Basic reinfection number and backward bifurcation

Baojun Song; Baojun Song

doi:10.3934/mbe.2021400

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 6: 8064-8083. doi: 10.3934/mbe.2021400

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Basic reinfection number and backward bifurcation

Baojun Song ^,

Department of Applied Mathematics and Statistics, Montclair State University, 1 Normal Avenue, Montclair, NJ 07043, USA

Received: 22 June 2021 Accepted: 06 September 2021 Published: 16 September 2021

Some epidemiological models exhibit bi-stable dynamics even when the basic reproduction number $ {{{\cal R}_{0}}} $ is below $ 1 $, through a phenomenon known as a backward bifurcation. Causes for this phenomenon include exogenous reinfection, super-infection, relapse, vaccination exercises, heterogeneity among subpopulations, etc. To measure the reinfection forces, this paper defines a second threshold: the basic reinfection number. This number characterizes the type of bifurcation when the basic reproduction number is equal to one. If the basic reinfection number is greater than one, the bifurcation is backward. Otherwise it is forward. The basic reinfection number with the basic reproduction number together gives a complete measure for disease control whenever reinfections (or relapses) matter. We formulate the basic reinfection number for a variety of epidemiological models.
- disease dynamics,
- differential equations,
- bifurcation,
- basic reinfection number
Citation: Baojun Song. Basic reinfection number and backward bifurcation[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 8064-8083. doi: 10.3934/mbe.2021400

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Abstract

Some epidemiological models exhibit bi-stable dynamics even when the basic reproduction number $ {{{\cal R}_{0}}} $ is below $ 1 $, through a phenomenon known as a backward bifurcation. Causes for this phenomenon include exogenous reinfection, super-infection, relapse, vaccination exercises, heterogeneity among subpopulations, etc. To measure the reinfection forces, this paper defines a second threshold: the basic reinfection number. This number characterizes the type of bifurcation when the basic reproduction number is equal to one. If the basic reinfection number is greater than one, the bifurcation is backward. Otherwise it is forward. The basic reinfection number with the basic reproduction number together gives a complete measure for disease control whenever reinfections (or relapses) matter. We formulate the basic reinfection number for a variety of epidemiological models.

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