Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models

Guillaume Cantin; Cristiana J. Silva; Guillaume Cantin; Cristiana J. Silva

doi:10.3934/math.2019.4.1145

AIMS Mathematics

2019, Volume 4, Issue 4: 1145-1169. doi: 10.3934/math.2019.4.1145

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Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models

Guillaume Cantin ^{1
,
,},
Cristiana J. Silva ²

1.
Laboratoire de Mathématiques Appliquées du Havre, Normandie Université, FR CNRS 3335, ISCN, 76600 Le Havre, France
2.
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received: 26 May 2019 Accepted: 23 July 2019 Published: 19 August 2019
MSC : 34A34, 34C60, 92B05

In this paper, we propose an original complex network model for an epidemic problem in an heterogeneous geographical area. The complex network is constructed by coupling nonidentical instances of a HIV/AIDS epidemiological model for which a disease-free equilibrium and an endemic equilibrium can coexist. After proving the existence of a positively invariant region for the solutions of the complex network problem, we investigate the effect of the coupling on the dynamics of the network, and establish the existence of a unique disease-free equilibrium for the whole network, which is globally asymptotically stable. We prove the existence of an optimal topology that minimizes the level of infected individuals, and apply the theoretical results to the case of the Cape Verde archipelago.
- complex network,
- epidemiological model,
- basic reproduction number,
- graph topology,
- HIV/AIDS,
- Cape Verde
Citation: Guillaume Cantin, Cristiana J. Silva. Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models[J]. AIMS Mathematics, 2019, 4(4): 1145-1169. doi: 10.3934/math.2019.4.1145

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Abstract

In this paper, we propose an original complex network model for an epidemic problem in an heterogeneous geographical area. The complex network is constructed by coupling nonidentical instances of a HIV/AIDS epidemiological model for which a disease-free equilibrium and an endemic equilibrium can coexist. After proving the existence of a positively invariant region for the solutions of the complex network problem, we investigate the effect of the coupling on the dynamics of the network, and establish the existence of a unique disease-free equilibrium for the whole network, which is globally asymptotically stable. We prove the existence of an optimal topology that minimizes the level of infected individuals, and apply the theoretical results to the case of the Cape Verde archipelago.

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