This paper is devoted to the eigenvalue complementarity problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differentiable optimization program involving the Rayleigh quotient on a simplex (Queiroz et al., Math. Comput. 73, 1849-1863). We discuss a logarithmic function and a quadratic programming formulation to find a complementarity eigenvalue by computing a stationary point of an appropriate merit function on a special convex set. A variant of the spectral projected gradient algorithm with a specially designed line search is introduced to solve the EiCP. Computational experience shows that the application of this algorithm to the logarithmic function formulation is a quite efficient way to find a solution to the symmetric EiCP.

A human respiratory syncytial virus surveillance system was implemented in Florida in 1999, to support clinical decision-making for prophylaxis of premature newborns. Recently, a local periodic SEIRS mathematical model was proposed in [Stat. Optim. Inf. Comput. 6 (2018), no. 1, 139-149] to describe real data collected by Florida's system. In contrast, here we propose a non-local fractional (non-integer) order model. A fractional optimal control problem is then formulated and solved, having treatment as the control. Finally, a cost-effectiveness analysis is carried out to evaluate the cost and the effectiveness of proposed control measures during the intervention period, showing the superiority of obtained results with respect to previous ones.

In this paper, we discuss the Eigenvalue Complementarity Problem (EiCP) where at least one of its defining matrices is asymmetric. A sufficient condition for the existence of a solution to the EiCP is established. The EiCP is shown to be equivalent to finding a global minimum of an appropriate merit function on a convex set Ω defined by linear constraints. A sufficient condition for a stationary point of this function on Ω to be a solution of the EiCP is presented.A branch-and-bound procedure is developed for finding a global minimum of this merit function on Ω. In addition, a sequential enumerative algorithm for the computation of the minimum and the maximum eigenvalues is also discussed. Computational experience is included to highlight the efficiency and efficacy of the proposed methodologies to solve the asymmetric EiCP.

We study an optimal control problem for a non-autonomous SEIRS model with incidence given by a general function of the infective, the susceptible and the total population, and with vaccination and treatment as control variables. We prove existence and uniqueness results for our problem and, for the case of mass-action incidence, we present some simulation results designed to compare an autonomous and corresponding periodic model, as well as the controlled versus uncontrolled models.2010 Mathematics Subject Classification. Primary: 34H05, 92D30; Secondary: 37B55, 49M05.