Table 1 Description of parameters in system (1).

From: Bang-bang control optimization in infectious disease model with incorporating breakthrough and reinfection

Parameters

Definition

\(\beta\)

The infection rate of infected

\(\epsilon\)

The failure rate of the natural immunity of recovered

\(\sigma\)

The failure rate of vaccinated

\(\alpha\)

The vaccination rate for susceptible

\(\theta\)

The immunization loss rate of vaccinated

\(\gamma\)

The rate at which recovered become susceptible after natural immunity fails

\(\eta\)

Recovery rate of the infected