A Mathematical Model of Vaccination Control for Diphtheria in Nigeria

Diphtheria remains a public health concern in Nigeria, where interruptions in routine immunisation, uneven access to health services, waning immunity, and surveillance limitations may leave sizeable groups susceptible to infection. This study develops a deterministic susceptible–infectious–recovered–vaccinated (SIRV) model to examine the influence of vaccination on diphtheria transmission. The model incorporates demographic turnover, disease-induced mortality, recovery, vaccination, and waning infection-acquired and vaccine-induced immunity. The basic reproduction number, R0, is derived using the next-generation method, and the disease-free and endemic equilibria are characterised. Local stability analysis shows that the disease-free equilibrium is locally asymptotically stable when R0 < 1 and unstable when R0 > 1, while a biologically meaningful endemic equilibrium exists above the invasion threshold. Sensitivity analysis indicates that transmission increases R0, whereas vaccination and recovery reduce it; faster loss of vaccine-induced immunity increases transmission potential. Numerical illustrations show a monotonic decline in R0 as the vaccination rate increases and demonstrate how waning protection can replenish the susceptible population. The model therefore highlights the importance of sustained routine immunisation, appropriate booster delivery, surveillance, and rapid outbreak response. Because the numerical parameter values are illustrative rather than fitted to a unified Nigerian dataset, the framework requires calibration and validation with current Nigerian surveillance, demographic, and immunisation data before use for forecasting or resource allocation.