Theoretical advancement in dengue transmission model in Malaysia using fractional order differential equations

This thesis aims to deploy and develop fractional-order differential equations that possess hereditary properties in modelling the dengue transmission dynamics. Dengue models are generally of integer-order derivative systems, that cannot fully explain the behaviour of dengue transmission, which involves memory effect. In this study, three different deterministic fractional-order models are constructed, including the basic framework model, temperature-driven model, and dengue control model, using Caputo’s derivative definition. The susceptible-infected-recovered (SIR) model is considered in the formulation. The significant differences between the integer-order model and the fractional-order model, and the relation of the order of the derivative with the dynamical behaviour of the dengue epidemic are addressed. Furthermore, this thesis discusses the effect of the temperature in the dengue transmission, and the efficacy of current dengue control measures, particularly in Malaysia. The theoretical analysis of the existence and stability of the equilibrium point is presented in detail. Additionally, sensitivity analysis is performed to assess the importance of model parameters in disease transmission and disease prevalence. The recorded dengue cases in Malaysia are used in numerical simulations. Numerical results reveal that the convergence rate of fractional-order models is more gradual compared to the integer-order model. A lower value of the order corresponds to a slower decaying time and reduction in the size of epidemics. The temperature-driven models show that the fractional-order model is more stable since there is no oscillatory behaviour observed in the solutions, unlike in the integer-order model. These models also predict that dengue can be persisted even in the non-optimal temperature condition. The dengue control model shows that vector control tools are the most efficient way to combat the spread of dengue viruses, and the combination of them with individual protection makes it more effective. In fact, with the massive application of individual protection only, the number of cases can be reduced. Conversely, mechanical control alone cannot suppress the excessive number of cases in the population, although it can significantly reduce the number of Aedes mosquitoes. Overall, these findings have significant implications in understanding the transmission dynamics of dengue, and the proposed fractional-order models are found to be a great alternative in describing the real epidemic of dengue transmission.

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