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 [Formula: see text]. 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 [Formula: see text], the investigated discrete formulations will be [Formula: see text]-nonincreasing or [Formula: see text]-nondecreasing by examining [Formula: see text]-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.

Keywords: Airborne respiratory tract infection; Lyapunov functional; Modified Atangana-Baleanu-Caputo fractional operator; Numerical and epidemiological modeling; Optimum control; Parameter estimation; Vaccination.

Fig. 1

Schematic diagram for the spread of the influenza virus across the age group.

Fig. 2

Surface plots for with influenza transmission (3.3) for varying parameters for .

Fig. 3

Overall influenza occurrences and temperature information indicated by the red histogram show the prevalence of incidents of influenza in Madrid province between January 2012 and December 2023. The gray area represents November, December, and January annually. According to January 2012 to December 2023, the monthly province was purple, whereas the monthly , was blue. The green line shows the entirety of the monthly from January 2012 to December 2023. The cyan line represents the monthly total in the Madrid province from January 2012 to December 2023.

Fig. 4

Contact matrix for all three generations with validity of correlation coefficients other than zero: (a) indicates (b) indicates . The fitted plots show a monthly mean (c) and (d) , respectively.

Fig. 5

(a) The appropriate findings for the amount of additional cases documented between the start of 2012 and at the end of 2023. This signifies the estimated information, while the red dots indicate the real facts. The regions that go from thickest to lightest represent the 50%, 90%, 95%, and 99% probability ranges of the confidence accuracy. (b) The appropriate outcome of the amount of incidents that were not documented between January 2012 and December 2023. The blue curve depicts the elimination fraction , whereas the light-red shaded region shows the confidence level of the 95% range. The gray area represents November, December and January cases for each year for age heterogeneity.

Fig. 6

The M-C-M-C for the latest 1000 data points of . In other figure, the purple dot indicates the measurements associated with the element. The probability histogram for . The red curve is the normal distribution curve for .

Fig. 7

Performing a sensitivity assessment on the parameters used that impact and their correlation for PRCC outcomes.

Fig. 8

Time development mechanism for the discrete fractional-order influenza model (3.3) is shown concerning and . For combining (a,b) and (c,d) and and (e,f) and and (g,h) and respectively.

Fig. 9

Time development mechanism for the discrete fractional-order influenza model (3.3) is shown concerning and . For combining (a,b) and (c,d) and and (e,f) and and and and respectively, with the risk factor for acquiring the infection between and .

Fig. 10

Time development mechanism for the discrete fractional-order influenza model (3.3) is shown concerning to . For combining (a) and (b) and and (c) and respectively, with the possibility that will be infested by a contaminated individual .

Fig. 11

Time development mechanism for the discrete fractional-order influenza model (3.3) is shown concerning . For combining (a) and with immunization rate (b) and with recovery rate factor and (c) and respectively, with the self-recovery factor with .

Fig. 12

Time development mechanism for the discrete fractional-order influenza model (3.3) is shown concerning to . For combining (a) and (b) and and (c) and respectively, with probability of inducing after being infested.

Fig. 13

Plots for the impact of only using vaccines and promotional activities, when the scenario with and without controls uses technique (1) due to the discrete fractional-order of the mABC difference operator for the influenza model (6.1) in terms of age heterogeneity.

Fig. 14

Plots for the impact of only using vaccines and promotional activities, when the scenario with and without controls uses technique (2) due to the discrete fractional-order of the mABC difference operator for the influenza model (6.1) in terms of age heterogeneity.

Fig. 15

Plots for the impact of only using vaccines and promotional activities, when the scenario with and without controls uses technique (3) due to the discrete fractional-order of the mABC difference operator for the influenza model (6.1) in terms of age heterogeneity.

Fig. 16

Plots for the impact of only using vaccines and promotional activities, when the scenario with and without controls uses technique (4) due to the discrete fractional-order of the mABC difference operator for the influenza model (6.1) in terms of age heterogeneity.

Fig. 17

Plots for the impact of without using vaccines and promotional activities, when the scenario with and without controls uses technique (4) due to the discrete fractional-order of the mABC difference operator for the influenza model (6.1) in terms of age category for the 0-19 age group and the 20-70 are purposed.

Fig. 18

Plots for the impact of only using vaccines and promotional activities, when the scenario with and without controls uses technique (5) due to the discrete fractional-order of the mABC difference operator for the influenza model (6.1) in terms of age category for the 0-19 age group and the 20-70 are purposed.