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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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