Eur. Phys. J. Plus (2017) 132: 363
https://doi.org/10.1140/epjp/i2017-11625-3

Regular Article

On the dynamics of HIV-AIDS and cryptosporidiosis

¹ Department of Mathematics, Vaal University of Technology, Vanderbijlpark, South Africa
² Department of Mathematics, City University of Science and Information Technology, 2500, Khyber Pakhtunkhwa, Pakistan
³ Department of Mathematics and Statistics, Kumasi Technical University, Kumasi, Ghana
⁴ Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria

^* e-mail: altafdir@gmail.com

Received: 10 June 2017
Accepted: 30 June 2017
Published online: 25 August 2017

Abstract

This paper seeks to examine a mathematical model for cryptosporidiosis-HIV co-infection, in order to explore their synergistic relationship in the presence of prevention and treatment. Firstly, we investigate the sub-models and their steady states properties. They are found to be locally and globally stable. Thereafter, the steady states of the co-infected model waswere studied and itthey proved to exhibit backward bifurcation phenomena. Furthermore, we incorporate time-dependent controls in the model and apply Pontryagin’s maximum principle, so that we can determine the best optimal strategy to control the co-infected diseases. Finally, we present the numerical results, which show that the best strategy to control co-infection is to combine all the five controls at the same time.