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Uncertainties due to randomness and fuzziness comprehensively exist in control and decision support systems. In the present study, we introduce notion of occurring probability of possible values into hesitant fuzzy linguistic element (HFLE) and define hesitant probabilistic fuzzy linguistic set (HPFLS) for ill structured and complex decision making problem. HPFLS provides a single framework where both stochastic and non-stochastic uncertainties can be efficiently handled along with hesitation. We have also proposed expected mean, variance, score and accuracy function and basic operations for HPFLS. Weighted and ordered weighted aggregation operators for HPFLS are also defined in the present study for its applications in multi-criteria group decision making (MCGDM) problems. We propose a MCGDM method with HPFL information which is illustrated by an example. A real case study is also taken in the present study to rank State Bank of India, InfoTech Enterprises, I.T.C., H.D.F.C. Bank, Tata Steel, Tata Motors and Bajaj Finance using real data. Proposed HPFLS-based MCGDM method is also compared with two HFL-based decision making methods.

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A stochastic SIR influenza vertical transmission model is examined in this paper where vaccination and an incidence rate that is not linear are considered. To determine whether testosterone regulates lower sintering HPA axis function in males, we used a stochastic SIR epidemic procedure with divergent influences on ACTH and cortisol. The suppressive effects on cortisol can be attributed to a peripheral (adrenal) locus. Following that, we came to the conclusion that experimental solutions have been discovered and the requisite statistical findings have been examined. Finally, we deduce that the given mathematical model and the results are relevant to medical research. In the future, this research can be further extended to simulate more results in the medical field.