Generalized Euler method to study the vaccination effects on dynamics of measles infection model under non-singular kernel

Why this study matters for fighting measles

Measles is one of the most contagious viruses known, yet millions of children still miss their shots each year. This paper uses advanced mathematics to explore how vaccination, quarantine, and treatment can work together to curb measles outbreaks, especially in regions with limited health resources. By building a model that remembers the past—such as earlier infections or vaccination drives—the authors aim to better capture how real communities respond over time and to guide long‑term decisions about investing in vaccines.

Looking at measles through a population lens

The researchers start from a standard way of representing disease spread that divides people into groups: those who are susceptible to infection, recently exposed, vaccinated, currently infectious, quarantined, or recovered. People move between these groups as they are born, get vaccinated, fall ill, are isolated, or recover. The model reflects a setting like Ghana, where vaccine shortages have led to recent measles flare‑ups. Each process—such as how quickly exposed people become infectious, or how fast treatment helps them recover—is represented by a rate based on published health data.

Adding memory to how measles spreads

Traditional models assume that the future depends only on what is happening right now. In reality, past events leave a long shadow: immunity can fade slowly, behavior changes after earlier outbreaks, and health systems react with delays. To capture this, the authors use a “fractional” version of calculus, which smoothly blends present conditions with a weighted history of the system. This allows the model to remember earlier infections, vaccinations, and interventions. Mathematically, they employ a modern type of fractional derivative with a non‑singular kernel, which avoids some technical issues and is well suited for problems with memory.

Testing stability and the tipping point for outbreaks

With this memory‑rich model in hand, the team analyzes when measles dies out and when it can persist. They identify a key threshold, the basic reproduction number, which measures how many new infections a typical case will cause in a susceptible community. If this number falls below one, the disease cannot sustain itself. Using tools from dynamical systems theory, they show that when vaccination and other measures push this threshold below one, the system naturally drifts toward a disease‑free state. When the threshold is above one, the model predicts a steady level of ongoing infection, but still shows how quarantine and treatment can limit its impact.

Simulating vaccination, quarantine, and memory effects

To explore realistic scenarios, the authors implement a specialized numerical scheme called the generalized Euler method, adapted to handle the model’s memory terms without losing stability. They simulate outbreaks for different degrees of memory and for various public‑health strategies. When the model gives more weight to past history, epidemic peaks are smaller and delayed, mimicking how previous vaccination campaigns and lingering immunity can soften new waves. Increasing the vaccination rate sharply reduces the numbers of exposed and infectious people and boosts the vaccinated and recovered groups. Stronger quarantine—moving infectious people quickly into isolation—lowers and shortens the peak of both infectious and quarantined individuals, reducing the total toll of the disease.

What this means for public health decisions

The study concludes that maintaining high vaccination coverage, combined with timely quarantine and treatment, is essential for long‑term measles control. By explicitly accounting for how past events shape present risk, the fractional model provides a more realistic picture than classical approaches, especially in places where immunity and behavior change slowly over time. The results support policies that prioritize stable funding for vaccine supply chains and rapid isolation of cases, showing that these investments can prevent major outbreaks and keep the basic reproduction number below the danger threshold.

Citation: Yadav, L.K., Gour, M.M., Purohit, S.D. et al. Generalized Euler method to study the vaccination effects on dynamics of measles infection model under non-singular kernel. Sci Rep 16, 10429 (2026). https://doi.org/10.1038/s41598-026-40951-3

Keywords: measles vaccination, disease modeling, fractional calculus, quarantine strategies, epidemic dynamics