Optimal vaccination model of airborne infection under variable humidity and demographic heterogeneity for hybrid fractional operator technique

Abstract Airborne respiratory tract infection typically occurs seasonally in subtropical countries, particularly during winter, when transmission and fatality rates considerably rise, indicating that low humidity and freezing temperatures facilitate the transmission of viral strains in age heterogeneity. Despite this, the atmospheric elements that contribute to periodic influenza occurrences and their critical influence on the spread of influenza stay ambiguous in various age groups. The oversight of undetected cases amid a widespread outbreak of transmissible illnesses results in an underappreciation of the prevalence of infection and the basic recurrence rate. This study proposes the dynamics of the influenza epidemic in the province of Madrid, Spain, with an emphasis on the effects of control employing actual data. The main challenge is accurately estimating the virus’s rate of transmission and assessing the effectiveness of vaccination campaigns. By taking into account the modified Atangana-Baleanu-Caputo (mABC) fractional difference operator, we develop an analytical framework for an outbreak caused by influenza and broaden it to accommodate the fractional scenario. The non-negativity and boundedness are guaranteed by the computation of the fractional-order influenza system. At the disease-free equilibrium (DFE), we perform a local asymptotic stability analysis (LAS) and display the outcome for $$\mathbb {R}_{0}<1$$ . In addition, periodic solutions and the model’s uniform permanence are proved. Environmental factors to decrease interaction between different ages, increase immunization protection, and minimize vaccine refusal risks are the most efficient way to meet preventative and surveillance targets. Our system’s best-fit parameter settings were detected using the Markov Chain Monte Carlo (M-C-M-C) technique with influenza information collected in Spain. We predict a basic reproduction number of 1.3645 (96% C.I: (1.3644, 1.3646)). The framework’s essential variables are determined using unpredictability and sensitivity evaluation. To further bolster the operator’s effectiveness, a number of tests of this novel kind of operator were conducted. We remark that in various time scale domains $$\mathbb {N}_{\mathbbm {k}}$$ , the investigated discrete formulations will be $$\Gamma _{1}^{2}$$ -nonincreasing or $$\Gamma _{1}^{2}$$ -nondecreasing by examining $$\Gamma _{1}$$ -monotonicity formulations and the basic properties of the suggested operator. Algorithms are constructed in the discrete generalized Mittag-Leffler (GML) kernel for mathematical simulations, emphasizing the effects of the infection resulting from multiple factors. The dynamical technique used to build the influenza framework was significantly impacted by fractional-order. In order to lessen the infections, time-dependent control factors are also implemented. The optimality criteria are produced by applying Pontryagin’s maximal argument to prove the validity of the most effective control. If vaccine penetration and immunity rates have been resurrected, achieving the control objective requires 12 months longer and costs less than the previous scenario.