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Qualitative analyses of SIS epidemic model with vaccination and varying total population size. (English) Zbl 1045.92039
Summary: An SIS epidemic model with vaccination, temporary immunity, and varying total population size is studied. Three threshold parameters $$R_0$$, $$R_1$$, and $$R_2$$ are identified. The disease-free equilibrium is globally stable if $$R_0 \leq 1$$ and unstable if $$R_0 > 1$$, the endemic equilibrium is globally stable if $$R_0 > 1$$. The disease cannot break out if $$R_1 < 1$$, the disease may break out when the fractions of the susceptible and the infectious satisfy some condition if $$R_1 > 1$$ and $$R_0 \leq 1$$. The population becomes extinct ultimately and the disease always exists in the population if $$R_0 > 1$$ and $$R_2 \leq 1$$. There is a really endemic disease if $$R_0 > 1$$ and $$R_2 > 1$$.
The global stability of the disease-free equilibrium and the existence and global stability of the endemic equilibrium are proved by means of LaSalle’s invariance principle, the method of estimating values and Stokes’ theorem, respectively. The results with vaccination and without vaccination are compared, and the measures and effects of vaccination are discussed.

##### MSC:
 92D30 Epidemiology 34D23 Global stability of solutions to ordinary differential equations
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