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Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model's dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if R v is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter β 2 that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when ψ = 1 for the imperfect vaccine case; the model is both stable globally asymptotically at the disease-free ( R 0 ≤ 1 ) and endemic cases ( R 0 > 1 ). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number R 0 ≈ 1.2591 . We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail. [ABSTRACT FROM AUTHOR]

Clay minerals are one of the most utilized minerals among non-metals. These are hydrous aluminum silicates with a layer (sheet-like) structure. Kaolin is a hydrous aluminosilicate mineral with a thin platelet structure. Kaolin is extensively used in paper, paint, and many other industries. Wet processing of kaolin will not be sustainable over the long term because global freshwater resources are becoming scarce. Hence, a process is necessary that does not consume water during the beneficiation of kaolin. This study developed a dry beneficiation process for low-grade kaolin of 59.6%, with 12% quartz and about 6% titaniferous impurities from Nagar Parkar, Sindh province, Pakistan. To develop a size difference between kaolinite and impurities, steel balls clad with rubber were used as the grinding media in a selective grinding unit. Screens of 60 and 400 mesh were employed to classify the feed of air classifier. Oversize +60 mesh was reground, 400 to 60 mesh fractions were sent to an air classifier, and −400 mesh was considered to be a product with the grade and recovery of 90.6% and 20.5%, respectively. Air classifier experiments were designed using central composite design. An experiment using a fan speed of 1200 revolutions per minute (rpm) and a shutter opening of 4.0 showed optimum results, with maximum kaolinite grade and recovery of 91.5% and 35.9%, respectively. The statistical models developed for grade and recovery predicted the optimum results at a fan speed of 1251 rpm and shutter opening of 3.3 with the maximum kaolinite grade and recovery of 91.1% and 24.7%, respectively. The differences between experimental and predicted grade and recovery were 0.1% and 2.4%, respectively. The characterization results showed the total upgrade of kaolin from 59.6% to 91.2%, with 27.1% recovery during the process. The designed methodology has the potential to improve the yield of the product by focusing on its recovery. Furthermore, the designed process can be improved by using different sized balls in the selective grinding unit. This beneficiation process can utilize more than one air classifier in series to achieve the targeted results. [ABSTRACT FROM AUTHOR]