ABSTRACT
We aim to provide a detailed analysis of the dynamic process of the spread of one of the more fatal infectious diseases known as HIV-AIDS with the inclusion of cryptosporidiosis using different approaches. On one hand, we investigated the model using different concepts of differentiation and integration. On the other hand, we investigated a model within the framework of stochastic analysis using an unusual distribution called log-normal distribution. We provided a motivation for using each concept; for instance, we argued that the spread can follow at the same time of a Gaussian and non-Gaussian distribution but with a steady state, which corresponds to the probability distribution associated to the exponential decay law used to construct the well-known Caputo-Fabrizio fractional differential operator. However, if the distribution does not have a steady state, from Gauss to non-Gauss, the distribution is linked to that of the generalised Mittag-Leffler function used in the Atangana-Baleanu fractional derivative, thus the model with a non-singular and non-local kernel. Finally, if the model displays some random behaviours but yet follows the Markovian process, then the stochastic approach is adopted. For each model, we provided a detailed analysis underpinning the determination of conditions under which the existence of a unique set of exact solutions are guaranteed and finally provide numerical solutions using a newly established numerical scheme similar to Adams-Bashforth but yet more efficient and user-friendly. Theoretical numerical simulations suggest that these mathematical applications could be used to better understand the dynamics behind the spread of HIV/AIDS.
