Optimal Control of a SEIR Model with Mixed Constraints and L 1 Cost

Abstract

Abstract. Optimal control can help to determine vaccination policies for infectious diseases. For diseases transmitted horizontally, SEIR compartment models have been used. Most of the literature on SEIR models deals with cost functions that are quadratic with respect to the control variable, the rate of vaccination. Here, we propose the introduction of a cost of L 1 type which is linear with respect to the control variable. Our starting point is the recent work [1], where the number of vaccines at each time is assumed to be limited. This yields an optimal control problem with a mixed control-state constraint. We discuss the necessary optimality conditions of the Maximum Principle and present numerical solutions that precisely satisfy the necessary conditions.

On a New Epidemic Model with Asymptomatic and Dead-Infective Subpopulations with Feedback Controls Useful for Ebola Disease

“…See, for instance, a related UK medical report [29] and see also [27]. Recent work on the incorporation of infective corpses and asymptomatic infectious type as new subpopulation is discussed, for instance, in [26,28]. The epidemic SEIADR model with vaccination and antiviral treatment together with infective corpses culling is as follows:…”

“…On the other hand, it turns out as known due to medical experience that there are individuals who are infective but do not have significant external symptoms, that is, the socalled the "asymptomatic" ( ) subpopulation, [26]. This occurs even in the common known influenza disease.…”

On a New Discrete SEIADR Model with Mixed Controls: Study of Its Properties

“…This makes the model interesting for its application for studies in the Ebola disease and similar diseases in which the re-emergences of the presence of infectious individuals cannot be explained otherwise [8]. The infected stage of the alive population is split into three different subpopulations, namely, the exposed, the symptomatic infectious and the asymptomatic infectious subpopulation, where an individual may be contagious even if he/she/it does not show the symptoms of the disease [9,10]. Furthermore, we have designed our model in order to take into account the deceased infectious individuals as a new infectious subpopulation, so we can prove their importance in dynamics of the population as the contagiousness is still present on some of the corpses, that must be disposed with great care [9,[11][12][13].…”

“…The infected stage of the alive population is split into three different subpopulations, namely, the exposed, the symptomatic infectious and the asymptomatic infectious subpopulation, where an individual may be contagious even if he/she/it does not show the symptoms of the disease [9,10]. Furthermore, we have designed our model in order to take into account the deceased infectious individuals as a new infectious subpopulation, so we can prove their importance in dynamics of the population as the contagiousness is still present on some of the corpses, that must be disposed with great care [9,[11][12][13]. Thus, the final stages of the illness under study can be split into live and immune individuals, i.e., recovered (R) subpopulation, and infectious and dead individuals, i.e., dead (D) subpopulation.…”

“…Reducing the disease within the population should be main objective of the control measures proposed i.e., vaccination strategies and the medical treatment of the infectious subpopulation through antiviral treatment or care of the symptoms, which we will also include in our equations. This type of control strategies have been proposed in many models in the literature [3,9,14,15,[23][24][25][26][27]. The stability analysis and control optimization of a a subpopulation of infectious individuals in treatment with vaccination is addressed in [28], and a time-delay model with a strategy of impulsive vaccination and a saturated incidence rate is addressed in [29].…”