Complex dynamics and pattern formation in a diffusive epidemic model with an infection-dependent recovery rate
Wael El Khateeb1, Chanaka Kottegoda2, Chunhua Shan1
1Department of Mathematics and Statistics, The University of Toledo, Toledo, OH, 43606, USA.
Mathematical Biosciences
|December 20, 2025
Summary
This study introduces a diffusive epidemic model where recovery depends on infection levels. Faster susceptible movement drives spatial patterns, highlighting the need for targeted strategies against regional disease waves.
Area of Science:
- Epidemiology
- Mathematical Biology
- Dynamical Systems
Background:
- Epidemic models are crucial for understanding disease spread.
- Infection-dependent recovery rates and spatial dynamics are key factors influencing disease transmission.
- Bifurcation analysis is a powerful tool for studying complex behaviors in mathematical models.
Purpose of the Study:
- To formulate and analyze a diffusive epidemic model with an infection-dependent recovery rate.
- To investigate the emergence of spatial and spatiotemporal patterns driven by diffusion.
- To understand the role of susceptible and infected populations in pattern formation.
Main Methods:
- Bifurcation analysis of reaction kinetics to identify steady states and periodic solutions.
- Analysis of diffusion-driven instability, with susceptible individuals as inhibitors and infected individuals as activators.
- Investigation of k-mode Turing instability and (k1, k2)-mode Turing-Hopf bifurcation for pattern formation.
- Examination of transient dynamics from temporal oscillations to spatial patterns.
Main Results:
- The model exhibits multiple constant steady states and spatially homogeneous periodic solutions.
- Diffusion-driven instability is observed, leading to pattern formation.
- Faster movement of susceptible populations induces spatial and spatiotemporal patterns.
- Asynchronous disease recurrence, spatially patterned waves, and localized hotspots are identified.
Conclusions:
- The model demonstrates complex transmission dynamics, including pattern formation and localized outbreaks.
- Spatially targeted intervention strategies are essential for controlling regionally varying and cyclical disease waves.
- Understanding the interplay between diffusion and reaction kinetics is vital for effective epidemic control.
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