Chaos detection in SIR model with modified Beddington–De Angelis type incidence rate and saturated treatment

Abstract

In this paper, we construct an SIR epidemic model with a modified Beddington–De Angelis type incidence rate and saturated treatment rate. We modify the incidence rate to incorporate the isolation of infected individuals after detection, and separation of some susceptible individuals from the rest to avoid the infection, without an increase in the number of classes. We find that the system has a unique disease-free equilibrium (DFE) which is locally asymptotically stable when the reproduction number is less than unity. The multiple endemic equilibria may exist irrespective of the basic reproduction number. The existence of bistability is encountered. Supercritical transcritical (forward), as well as subcritical transcritical (backward) bifurcation, may occur at [Formula: see text] where contact rate, [Formula: see text] acts as the bifurcation parameter. Therefore, DFE need not be globally stable. The conditions for the existence of Andronov–Hopf bifurcation are deduced with maximum treatment capacity, [Formula: see text] as the bifurcation parameter. The impacts of isolation of confirmed infected cases and separation of some susceptible from rest are studied numerically as well as the effect of saturation in treatment. The existence of chaotic behavior is deduced by showing the maximum Lyapunov exponent to be positive as well as the sensitivity to initial conditions. The computation of the Kalpan–Yorke dimension to be fractional confirms the existence of fractal-type strange attractor. The positive Kolmogorov–Sinai entropy further strengthens the claim of the existence of chaos.