Abstract
A model for a generic disease with incubation and recovered stages is proposed. It incorporates a vaccinated subpopulation which presents a partial immunity to the disease. We study the stability, periodic solutions and impulsive vaccination design in the generalized modeled system for the dynamics and spreading of the disease under impulsive and non-impulsive vaccination. First, the effect of a regular impulsive vaccination on the evolution of the subpopulations is studied. Later a non-regular impulsive vaccination strategy is introduced based on an adaptive control law for the frequency and quantity of applied vaccines. We show the later strategy improves drastically the efficiency of the vaccines and reduce the infectious subpopulation more rapidly over time compared to a regular impulsive vaccination with constant values for both the frequency and vaccines quantity.
Highlights
There is a network of interactions that define the spreading of any infectious disease
We focus our study when the disease-free equilibrium point is unstable and a regular impulsive vaccination is added to the regular non-impulsive one in order to avoid the permanence of the infectious subpopulation
We introduce the concepts of vaccination cost (VC), directly related to the treatment and the number of consumed vaccines, and the disease cost (DC), related to the quantity of infected subpopulation over time
Summary
There is a network of interactions that define the spreading of any infectious disease. When only a regular non-impulsive vaccination is applied the dynamics of the SVEIR model asymptotically leads the state variables of the system (subpopulations) to either a disease-free equilibrium (DFE) regime or an endemic one. A regime where the infectious subpopulation tends to zero is obtained Such vaccination strategies are based on adaptive control techniques since the rules for generating the impulses are updated based on those used formerly for signal adaptation [13,14,15,16,17,18,19], but whose application in disease control and vaccination is novel. The efficiency of the previous adaptive sampling laws with respect to the regular impulsive vaccination will be compared, and, the SVEIR model will be used to describe a possible outbreak of pertussis and the evolution of the disease applying different vaccination strategies.
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