## A simple vaccination model with multiple endemic states.(English)Zbl 0954.92023

A simple two-dimensional SIS model with vaccination is considered and a backward bifurcation for some parameter values. They divide the population into three classes – susceptibles (S), infectives (I) and vaccinated (V): $\begin{cases} \dot S\;=\;\mu N-\beta SI/N-(\mu+ \varphi)S+ cI+\theta V,\\ \dot I\;=\;\beta(S+ \sigma V)I/N- (\mu+c)I,\\ \dot V\;=\;\varphi S-\sigma \beta VI/N- (\mu+ \theta)V, \end{cases}$ where $$\beta$$ is the infectious contact rate, $$c$$ the recovery rate for the disease, $$\mu$$ the natural birth/death rate, $$N$$ the population size, $$\varphi$$ the vaccination rate, $$\theta$$ the rate at which the vaccination wears off and $$\sigma$$ measures the effenciency of the vaccine. A complete bifurcation analysis of the model in terms of the vaccine-reduced reproduction number is given, and some extensions are considered. The results of mathematical analysis indicate that a vaccination campaign $$\varphi$$ meant to reduce a disease’s reproduction number $$R(\varphi)$$ below one may fail to control the disease. If the aim is to prevent an epidemic outbreak, a large initial number of infective persons can cause a high endemicity level to arise rather suddenly even if the vaccine-reduced reproduction number is below the threshold. If the aim is to eradicate an already established disease, bringing the vaccine-reduced reproduction number below one this may also fail.

### MSC:

 92D30 Epidemiology 34C23 Bifurcation theory for ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models
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### References:

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