A threshold policy to curb WNV transmission to birds with seasonality

Seasonality is pervasive in nature and WNV is a complex disease which appears to be transmitted periodically. Lacking of vaccine and anti-virus treatment brings culling to be an effective strategy of controlling the spread of WNV. This paper proposes a piecewise smooth model of WNV transmission between mosquitoes and birds with periodic forcing by employing the threshold policy of culling mosquitoes and birds when the number of infected birds exceeds the critical level. We firstly investigate the dynamical behaviours of the periodic subsystems and illustrate the existence and global stability of the disease-free and positive periodic solutions. Then by assuming a periodic variation in the total number of mosquitoes and birds with and without culling strategy respectively in the ecosystem, the subsystems are reduced, respectively. Further, the sliding mode domain and sliding mode dynamics are derived by using the Filippov convex method and the existence of the sliding periodic solution is illustrated. Finally, numerical investigations are carried out to address how the periodic forcing and culling strategy affect the pattern and amplitude of the novel periodic solution. Our results show that a suitable combination of threshold value and culling rates should be properly chosen to ensure that the ideal periodic solution with a low number of infected birds/mosquitoes oscillating in a small range is globally stable and consequently is approximated, which suggests useful strategies of curbing WNV transmission by setting proper threshold level and culling rates.