A Mathematical Model of Anthrax Transmission in Animal Populations

A general mathematical model of anthrax (caused by Bacillus anthracis) transmission is formulated that includes live animals, infected carcasses and spores in the environment. The basic reproduction number (mathcal {R}_0) is calculated, and existence of a unique endemic equilibrium is established for (mathcal {R}_0) above the threshold value 1. Using data from the literature, elasticity indices for (mathcal {R}_0) and type reproduction numbers are computed to quantify anthrax control measures. Including only herbivorous animals, anthrax is eradicated if (mathcal {R}_0 < 1). For these animals, oscillatory solutions arising from Hopf bifurcations are numerically shown to exist for certain parameter values with (mathcal {R}_0>1) and to have periodicity as observed from anthrax data. Including carnivores and assuming no disease-related death, anthrax again goes extinct below the threshold. Local stability of the endemic equilibrium is established above the threshold; thus, periodic solutions are not possible for these populations. It is shown numerically that oscillations in spore growth may drive oscillations in animal populations; however, the total number of infected animals remains about the same as with constant spore growth.