Modeling the transmission dynamics of endemic meningitis: stability analysis and long-term persistence.

This study develops and analyzes a deterministic compartmental model to understand the transmission dynamics of Neisseria meningitidis under endemic conditions. Compartment profile plots illustrate the progression of the disease to an endemic level over time. Explicit expressions for the endemic equilibrium points and the basic reproductive number are derived. The basic reproductive number is decomposed into three interpretable components: contributions from (i) carriers, (ii) carriers progressing to symptomatic cases, and (iii) direct symptomatic cases (i.e., infectives bypassing carriership), highlighting the role of both carriers and infectives in long-term disease persistence. Sensitivity analysis identifies the transmission rate, the odds in favor of carrier infectivity, and the rate of loss of carriership as critical parameters influencing the basic reproductive number. Phase-plane plots (S-C and S-I) illustrate how variation in the basic reproductive number shapes the trajectory of convergence to the endemic equilibrium. Stability analysis shows that the endemic equilibrium is unstable when the product of the basic reproductive number and the susceptible population at equilibrium exceeds unity. Using Lyapunov functions and LaSalle's invariance principle, we further show that the long-term dynamics are fully determined by this threshold: the disease-free equilibrium is globally asymptotically stable when the basic reproductive number is less than one, whereas a unique endemic equilibrium emerges and is globally stable when it is greater than one, demonstrating the persistence or elimination of meningitis depending on the threshold.