Abstract
Vaccines are not administered on a continuous basis, but injections are practically introduced at discrete times often separated by an important number of time units, and this differs depending on the nature of the epidemic and its associated vaccine. In addition, especially when it comes to vaccination, most optimization approaches in the literature and those that have been subject to epidemic models have focused on treating problems that led to continuous vaccination schedules but their applicability remains debatable. In search of a more realistic methodology to resolve this issue, a control modeling design, where the control can be characterized analytically and then optimized, can definitely help to find an optimal regimen of pulsed vaccinations. Therefore, we propose a susceptible-infected-removed (SIR) hybrid epidemic model with impulse vaccination control and a compartment that represents the number of vaccinated individuals supposed to not acquire sufficient immunity to become permanently recovered due to the short-term effect of vaccines. A basic reproduction number, when the control is defined as a constant parameter, is calculated. Since we also need to find the optimal values of this impulse control when it is defined as a function of time, we start by stating a general form of an impulse version of Pontryagin’s maximum principle that can be adapted to our case, and then we apply it to our model. Finally, we provide our numerical simulations that are obtained via an impulse progressive-regressive iterative scheme with fixed intervals between impulse times (theoretical example of an impulse at each week), and we conclude with a discussion of our results.
Highlights
In 1998, Shulgin et al [1], highlighted the potential of pulse vaccination strategy in eradicating some epidemics at relatively low values of vaccination compared to conventional strategies as theoretically proven in [2]
After the promising results presented in these references, many researchers showed more interest in the study of the complexity of epidemic models that involved this type of control measures, as in work of Alberto d’Onofrio in [8] where it was numerically observed through the use of SEIR model that pulse vaccination program is slightly
Mathematics 2019, 7, 420 more efficient than the traditional continuous one, Zhou and Liu in [9] with similar results in case of SIS model, Zeng and Chen in [10] where it is understood that SIRS model becomes much more complex under such vaccination policies while these authors concluded the same with Sun in [11] in case of SIR model
Summary
In 1998, Shulgin et al [1], highlighted the potential of pulse vaccination strategy in eradicating some epidemics at relatively low values of vaccination compared to conventional strategies as theoretically proven in [2]. The optimization subject to such epidemic models, has mostly aimed to minimize the control policy cost associated with vaccination; as far as we know, there have been no studies that tried to resolve this problem in case of pulses with clearest and simplest methodology for better practicability. This means there is a need for a numerical method that applies exactly most findings of the theoretical calculus related to the necessary conditions of optimality with possibly clear analytical formulation of the sought optimal impulse control and without the help of any type of gradient methods. Impulse progressive-regressive iterative schemes are used for finding the results
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