Impacts of non-locality and memory kernel of fractional derivative, awareness and treatment strategies on HIV/AIDS prevalence

Abstract

We propose a Liouville–Caputo non-linear fractional order α(0<α≤1) epidemic model to understand the transmission dynamics of infectious diseases, like HIV/AIDS, and explore the presence of memory kernel of fractional derivative for the precautionary measure against the transmission of disease to control HIV/AIDS prevalence. A proper sex education, which depends on number of HIV and AIDS, to unaware susceptible as well as fear of infection among these aware susceptible are employed here. Treatment compartment is incorporated to treat the symptomatic HIV and AIDS infected individuals.Using the Banach fixed point theorem, we first ensure the positivity, existence and uniqueness of solution, and biologically feasible region. Next after defining the basic reproduction number (R0), its uses on stability (local and global) and direction of bifurcation are manifested here. A quantitative assessment is conducted onR0and then we explore the sensitive parameter/s, which control the HIV/AIDS prevalence.Numerical experiment is conducted based on the Adams–Bashforth–Moulton predictor corrector method to obtain the approximate solution of the proposed model. Impacts of disease transmission rate, awareness, psychological fear of infection and treatment strategies on the HIV/AIDS prevalence are analysed based on non-linear dynamical tools, like, time series, phase diagram and spectra analysis. An effort is made to see how the order and kernel of the fractional derivative play a crucial role on the dynamics of the system and in the precautionary measure against the transmission of disease during transition as well as post-transition phases. We employ intervention strategies for controlling the spread as well as possibility of eradicating the HIV/AIDS prevalence. It is seen that the reduction ofαis comparable to awareness, psychological fear of infection and treatment strategies to control disease. Finally, it is shown that the use of fractional derivative in disease transmission dynamics is one of the most efficient applied mathematics content compatible with the natural characteristics of human memory.