Bifurcation and Stability Analysis of a Discrete Time Sir Epidemic Model with Vaccination

Özlem Ak Gümüş ,A George Maria Selvam ,D Abraham Vianny

doi:10.28924/2291-8639-17-2019-809

Abstract

In this paper, we study the qualitative behavior of a discrete-time epidemic model with vaccination. Analysis of the model shows forth that the Disease Free Equilibrium (DFE) point is asymptotically stable if the basic reproduction number R0 is less than one, while the Endemic Equilibrium (EE) point is asymptotically stable if the basic reproduction number R0 is greater than one. The results are reinforced with numerical simulations and enhanced with graphical representations like time trajectories, phase portraits and bifurcation diagrams for different sets of parameter values.

Highlights

Mathematical models defining biological events has an important place in the study of population dynamics
We study the qualitative behavior of a discrete-time epidemic model with vaccination
Analysis of the model shows forth that the Disease Free Equilibrium (DFE) point is asymptotically stable if the basic reproduction number R0 is less than one, while the Endemic Equilibrium (EE) point is asymptotically stable if the basic reproduction number R0 is greater than one

Summary

Introduction

Mathematical models defining biological events has an important place in the study of population dynamics. The work of Kermack and McKendrick have the basic foundation for epidemics, the first attempt in explaining, predicting or modeling of epidemics dates back to over a century is made by Hamer (1906), Ross (1911). These early models operate on the principle where individuals can be classified by their epidemiological status which are susceptible to the infection, infected and recovered (immune) ( [12], [13], [14]). Statistical data on diseases are collected at a specific time In this case, the appropriate model defining the disease will be the discrete time model [19]. The continuous-time logistic equation has only equilibrium dynamics, the well known discrete logistic equation which is discrete counterpart of it exhibits period doubling bifurcation to chaos ( [22], [23])

SIR Epidemic Model with Vaccination

The Discrete Time System

Stability of Equilibrium points and Numerical Simulations

Findings

Conclusion