An epidemic prediction from analysis of a combined HIV-COVID-19 co-infection model via ABC-fractional operator

The whole world is still shaken by the new corona virus and many countries are starting opting for the lockdown again after the first wave that already killed thousands of people. New observations also show that the virus spreads quickly during the cold period closer to winter season. On the other side, the number of new infections decreases considerably during hot period closer to summer time. The geographic structure of our planet is such that when some countries (in a hemisphere) are in their winter season, others in the other hemisphere are in their summer season. However, we have observed in the world some countries undertaking national lockdown during their summer time, which result in their economy to be hugely hit. Other factors, beside the lockdown, have also impacted negatively the socio-economic situation in affected countries. These include, among others, the human immunodeficiency virus (HIV) susceptible to combine to the new corona virus. The new corona virus is indeed recent and many of its effect and impact on the society are still unknown and are still to be uncovered. Hence we use here the of Atangana-Baleanu fractional derivative to mathematically express and analyses a model of HIV disease combined with COVID-19 to assess the pandemic situation in many countries affected, such as South Africa, United Kingdom (UK), China, Spain, United States of America (USA), and Italy. A way to achieve that is to perform stability and bifurcation analysis. It is also possible to investigate in which conditions the combined model contains a forward and a backward bifurcation. Moreover, utilizing the techniques of Schaefer and Banach fixed point theorems, existence and uniqueness of solutions of the generalized fractional model were presented. Also, the Atangana-Baleanu fractional (generalized) HIV-COVID-19 con-infection model is solved numerically via well-known and effective numerical scheme and a predicted prevalence for the COVID-19 is provided. The global trend shown by the numerical simulation proves that the disease will stabilize at a later stage when adequate measures are taken.