ABSTRACT The outbreak of COVID-19 has made scientists from all over the world do not measure efforts to understand the dynamics of the disease caused by this coronavirus. Several mathematical models have been proposed to describe the dynamics and make predictions. This work proposes a mathematical model that includes social isolation of susceptible individuals as a strategy of suppression and mitigation of the disease. The Susceptible-Infectious-Isolated-Recovered-Dead (SIQRD) model is proposed to analyze three important issues about the dynamics of the disease taking into account social isolation: when the isolation should begin? How long to keep the isolation? How to get out of this isolation? To get answers, computer simulations are provided and their results discussed. The results obtained show that beginning social isolation on the 10th or 15th days, after confirmation of the 50th case, and with 70% of the population in isolation, seems to be promising, since the infected curve does not grow much until it enters the isolation and remains at a stable level during the isolation. On the other hand an abrupt release of the social isolation will imply a second peak of infected individuals above the first one, which is not desired. Therefore, the release from social isolation should be gradual.

RESUMO A infecção pelo vìrus da dengue constitui atualmente um dos maiores problemas da saúde pública pelo mundo inteiro. Como ainda não existem vacinas contra os vìrus da dengue, é importante ter modelos matemáticos que representem a dinâmica da epidemia visando o controle da propagação da doença. Neste trabalho, utilizamos um sistema SEIR contìnuo que modela a dinâmica da transmissão da dengue e fizemos um estudo da estabilidade do modelo com os pontos de equilìbrio biologicamente viáveis. O objetivo deste trabalho é aplicar estratégias de controle alternativas para diminuir a propagação desta doença infecciosa. A estratégia utilizada foi o controle ótimo linear para sistemas não-lineares. Simulações numéricas mostram a eficiência do controle proposto.

ABSTRACT Dengue virus infection is nowadays one of the major worldwide public health problems. As there is no vaccine against dengue virus, it is important to have mathematical models that represent the dynamics of the epidemic aiming to control the spread of disease. In this work, we use a system SEIR continuous modeling the dynamics of dengue transmission and made a study of the stability of model with the biologically feasible equilibria points. The aim of this work is to apply alternative control strategies to reduce the spread of this infectious disease. The strategy used was the optimal control for non linear systems. Numerical simulations show the effectiveness of the proposed control.

O propósito deste trabalho é encontrar estratégias ótimas de controle de pragas no sistema biológico que apresenta comportamento não-linear. O controle, baseado no modelo de Lotka - Volterra de duas presas e um predador, é aplicado em um agroecossistema de plantação de soja. O objetivo desta estratégia de controle é manter a população de pragas abaixo de nível de danos econômicos. Este problema do controle ótimo é resolvido através do método das Equações de Riccati Dependentes do Estado (SDRE). Simulações numéricas para as estratégias do controle de pragas propostas, baseadas no modelo de Lotka-Volterra, foram realizadas para mostrar eficácia deste método.

The purpose of this work is to choose optimal control strategies of the biological system which exhibits the nonlinear behavior. The control, based on two preys - one predator Lotka-Volterra model, is applied in soybean crop agroecosystems. The goal of this pest control strategy is to maintain the pest population density at level below the economic injury level. This optimal control problem is solved by State Dependent Riccati Equation (SDRE) method. Numerical simulations for proposed strategies of biological pest control based on the Lotka-Volterra model are provided to show the effectiveness of this method.