A reaction-diffusion system with cross-diffusion modeling the spread of an epidemic disease

Bendahmane, Mostafa; Langlais, Michel

doi:10.1007/s00028-010-0074-y

Cited by 42 publications

(38 citation statements)

References 25 publications

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“…In this proof we adapt the result obtained in [13] (and the references therein) to the system (1.1)-(1.3) with (2.1)-(2.3) and Observe that for any fixed u i,0 ∈ W 1,p (Ω 1 ) (p > 3), there exists a maximal existence time T ∈ (0, +∞] such that the system (1.1) has a unique solution u i ∈ C (0, T ; W 1,p (Ω 1 )) ∩ C ∞ (Ω) (see the result of Amann [44] (see also [45,46] …”

Section: Existence and Uniqueness Of The Classical Solution (Proof Ofmentioning

confidence: 85%

“…Then, Mimura and Kawasaki [4] introduced the notion of cross-diffusion to describe numerically the segregation effects for large time of a two-component competitive system, starting from heterogeneous distributions initially (see also Mimura and Yamaguti [5]). Later, Galiano et al [6,7] perform the numerical analysis of solutions for a one-dimensional nonlinear cross-diffusion population model (see also [8][9][10][11][12][13][14]). …”

Section: Introductionmentioning

confidence: 99%

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A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion

Anaya

Bendahmane

Langlais

et al. 2015

Computers & Mathematics with Applications

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a b s t r a c tIn this paper, we are concerned with a model of the indirect transmission of an epidemic disease between two spatially distributed host populations having non-coincident spatial domains with nonlocal and cross-diffusion, the epidemic disease transmission occurring through a contaminated environment. The mobility of each class is assumed to be influenced by the gradient of the other classes. We address the questions of existence of weak solutions and existence and uniqueness of classical solution by using, respectively, a regularization method and an interpolation result between Banach spaces. Moreover, we propose a finite volume scheme and proved the well-posedness, nonnegativity and convergence of the discrete solution. The convergence proof is based on deriving a series of a priori estimates and by using a general L p compactness criterion. Finally, the numerical scheme is illustrated by some examples.

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Section: Existence and Uniqueness Of The Classical Solution (Proof Ofmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion

Anaya

Bendahmane

Langlais

et al. 2015

Computers & Mathematics with Applications

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“…Moreover, one can show that k@ t u " n k L 2 ð0,T;H À1 ðÞÞ c and k@ t v " n k L 2 ð0,T;H À1 ðÞÞ c, for some constant c > 0. Hence the similar approach in [16,30] and with the above estimates one can prove the global existence of approximate solutions of the problem (2.1). Therefore as n !…”

Section: Approximation Problemmentioning

confidence: 90%

“…Our assumptions (H 1 À H 3 ) is standard one and valid for nonlinear diffusion operator, for more detail one can see [15]. To improve the regularity requirements of solutions we assume from the definition of (u, v) and [16], one can easily understand that it satisfies ðu, vÞ p q minðu, vÞ. Under these assumptions, establishing the weak solutions to the system (1.1) in distribution sense is really tough task.…”

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confidence: 99%

Weak-renormalized solutions for predator–prey system

Shangerganesh

Balan

Balachandran

2013

Applicable Analysis

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“…Reaction–diffusion models have been well defined and numerically tested (e.g. Bendahmane & Langlais ), particularly for disease spread (e.g. Wang & Zhao ), and have been shown in some simplified cases to agree with field observations (Dwyer ).…”

Section: Introductionmentioning

confidence: 99%

Using the Spatial Population Abundance Dynamics Engine for conservation management

et al. 2015

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Summary1. An explicit spatial understanding of population dynamics is often critical for effective management of wild populations.Sophisticated approaches are available to simulate these dynamics, but are largely either spatially homogeneous or agentbased, and thus best suited to small spatial or temporal scales. These approaches also often ignore financial decisions crucial to choosing management approaches on the basis of cost-effectiveness. 2.We created a user-friendly and flexible modelling framework for simulating these population issues at large spatial scalesthe Spatial Population Abundance Dynamics Engine (SPADE). SPADE is based on the STAR model (McMahon et al. 2010) and uses a reaction-diffusion approach to model population trajectories and a cost-benefit analysis technique to calculate optimal management strategies over long periods and across broad spatial scales. It expands on STAR by incorporating species interactions and multiple concurrent management strategies, and by allowing full user control of functional forms and parameters.3. We used SPADE to simulate the eradication of feral domestic cats Felis catus on sub-Antarctic Marion Island (Bester et al. 2002) and compared modelled outputs to observed data. The parameters of the best-fitting model reflected the conditions of the management programme, and the model successfully simulated the observed movement of the cat population to the southern and eastern portion of the island under hunting pressure. We further demonstrated that none of the management strategies would likely have been successful within a reasonable timeframe if performed in isolation. 4. SPADE is applicable to a wide range of population management problems, and allows easy generation, modification and analysis of management scenarios. It is a useful tool for the planning, evaluation and optimisation of the management of wild populations, and can be used without specialised training.

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A reaction-diffusion system with cross-diffusion modeling the spread of an epidemic disease

Cited by 42 publications

References 25 publications

A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion

A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion

Weak-renormalized solutions for predator–prey system

Using the Spatial Population Abundance Dynamics Engine for conservation management

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