Search

An inverse problem to identify parameters for the single-term (and multi-term) fractional-order system of an outbreak of dengue fever is considered. Firstly, we propose a numerical method for the fractional-order dengue fever system based on the Gorenflo-Mainardi-Moretti-Paradisi (GMMP) scheme and the Newton method. Secondly, two methods, the modified grid approximation method (MGAM) and the modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) algorithm are expanded to estimate the fractional orders and coefficients for fractional differential equations. Then, we use GMMP and MH-NMSS-PSO to estimate the parameters of the fractional-order dengue fever system. With the new fractional orders and parameters, our fractional-order dengue fever system is capable of providing numerical results that agree very well with the real data. Furthermore, for searching a better dengue fever system, a multi-term fractional-order epidemic system of dengue fever is proposed. We also use the MGAM and MH-NMSS-PSO to estimate the fractional orders and coefficients of the multi-term fractional-order system. To verify the efficiency and accuracy of the proposed methods in dealing with the fractional inverse problem, a numerical example with real data is investigated. Using the statistics from the 2009 outbreak of the disease in the Cape Verde islands, we are able to predict the fractional orders and parameters of the fractional dengue fever system. With the new fractional orders and parameters, our multi-term fractional-order dengue fever system is capable of providing numerical results that agree better with the real data than other integer-order models. [ABSTRACT FROM AUTHOR]

This paper deals with a new mathematical model for a dengue fever outbreak based on a system of fractional differential equations. The equilibrium points and stability of the new system are studied. To simulate this model, a new and efficient numerical method is provided and its stability and convergence are proved. According to a real outbreak on the Cape Verde Islands occurred in year 2009, the new model is examined for a period of three months by using singular or nonsingular kernels in the definition of derivative operator. Simulation results show that the proposed formalism with exponential kernel agrees well with the real data in the early stage of the epidemic while the Mittag-Leffler kernel fits the reality for the later part of the time interval. Hence, the new framework in a hybrid manner can properly simulate the dynamics of the disease in the whole of the time interval. In order to stabilize the disease-free equilibrium point of the system under investigation, two control strategies are suggested. Numerical simulations verify that the proposed stabilizing controllers are efficient and provide significantly remarkable results. • A novel fractional model of the dengue fever outbreak is proposed. • A new numerical method is provided and its stability and convergence are proved. • Two stabilizing controllers are suggested for the disease-free equilibrium point. • The new framework can simulate the dynamics of the disease properly. • The stabilizing controllers are efficient and provide satisfactory results. [ABSTRACT FROM AUTHOR]