A mathematical interpretation for outbreaks of bacterial meningitis under the effect of time-dependent transmission parameters

We consider a SIR-type compartmental model divided into two age classes to explain the seasonal exacerbations of bacterial meningitis, especially among children outside of the meningitis belt. We describe the seasonal forcing through time-dependent transmission parameters that may represent the outbreak of the meningitis cases after the annual pilgrimage period (Hajj) or uncontrolled inflows of irregular immigrants. We present and analyse a mathematical model with time-dependent transmission. We consider not only periodic functions in the analysis but also general non-periodic transmission processes. We show that the long-time average values of transmission functions can be used as a stability marker of the equilibrium. Furthermore, we interpret the basic reproduction number in case of time-dependent transmission functions. Numerical simulations support and help visualize the theoretical results.