Algorithmic methods for investigating equilibria in epidemic modeling

Brown, Christopher W.; Kahoui, M’hammed El; Novotni, Dominik; Weber, Andreas

doi:10.1016/j.jsc.2005.09.011

Cited by 41 publications

(38 citation statements)

References 25 publications

(41 reference statements)

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“…(7-9). The systems is widely used and well studied [23][24][25][26][27]. So we will not provide new insights into the structure of the system, but it is well suited as a test object for our algorithmic methods.…”

Section: The Sirs Epidemiological Modelmentioning

confidence: 99%

On Muldowney’s Criteria for Polynomial Vector Fields with Constraints

Errami

Seiler

Sturm

et al. 2011

Computer Algebra in Scientific Computing

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Abstract. We study Muldowney's extension of the classical BendixsonDulac criterion for excluding periodic orbits to higher dimensions for polynomial vector fields. Using the formulation of Muldowney's sufficient criteria for excluding periodic orbits of the parameterized vector field on a convex set as a quantifier elimination problem over the ordered field of the reals we provide case studies of some systems arising in the life sciences. We discuss the use of simple conservation constraints and the use of parametric constraints for describing simple convex polytopes on which periodic orbits can be excluded by Muldowney's criteria.

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Section: The Sirs Epidemiological Modelmentioning

confidence: 99%

On Muldowney’s Criteria for Polynomial Vector Fields with Constraints

Errami

Seiler

Sturm

et al. 2011

Computer Algebra in Scientific Computing

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“…Since many epidemiological models exhibit the threshold phenomenon one expects in general these conditions to present some kind of redundance, and so need a simplification work. In [5] the factorization of the characteristic polynomial at the disease free equilibrium was used as a "divide and conquer" strategy to simplify the Routh-Hurwitz conditions and this was successful for several models.…”

Section: Finding Equilibria and Factorizing Their Characteristic Polymentioning

confidence: 99%

“…One may indeed use a Routh-Hurwitz like criterion to formulate the problem as a first order formula in the language of ordered fields and then resort to real algebra algorithms as a simplification tool, see e.g., [2,5,12,14,16]. This approach has the advantage of being completely algorithmic, but due to the very complex combinatorial nature of simplification of first order formulae it may produce a quite involved condition for some models whereas hand calculations are successful in producing a simple equivalent condition, see e.g., [5].…”

Section: Introductionmentioning

confidence: 99%

Stability of Disease Free Equilibria in Epidemiological Models

Kahoui

Otto

2009

Math.Comput.Sci.

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International audienceIn this paper we study the structural properties of epidemiological models and we derive a necessary and sufficient condition for the stability of their disease free equilibria. We then show that for a large class of models our condition may be explicitly obtained by using symbolic computation tools. We also give a sufficient condition for the global stability of the disease free equilibrium

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“…Specifically we will use Maple [10], Reduce [13] and Z3 [12]. Using all these computational tools the endemic equilibrium configuration and the basic reproductive number [3,4,5,6,7,8] will be derived. From the algebraic expression for the basic reproductive number some control measures will be proposed.…”

Section: Introductionmentioning

confidence: 99%

Using computer algebra and SMT-solvers to analyze a mathematical model of cholera propagation

Arredondo

2014

SPIE Proceedings

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We analyze a mathematical model for the transmission of cholera. The model is already defined and involves variables such as the pathogen agent, which in this case is the bacterium Vibrio cholera, and the human population. The human population is divided into three classes: susceptible, infectious and removed. Using Computer Algebra, specifically Maple we obtain two equilibrium states: the disease free state and the endemic state. Using Maple it is possible to prove that the disease free state is locally asymptotically stable if and only if R0 < 1. Using Maple it is possible to prove that the endemic equilibrium state is locally stable when it exists, it is to say when R0 > 1. Using the package Red-Log of the Computer algebra system Reduce and the SMT-Solver Z3Py it is possible to obtain numerical conditions for the model. The formula for the basic reproductive number makes a synthesis with all epidemic parameters in the model. Also it is possible to make numerical simulations which are very illustrative about the epidemic patters that are expected to be observed in real situations. We claim that these kinds of software are very useful in the analysis of epidemic models given that the symbolic computation provides algebraic formulas for the basic reproductive number and such algebraic formulas are very useful to derive control measures. For other side, computer algebra software is a powerful tool to make the stability analysis for epidemic models given that the all steps in the stability analysis can be made automatically: finding the equilibrium points, computing the jacobian, computing the characteristic polynomial for the jacobian, and applying the Routh-Hurwitz theorem to the characteristic polynomial. Finally, using SMT-Solvers is possible to make automatically checks of satisfiability, validity and quantifiers elimination being these computations very useful to analyse complicated epidemic models.

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Algorithmic methods for investigating equilibria in epidemic modeling

Cited by 41 publications

References 25 publications

On Muldowney’s Criteria for Polynomial Vector Fields with Constraints

On Muldowney’s Criteria for Polynomial Vector Fields with Constraints

Stability of Disease Free Equilibria in Epidemiological Models

Using computer algebra and SMT-solvers to analyze a mathematical model of cholera propagation

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