A Model of Spatial Spread of an Infection with Applications to HIV/AIDS in Mali

Ouaténi Diallo, Yaya Koné, Jérôme Pousin

Faculté des Sciences et Techniques, USTTB, Bamako, Mali.
Institut Camille Jordan, INSA de Lyon, Lyon, France.

DOI: 10.4236/am.2012.312256                

Abstract

In this paper we introduce a classical SI model to capture the spread of an infectious disease within a population. More precisely, the spatial diffusion of HIV/AIDS in a population is modeled. For that, we assume that the spread is due to the anarchical comportment of infected individuals along a road, especially, “lorry drivers”. The question which consists of the control of the infection is also addressed. Infected individuals moving from a town to another one, the diffusion is then anisotropic with a main direction of propagation, namely the road direction. Using a semi-group argument and a maximum principle, the uniqueness of a solution to the problem is established. This solution is also estimated. We end this paper by considering some numerical experiments in the case of HIV/AIDS spread in Mali along a road connecting two towns.

Keywords

Spacial Spread of Infections; Controlability; Maximum Principle

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	E. Maisse, “Analyse et Simulation Numérique de Phénomènes de Diffusion-Dissolution/Précipitation en Milieu Poreux, Appliquuées au Stockage de Déchets,” Thèse de doctorat, Université Claude Bernard-Lyon1, Lyon, 1998.
[2]	G. F. Webb, “A Reaction-Difusion Model for a Deterministic Diffusive Epidemic,” Journal of Mathematical Analysis and Applications, Vol. 84, No. 1, 1981, pp. 150-161. doi:10.1016/0022-247X(81)90156-6
[3]	L. Melkemi, A. Z. Mokrane and A. Youkana, “On the Uniform Boundedness of the Solutions of Systems of Reaction-Diffusion Equations,” Electronic Journal of Qualitative Theory of Differential Equations, Vol. 2005, No. 24, 2005, pp. 1-10.
[4]	F. A. Milner and R. Zhao, “Analysis of an S-I-R Model of Epidemics with Directed Spatial Diffusion,” 2011. http://biblioteca.universia.net/html_bura/ficha/params/title/analysis-of-an-s-i-r-model-of-epidemics-with/id/46036797.html
[5]	H. Brezis, “Analyse fonctionnelle,” Théorie et applications, Dunod, 2002.
[6]	R. Dautray and J. L. Lions, “Analyse Mathématique et Calcul Numérique Pour les Sciences et les Techniques,” Vol. 3, Masson, Paris, 1985.
[7]	A. Pazy, “Semigroup of Linear Operators and Applications to Partial Differential Equations,” Springer-Verlag, New York, 1983. doi:10.1007/978-1-4612-5561-1