Applied Mathematics and Computation 339 (2018) 488–506 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Modeling the within-host co-infection of influenza A virus and pneumococcus Fulgensia Kamugisha Mbabazi a,∗ , J.Y.T. Mugisha a,b , M. Kimathi c a Pan African University Institute of Basic Sciences Technology and Innovation, Nairobi P.O. Box 62000–00200, Kenya b Department of Mathematics, Makerere University, Kampala P.O. Box 7062, Uganda c Department of Mathematics, Statistics and Actuarial Sciences, Machakos University, Machakos P.O. Box, 136–90100, Kenya a r t i c l e i n f o 2010 MSC: 92B05 Keywords: Within-host model Co-infection Global stability Pathogen fitness Sensitivity a b s t r a c t In this paper a nonlinear mathematical model for a within-host co-infection of influenza A virus and pneumococcus is investigated. Conditions that explain the relations amid R IP and its relationship to the global asymptotic stability of the infection-free steady state are dis- cussed. A graph-theoretic method shows that, the unique endemic steady state is globally asymptotically stable. The sensitivity analysis show that, the pathogen fitness for pneu- mococcus and influenza A virus are most sensitive to maximum number of bacteria an alveolar macrophage can catch, phagocytosis rate, number of infectious IAV and pneumo- coccus particles liberated from lysis of infected cells and infection rates of influenza A virus and pneumococcal. Numerical results of the model show that, there exists a biolog- ically important steady state where the two infectious pathogens of unequal strength co- exist and replace each other in the epithelial cell population (with pneumococcus leading) when the pathogen fitness for each infection exceeds unity, and we find that this endemic steady state is globally asymptotically stable. Further, the impact of influenza A virus on pneumococcus and vice-visa leads to a bifurcation state. © 2018 Elsevier Inc. All rights reserved. 1. Introduction Infectious diseases are significant and frequently cause human illness that lead to mortality across the globe. Influenza A commonly known as ’flu’ is an infectious disease caused by a virus that is categorized in four different types A, B, C and D (IAV, IBV, ICV and IDV) [71], but only influenza A and B viruses cause clinically significant human disease and seasonal epi- demics [77]. It causes yearly chronic epidemic outbreaks, and individuals become infected several times over their lifetime [4]. They are distinguished by differences in two major virus surface proteins (Hemagglutinin (HA) and Neuraminidase (NA)) [75]. There are 16 diverse types of HA and 9 diverse types of NA, thus, there are potentially 144 diverse subtypes of influenza A viruses [76]. With these types, virus A is epidemiologically essential for humans because it can recombine its genes with those of strains circulating in animal populations (birds, swine and horses). In Africa, surveillance data on Influenza-Like Ill- ness (ILI) and Severe Acute Respiratory Infection (SARS) presented at the 6th African Network for influenza surveillance and epidemiology (AISE) confirmed influenza A virus in most African countries [84]. Influenza is highly infectious, transmitted ∗ Corresponding author. E-mail addresses: fmbaba1zi@gmail.com, fmbabazi1@sci.busitema.ac.ug (F.K. Mbabazi). https://doi.org/10.1016/j.amc.2018.07.031 0096-3003/© 2018 Elsevier Inc. All rights reserved.