Dynamical behaviors of a stochastic SIVS epidemic model with the Ornstein-Uhlenbeck process and vaccination of newborns

Abstract

In this paper, we study a stochastic SIVS infectious disease model with the Ornstein-Uhlenbeck process and newborns with vaccination. First, we demonstrate the theoretical existence of a unique global positive solution in accordance with this model. Second, adequate conditions are inferred for the infectious disease to die out and persist. Then, by classic Lynapunov function method, the stochastic model is allowed to obtain the sufficient condition so that the stochastic model has a stationary distribution represents illness persistence in the absence of endemic equilibrium. Calculating the associated Fokker-Planck equations yields the precise expression of the probability density function for the linearized system surrounding the quasi-endemic equilibrium. In the end, the theoretical findings are shown by numerical simulations.

Fig 1. Computer simulations of S(t), I(t) and V(t) for system Eq (5) in case of disease extinction.

Fig 2. Computer simulations of S(t), I(t) and V(t) for system Eq (5) in the presence of disease persistence.

Fig 3. Computer simulations of expectation and standard deviation for the model Eq (5).

Fig 4. Computer simulations of the histograms of frequencies for the model Eq (5).

Fig 5. Comparison plot between the stochastic model Eq (5) with the mean-reverting process and deterministic model Eq (1).

Fig 6. Computer simulations for different values of k for the model Eq (5).

Fig 7. Computer simulations for different values of θ for the model Eq (5).