Abstract
This paper discusses the disease-free and endemic equilibrium points of a SVEIRS propagation disease model which potentially involves a regular constant vaccination. The positivity of such a model is also discussed as well as the boundedness of the total and partial populations. The model takes also into consideration the natural population growing and the mortality associated to the disease as well as the lost of immunity of newborns. It is assumed that there are two finite delays affecting the susceptible, recovered, exposed, and infected population dynamics. Some extensions are given for the case when impulsive nonconstant vaccination is incorporated at, in general, an aperiodic sequence of time instants. Such an impulsive vaccination consists of a culling or a partial removal action on the susceptible population which is transferred to the vaccinated one. The oscillatory behavior under impulsive vaccination, performed in general, at nonperiodic time intervals, is also discussed.
Highlights
Important control problems nowadays related to Life Sciences are the control of ecological models like, for instance, those of population evolution Beverton-Holt model, Hassell model, Advances in Difference EquationsRicker model, etc. 1–5 via the online adjustment of the species environment carrying capacity, that of the population growth or that of the regulated harvesting quota as well as the disease propagation via vaccination control
Some extensions are given for the case when impulsive nonconstant vaccination is incorporated at, in general, an aperiodic sequence of time instants
In a set of papers, several variants and generalizations of the Beverton-Holt model standard time-invariant, time-varying parameterized, generalized model or modified generalized model have been investigated at the levels of stability, cycle-oscillatory behavior, permanence, and control through the manipulation of the carrying capacity see, e.g., 1–5
Summary
Important control problems nowadays related to Life Sciences are the control of ecological models like, for instance, those of population evolution Beverton-Holt model, Hassell model, Advances in Difference Equations. We focus on the existence and some properties of disease-free and endemic equilibrium points of a SVEIRS model subject to an eventual constant regular vaccination rather than to an impulsive vaccination type. This impulsive vaccinations strategy will be performed as follows at a sequence of time instants {tk}k∈Z0 ran in general at a nonperiodic “in-between” sampling interval sequence:. In , an impulsive vaccination strategy is discussed for a SVEIR epidemic model whose latent period is a point delay while the existence of an immune period is not assumed. The disease-free equilibrium point results to be locally asymptotically stable if the reproduction number is less than unity while the endemic equilibrium point is locally asymptotically stable if such a number exceeds unity
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