Analyzing steady-state equilibria and bifurcations in a time-delayed SIR epidemic model featuring Crowley-Martin incidence and Holling type II treatment rates

A Venkatesh; M Prakash Raj; B Baranidharan; Mohammad Khalid Imam Rahmani; Khawaja Tauseef Tasneem; Mudassir Khan; Jayant Giri

doi:10.1016/j.heliyon.2024.e39520

Analyzing steady-state equilibria and bifurcations in a time-delayed SIR epidemic model featuring Crowley-Martin incidence and Holling type II treatment rates

Heliyon. 2024 Oct 21;10(21):e39520. doi: 10.1016/j.heliyon.2024.e39520. eCollection 2024 Nov 15.

Authors

A Venkatesh¹, M Prakash Raj¹, B Baranidharan², Mohammad Khalid Imam Rahmani³, Khawaja Tauseef Tasneem³, Mudassir Khan⁴, Jayant Giri^{5

6}

Affiliations

¹ Department of Mathematics, A.V.V.M. Sri Pushpam College (Affilated to Bharathidasan University, Tiruchirappalli), Poondi, Thanjavur, 613503, Tamilnadu, India.
² Department of Mathematics, National Institute of Technology Puducherry, Karaikal, 609609, Puducherry, India.
³ College of Computing and Informatics, Saudi Electronic University, Riyadh, Saudi Arabia.
⁴ Department of Computer Science, College of Science & Arts, Tanumah, King Khalid University, Abha, Saudi Arabia.
⁵ Department of Mechanical Engineering, Yeshwantrao Chavan College of Engineering, Nagpur, India.
⁶ Department of VLSI Microelectronics, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai, India.

Abstract

This article presents a time-delayed SIR epidemiological model that has been quantitatively examined. The model incorporates a logistic growth function for the susceptible population, a Crowley-Martin type incidence, and Holling type II treatment rates. We investigated two separate time delays. The first delay refers to the rate at which new infections occur, allowing us to evaluate the impact of the latent period. The second delay relates to the rate of treatment for those who have contracted the infection, which allows us to examine the consequences of postponed access to therapy. The investigation of the steady-state behavior of the model emphasizes two equilibria, namely, the infection-free equilibrium and the endemic equilibrium. The determination of critical values involves the use of the fundamental reproduction number, denoted $R_{0}$ , which serves as a predictive measure to determine the potential elimination of a disease within a specific population. Using the fundamental reproduction number, it can be shown that infection-free equilibrium exhibits local asymptotic stability when the value of $R_{0}$ is less than 1. In contrast, when $R_{0}$ exceeds 1, the infection-free equilibrium becomes unstable in the context of the time-delayed system. Furthermore, an analysis of the steady-state dynamics of the endemic equilibrium indicates the appearance of oscillations and periodic solutions with the Hopf bifurcation for all feasible combinations of two-time delays as the bifurcated parameter. In sensitivity analysis, a sensitivity index is utilized to evaluate the relative modification in the fundamental reproduction number caused by each parameter. In summary, numerical simulations are employed to offer empirical evidence for the theoretical findings.

Keywords: 34D20; 34D23; 37B25; Bifurcation; Holling type II treatment rate; Logistic growth; Mathematical modeling; Reproduction number; Sensitivity analysis; Stability analysis; Time delay.