Optimization is central to any problem involving decision making. The area
of optimization has received enormous attention for over 30 years and it is still popular
in research field to this day. In this paper, a global optimization method called Improved
Homotopy with 2-Step Predictor-corrector Method will be introduced. The method in-
troduced is able to identify all local solutions by converting non-convex optimization
problems into piece-wise convex optimization problems. A mechanism which only consid-
ers the convex part where minimizers existed on a function is applied. This mechanism
allows the method to filter out concave parts and some unrelated parts automatically.
The identified convex parts are called trusted intervals. The descent property and the
global convergence of the method was shown in this paper. 15 test problems have been
used to show the ability of the algorithm proposed in locating global minimizer.
This study presents a two-strain deterministic model which incorporates Dengvaxia vaccine and insecticide (adulticide) control strategies to forecast the dynamics of transmission and control of dengue in Madeira Island if there is a new outbreak with a different virus serotypes after the first outbreak in 2012. We construct suitable Lyapunov functions to investigate the global stability of the disease-free and boundary equilibrium points. Qualitative analysis of the model which incorporates time-varying controls with the specific goal of minimizing dengue disease transmission and the costs related to the control implementation by employing the optimal control theory is carried out. Three strategies, namely the use of Dengvaxia vaccine only, application of adulticide only, and the combination of Dengvaxia vaccine and adulticide are considered for the controls implementation. The necessary conditions are derived for the optimal control of dengue. We examine the impacts of the control strategies on the dynamics of infected humans and mosquito population by simulating the optimality system. The disease-free equilibrium is found to be globally asymptotically stable whenever the basic reproduction numbers associated with virus serotypes 1 and j (j ∈ {2,3,4}), respectively, satisfy R01,R0j ≤ 1, and the boundary equilibrium is globally asymptotically stable when the related R0i (i = 1,j) is above one. It is shown that the strategy based on the combination of Dengvaxia vaccine and adulticide helps in an effective control of dengue spread in the Island.