Modeling the dynamics of rubella disease with vertical transmission.

Rubella is a highly contagious and serious human disease caused by the rubella virus. It affects everyone around the world, but it is especially common in pregnant women and children. In particular, when pregnant women are infected with the rubella virus, it causes Congenital Rubella Syndrome (it transmit vertically from mother to fetus, which causes that the new born baby to inherit birth defect disease). In order to prevent this viral disease, children must receive an MMR (measles, mumps and rubella) vaccine twice. If children receive two doses of the vaccine, then they develop long life immunity (protected against rubella). Based on the biological behavior of rubella disease, the SVPEIRS (susceptible, vaccinated, protected, exposed, infected, recovered) deterministic mathematical model of rubella disease dynamics is proposed. From the perspective of the qualitative behavior of the model, it is bounded in the invariant region and all the solutions of the compartment are positive. In addition, the equilibrium points and the stability of the equilibrium points (local and global) are also analyzed. The basic reproductive number is determined using a next-generation matrix. The results of the sensitivity analysis show that rubella is spread in a community if the values of contact rate, vertical transmission (neonatal infection) rate, exposure rate and rate of waning out of the first vaccinating dose are increase by keeping other parameters constant. On the other hand, increasing the first and second vaccination rate and treatment rate can help to control rubella in the community. Numerical simulation results show that due to the lack of protection for women before pregnancy, the number of infections increases with the birth of infected children, and the two doses of vaccine play a significant role in reducing and eliminating rubella. Therefore, to eliminate rubella in the community, healthcare and policymakers must pay attention to these parameters.