ABSTRACT Aedes aegypti is the main vector of multiple diseases, such as dengue, yellow fever, Zika, and chikungunya. Diseases associated with mosquitoes have been growing in recent years, with more than one-third of the world population at risk. Control techniques have been studied to prevent the spread of the Aedes aegypti, such that new formula and how to use adulticides and larvicides, among others. This paper proposes a novel approach in the field of partial differential equations and optimization. We consider a two-dimensional diffusion-reaction model that describes the interaction between aquatic and adult female stages spreading across a domain with parameters dependent on rainfall and temperature. We also formulate a mono-objective and multiobjective optimization problem to minimize the Aedes aegypti populations and the control, considering the application of adulticides and larvicides, using actual data from the city of Lavras/Brazil. The operator splitting technique is used to solve the diffusion-reaction system, coupling finite difference and the fourth-order Runge-Kutta method and optimal solutions were searched by using the Real-Biased Genetic Algorithm and Non-dominated Sorting Genetic Algorithm -II. Numerical results show significant reduction of the Aedes aegypti population. diseases dengue fever Zika chikungunya years onethird one third risk larvicides others twodimensional two dimensional diffusionreaction diffusion reaction temperature monoobjective mono objective control LavrasBrazil Lavras Brazil Lavras/Brazil system fourthorder fourth order RungeKutta Runge Kutta RealBiased Real Biased Nondominated Non dominated II. II -II

ABSTRACT A multiobjective impulsive control scheme is proposed to answer how optimal vaccination campaigns should be implemented, regarding two conflicting targets: making the total number of infecteds small and the vaccination campaign as handy as possible. In this paper, a stochastic SIR model is used to better depict the characteristics of a disease in practical terms, where little influences may lead to sudden and unpredictable changes in the behavior of transmissions. This model is extended to analyze the effects of impulsive vaccinations in two phases: the transient regime control, taking into account the necessity to reduce the number of infected individuals to an acceptable level in a finite time, and the permanent regime control, that will act with fixed vaccinations to avoid another outbreak. A parallel version of NSGA-II is used as the multiobjective optimization machinery, considering both the probability of eradication and the vaccination campaign costs. The final result using the proposed framework shows nondominated policies that can guide public managers to decide which is the best procedure to be taken depending on the present situation.

RESUMO Devido aos progressos recentes na manipulação genética, os mosquitos transgênicos podem ser uma alternativa viável para reduzir algumas doenças. Condições de viabilidade são obtidas pela simulação e análise de modelos matemáticos que descrevem o comportamento de populações de mosquitos selvagens e transgênicos convivendo em uma mesma área geográfica. Neste trabalho, apresentamos um modelo de reação-difusão com um termo de reação não linear, uma função que descreve a interação entre mosquitos selvagens e transgênicos, levando em consideração sua zigosidade. O termo difusivo representa uma dispersão espacial uniforme caracterizada por um parâmetro de difusão fixo. O sistema de equações diferenciais parciais obtidas é resolvido numericamente combinando um método implícito de Runge-Kutta e um método de elementos finitos, através da técnica de divisão sequencial de operadores. Vários cenários são analisados, simulando a liberação espacial de mosquitos transgênicos, e levam ao entendimento de uma relação intrínseca entre as variedades transgênica e selvagem para diferentes condições iniciais.

ABSTRACT Due to recent advances in genetic manipulation, transgenic mosquitoes can be a viable alternative to reduce some diseases. Viability conditions are obtained by the simulation and analysis of mathematical models that describe the behavior of wild and transgenic mosquitoes population living in the same geographic area. In this work, we present a reaction-diffusion model with a nonlinear reaction term, a function that describes the interaction between wild and transgenic mosquitoes taking into account their zygosity. The diffusive term represents a uniform spatial spread characterized by a fixed diffusion parameter. The system of partial differential equations obtained is solved numerically by combining a implicit Runge-Kutta method and finite elements method, through the sequential operator splitting technique. Several scenarios are analyzed, simulating the spatial release of transgenic mosquitoes, and lead to an understanding of an intrinsic relationship between the transgenic and wild varieties for different initial conditions.