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

Abstract

Background The 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. Results A 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 $$\mathcal {R}_0$$ R 0 . Then the stability of the equilibria is given in terms of $$\mathcal {R}_0$$ 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. Conclusion It is proven that the disease-free equilibrium and endemic equilibrium are stable if $$\mathcal {R}_0<1$$ R 0 < 1 and $$\mathcal {R}_0>1$$ R 0 > 1 , respectively. Also, the disease-free equilibrium is globally stable when $$\mathcal {R}_0\le 1$$ R 0 ≤ 1 . The system has a transcritical bifurcation when $$\mathcal {R}_0=1$$ 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.

MODELS: a six-step framework for developing an infectious disease model

“…Model stability refers to the degree of consistency in the output of a model when slight variations in the epidemic data are observed [ 5 ]. In epidemiological research, models are often used to predict disease transmission trends, assess the effectiveness of interventions, and provide a theoretical basis for public health decision-making.…”

On spatio-temporal dynamics of COVID-19 epidemic 𝑆𝐸_1𝐸_2𝐼_1𝐼_2𝑅𝑆 model incorporating virus mutations and vaccinations effects

“…The fractional mathematical model of spread of vector-bone diseases was analysed in [38]. The authors of [39] studied discrete SIS model by assuming that the disease will not cause death where the vaccination program is considered.…”

Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity

“…For example, in [11] the authors studied the dynamical behavior of a discrete SIR epidemic model with a constant vaccination strategy. The stability and bifurcations of a discrete susceptible-infected-susceptible (SIS) epidemic model with vaccination were investigated in [12], while Xiang et al in [13] developed a discrete SIRS model that included vaccination and examined its dynamical behavior.…”