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Delay Induced Oscillations in a Dynamical Model for Infectious Disease

In: Trends in Biomathematics: Modeling, Optimization and Computational Problems

Author

Listed:
  • A. Kumar

    (Indian Institute of Technology Patna, Department of Mathematics)

  • P. K. Srivastava

    (Indian Institute of Technology Patna, Department of Mathematics)

Abstract

When a disease outbreaks in a population the information about disease prevalence induces behaviour changes in the susceptible population. This causes the susceptible individuals to adopt protective and precautionary measures. Thus this fraction of susceptible population may be virtually immune to the infection for some time. As the immunity degrades with passage of time, a time delay in waning of immunity is also accounted in the model. Thus, in this study, a delay differential equation model for the dynamics of infectious diseases is proposed and analysed which accounts for the effect of information on the susceptible population. The model analysis is carried out and it is found that the disease free equilibrium exists unconditionally, whereas a unique infected equilibrium is obtained when the basic reproduction number (R 0) is greater than one. Also, the disease free equilibrium is locally stable independent of delay when R 0 1, the model exhibits rich and complex dynamics in presence of time delay. The occurrence of oscillatory behaviour of the population around the infected equilibrium via Hopf bifurcation is observed analytically. Numerical simulation is performed to discuss and analyse the analytical results.

Suggested Citation

  • A. Kumar & P. K. Srivastava, 2018. "Delay Induced Oscillations in a Dynamical Model for Infectious Disease," Springer Books, in: Rubem P. Mondaini (ed.), Trends in Biomathematics: Modeling, Optimization and Computational Problems, pages 313-324, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-91092-5_21
    DOI: 10.1007/978-3-319-91092-5_21
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