Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers

Martin Luther Mann  Manyombe; Joseph  Mbang; Jean  Lubuma; Berge  Tsanou

doi:10.3934/mbe.2016019

Article Contents

2016, Volume 13, Issue 4: 813-840. Doi: 10.3934/mbe.2016019

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Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers

1.
Department of Mathematics, Faculty of Science, University of Yaounde 1, P.O. Box 812 Yaounde
2.
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002

Received: August 2015

Revised: January 2016

Published: April 2016

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Abstract

In this paper, an epidemic model is investigated for infectious diseases that can be transmitted through both the infectious individuals and the asymptomatic carriers (i.e., infected individuals who are contagious but do not show any disease symptoms). We propose a dose-structured vaccination model with multiple transmission pathways. Based on the range of the explicitly computed basic reproduction number, we prove the global stability of the disease-free when this threshold number is less or equal to the unity. Moreover, whenever it is greater than one, the existence of the unique endemic equilibrium is shown and its global stability is established for the case where the changes of displaying the disease symptoms are independent of the vulnerable classes. Further, the model is shown to exhibit a transcritical bifurcation with the unit basic reproduction number being the bifurcation parameter. The impacts of the asymptomatic carriers and the effectiveness of vaccination on the disease transmission are discussed through through the local and the global sensitivity analyses of the basic reproduction number. Finally, a case study of hepatitis B virus disease (HBV) is considered, with the numerical simulations presented to support the analytical results. They further suggest that, in high HBV prevalence countries, the combination of effective vaccination (i.e. $\geq 3$ doses of HepB vaccine), the diagnosis of asymptomatic carriers and the treatment of symptomatic carriers may have a much greater positive impact on the disease control.

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Mathematics Subject Classification: 34A34, 92D30, 65L03.

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