Problem statement: Mathematical models are a useful tool for understanding and describing the transmission of diseases such as dengue fever, one of the most prevalently emerging diseases common to tropical and subtropical areas throughout South East Asia. By taking into account human susceptibility to disease, the dynamics of a dengue disease model is proposed. Approach: Using standard methods for analyzing a system, the stability of the model is determined by using Routh-Hurwitz criteria. Results and Conclusion: We can show that the basic reproductive number (R 0 ), the threshold parameter, when R 0 <1, the disease-free state is locally asymptotically stable. If R 0 >1, the endemic equilibrium state is locally asymptotically stable. Numerical results illustrate the dynamics of the disease within the context of varying parameter values.

The generalized time fractional Kolmogorov-Petrovsky-Piskunov equation (FKPP),, which plays an important role in engineering, chemical reaction problem is proposed by Caputo fractional order derivative sense. In this paper, we develop a framework wavelet, including shift Chebyshev polynomial of the first kind as a mother wavelet, and also construct some operational matrices that represent Caputo fractional derivative to obtain analytical solutions for FKPP equation with three different types of Initial Boundary conditions (Dirichlet, Dirichlet-Neumann, and Neumann-Robin). Our results shown that the Chebyshev wavelet is a powerful method, due to its simplicity, efficiency in analytical approximations, and its fast convergence. The comparison of the Chebyshev wavelet results indicates that the proposed method not only gives satisfactory results but also do not need large amount of CPU times.

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