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

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 [Formula: see text]. Then the stability of the equilibria is given in terms of [Formula: see text], 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 [Formula: see text] and [Formula: see text], respectively. Also, the disease-free equilibrium is globally stable when [Formula: see text]. The system has a transcritical bifurcation when [Formula: see text] 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.