Influence of fractal–fractional differentiation and independent quadratic Lévy jumps on the dynamics of a general epidemic model with vaccination strategy

The emergence and reemergence of various infectious diseases have adversely affected human life both from a social and economic point of view, and thus the study of epidemic diseases has diverted the attention of scientists from different fields. Particularly, to understand the dynamics and control mechanisms of the novel infections, one must focus on mathematical modeling, which is assumed to be the most important tool for analyzing the new epidemics. In this sense, the most common strategy used in the literature is to characterize the dynamics of infections via compartmental models. In this work, we formulated a general mathematical model along with a vaccination strategy to reveal the novel findings on the asymptotic properties of the underlying model. The complicated rapid fluctuations of the problem are taken into consideration, and second-order quadratic jumps with four independent compensated Poisson processes are studied. The model was further enhanced by including the two general interference functions. The key threshold determinant R0⋆ is calculated and it is proved that for R0⋆>1, the model has the properties of stationarity and ergodicity (i.e., permanence scenario). However, if R0⋆ is less than one, the infection will disappear exponentially (i.e., disappearance scenario). These results exhibit that noise sources play a dominant role in explaining the asymptotic behavior of any biological phenomenon. Numerically, we confirm the above-mentioned illustrations, and the simulation suggests the following results: (a) the mean-time of the solution depends on the noise intensities (b) the second-order Poisson jumps produce a negative effect on the time required for the survival of the infection. The study was further extended by considering non-integer differential equations, and the underlying model was studied both from fractional and fractal dimensions. This combination tries to give a physical explanation of infectious pathways. The stability of fractional ordered model was studied by using the Hyers–Ulam (HU) approach. To find the numerical solution of the fractional model, we presented and developed a fractal–fractional numerical scheme by employing the Adams Bashforth’s method. We find that studying the models from a fractal and fractional point of view has a major impact on the infection’s development. The dynamics of the times-series and chaotic behaviors vary by merely modifying the fractal or fractional orders of the model. More generally, the theoretical and numerical results of this paper provide excellent insight into the long-run behavior of the epidemic under a vaccination strategy.