Transmission Dynamics of an Influenza Model with Age of Infection and Antiviral Treatment

Abstract

Age of infection (the time lapsed since infection) is an important factor to consider when modeling the transmission dynamics of influenza under the influence of antiviral treatment and drug-resistance. In this paper, we consider an influenza model which includes an age of infection. The model includes partial differential equations (PDEs) in order to describe the variable infectiousness and effect of antivirals during the infectious period. We derived the formulas for various reproduction numbers (RN) including \({\mathcal R_{SC}}\) (the controlled RN by one sensitive case), \({\mathcal R_{TC}}\) (the controlled total RN by one sensitive case), and \({\mathcal R_R}\) (the RN by one resistant case). The model analysis shows that \({\mathcal R_{SC}}\) and \({\mathcal R_R}\) determine both the global stability of the disease free equilibrium and the existence of the non-trivial equilibria. Local stabilities of the non-trivial equilibria are also discussed. Numerical simulations are conducted to not only confirm or extend the analytic results on qualitative behaviors of the system, but also reveal important quantitative properties of the disease dynamics influenced by antiviral treatment. These results are then used to assess the effectiveness of treatment programs in terms of both the RNs and the epidemic size. Our findings illustrate possibility that a higher level of antiviral use may lead to an increase of the epidemic size, despite the fact that there is a fitness cost of the drug-resistant strains. This suggests that programs for antiviral use should be designed carefully to avoid the adverse effect.