A mathematical model for managing diphtheria outbreaks in Nigeria via vaccination, enhanced surveillance, effective quarantine, and social distancing measures

Source abstract

Diphtheria is a highly infectious respiratory and skin disease that poses a significant threat to global public health. In Nigeria, children and adults with low immunity remain at high risk due to recurring annual outbreaks in the last decade, with a death rate of as high as 68.8\%. This paper presents a deterministic compartmental model to evaluate the effectiveness of interventions implemented by the Nigerian government. The model's mathematical and epidemiological validity is established through proofs of non-negativity and boundedness. The basic reproduction number ($R_0$) is derived, and stability analysis confirms that the diphtheria-free equilibrium is locally and globally asymptotically stable if $R_0 < 1$ and unstable if $R_0 > 1$. The model is calibrated to the cumulative confirmed cases of diphtheria in Nigeria (2023--2025) using nonlinear least-squares optimization and parametric bootstrap methods. The fitting estimates a high intrinsic transmission rate ($\beta \approx 17.91 \text{ day}^{-1}$) and a high existing social distancing factor ($d \approx 0.7801$), indicating that the outbreak was under significant NPI-based control, but still needs stricter enforcement, as the substantial disease-induced death rate fitted ($\delta \approx 0.3253 day^{-1}$) confirms high virulence. The sensitivity analysis identified social distancing ($d$) as the most effective intervention to reduce the $R_0$ threshold, followed by the symptomatic treatment rate ($\tau$) and the vaccination rate ($\nu$). A key finding is that the enhanced surveillance rate ($\psi$) has a negligible impact on the $R_0$ threshold, indicating its role is in case management rather than preventing initial spread. 3D surface analysis highlights strong synergistic effects when combining social distancing with either treatment or vaccination, offering optimized policy options. Numerical simulations show that an aggressive vaccination campaign ($\nu = 0.5 day^{-1}$) alongside targeted community screening to clear the significant asymptomatic "silent reservoir" is the most effective strategy for long-term elimination. These findings provide quantitative evidence for prioritizing social distancing and vaccination over intensive surveillance investments in resource-constrained settings.