**Global Stability Analysis of Logistically Grown SIR Model with Loss of Immunity, Inhibitory Effect, Crowding Effect and its Protection Measure**

Department of Applied Mathematics, University of Calcutta, Kolkata, India

∗E-mail: uttam_math@yahoo.co.in

### Received:

Received: 12 December 2016; revised: 30 March 2018; accepted: 04 April 2018; published online: 30 May 2018

### DOI: 10.12921/cmst.2016.0000071

### Abstract:

In this paper we have considered an SIR model with logistically grown susceptible in which the rate of incidence is directly affected by the inhibitory factors of both susceptible and infected populations and the protection measure for the infected class. Permanence of the solutions, global stability and bifurcation analysis in the neighborhood of equilibrium points has been investigated here. The Center manifold theory is used to find the direction of bifurcations. Finally numerical simulation is carried out to justify the theoretical findings.

### Key words:

center manifold theory, Hopf bifurcation, inhibition effect, logistic growth rate, loss immunity, normal forms

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