The basic reproduction number in some discrete-time epidemic models

Linda J.S. Allen, P. Van Den Driessche

Research output: Contribution to journalArticlepeer-review

157 Scopus citations


The next generation matrix approach for calculating the basic reproduction number [image omitted] is summarized for discrete-time epidemic models. This approach is applied to six disease models developed for the study of two emerging wildlife diseases: hantavirus in rodents and chytridiomycosis in amphibians. Two of the models include discrete spatial patches. For each model, [image omitted] is calculated in terms of the model parameters. For [image omitted], if a small number of infectives is introduced, then the wildlife disease dies out. Global stability of the disease-free equilibrium is verified for a special case of the SI hantavirus model when [image omitted]. In addition, a numerical example indicates that there is a transcritical bifurcation at [image omitted], with the disease dying out if [image omitted] but tending to an endemic level if [image omitted].

Original languageEnglish
Pages (from-to)1127-1147
Number of pages21
JournalJournal of Difference Equations and Applications
Issue number10-11
StatePublished - Oct 2008


  • Basic reproduction number
  • Chytridiomycosis
  • Discrete-time epidemic model
  • Epidemic model on patches
  • Hantavirus
  • Next generation matrix


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