A simple mathematical model for genetic effects in pneumococcal carriage and transmission

Streptococcus pneumoniae (S. pneumoniae) is a bacterium commonly found in the throat of young children. Pneumococcal serotypes can cause a variety of invasive and non-invasive diseases such as meningitis and pneumonia. In 2000 a vaccine was introduced in the USA that not only prevents vaccine type disease but has also been shown to eliminate carriage of the vaccine serotypes. One key problem with the vaccine is that it has been observed that the same sequence types (genetic material found in the serotypes) are able to manifest in more than one serotype. This is a potential problem if sequence types associated with invasive disease may express themselves in multiple serotypes.We present a basic differential equation mathematical model for exploring the relationship between sequence types and serotypes where a sequence type is able to manifest itself in one vaccine serotype and one non-vaccine serotype. An expression for the effective reproduction number is found and an equilibrium and then a global stability analysis carried out. We illustrate our analytical results by using simulations with realistic parameter values.