The development of a deterministic dengue epidemic model with the influence of temperature: A case study in Malaysia.

• Dengue model with the temperature effect, using integer-order and the fractional-order differential equations is proposed. • The optimal temperature for a maximum dengue transmission is around 32∘C. • The backward bifurcation occurs in the presence of dengueinduced death rate. • The solutions of the aquatic stage of the integer-order model exhibit oscillatory behaviour whenever dengue is endemic. • Fractional-order dengue model provides stable solutions compared to the integer-order model. In this study, a deterministic mathematical model of the dengue transmission is developed by considering the effect of temperature on the transmission dynamics. It has a locally asymptotically stable disease-free equilibrium (DFE) point whenever the basic reproduction number (R 0) is less than unity. This model has shown a possibility of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium point when R 0 < 1. Using the entomological data of the Aedes mosquito population and the experimental data of the dengue transmission in Malaysia, R 0 is evaluated at different temperatures, given that at 32∘C, R 0 attain its maximum value. The solutions of the model show an oscillatory behaviour. Although the oscillations are unobservable in the state variable, they are present in the numerical solution. Finally, a fractional analogue of the dengue model is presented, and numerical solutions are performed to make a comparison with the integer-order model. A comparison of the two results reveals that the fractional-order model provides stable solutions as the oscillatory behaviour can be dampened. This paper provides numerical evidence that the dengue virus can spread in temperate areas where it is no longer limited to tropical and subtropical regions, and suggests that the fractional-order approach can be a great alternative in modelling dengue transmission dynamics. [ABSTRACT FROM AUTHOR]