This article deals with the study of a biological feasible two-strain epidemic model considering six compartments with non-monotone incidence and saturated treatment. The model has four types of equilibria, namely disease-free equilibrium, strain-1 endemic equilibrium, strain-2 endemic equilibrium and co-infected endemic equilibrium. We have shown local and global stability of different equilibria in terms of basic reproduction numbers of the system (R0\documentclass[12pt]{minimal}
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\begin{document}$$ R_0 $$\end{document}) and for two strains (namely R0′\documentclass[12pt]{minimal}
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\begin{document}$$R_0^{'}$$\end{document} and R0′′\documentclass[12pt]{minimal}
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\begin{document}$$R_0^{''}$$\end{document}). If both R0′\documentclass[12pt]{minimal}
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\begin{document}$$R_0^{'}$$\end{document} and R0′′\documentclass[12pt]{minimal}
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\begin{document}$$R_0^{''}$$\end{document} are less than unity, then the disease eradicates from the community. It has been also established that global stability of different endemic equilibria depends on values of both R0′\documentclass[12pt]{minimal}
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\begin{document}$$R_0^{'}$$\end{document}, R0′′\documentclass[12pt]{minimal}
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\begin{document}$$R_0^{''}$$\end{document} and also on strain inhibitory effect reproduction numbers (Rα1\documentclass[12pt]{minimal}
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\begin{document}$$R_{\alpha _1}$$\end{document} or/and Rα2\documentclass[12pt]{minimal}
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\begin{document}$$R_{\alpha _2}$$\end{document}). We have also verified the principle of exclusive competition, i.e., strain with a higher reproduction number dominates other strains with a lower reproduction number. We have established that the model experiences transcritical bifurcation depending on values of the reproduction numbers. Lastly, we have formulated a time-dependent optimal control problem using Pontryagin’s maximum principle to minimize the number of infected populations and also to reduce the implemented cost for applied treatment control. Numerical simulations are carried out to establish the effect of model parameters on disease spreading as well as the effect of controls.
