Modeling and analysis of the secondary routine dose against measles in China

Abstract

Abstract Measles remain to be an important global public health issue in China. In spite of large coverage rates of the first dose of Measles mumps rubella (MMR) combination vaccines (MMR1), large numbers of measles cases continue to be reported in China in recent years due to the high incidence and the low coverage of the second MMR vaccine dose (MMR2). This paper is devoted to modeling the combined effects of MMR1 and MMR2 coverage rates on the controlling of measles. To do that, we propose and study a robust time-delayed compartment measles infection model where MMR2 is followed after a fixed time interval of MMR1, and the combined elements of infection and mass immunization are also considered. By using the methods of Lyapunov functional and the uniform theory for infinite-dimensional dynamical systems, a threshold dynamics determined by the basic reproduction number ℜ 0 $\Re_{0}$ is established: the measles can be eradicated if ℜ 0 1 $\Re_{0}>1$ . Moreover, it is shown that the endemic equilibrium is locally asymptotically stable once ℜ 0 > 1 $\Re_{0}>1$ . Numerical simulations are performed to support the theoretical results and to consider the effects of MMR2 on the controlling of measles. Our results show that to eliminate measles in China, we should have MMR1 coverage rates larger than 88.01% based on perfect MMR2 coverage, and have MMR2 coverage rates larger than 92.63% based on perfect MMR1 coverage; Moreover, our simulations suggest that there is a risk that paradox of vaccination against measles in China may occur: that is, the final size of infected individuals may even increase in spite of the increase of MMR2 coverage rates.

Keywords

• measles model
• second dose
• Lyapunov function
• globally asymptotic stability
• basic reproductive number