M. Teresa T. Monteiro

Vaccination models and optimal control strategies to dengue

As the development of a dengue vaccine is ongoing, we simulate an hypothetical vaccine as an extra protection to the population. In a first phase, the vaccination process is studied as a new compartment in the model, and different ways of distributing the vaccines investigated: pediatric and random mass vaccines, with distinct levels of efficacy and durability. In a second step, the vaccination is seen as a control variable in the epidemiological process. In both cases, epidemic and endemic scenarios are included in order to analyze distinct outbreak realities.

Dengue disease, basic reproduction number and control

Dengue is one of the major international public health concerns. Although progress is underway, developing a vaccine against the disease is challenging. Thus, the main approach to fight the disease is vector control. A model for the transmission of dengue disease is presented. It consists of eight mutually exclusive compartments representing the human and vector dynamics. It also includes a control parameter (insecticide) in order to fight the mosquito. The model presents three possible equilibria: two disease-free equilibria (DFE) and another endemic equilibrium. It has been proved that a DFE is locally asymptotically stable, whenever a certain epidemiological threshold, known as the basic reproduction number, is less than one. We show that if we apply a minimum level of insecticide, it is possible to maintain the basic reproduction number below unity. A case study, using data of the outbreak that occurred in 2009 in Cape Verde, is presented.

Dynamics of Dengue epidemics when using optimal control

We present an application of optimal control theory to Dengue epidemics. This epidemiologic disease is important in tropical countries due to the growing number of infected individuals. The dynamic model is described by a set of nonlinear ordinary differential equations, that depend on the dynamics of the Dengue mosquito, the number of infected individuals, and peoples motivation to combat the mosquito. The cost functional depends not only on the costs of medical treatment of the infected people but also on the costs related to educational and sanitation campaigns. Two approaches for solving the problem are considered: one using optimal control theory, the other carried out by first discretizing the problem and then solving it with nonlinear programming. The results obtained with OC-ODE and IPOPT solvers are given and discussed. We observe that with current computational tools it is easy to obtain, in an efficient way, better solutions to Dengue problems, leading to a decrease in the number of infected mosquitoes and individuals in less time and with lower costs.

Dengue in Cape Verde: vector control and vaccination

In 2009, for the first time in Cape Verde, an outbreak of dengue was reported and more than 20,000 people were infected. Only a few prophylactic measures were taken. The effects of vector control on disease spreading, such as insecticide (larvicide and adulticide) and mechanical control, as well as an hypothetical vaccine, are estimated through simulations with the Cape Verde data.

Insecticide Control in a Dengue Epidemics Model

A model for the transmission of dengue disease is presented. It consists of eight mutually‐exclusive compartments representing the human and vector dynamics. It also includes a control parameter (insecticide) in order to fight the mosquitoes. The main goal of this work is to investigate the best way to apply the control in order to effectively reduce the number of infected humans and mosquitoes. A case study, using data of the outbreak that occurred in 2009 in Cape Verde, is presented.

Sensitivity Analysis in a Dengue Epidemiological Model

Epidemiological models may give some basic guidelines for public health practitioners, allowing the analysis of issues that can influence the strategies to prevent and fight a disease. To be used in decision making, however, a mathematical model must be carefully parameterized and validated with epidemiological and entomological data. Here an SIR (S for susceptible, I for infectious, and R for recovered individuals) and ASI (A for the aquatic phase of the mosquito, S for susceptible, and I for infectious mosquitoes) epidemiological model describing a dengue disease is presented, as well as the associated basic reproduction number. A sensitivity analysis of the epidemiological model is performed in order to determine the relative importance of the model parameters to the disease transmission.

Optimization of Dengue epidemics: a test case with different discretization schemes

The incidence of Dengue epidemiologic disease has grown in recent decades. In this paper an application of optimal control in Dengue epidemics is presented. The mathematical model includes the dynamic of Dengue mosquito, the affected persons, the people’s motivation to combat the mosquito and the inherent social cost of the disease, such as cost with ill individuals, educations and sanitary campaigns. The dynamic model presents a set of nonlinear ordinary differential equations. The problem was discretized through Euler and Runge Kutta schemes, and solved using nonlinear optimization packages. The computational results as well as the main conclusions are shown.

Seasonality effects on Dengue: basic reproduction number, sensitivity analysis and optimal control

This work was supported by Portuguese funds through The Portuguese Foundation for Science and Technology (FCT). Rodrigues and Torres are supported by the Center for Research and Development in Mathematics and Applications (CIDMA) within project PEst-OE/MAT/UI4106/2014; Monteiro is supported by the ALGORITMI R&D Center and project PEST-OE/EEI/UI0319/2014.

Modeling some real phenomena by fractional differential equations

- This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundacao para a Ciencia e a Tecnologia), within project UID/MAT/04106/2013; third author by the ALGORITMI R&D Center and project PEst-UID/CEC/00319/2013. The authors are very grateful to two referees for many constructive comments and remarks.

Bioeconomic perspectives to an optimal control dengue model

A model with six mutually exclusive compartments related to dengue is studied. Three vector control tools are considered: insecticides (larvicide and adulticide) and mechanical control. The basic reproduction number associated to the model is presented. The problem is studied using an optimal control approach. The human data are based on the dengue outbreak that occurred in Cape Verde. Control measures are simulated in different scenarios and their consequences analysed.