Stochastic and Deterministic Models for Rubella Dynamics with Two Doses of Vaccination and Vertical Transmission

Rubella is a highly contagious viral disease that can be transmitted through contact with infected people or objects. If a pregnant woman is infected with rubella, the virus can be transmitted to the fetus, which can cause birth defects. To prevent rubella, children need to receive the MMR vaccine twice. In this study, we developed a deterministic and stochastic mathematical model to investigate the dynamics of rubella infection, considering vertical transmission and environmental factors. The model divides the population into five groups: susceptible, first vaccinated, protected, infected, and recovered. From the perspective of the qualitative behavior of the model, the model is bound to an invariant domain, and all solutions within the compartment are positive. We analyzed the equilibrium point of the model and its stability, and we determined the basic reproductive number using the next-generation matrix. The basic reproductive number for the stochastic model is smaller than that for the deterministic model due to randomness. By comparing the two models in terms of the basic reproductive number, extending the deterministic model to a stochastic model can provide more realistic information about the disease, predict its dynamic propagation, and give suggestions for control mechanisms. Numerical simulations and sensitivity analyses were performed to study the influence of fundamental parameters, and the simulation results were displayed graphically using the MATLAB computer software. Parameter values were obtained from published papers, some of which were assumed. The simulation results show that the stochastic curve is not as smooth as the deterministic curve due to the randomness of the rubella. The consideration of the double vaccination dose and white noise intensity plays an important role in predicting disease status and minimizing and eradicating rubella. Therefore, reducing vertical infection and transmission rates and increasing vaccination and recovery rates will help to prevent and control the spread of rubella.