Stability Analysis for Measles Infection with SEIRV+D Model

Measles, a highly contagious disease, remains a significant public health challenge despite the availability of effective vaccines whose spread dynamics are influenced by population density, contact rates, and vaccination coverage. Mathematical modeling is essential for forecasting outbreaks and evaluating intervention strategies. Thus, we aimed to analyze the threshold dynamics of measles in Algeria.

This study used the SEIRV+D (Susceptible–Exposed–Infected-Recovered–Vaccinated+Deceased) model, which tracks six key groups, to analyze measles transmission. We performed equilibrium analysis, the Disease-Free Equilibrium (DFE) and derived Endemic Equilibrium (EE). The stability of these equilibria was analyzed through the basic reproduction number ($\mathcal{R}_0$). Stability analysis confirms that the DFE is locally asymptotically stable when $\mathcal{R}_0<1$, while the EE becomes locally asymptotically stable when $\mathcal{R}_0>1$, and the DFE is globally asymptotically stable when $\mathcal{R}_0<1$ and unstable if $\mathcal{R}_0>1$.

Furthermore, the detailed bifurcation analysis reveals the occurrence of a forward bifurcation at $\mathcal{R}_0=1$, and sensitivity analysis helped identify the parameters that most influence the basic reproduction number $\mathcal{R}_0$, highlighting the critical role of vaccination-related factors. Numerical simulations for Algeria demonstrate that increasing the vaccination rate from suboptimal coverage to the WHO-recommended threshold of $95 \%$ significantly reduces both the cumulative number of infections and measles-attributable deaths. The model clearly shows the direct relationship between vaccination coverage, the magnitude of $\mathcal{R}_0$, and the long-term disease burden. The main results show that measles elimination in Algeria was achieved through modest vaccination coverage.