Mathematical modeling and nonstandard finite difference scheme analysis for the environmental and spillover transmissions of Avian Influenza A model

This work models, analyzes and assesses the impacts of environmental and spillover transmissions on Avian Influenza Virus (AIV) type A infection formulated in terms of nonlinear ordinary differential system that takes into account five spreading pathways: poultry-to-poultry; environment-to-poultry; poultry-to-human (spillover event); environment-to-human and poultry-to-environment. An in-depth theoretical and numerical analysis of the model is performed as follows. The basic reproduction number is computed and shown to be a sharp threshold for the global asymptotic dynamics of the submodel without recruitment of infected poultry. These results are obtained through the construction of suitable Lyapunov functions and the application of Poincare-Bendixson combined with Lyapunov-LaSalle techniques. When the infected poultry is brought into the population, the model exhibits only a unique endemic equilibrium whose global asymptotic stability is established using the same techniques mentioned earlier. Further, the model is shown to exhibit a transcritical bifurcation with the value one of the basic reproduction number being the bifurcation parameter threshold. We further prove that during avian influenza outbreaks, the recruitment of infected poultry increases the disease endemic level. We show that the classical Runge-Kutta numerical method fails to preserve the positivity of solutions and alternatively design a nonstandard finite difference scheme (NSFD), which preserves the essential properties of the continuous system. Numerical simulations are implemented to illustrate the theoretical results and assess the role of the environmental and spillover transmissions on the disease.