## Mathematical analysis of a cholera model with vaccination.(English)Zbl 1406.92351

Summary: We consider a $$SVR-B$$ cholera model with imperfect vaccination. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. We calculate the certain threshold known as the control reproduction number $$\mathscr{R}_v$$. If $$\mathscr{R}_v<1$$, we obtain sufficient conditions for the global asymptotic stability of the disease-free equilibrium; the diseases will be eliminated from the community. By comparison of arguments, it is proved that if $$\mathscr{R}_v>1$$, the disease persists and the unique endemic equilibrium is globally asymptotically stable, which is obtained by the second compound matrix techniques and autonomous convergence theorems. We perform sensitivity analysis of $$\mathscr{R}_v$$ on the parameters in order to determine their relative importance to disease transmission and show that an imperfect vaccine is always beneficial in reducing disease spread within the community.

### MSC:

 92C60 Medical epidemiology
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