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The effect of backward bifurcation in controlling measles transmission by vaccination. (English) Zbl 1448.92327

Summary: A deterministic model for measles transmission, which is incorporating logistic growth rate and vaccination, is formulated and rigorously analyzed. The certain epidemiological threshold, known as the basic reproduction number, is derived. The proposed model has a locally asymptotically stable disease-free equilibrium whenever the basic reproduction number, is less than unity. Further, the proposed model exhibits the phenomenon of backward bifurcation, where stable disease-free equilibrium of the model coexists with a stable endemic equilibrium, whenever the basic reproduction number is less than unity. This study is suggested that decreasing the basic reproduction number is insufficient for disease eradication due to schedule vaccination is the cause of the occurrence of backward bifurcation. Furthermore, the study results are shown that the backward bifurcation in the formulated model is removed if increasing the efficacy of vaccine, coverage of primary vaccination, boosting second dose vaccination and decreasing waning of vaccine. When the basic reproduction number is greater than unity, the models have a unique endemic equilibrium which is globally asymptotically stable. The study results can be helpful in providing the information to public health authorities and policy maker in controlling the spread of measles by vaccination.

MSC:

92D30 Epidemiology
34C60 Qualitative investigation and simulation of ordinary differential equation models
34D05 Asymptotic properties of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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