A mathematical model of anthrax transmission in animal populations
This paper studies a general mathematical model for the transmission of anthrax in animals. The model is formulated as a set of ordinary differential equations as follows: \[ \begin{aligned}\frac{ds}{dt}& =r(s+i)\left(1-\frac{s+i}{K}\right)-\eta_a as-\eta_c sc-\eta_i\frac{si}{s+i}-\mu s+\tau i,\\ \frac{di}{dt} & =\eta_a as+\eta_c sc+\left(\eta_i \frac{s}{s+i}-(\gamma+\mu+\tau)\right)i,\\ \frac{da}{dt}& =-\alpha a+\beta c,\\ \frac{dc}{dt}& =(\gamma+\mu)i-\delta(s+i)c-\kappa c,\end{aligned} \] where the parameters \(r,\mu,\tau,\gamma,\delta,\kappa,\beta,\alpha\) are birth rate, death rate, recovery rate, disease death rate, carcass consumption rate, carcass decay rate, spore growth rate, and spore decay rate, respectively; \(K\) represents the carrying capacity, and \(\eta_a,\eta_i,\eta_c\) are the transmission rates for environment to susceptible animals, infected animals to susceptible animals, infected carcasses to susceptible animals, respectively. Under the assumption of \(\eta_i/(\gamma+\mu+\tau)<1\) and \(n=s+i\), the authors demonstrate the existence of two disease-free equilibria, compute the basic reproduction number, and prove the existence of a unique endemic equilibrium under some conditions. The type reproduction numbers for eradication of anthrax in animal populations are compared. The model of herbivores is investigated numerically and periodic solutions are shown for some parameter values. Numerical results reveal that oscillations in spore growth could drive oscillations in animal populations.
scientific article; zbMATH DE number 6715425