Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics

Mahmood Parsamanesh,Majid Erfanian,Saeed Mehrshad

doi:10.1186/s12859-020-03839-1

Mahmood Parsamanesh, Majid Erfanian + Show 1 more

Open Access

https://doi.org/10.1186/s12859-020-03839-1

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Abstract

BackgroundThe spread of infectious diseases is so important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and to investigate under which conditions it will be wiped out or continued.ResultsA discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number mathcal {R}_0. Then the stability of the equilibria is given in terms of mathcal {R}_0, and the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of the model, a discretized model is obtained and analyzed.ConclusionIt is proven that the disease-free equilibrium and endemic equilibrium are stable if mathcal {R}_0<1 and mathcal {R}_0>1, respectively. Also, the disease-free equilibrium is globally stable when mathcal {R}_0le 1. The system has a transcritical bifurcation when mathcal {R}_0=1 and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.

Highlights

The spread of infectious diseases is so important that changes the demography of the population
Allen [3] studied some discretetime SI, SIR, and susceptible- infected- susceptible (SIS) epidemic models and showed that the simple discrete-time SI and SIR epidemic models without births or deaths mimic the behavior of the continuoustime models, while the behavior in the discrete-time SI, SIR, and SIS models with recovery or births differ from their continuous analogs
In the theorem we prove the global stability of the disease-free equilibrium

Summary

Introduction

The spread of infectious diseases is so important that changes the demography of the population. Mathematical models are useful to study the behavior of an infection when it enters a population and to investigate under which conditions it will be wiped out or continued. The literature about mathematical epidemic models that have been constructed and analyzed for various types of diseases is very rich; see, for example, [1, 2]. Such models can be formulated either as continuous-time models by differential equations or as Parsamanesh et al BMC Bioinformatics (2020) 21:525 discrete-time ones by difference equations. Parsamanesh and Mehrshad [7] performed a similar investigation on an SIS model with a temporary vaccination program with standard incidence. The local and global stabilities of disease-free equilibrium were derived as well as the local stability of the endemic equilibrium

Results

Discussion

Conclusion