A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion. (July 2015)

• Record Type: Journal Article
• Authors: Anaya, Verónica; Bendahmane, Mostafa; Langlais, Michel; Sepúlveda, Mauricio
• Is Part Of: Computers & mathematics with applications. Volume 70:issue 2(2015)
• Page Start: 132
• Page End: 157
• Publication Date: 2015-07
• Subjects: Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2015.04.021
• Languages: English
• ISSNs: 0898-1221

Abstract

<abstract xml:lang="en" abstract-type="author" id="a000005"> <title id="st000005">Abstract</title> <sec> <p id="sp000125">In 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 <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjq1wrf04" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si25.gif" display="inline" overflow="scroll" id="d13e497" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> compactness criterion. Finally, the numerical scheme is illustrated by some examples.</p> </sec> </abstract>