Basic reinfection number and backward bifurcation

Area of Science:

• Epidemiology
• Mathematical Biology
• Infectious Disease Dynamics

Background:

• Some epidemiological models exhibit bi-stable dynamics, a phenomenon known as backward bifurcation, even when the basic reproduction number ($R_0$) is below 1.
• Causes for backward bifurcation include exogenous reinfection, super-infection, relapse, vaccination, and subpopulation heterogeneity.

Purpose of the Study:

• To define a new threshold, the basic reinfection number, to quantify reinfection forces.
• To characterize the type of bifurcation when the basic reproduction number equals one.
• To provide a comprehensive measure for disease control when reinfections or relapses are significant.

Main Methods:

• Definition of the basic reinfection number as a second threshold.
• Analysis of epidemiological models to formulate the basic reinfection number.
• Comparison of backward and forward bifurcation based on the basic reinfection number.

Main Results:

• The basic reinfection number characterizes the bifurcation type when $R_0 = 1$.
• A backward bifurcation occurs if the basic reinfection number is greater than one.
• A forward bifurcation occurs if the basic reinfection number is less than or equal to one.

Conclusions:

• The basic reproduction number and basic reinfection number together offer a complete measure for disease control in models with reinfection or relapse.
• The study formulates the basic reinfection number for various epidemiological models.
• This new metric enhances understanding and management of infectious diseases influenced by reinfection dynamics.