Some discrete-time SI, SIR, and SIS epidemic models

Area of Science:

• Mathematical Biology
• Epidemiology
• Dynamical Systems

Background:

• Continuous-time epidemic models (SI, SIR, SIS) are well-established.
• Discrete-time models offer alternative perspectives on disease spread.
• Understanding deviations from continuous models is crucial for accurate predictions.

Purpose of the Study:

• To analyze discrete-time versions of SI, SIR, and SIS epidemic models.
• To investigate the impact of positivity restrictions on model behavior.
• To explore the potential for complex dynamics like period-doubling and chaos in discrete models.

Main Methods:

• Formulation of nonlinear difference equations for SI, SIR, and SIS models.
• Analysis of model behavior under the restriction of positive solutions (S>0, I>0).
• Examination of systems including births and deaths to assess additional feedback effects.

Main Results:

• Discrete SI and SIR models with positivity restrictions exhibit behavior similar to continuous analogues.
• Positivity alone is insufficient for asymptotic convergence in discrete SIS models.
• Discrete SIS models, and SI/SIR models with births/deaths, can display period-doubling and chaotic dynamics for specific parameter values.

Conclusions:

• Discrete-time epidemic models can exhibit richer dynamics than their continuous counterparts.
• The inclusion of factors like births and deaths, and the nature of feedback loops, significantly influence model behavior.
• Careful consideration of discrete dynamics is necessary for a comprehensive understanding of epidemic spread.