Chaos, Solitons and Fractals 134 (2020) 109706 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Positivity and boundedness preserving numerical algorithm for the solution of fractional nonlinear epidemic model of HIV/AIDS transmission Zafar Iqbal a,f,∗ , Nauman Ahmed a,f , Dumitru Baleanu b,c , Waleed Adel d , Muhammad Rafiq e , Muhammad Aziz-ur Rehman a , Ali Saleh Alshomrani g a Department of Mathematics, University of Management and Technology, Lahore 54000, Pakistan b Department of Mathematics, Cankaya University, Balgat, Ankara 06530, Turkey c Institute of Space Sciences, Magurele-Bucharest, Romania d Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, 35516, Egypt e Faculty of Engineering, University of Central Punjab, Lahore, 54000, Pakistan f Department of Mathematics and Statistics, The University of Lahore, Lahore, 54000, Pakistan g Faculty of Science, Department of Mathematics, King Abdulaziz University, Jaddah, Saudi Arabia a r t i c l e i n f o Article history: Received 28 November 2019 Revised 23 January 2020 Accepted 17 February 2020 Keywords: HIV/AIDS disease transmission Structure preserving algorithm Fractional order epidemic model Simulations a b s t r a c t In this article, an integer order nonlinear HIV/AIDS infection model is extended to the non-integer nonlin- ear model. The Grunwald Letnikov nonstandard finite difference scheme is designed to obtain the numer- ical solutions. Structure preservence is one of the main advantages of this scheme. Reproductive number R 0 is worked out and its key role in disease dynamics and stability of the system is investigated with the following facts, if R 0 < 1 the disease will be diminished and it will persist in the community for R 0 > 1. On the other hand, it is sought out that system is stable when R 0 < 1 and R 0 > 1 implicates that system is locally asymptotically stable. Positivity and boundedness of the scheme is also proved for the generalized system. Two steady states of the system are computed and verified by computer simulations with the help of some suitable test problem. © 2020 Published by Elsevier Ltd. 1. Introduction The appearance of the HIV/AIDS disease in 1981 and its incur- ability has made it a nightmare to the mankind, especially for the biologists and health policy makers. The terror of the disease can be observed by the WHO report which describes the fright of this unstoppable disease. Acquired immunodeficiency syndrome (AIDS) is a contagious life threatening disease due to immunodeficiency virus named as human immunodeficiency virus (HIV). Both dis- eases are responsible for the destruction of the CD4+ helper T - cells in the immune system. HIV reduces the functionality of the immune system, that fights the disease causing organisms. HIV is a communicable disease that communicates through sexual contacts. ∗ Corresponding author at: Department of Mathematics, University of Manage- ment and Technology, Lahore 54000, Pakistan. E-mail addresses: zafariqbal0092@yahoo.com (Z. Iqbal), nauman.ahmd01@gmail.com (N. Ahmed), dumitru.baleanu@gmail.com (D. Baleanu), waleedadel85@yahoo.com (W. Adel), m.rafiq@ucp.edu.pk (M. Rafiq), aziz.rehman@umt.edu.pk (M. Aziz-ur Rehman), aszalshomrani@ku.edu.sa (A.S. Alshomrani). It can also be transmitted through infected blood transfusion, from mother to child before birth (during pregnancy). Even at the stage of child birth and mother feeding, baby can receive the disease. HIV takes a long time to progress into AIDS by weakening the im- mune system. The HIV and AIDS diseases are incurable so far, but there are medicines that slow down the progress of the disease. This virus infection can be studied by considering two stages as primary infection also termed as acute HIV. This stage may remain for a few weeks. Generally at this stage, viral load (i.e the amount of virus in the blood stream) is considerably high. Consequently, the infection spreads more rapidly than the next stage. The second stage is called clinical latent infection (Chronic HIV). At this stage, some symptoms such as swollen lymph nodes, weight loss, oral yeast and shingles etc., may appear. These symp- toms are not final to detect the disease. In some cases, there are no particular signs and symptoms that may be reflected by the pa- tient. These types of infected cases are called asymptomatic, others are called symptomatic. It takes decades for HIV to develop into AIDS. When the AIDS develops, it severely damages the immune system, making a person susceptible to different types of cancers. https://doi.org/10.1016/j.chaos.2020.109706 0960-0779/© 2020 Published by Elsevier Ltd.