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Mathematical Modelling of the Spread of Avian Influenza (Bird flu) Using Reaction Diffusion Equation and Travelling Wave

Benneth Peter and, Saheed K. Olarewaju

Abstract

Avian influenza remains a major threat to both wild and domestic bird populations, largely because its spatial spread through migratory movement is not fully understood. Developing accurate spatial models of transmission is essential for predicting outbreak patterns and guiding effective control strategies. To address this gap, this study constructs a reaction diffusion SIR model to describe how avian influenza spreads through a homogeneous bird population. The main objective is to understand the dynamics of disease invasion by determining the travelling-wave speed, finding the final proportion of susceptible birds after the outbreak, and analyzing the stability conditions under which the infection can either spread or die out. The model assumes that only infected birds move across space, while susceptible and recovered birds remain stationary. After formulating the system, the equations were nondimensionalized and transformed into ordinary differential equations using a travelling-wave approach. Linearization around the disease-free equilibrium yields a characteristic expression that determines the wave speed. The analysis shows that a travelling wave exists only when the speed satisfies c?2?(1 ? 1 R0) The final number of susceptible birds is determined by the implicit relation(send) ? 1 R0 ln(send) = 1 Stability analysis shows that the disease-free equilibrium is stable when 0 R <1 and unstable when 0 R >1. Overall, the results demonstrate that the movement of infected birds strongly influences both the speed and severity of epidemic spread. The model provides a useful theoretical framework for understanding spatial disease dynamics in migratory bird populations and offers insights that can support control measures such as quarantine, vaccination, and restrictions on bird movement.

Keywords

Mathematical ModelingAvian InfluenzaNondimensionalizationlinearization

References

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