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

Ankur Jyoti Kashyap; Arnab Jyoti Bordoloi; Fanitsha Mohan; Anuradha Devi; Ankur Jyoti Kashyap; Arnab Jyoti Bordoloi; Fanitsha Mohan; Anuradha Devi

doi:10.3934/mmc.2023030

Mathematical Modelling and Control

2023, Volume 3, Issue 4: 370-386. doi: 10.3934/mmc.2023030

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Research article

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

1.
Department of Mathematics, The Assam Royal Global University, Assam, India
2.
Department of Mathematics, Girijananda Chowdhury University, Assam, India

Received: 25 May 2023 Revised: 25 May 2023 Accepted: 11 October 2023 Published: 22 December 2023

Anthrax is a bacterial infection caused by Bacillus anthracis, 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.
- anthrax disease,
- basic reproduction number,
- asymptotic stability,
- bifurcation analysis,
- fractional-order system
Citation: Ankur Jyoti Kashyap, Arnab Jyoti Bordoloi, Fanitsha Mohan, Anuradha Devi. Dynamical analysis of an anthrax disease model in animals with nonlinear transmission rate[J]. Mathematical Modelling and Control, 2023, 3(4): 370-386. doi: 10.3934/mmc.2023030

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Abstract

Anthrax is a bacterial infection caused by Bacillus anthracis, 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.

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