We consider an optimal control problem for a non-autonomous model of ODEs that describes the evolution of the number of customers in some firm. Namely we study the best marketing strategy. Considering a L2 cost functional, we establish the existence and uniqueness of optimal solutions, using an inductive argument to obtain uniqueness on the whole interval from local uniqueness. We also present some simulation results, based on our model, and compare them with results we obtain for an L1 cost functional. For the L1 cost functional the optimal solutions are of bang-bang type and thus easier to implement, because at every moment possible actions are chosen from a finite set of possibilities. For the autonomous case of L2 problem, we show the effectiveness of the optimal control strategy against other formulations of the problem with simpler controls.

Infectious diseases are caused by pathogenic microorganisms and can spread through different ways. Mathematical models and computational simulation have been used extensively to investigate the transmission and spread of infectious diseases. In other words, mathematical model simulation can be used to analyse the dynamics of infectious diseases, aiming to understand the effects and how to control the spread. In general, these models are based on compartments, where each compartment contains individuals with the same characteristics, such as susceptible, exposed, infected, and recovered. In this paper, we cast further light on some classical epidemic models, reporting possible outcomes from numerical simulation. Furthermore, we provide routines in a repository for simulations.

ABSTRACT BACKGROUND: The fragility of healthcare systems worldwide had not been exposed by any pandemic until now. The lack of integrated methods for bed capacity planning compromises the effectiveness of public and private hospitals’ services. OBJECTIVES: To estimate the impact of the COVID-19 pandemic on the provision of intensive care unit and clinical beds for Brazilian states, using an integrated model. DESIGN AND SETTING: Experimental study applying healthcare informatics to data on COVID-19 cases from the official electronic platform of the Brazilian Ministry of Health. METHODS: A predictive model based on the historical records of Brazilian states was developed to estimate the need for hospital beds during the COVID-19 pandemic. RESULTS: The proposed model projected in advance that there was a lack of 22,771 hospital beds for Brazilian states, of which 38.95% were ICU beds, and 61.05% were clinical beds. CONCLUSIONS: The proposed approach provides valuable information to help hospital managers anticipate actions for improving healthcare system capacity.