Normalized Caputo–Fabrizio SVIR modeling and bifurcation analysis

Why this matters for understanding outbreaks

When we think about epidemics, we often picture simple curves that rise and fall as a disease spreads and then fades. But real outbreaks remember their past: how fast people got sick before, when vaccines were rolled out, and how long immunity lasts all shape what comes next. This paper introduces a new way to build “memory” directly into epidemic models that include vaccination, aiming to capture more realistic waves of infection without making the mathematics unstable or misleading.

A new way to let epidemics remember

The authors work within a classic framework that divides a population into four groups: people who can still catch the disease (susceptible), those who are vaccinated, those currently infectious, and those who have recovered. Traditional models describe how people move between these groups using standard calculus, which treats the present rate of change as depending only on the current state. Here, the authors replace the usual time derivative with a “normalized Caputo–Fabrizio” operator, a special mathematical tool that lets the model weigh the entire history of the outbreak while avoiding infinite spikes or arbitrary scaling. Normalization ensures that past events influence the present like an average, rather than piling up in an unrealistic way.

How the model behaves in theory

With this memory-aware setup, the team first checks that the model behaves sensibly. They prove that, for any reasonable starting conditions, there is a single, well-defined solution that keeps all four population groups non‑negative and preserves the total population over time. They identify a family of disease‑free end states in which everyone is either vaccinated or recovered and show that, mathematically, these states are stable: small introductions of infection die out instead of exploding, provided the effective reproduction number is below one. Even when this threshold is exceeded, the model only allows outbreaks to grow temporarily, not to settle into strange or unphysical long‑term patterns.

What simulations reveal about memory and vaccination

To see what the equations mean in practice, the authors run computer experiments across different levels of “memory strength,” controlled by a fractional order parameter. When memory is strong, infection curves rise more slowly, peak later, and reach lower maximum levels, while the susceptible group declines more gently. Vaccinated and recovered groups build up more gradually but can still reach similar final proportions. Varying infection and vaccination rates shows how memory softens otherwise sharp, high peaks typical of classical models. The numerical scheme they design mimics the model’s history‑dependent behavior by summing contributions from all previous time steps, and they verify that their method converges reliably and reproduces the familiar classical model when memory is turned off.

When complex patterns cannot occur

Many modern studies look for bifurcations—sudden qualitative changes in epidemic behavior, such as the appearance of multiple stable outcomes or sustained oscillations that resemble recurring waves. The authors carry out a detailed bifurcation analysis and reach a clear conclusion for the setting they study: in a closed population with constant vaccination and no births, deaths, or vaccine failure, the model cannot support either backward bifurcation (where the disease can persist even when the reproduction number is below one) or Hopf bifurcation (which would generate endless cycles). Even when they replace simple infection terms with a saturated form that usually encourages richer behavior, the only long‑term outcomes remain infection‑free states. Any wiggles seen in the simulations are transient echoes of initial conditions amplified by memory, not true repeating waves.

What this means for future epidemic modeling

In everyday terms, this work shows how to build epidemic models that remember their past in a controlled and physically meaningful way, while still remaining mathematically well behaved. The new approach smooths and stabilizes outbreak curves under vaccination, but in the simplified setting studied, it cannot on its own generate multiple long‑term scenarios or permanent cycles. To capture phenomena like recurring seasonal waves or the coexistence of high‑ and low‑infection states, the authors argue that modelers must add real‑world complications such as births, deaths, or imperfect vaccines on top of this memory structure. Their framework provides a solid starting point for those richer models, promising more realistic tools for planning and evaluating vaccination policies.

Citation: Shafqat, R., Al-Quran, A., Alsaadi, A. et al. Normalized Caputo–Fabrizio SVIR modeling and bifurcation analysis. Sci Rep 16, 8193 (2026). https://doi.org/10.1038/s41598-026-38301-4

Keywords: epidemic modeling, fractional calculus, vaccination dynamics, disease memory effects, bifurcation analysis