Optimal Control for an SEIR Epidemic Model with Nonlinear Incidence Rate

Abstract

The objective of this paper is to explore possible impacts of nonlinearity of functional responses and a number of compartments of an infection disease model on principal qualitative properties of the optimal controls. To address this issue, we consider optimal controls for an Susceptible-Exposed-Infectious-Removed (SEIR) model of an endemically persisting infectious disease. We assume that the incidence rate is given by an unspecified nonlinear function constrained by a few biologically motivated conditions. For this model, we consider five controls (which comprise all controls that are possible for this model) with a possibility of acting simultaneously, and establish principal qualitative properties of the controls. A comparison with a similar SIR model is provided.

Optimal Controls of the Highly Active Antiretroviral Therapy

“…Finally, we note that the qualitative descriptions of the optimal controls (6.1)-(6.7) assuming that constrains (2.7)-(2.10) of Assumption 2 are held. However, while these constrains significantly simplify the analysis, they are not necessary: by arguments that were used in [29,34], it is possible to establish the validity of relationships (6.1)-(6.7) for the cases where conditions (2.7)-(2.10) of Assumption 2 are violated.…”

“…. Finally, arguments similar to these used by the authors in [34] allow to conclude that the functions 푤 * (푡) − 푟 * (푡) and 푝 * (푡) + 푞 * (푡) − 푟 * (푡) are also differentiable almost everywhere on the interval [0, 푇]. en, using the second and third equations of system (3.16), we formulate the required differential equation:…”

“…Nevertheless, despite this complexity, there is a certain progress in application of the control-independent functionals in biomedical control problems. Results that affirmatively demonstrate that the use of such functionals is possible and potentially highly beneficial, include publications of Ledzewicz, Schättler and their coauthors on the optimal control in anticancer therapy (see, e.g., [23,24] and bibliography therein), and of the authors on the optimal waste water biotreatment [25,[26][27][28] and on the optimal control of epidemics [29][30][31][32][33][34]. e objective functionals that include squares of the controls naturally occur in problems originated in mechanics and engineering, where square of a control is usually proportional to the energy required for the control action.…”

“…Moreover, comparison of these results with the controls found for the same model with the standard bilinear transmission rate in [30,32] shows that the nonlinearity of incidence rate does not affect the principal properties of the control. A more complicated 3-dimensional SEIR model with an unspecified nonlinear incidence rate and five simultaneously acting controls (which comprise all controls possible for this model) was considered in [34].…”

Optimal Control against the Human Papillomavirus: Protection versus Eradication of the Infection

“…Furthermore, for such functionals, the optimal controls can be obtained as explicit functions of the state and adjoint variables via the Pontryagin maximum principle. Although the financial cost of public health interventions to prevent epidemics most likely grows linearly with the magnitude of the corresponding control [24], we choose objective functionals with a quadratic dependence on the controls following the current trend of the literature [17,18,[25][26][27].…”