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Discontinuity, Nonlinearity, and Complexity 13(4) (2024) 675--688 | DOI:10.5890/DNC.2024.12.008

Subrata Paul$^{1}$, Animesh Mahata$^2$, Supriya Mukherjee$^3$, Prakash Chandra Mali$^4$, Banamali Roy$^5$

$^{1}$ Department of Mathematics, Arambagh Government Polytechnic, Arambagh-712602, India

$^{2}$ Department of Mathematics, Sri Ramkrishna Sarada Vidya Mahapitha, Kamarpukur, Hooghly- 712612, India

$^{3}$ Department of Mathematics, Gurudas College, 1/1, Suren Sarkar Road, Kolkata-700054, India

$^{4}$ Department of Mathematics, Jadavpur University, 188, Raja S.C. Mallik Road, Kolkata-700032, India

$^{5}$ Department of Mathematics, Bangabasi Evening College, 19, Rajkumar Chakraborty Sarani, Kolkata-700009, India

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Abstract

A nonlinear delayed fractional order SIR compartmental model with a Holling type II saturated incidence rate and treatment rate are explored in this manuscript in the Caputo order fractional derivative approach. A few findings for the new model's existence and uniqueness criterion, as well as non-negativity and boundedness of the solution, have been established. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium point $E_0$ when $R_0< 1$ and at epidemic equilibrium $E_1$ when $R_0 >1$. We have studied forward bifurcation at $E_0$ and Hopf bifurcation at $E_1$ theoretically as well as numerically of our proposed model. The stability behavior of the endemic equilibrium is also discussed, revealing that oscillatory and periodic solutions may appear via Hopf bifurcation when regarding delay as the bifurcation parameter. Analytical correlations between delay and other system characteristics are built to ensure the presence of stability situations. Additionally, the suggested model's solution is approximated using the fractional-order Taylor's technique. Both graphical presentations and numerical simulations might be carried out with the help of MATLAB.

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