On improving public health after COVID-19 epidemic: A fractal-fractional mathematical solutions with short memory effect and efficient optimal strategies.

As per the report of W.H.O. about 7 million people died in India till date due to COVID-19 infection. The transmission of COVID-19 infection can affect the temporal and geographic diversity of environmental pollution, thereby disrupting "planetary health" and livelihood. The consensus is that COVID-19 could have significant long-lasting effects on ecosystem and society. It is possible to reach an agreement to create and maintain an ecologically sound environment and a circular bio-economy to try to solve these issues. For the first time, a fractional mathematical model is formulated where the infection is considered due to unhygienic environment with a synergy between mathematical fractal parameters and biology of the disease transmission. Other mathematical analysis such as the boundedness of solutions, the wellposedness of the proposed model concerning existence results, etc. are investigated. Additionally, evaluation of vaccine-clearance equilibrium point is performed. Sensitivity parameters analysis and model's stability also steps in. To get numerical results, the "Adams-Bashforth-Moulton" method with slight modification in the kernel is used. The fractional parameters: memory effect and fractional diffusion shows a good performance of the proposed model in depicting the disease dynamics. Consequences of follow-up optimal control functions in Susceptives and Vaccinated individuals, where feasible strategies in terms of the control maps are presented.