Delay Induced Oscillations in a Dynamical Model for Infectious Disease
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 (R0) is greater than one. Also, the disease free equilibrium is locally stable independent of delay when R0 < 1. For R0 > 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.
Anuj Kumar is thankful for the financial support of Council of Scientific and Industrial Research, India (Grant No.: 09∕1023(0009)∕2012–EMR–I). Prashant Srivastava thanks to the International Union of Biological Sciences (IUBS) for partial support of living expenses in Moscow, during the 17th BIOMAT International Symposium, October 29-November 04, 2017.
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