Caputo fractional order derivative model of Zika virus transmission dynamics

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The Zika Virus (ZIKV) is a highly contagious disease, and several outbreaks have occurred since it emerged. It is transmitted from one to another human via a mosquito Aedes aegypti. There is no vaccine or established medicine available for ZIKV to date. There is an urgent need to enhance an understanding of the progression mechanism of the disease when drugs or vaccines are not available. Mathematical modeling is a tool that might be helpful to understand the progression dynamics of ZIKV which can enable us to make control strategies for invading the progression dynamics of disease. SEIR-SEI is a famous compartmental deterministic modeling based on integer-order derivative calculus. Nowadays, conversion from integer to fractional order-based derivative modeling is in trend, and it is a very effective and high degree of accuracy. In this paper, we proposed a Caputo fractional-order based susceptible-exposed-infected-recovered (SEIR) structure for hosts and a susceptible-exposed-infected (SEI) structure for mosquitoes for transmission dynamics of ZIKV. For this purpose, we modified the classical compartmental model used in the study of progression dynamics of the Zika fever outbreak in El-Salvador during 2015-16. The modified model involves nonlinear differential equations of fractional (non-integer) order which has an advantage over the classical model due to its memory effect property. Our study includes eight regions across the globe where the Zika outbreak has occurred during the year 2013-2016 including six major archipelagos of French Polynesia, i.e., Tahiti, Sous-le-vent, Moorea, Tuamotu, Marquises, and Australes. The other two regions included Costa Rica and Colombia. The outbreak in selected regions was studied first using a classical model and then compared by a fractional-order model. The data of outbreaks are best fitted with the fractional-order model which enables us to estimate the best parameters values for the outbreaks. Using this modeling, the epidemic threshold parameter $R_0$ was computed which is more accurate than the classical one. Hence, the fractional-order model for ZIKV transmission dynamics is much better prediction, analysis, and disease parameters estimation than the classical model. This modeling enhances the knowledge in the field of fractional order and understanding the ZIKV transmission accurately.