Co-infection mathematical model for HIV/AIDS and tuberculosis with optimal control in Ethiopia.

The co-epidemics of HIV/AIDS and Tuberculosis (TB) outbreak is one of a serious disease in Ethiopia that demands integrative approaches to combat its transmission. In contrast, epidemiological co-infection models often considered a single latent case and recovered individuals with TB. To bridge this gap, we presented a new optimal HIV-TB co-infection model that considers both high risk and low risk latent TB cases with taking into account preventive efforts of both HIV and TB diseases, case finding for TB and HIV/AIDS treatment. This study aimed to develop optimal HIV/AIDS-TB co-infection mathematical model to explore the best cost-effective measure to mitigate the disease burden. The model is analysed analytically by firstly segregating TB and HIV only sub models followed by the full TB-HIV co-infection model. The Disease Free Equilibrium (DFE) and Endemic Equilibrium (EE) points are found and the basic reproduction number R0 is obtained using the next generation matrix method (NGM). Based on the threshold value R0, the stabilities of equilibria for each sub-model are analysed. The DFE point is locally asymptotically stable when R0 < 1 and unstable when R0 > 1. The EE point is also asymptotically stable when R0 > 1 and does not exist otherwise. At R0 = 1, the existence of backward bifurcation phenomena is discussed. To curtail the cost and disease fatality, an optimal control model is formulated via time based controlling efforts. The optimal mathematical model is analysed both analytically and numerically. The numerical results are presented for two or more control measures at a time. In addition, the Incremental Cost-Effectiveness Ratio(ICER) has identified the best strategy which is crucial in limited resource. Hence, the model outcomes illustrated that applying HIV/AIDS prevention efforts and TB case finding concurrently is the most cost-effective strategy to offer substantial relief from the burden of the pandemic in the community. All results found in this study have significant public health lessons. We anticipated that the results will notify evidence based approaches to control the disease. Thus, this study will aids in the fight against HIV/AIDS, TB, and their co-infection policy-makers and other concerned organizations.