The role of prostitution on HIV transmission with memory: A modeling approach

HIV is a topic that has been greatly discussed and researched due to its impact on human population. Many campaigns have been put into place, and people have been made aware of the various effects of the disease. This paper considers a fractional-order HIV epidemic model with the inclusion of prostitution in the population and its consequences on the disease transmission. The model describes the spread of HIV disease in a system consisting of a population of susceptible males and the female sex workers. The focus is on the spread of HIV by female sex workers through prostitution, because in the present world sexual transmission is the major cause of the HIV transmission. The fractional derivatives are taken in Caputo sense and the numerical solution of the model is obtained by L1 scheme which involves the memory trace that can capture and integrate all past activity. Positivity and boundedness of the solution and the stability conditions of the fractional-order system are determined. Moreover, model equilibria are determined, and their stability analysis are considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle's invariant principle. The threshold quantity namely basic reproduction number R0 is calculated and analyzed for the disease status. On the basis of R0, the disease progress can be determined i.e., the population is free from the disease if R01. Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Further, numerical simulations of the model and their graphical presentations are performed to visualize the effectiveness of our theoretical results and to observe the impact of the arbitrary order derivative. The results obtained show the effectiveness and strength of the applied L1 scheme.