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.
Abstract. For a family of periodic SEIRS models with general incidence, we prove the existence of at least one endemic periodic orbit when R 0 > 1. Additionally, we prove the existence of a unique disease-free periodic orbit, that is globally asymptotically stable when R 0 < 1. In particular, our main result establishes a sharp threshold between existence and non-existence of endemic periodic orbits for this family of models.
a b s t r a c tFor a non-autonomous SEIRS model with general incidence, that admits (Kuniya and Nakata, 2012) as a very particular case, we obtain conditions for extinction and strong persistence of the infectives. Our conditions are computed for several particular settings and extend the hypothesis of several proposed non-autonomous models. Additionally we show that our conditions are robust in the sense that they persist under small perturbations of the parameters in some suitable family. We also present some simulations that illustrate our results.
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