Bifurcation, stability, and cluster formation of multi-strain infection models
1Department of Applied Mathematics, The University of Western Ontario, London, ON, N6A 5B7, Canada, chan.bernard@gmail.com.
Abstract:
Clustering behaviours have been found in numerous multi-strain transmission models. Numerical solutions of these models have shown that steady-states, periodic, or even chaotic motions can be self-organized into clusters. Such clustering behaviours are not a priori expected. It has been proposed that the cross-protection from multiple strains of pathogens is responsible for the clustering phenomenon. In this paper, we show that the steady-state clusterings in existing models can be analytically predicted. The clusterings occur via semi-simple double zero bifurcation from the quotient networks of the models and the patterns which follow can be predicted through the stability analysis of the bifurcation. We calculate the stability criteria for the clustering patterns and show that some patterns are inherently unstable. Finally, the biological implications of these results are discussed.
Insights
This study analytically predicts steady-state clustering in multi-strain pathogen transmission models. These clustering behaviors, driven by cross-protection, can now be mathematically anticipated and analyzed for stability.
Area of Science:
- Mathematical modeling of infectious diseases
- Epidemiology
- Dynamical systems theory
Background:
- Multi-strain pathogen transmission models exhibit complex dynamics, including self-organized clustering.
- Previous numerical studies observed steady-state, periodic, and chaotic motions within these clusters.
- Cross-protection between pathogen strains has been hypothesized as the cause of clustering.
Purpose of the Study:
- To analytically predict steady-state clustering behaviors in multi-strain transmission models.
- To investigate the mathematical mechanisms underlying observed clustering phenomena.
- To analyze the stability of predicted clustering patterns.
Main Methods:
- Analytical prediction of steady-state clusterings.
- Utilizing semi-simple double zero bifurcation from quotient networks.
- Stability analysis of bifurcation patterns.
Main Results:
- Analytical prediction of steady-state clusterings is demonstrated.
- Clustering occurs via semi-simple double zero bifurcation.
- Stability criteria for clustering patterns are calculated, revealing inherent instabilities in some patterns.
Conclusions:
- Steady-state clustering in multi-strain transmission models can be analytically predicted.
- Bifurcation analysis provides a framework for understanding and predicting clustering patterns.
- The stability analysis highlights critical factors influencing the persistence of these biological patterns.
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