Dynamics of low and high pathogenic avian influenza in wild and domestic bird populations

Abstract

This paper introduces a time-since-recovery structured, multi-strain, multi-population model of avian influenza. Influenza A viruses infect many species of wild and domestic birds and are classified into two groups based on their ability to cause disease: low pathogenic avian influenza (LPAI) and high pathogenic avian influenza (HPAI). Prior infection with LPAI provides partial immunity towards HPAI. The model introduced in this paper structures LPAI-recovered birds (wild and domestic) with time-since-recovery and includes cross-immunity towards HPAI that can fade with time. The model has a unique disease-free equilibrium (DFE), unique LPAI-only and HPAI-only equilibria and at least one coexistence equilibrium. We compute the reproduction numbers of LPAI ( $ {\mathcal R}_{\rm L} $ ) and HPAI ( $ {\mathcal R}_{\rm H} $ ) and show that the DFE is locally asymptotically stable when $ {\mathcal R}_{\rm L} 1 $ ( $ {\mathcal R}_{\rm H} >1 $ ) and it is locally asymptotically stable if HPAI (LPAI) cannot invade the equilibrium, that is, if the invasion number $ \hat {\mathcal {R}}_{\rm L}^{\rm H} <1 $ ( $ \hat {\mathcal {R}}_{\rm H}^{\rm L} <1 $ ). We show using numerical simulations that the ODE version of the model, which is obtained by discarding the time-since-recovery structures (making cross-immunity constant), can exhibit oscillations, and also that the pathogens LPAI and HPAI can coexist with sustained oscillations in both populations. Through simulations, we show that even if both populations (wild and domestic) are sinks when alone, LPAI and HPAI can persist in both populations combined. Thus, reducing the reproduction numbers of LPAI and HPAI in each population to below unity is not enough to eradicate the disease. The pathogens can continue to coexist in both populations unless transmission between the populations is reduced.

Keywords

• Mathematical models
• age-structured differential equations
• reproduction number
• invasion number
• LPAI
• HPAI
• H5N1
• avian influenza