MATHEMATICAL BIOSCIENCES doi:10.3934/mbe.2017024 AND ENGINEERING Volume 14, Number 2, April 2017 pp. 377–405 OPTIMAL CONTROL ANALYSIS OF MALARIA–SCHISTOSOMIASIS CO-INFECTION DYNAMICS Kazeem Oare Okosun Department of Mathematics, Vaal University of Technology Andries Potgieter Boulevard, Vanderbijlpark, 1911, South Africa Robert Smith? Department of Mathematics, The University of Ottawa 585 King Edward Ave, Ottawa ON K1N6N5, Canada (Communicated by Gerardo Chowell-Puente) Abstract. This paper presents a mathematical model for malaria–schistosom- iasis co-infection in order to investigate their synergistic relationship in the presence of treatment. We first analyse the single infection steady states, then investigate the existence and stability of equilibria and then calculate the basic reproduction numbers. Both the single-infection models and the co-infection model exhibit backward bifurcations. We carrying out a sensitivity analysis of the co-infection model and show that schistosomiasis infection may not be associated with an increased risk of malaria. Conversely, malaria infection may be associated with an increased risk of schistosomiasis. Furthermore, we found that effective treatment and prevention of schistosomiasis infection would also assist in the effective control and eradication of malaria. Finally, we apply Pontryagin’s Maximum Principle to the model in order to determine optimal strategies for control of both diseases. 1. Introduction. Malaria and schistosomiasis often overlap in tropical and sub- tropical countries, imposing tremendous disease burdens [11, 19, 41]. The substan- tial epidemiological overlap of these two parasitic infections invariably results in frequent co-infections [16, 47]. The challenges facing the development of a highly effective malaria vaccine have generated interest in understanding the interactions between malaria and co-endemic helminth infections, such as those caused by Schis- tosoma, that could impair vaccine efficacy by modulating host-immune responses to Plasmodium infection and treatment [40, 41]. Both malaria and schistosomiasis are endemic to most African nations. However, the extent to which schistosomiasis modifies the risk of febrile malaria remains unclear. Malaria is an infectious disease that causes morbidity and mortality in the de- veloping world. There are an estimated 360 million cases [44], killing between one 2010 Mathematics Subject Classification. Primary: 92B05, 93A30; Secondary: 93C15. Key words and phrases. Malaria, schistosomiasis, optimal control. The authors are grateful to two anonymous reviewers whose comments greatly improved the manuscript. KOO acknowledges the Vaal University of Technology Research Office and the Na- tional Research Foundation (NRF), South Africa, through the KIC Grant ID 97192 for the finan- cial support to attend and present this paper at the AMMCS-CAIMS 2015 meeting in Waterloo, Canada. RS? is supported by an NSERC Discovery Grant. For citation purposes, please note that the question mark in “Smith?” is part of the author’s name. 377