Journal of Mathematical Biology

, Volume 30, Issue 7, pp 693-716

First online:

Dynamic models of infectious diseases as regulators of population sizes

  • Jaime Mena-LorcatAffiliated withInstituto de Matemáticas, Universidad Católica de Valparaíso
  • , Herbert W. HethcoteAffiliated withDepartment of Mathematics, University of Iowa

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Five SIRS epidemiological models for populations of varying size are considered. The incidences of infection are given by mass action terms involving the number of infectives and either the number of susceptibles or the fraction of the population which is susceptible. When the population dynamics are immigration and deaths, thresholds are found which determine whether the disease dies out or approaches an endemic equilibrium. When the population dynamics are unbalanced births and deaths proportional to the population size, thresholds are found which determine whether the disease dies out or remains endemic and whether the population declines to zero, remains finite or grows exponentially. In these models the persistence of the disease and disease-related deaths can reduce the asymptotic population size or change the asymptotic behavior from exponential growth to exponential decay or approach to an equilibrium population size.

Key words

Epidemiological models Population dynamics Thresholds Hopf bifurcation