A Study of the anthrax transmission model in herbivorous animals involving vaccination and harvesting

Abstract

Caused by the bacterium Bacillus anthracis, anthrax is a serious zoonotic disease with a mortality rate of up to 60%. This disease naturally occurs in soil and commonly affects both domestic and wild animals worldwide. Humans are often infected with anthrax by consuming contaminated animal products. This research focuses on the transmission of anthrax to herbivorous animals, particularly livestock, by examining animal harvesting practices and efforts to prevent its spread through vaccination. The SVICA deterministic model was developed to better understand the transmission of anthrax by categorizing the population into susceptible, vaccinated, and infected animals, as well as carcasses and anthrax spores. The basic reproduction number ($R_0$) is calculated using the NGM method to determine the outbreak threshold in a population. Additionally, the model analyzes the local stability of two disease-free equilibrium points when $R_0<1$ and shows the existence of an endemic equilibrium point when $R_0>1$. Numerical exploration was conducted to examine the outbreak dynamics both generally and in specific cases. By varying the infection rate as a bifurcation parameter, it was found that when $R_0>1$, there is a stable interval and an unstable interval for the endemic equilibrium point, separated by the Hopf bifurcation curve. When the endemic equilibrium is unstable, a limit cycle occurs. Two distinct limit cycle behaviors were observed with different limit cycle trends. One case exhibited a more even rate of change, while the other displayed a slow–fast limit cycle, indicating a situation where anthrax cases remain low for a time but then increase drastically.