Dynamical Study of Dengue Model with Caputo Time Fractional Derivative

Abstract

We discussed a viral disease that is named as Dengue which is caused by biting a female mosquito (Aedes). This disease is mild in the beginning and becomes severe with time. Dengue is a fever type epidermic disease spreading in over 100 countries nationwide. Now it is affecting about half of world population. Analysis is being done about its spread. A number of models are presented to control the spread of dengue. In the mid of 2005, dengue fever started spreading as the severe viral disease that transmit by means of a vector, periling 2.5 billion persons [14]. For the control of dengue, a number of mathematical models are being introduced [31]. In order to control female mosquito production, sterile male mosquito is being introduced instead of insecticides [32]. Apart from its control, a number of drugs have also been introduced for its treatment. Vaccines are also available for dengue. Fractional calculus is the study of an extended form of integrals and derivatives in frac tional orders. The most vital aspect of fractional derivatives is that, these models hold memory. In this thesis we have investigated two fractional order dengue models. First a differential equation of non integer order dengue hemorrhagic fever for the susceptible infected-recuperated (SIR) model is analysed. We find the disease free model of equilib rium and endemic model of equilibrium of the specific model analytically. Further we compute the R0 and show that R0 is less than unity, then disease-free equilibrium will be stable and for R0 > 1 the endemic equilibrium will be stable. Furthermore, fractional or der dengue model will be solved by trapezoidal method and simulations are presented. Secondly we have considered a dengue model with temperature effect where the man pop ulation split in three sections (susceptible, infected humans and resistant people), whereas the population of the female mosquito can be categorized in three parts (susceptible, in fected but not infectious and infectious) and pre-adult the population of female mosquito is distributed in two parts (infected mosquitos and the susceptible mosquitos). R0 be the num ber of basic reproduction and is derived using the method of next generation. We solved