Mathematical model for assessing the impact of vaccination and treatment on measles transmission dynamics

A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease‐free equilibrium coexists with a stabl...

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Published inMathematical methods in the applied sciences Vol. 40; no. 18; pp. 6371 - 6388
Main Authors Garba, S. M., Safi, M. A., Usaini, S.
Format Journal Article
LanguageEnglish
Published Freiburg Wiley Subscription Services, Inc 01.12.2017
Subjects
Bifurcations
equilibria
Equilibrium
Liapunov functions
Measles
reproduction number
stability
treatment
Unity
vaccination
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Summary:A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease‐free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed if either measles vaccine is assumed to be perfect or disease related mortality rates are negligible. In the latter case, the disease‐free equilibrium is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a nonlinear Lyapunov function of Goh‐Volterra type, to be globally asymptotically stable for a special case.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.4462