Abstract
In this study we consider a mathematical model of an SIR epidemic model with a saturated incidence rate. We used the optimal vaccination strategies to minimize the susceptible and infected individuals and to maximize the number of recovered individuals. We work in the nonlinear optimal control framework. The existence result was discussed. A characterization of the optimal control via adjoint variables was established. We obtained an optimality system that we sought to solve numerically by a competitive Gauss–Seidel like implicit difference method.
Highlights
The dynamics of the infectious diseases is an important research area in mathematical epidemiology
Most of them are interested in the formulation of the incidence rate, i.e. the infection rate of susceptible individuals through their contacts with infected individuals
We use in the second term in the functional objective, the quadratic term (1/2)τ u2, where τ is a positive weight parameter which is associated with the control u(t) and the square of the control variable reflects the severity of the side effects of the vaccination [13]
Summary
The dynamics of the infectious diseases is an important research area in mathematical epidemiology. The second one is the saturated incidence rate of the form βSI/(1 + α1S), where α1 is a positive constant. The third one is the saturated incidence rate of the form βSI/(1 + α2I), where α2 is a positive constant. In this incidence rate the number of the effective contacts between infected and susceptible individuals may saturate at high infecting levels due to the crowding of the infected individuals or due to the protection measures by the susceptible individuals. The fundamental characteristic of this model is: The modified saturated incidence rate βS(t)I(t)/(1 + α1S(t) + α2I(t)) (see, for example, [8,9,10]), which includes the three forms:.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
