Abstract
This paper analyses an SIRS epidemic model with the vaccination of susceptible individuals and treatment of infectious ones. Both actions are governed by a designed control system whose inputs are the subpopulations of the epidemic model. In addition, the vaccination of a proportion of newborns is considered. The control reproduction number Rc of the controlled epidemic model is calculated, and its influence in the existence and stability of equilibrium points is studied. If such a number is smaller than a threshold value R¯c, then the model has a unique equilibrium point: the so-called disease-free equilibrium point at which there are not infectious individuals. Furthermore, such an equilibrium point is locally and globally asymptotically stable. On the contrary, if Rc>R¯c, then the model has two equilibrium points: the referred disease-free one, which is unstable, and an endemic one at which there are infectious individuals. The proposed control strategy provides several free-design parameters that influence both values Rc and R¯c. Then, such parameters can be appropriately adjusted for guaranteeing the non-existence of the endemic equilibrium point and, in this way, eradicating the persistence of the infectious disease.
Highlights
The propagation of epidemic diseases within a host population has been studied since several decades ago
This control subsystem provides two more free-design parameters comparing with those available in the more usual SIRS models with vaccination and treatment. The intensity of both control actions is not directly proportional to the susceptible and/or infectious subpopulation, as it usually happens in the SIRS models. Both actions are provided by the control subsystem, and the parameters defining the dynamics of the controller are available to shape the vaccination and treatment actions
An appropriate adjustment of the control parameters guarantees the positivity of the controller model, as it has been mathematically proved in Theorem 1
Summary
The propagation of epidemic diseases within a host population has been studied since several decades ago. SIRS model as well as the subsystem providing the both proposed control actions, namely, the vaccination of the susceptible and treatment of the infectious subpopulations, respectively. The results of the proposed model are compared with an SIRS model with only a control action, either the vaccination of the susceptible subpopulation or the treatment of the infectious one. These comparisons are interesting from the viewpoint of the available resources relative to the existence of vaccines and/or medicaments to fight against the propagation of the disease
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