## Qualitative analysis of the SICR epidemic model with impulsive vaccinations.(English)Zbl 1318.92049

Summary: Control of epidemic infections is a very urgent issue today. To develop an appropriate strategy for vaccinations and effectively prevent the disease from arising and spreading, we proposed a modified susceptible-infected-removed model with impulsive vaccinations. For the model without vaccinations, we proved global stability of one of the steady states depending on the basic reproduction number $$R_{0}$$. As typically in the epidemic models, the threshold value of $$R_{0}$$ is 1. If $$R_{0}$$ is greater than 1, then the positive steady state called endemic equilibrium exists and is globally stable, whereas for smaller values of $$R_{0}$$, it does not exist, and the semi-trivial steady state called disease-free equilibrium is globally stable. Using impulsive differential equation comparison theorem, we derived sufficient conditions under which the infectious disease described by the considered model disappears ultimately. The analytical results are illustrated by numerical simulations for Hepatitis B virus infection that confirm the theoretical possibility of the infection elimination because of the proper vaccinations policy.

### MSC:

 92D30 Epidemiology 34A37 Ordinary differential equations with impulses 34C60 Qualitative investigation and simulation of ordinary differential equation models
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### References:

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