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A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data

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Stochastic epidemic models with a backward bifurcation.

Linda J S Allen¹, P van den Driessche

¹Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042. linda.j.allen@ttu.edu.

Mathematical Biosciences and Engineering : MBE

|March 10, 2010

Summary

Two new stochastic models for pertussis (whooping cough) reveal that imperfect vaccines can create bistability. In small populations, this phenomenon may not significantly alter disease dynamics.

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Area of Science:

Epidemiology
Mathematical Biology
Infectious Disease Modeling

Background:

Pertussis (whooping cough) dynamics are often modeled deterministically.
Deterministic models with imperfect vaccination can exhibit backward bifurcation, leading to bistability.
Stochastic models are crucial for understanding disease dynamics in smaller populations.

Purpose of the Study:

To formulate and analyze new stochastic epidemic models for pertussis.
To investigate the impact of bistability, arising from imperfect vaccination, on disease dynamics.
To compare the behavior of stochastic models with a deterministic model in a region of bistability.

Main Methods:

Development of a continuous-time Markov chain model.
Formulation of a stochastic differential equation model.
Analysis of model dynamics within a region of bistability.

Main Results:

The stochastic models exhibit bimodality in the infective population distribution within the bistability region.
For large populations (N≥1000), stochastic and deterministic models show agreement.
For small populations, stochastic models suggest limited impact of backward bifurcation on disease dynamics.

Conclusions:

Stochastic modeling provides a more nuanced understanding of pertussis dynamics, especially in smaller populations.
The presence of bistability due to imperfect vaccines may have less impact than predicted by deterministic models in small populations.
Further research into stochastic effects in epidemic modeling is warranted.