A mathematical model of anthrax transmission in animal populations
type: Item
instance of: scholarly article
title: A mathematical model of anthrax transmission in animal populations
zbMATH Open document ID: 1368.92191
author: Abdul-Aziz Yakubu
author: C. M. Saad-Roy
author: P. van den Driessche
published in: Bulletin of Mathematical Biology
publication date: 2017-05-15T00:00:00Z
review text: 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.
reviewed by: Yilun Shang
Mathematics Subject Classification ID: 92D30
Mathematics Subject Classification ID: 34D23
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6715425
zbMATH Keywords: anthrax
zbMATH Keywords: Hopf bifurcations
zbMATH Keywords: global stability
zbMATH Keywords: type reproduction number
MaRDI profile type: Publication
full work available at URL: s11538-016-0238-1
OpenAlex ID: W2565156843
Wikidata QID: Q39051464
cites work: Q4255467
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DOI: 10.1007/S11538-016-0238-1
description: scientific article; zbMATH DE number 6715425