A mathematical model of influenza with treatment and vaccination

The recent outbreaks of highly pathogenic avian influenza and associated human infec¬tions, arising primarily from direct contact with poultry in several regions of the world have highlighted the urgent need to prepare for the next influenza pandemic. Although measures such as closing schools, using face-masks, and keeping infected persons away from those susceptible (known as social distancing) may slow the effects of pandemic influenza, only vaccines and antiviral drugs are clearly efficacious in preventing infection or treating illness. Faced with the H5N1 pandemic threat and due to the lack of facilities for quarantine and isolation in some resource-poor countries (even though the availability of antivirals and the affordability of vaccine is also a challenging task), we construct a de¬terministic mathematical model with vaccination and treatment only in order to analyze their joint effect in curtailing an influenza epidemic. The results are interpreted in terms of the vaccination, treatment and vaccination and treatment-induced reproduction numbers, 71 v, 7&r, and 'JZvt, respectively. We observe that vaccinating and treating individuals concurrently is more effective in slowing down the epidemic than concentrating on a cohort vaccination campaign or treatment cam¬paign only. Population-level perversity cannot occur if the fitness ratio 0 < Hj < 1. Also, positivity and boundedness of solutions as well as persistence of the model are analyzed. We investigate the local and global stability of the steady states and observe that the treatment only submodel likewise the vaccination-only submodel exhibits the phenomenon of backward bifurcation, consequently, the full model also exhibits this phe¬nomenon. Sensitivity analysis on the key parameters that drive the disease dynamics is performed in order to determine their relative importance to disease transmission and prevalence. Numerical simulations are carried out to validate the model.