that if m is a positive integer, then…”

Section: Appendicesmentioning

confidence: 99%

“…Setting m = 2(k + 1) yields that C + · · · + C m >> 0, and then C is irreducible. If C is irreducible, then the Perron-Frobenius Theorem [7] implies that R(C) is a positive eigenvalue of C, and it has a corresponding positive eigenvector. Hence, there exist vectors x >> 0 and y >> 0, not both zero, such that:…”

Section: Appendicesmentioning

confidence: 99%

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Parasite sources and sinks in a patched Ross–Macdonald malaria model with human and mosquito movement: Implications for control

Ruktanonchai

Smith

Leenheer

2016

Mathematical Biosciences

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We consider the dynamics of a mosquito-transmitted pathogen in a multipatch Ross-Macdonald malaria model with mobile human hosts, mobile vectors, and a heterogeneous environment. We show the existence of a globally stable steady state, and a threshold that determines whether a pathogen is either absent from all patches, or endemic and present at some level in all patches. Each patch is characterized by a local basic reproduction number, whose value predicts whether the disease is cleared or not when the patch is isolated: patches are known as "demographic sinks" if they have a local basic reproduction number less than one, and hence would clear the disease if isolated; patches with a basic reproduction number above one would sustain endemic infection in isolation, and become "demographic sources" of parasites when connected to other patches. Sources are also considered focal areas of transmission for the larger landscape, as they export excess parasites to other areas and can sustain parasite populations. We show how to determine the various basic reproduction numbers from steady state estimates in the patched network and knowledge of additional model parameters, hereby identifying parasite sources in the process. This is useful in the context of control of the infection on natural landscapes, because a commonly suggested strategy is to target focal areas, in order to make their corresponding basic reproduction numbers less than one, effectively turning them into sinks. We show that this is indeed a successful control strategy -albeit a conservative and possibly expensive one-in case either the human host, or the vector does not move. However, we also show that when both humans and vectors move, this strategy may fail, depending on the specific movement patterns exhibited by hosts and vectors.

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Section: First What Does Irreducibility Of P Q and Qp Mean?mentioning

confidence: 99%

“…For a proof of this fact, see [7]. Below we use the notation x min = min i (x i ) and x max = max i (x i ) for a given vector x in R n .…”

Section: Bounds For R(s) and Implications For Controlmentioning

confidence: 99%

Section: Appendicesmentioning

confidence: 99%

Section: Appendicesmentioning

confidence: 99%

Section: Appendicesmentioning

confidence: 99%

See 3 more Smart Citations

Parasite sources and sinks in a patched Ross–Macdonald malaria model with human and mosquito movement: Implications for control

Ruktanonchai

Smith

Leenheer

2016

Mathematical Biosciences

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A parallel multisplitting solution of the least squares problem

Renaut

1998

Numer. Linear Algebra Appl.

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OnM-multisplittings of singularM-matrices with application to Markov chains

Bru

Cantó

Climent

1998

Numer. Linear Algebra Appl.

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Given a singular M–matrix of a linear system, convergent conditions under which iterative schemes based on M–multisplittings are studied. Two of those conditions, the index of the iteration matrix and its spectral radius are investigated and related to those of the M‐matrix. Furthermore, a parallel multisplitting iteration scheme for solving singular linear systems is suggested which can be applied to practical problems such as Poisson and elasticity problems under certain boundary conditions, the Neumann problem, and in Markov chains. A discussion of that multisplitting scheme, based on Gauss–Seidel type splittings is given for computing the stationary distribution vector of Markov chains. In this case a computational viable algorithm can be constructed, since only the nonsingularity of one weighting matrix of the multisplitting is needed. © 1998 John Wiley & Sons, Ltd.

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Nonnegative Matrices in the Mathematical Sciences

Cited by 2,693 publications

References 0 publications

Parasite sources and sinks in a patched Ross–Macdonald malaria model with human and mosquito movement: Implications for control

Parasite sources and sinks in a patched Ross–Macdonald malaria model with human and mosquito movement: Implications for control

A parallel multisplitting solution of the least squares problem

OnM-multisplittings of singularM-matrices with application to Markov chains

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