cba47f4e-b2fb-4790-a607-cebf723cc63220250314061802684wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051120202512020252410.37394/23206.2025.24https://wseas.com/journals/mathematics/2025.phpImekela D.EzekielDepartment of Mathematics, Covenant University, Ota, NIGERIASamuel A.IyaseDepartment of Mathematics, Covenant University, Ota, NIGERIATimothy A.AnakeDepartment of Mathematics, Covenant University, Ota, NIGERIAThis paper investigated the stability of the dynamical behavior of the susceptible (S), infectious (I) and recovered (R) (SIR) disease epidemic model with intracellular time delay that is unable to stabilize the unstable interior non-hyperbolic equilibrium. The study employed characteristics and bifurcation methods for investigating conditions of stability and instability of the SIR disease epidemic model using the dimensionless threshold reproduction value 𝑅0 for the disease-free equilibrium (DFE) point and the endemic equilibrium point. The study confirms that disease-free equilibrium (DFE) point and the endemic equilibrium point cannot coexist simultaneously. The paper equally investigated the local stability analysis of the reduced nonlinear SIR disease epidemic delay model when at least one of the characteristic roots has zero real parts while every other eigenvalue(s) has negative real parts. The result of the analysis of the model showed that the conditions for Hopf bifurcation obtained from the behavior of the systems are sufficient but not necessary since the model is unable to stabilize the unstable interior non-hyperbolic equilibrium. Specifically, the direction of Hopf bifurcation, the stability behavior and the period of the bifurcating periodic solutions of the interior nonhyperbolic equilibrium of the infectious disease model were explicitly determined using methods of the normal form concept (NFC) and the center manifold theorem (CMT) to investigate the transformed reduced operator differential equation (OpDE). The contribution of this paper is based on applications to assess the effectiveness of different control strategies of parameter values for stability properties of infectious disease models and can be found useful to bio-mathematicians, ecologists, biologists and public health workers for decision-making. Finally, a numerical example to verify the analytical finding was performed using the MATLAB software.3142025314202512614314https://wseas.com/journals/mathematics/2025/a285106-1985.pdf10.37394/23206.2025.24.14https://wseas.com/journals/mathematics/2025/a285106-1985.pdf10.11648/j.mma.20200503.14Edessa, G. K., and Koya, P. R. (2020). Modelling and stability analysis of a three species ecosystem with the third species response to the first species in sigmoid functional response form. Mathematical Modelling and Applications, Vol.5 No.3, pp. 156-166. doi: 10.11648/j.mma.20200503.14. 10.1007/s00285-019-01357-0Getto, P., Gyllenberg, M., Nakata, Y., and Scarabel, F. (2019). Stability analysis of a state dependent delay differential equation for cell maturation: analytical and numerical methods. Journal of Mathematical Biology, Vol.79 No.1, pp. 281–328. doi:10.1007/s00285-019-01357-0. Chena, D. W., Xua, Z., Leia, Z., Huanga, J., Liua, B., Gaoa, Z. and Penga, L. (2020). 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