Dynamical analysis of an anthrax disease model in animals with nonlinear transmission rate

Abstract

<abstract><p>Anthrax is a bacterial infection caused by <italic>Bacillus anthracis</italic>, primarily affecting animals and occasionally affecting humans. This paper presents two compartmental deterministic models of anthrax transmission having vaccination compartments. In both models, a nonlinear ratio-dependent disease transmission function is employed, and the latter model distinguishes itself by incorporating fractional order derivatives, which adds a novel aspect to the study. The basic reproduction number $ \mathcal{R}_0 $ of the epidemic is determined, below which the disease is eradicated. It is observed that among the various parameters, the contact rate, disease-induced mortality rate, and rate of animal recovery have the potential to influence this basic reproduction number. The endemic equilibrium becomes disease-free via transcritical bifurcations for different threshold parameters of animal recovery rate, disease-induced mortality rate and disease transmission rate, which is validated by utilizing Sotomayor's theorem. Numerical simulations have revealed that a higher vaccination rate contributes to eradicating the disease within the ecosystem. This can be achieved by effectively controlling the disease-induced death rate and promoting animal recovery. The extended fractional model is analyzed numerically using the Adams-Bashforth-Moulton type predictor-corrector scheme. Finally, it is observed that an increase in the fractional order parameter has the potential to reduce the time duration required to eradicate the disease from the ecosystem.</p></abstract>