Basic reinfection number and backward bifurcation

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Some epidemiological models exhibit bi-stable dynamics even when the basic reproduction number R0 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.

Original languageEnglish
Pages (from-to)8064-8083
Number of pages20
JournalMathematical Biosciences and Engineering
Issue number6
StatePublished - 2021


  • Basic reinfection number
  • Bifurcation
  • Differential equations
  • Disease dynamics


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