International Journal of Biomathematics Vol. 11, No. 2 (2018) 1850017 (38 pages) c  World Scientiﬁc Publishing Company DOI: 10.1142/S1793524518500171 Optimal control of intervention strategies in malaria–tuberculosis co-infection with relapse Peter Mpasho Mwamtobe School of Computer Science and Applied Mathematics University of Witwatersrand, Johannesburg Private Bag 3, Wits 2050, South Africa Mathematics and Statistics Department, University of Malawi The Malawi Polytechnic, Private Bag 303 Chichiri, Blantyre 3, Malawi pmwamtobe@gmail.com Simphiwe Mpumelelo Simelane Department of Applied Physics and Engineering Mathematics University of Johannesburg, South Africa simphinho@gmail.com Shirley Abelman ∗ and Jean Michel Tchuenche † School of Computer Science and Applied Mathematics University of Witwatersrand, Johannesburg Private Bag 3, Wits 2050, South Africa ∗ shirley.abelman@wits.ac.za † jmtchuenche@gmail.com Received 30 September 2016 Accepted 19 October 2017 Published 18 January 2018 A model which incorporates some of the basic epidemiological features of the co- dynamics of malaria and tuberculosis (TB) is formulated and the eﬀectiveness of current intervention strategies of these two diseases is analyzed. The malaria-only and TB-only models are considered ﬁrst. Global stability disease-free steady states of the two sub- models does not hold due to the co-existence of stable disease-free with stable endemic equilibria, a phenomenon known as backward bifurcation. The dynamics of the dual malaria–TB model with intervention strategies are also analyzed. Numerical simulations of the malaria–TB model are carried out to determine whether the two diseases can co- exist. Lastly, sensitivity analysis on key parameters that drive the disease dynamics is performed in order to identify their relative importance to disease transmission. Keywords : Tuberculosis; malaria; co-infection; intervention strategies; optimal control theory. Mathematics Subject Classiﬁcation 2010: 97M60, 92D30, 92D25 1850017-1