Propagation dynamics in an SIRS model with general incidence functions

Area of Science:

• Epidemiology
• Mathematical Biology
• Dynamical Systems

Background:

• SIRS models are crucial for understanding infectious disease dynamics.
• General incidence functions offer a more flexible approach to modeling disease transmission.
• Traveling wave solutions are essential for analyzing disease propagation patterns.

Purpose of the Study:

• To investigate initial value problems and traveling wave solutions in an SIRS model with general incidence.
• To establish a threshold criterion for disease spread based on the basic reproduction ratio.
• To determine the relationship between this threshold and the spreading speed of the disease.

Main Methods:

• Linearization of the infected equation at the disease-free steady state.
• Analysis of the basic reproduction ratio in a kinetic system.
• Proof of the threshold as the spreading speed for compactly supported initial conditions.
• Identification of the minimal wave speed for traveling wave solutions.

Main Results:

• A threshold is defined when the basic reproduction ratio exceeds unity.
• The threshold is proven to be the spreading speed for three unknown functions.
• This threshold represents the minimal wave speed for disease spreading traveling waves.
• If the basic reproduction ratio is less than unity, disease extinction and non-existence of nonconstant traveling waves are confirmed.

Conclusions:

• The basic reproduction ratio serves as a critical threshold for disease persistence and spread.
• The identified threshold accurately predicts the speed of disease propagation.
• The study confirms disease extinction under conditions of a low basic reproduction ratio.