Presentation Information
[MS14-03]Nonlinear study of SIR and UAIR models in temporal and Spatio-temporal frameworks
*Ravi Pratap Gupta1 (1. Banaras Hindu University (India))
Keywords:
Epidemic Model,Bifurcation,Chaos,Pattern Formation
In this presentation an SIR epidemic model with a saturated type incidence rate as well as treatment
is studied. Stability and bifurcation of all feasible equilibrium states are discussed. An endemic bubble is
observed through Hopf bifurcation in case of unique endemic equilibrium. Generalized Hopf bifurcation in
case of two endemic equilibrium is pointed out. In case of Bogdanov-Takens bifurcation there are two pairs
of feasible bifurcation thresholds. We also study the corresponding spatial model to understand the impact
of movement of individuals on the persistence and extinction of an infectious disease. It is evident that not
every susceptible person in a society is to blame for transmitting the disease. In such a situation, almost
all compartmental models, such as SI, SIS, SIR, SEIR, etc., are unable to predict the spread of the disease.
To overcome this unrealistic feature, in this study, we divide the susceptible population into unaware and
aware classes. This encourages us to extend the SIR model into an UAIR epidemic model with bilinear
incidence rate in the unaware class and a saturated type of incidence rate in the aware class. We discuss
the existence of Hopf bifurcation and Turing instability, which determines the Turing space in the spatial
domain. Extensive numerical simulations are performed to illustrate various type of pattern formation like
holes, strips, and holes-strips mixture inside the Turing space, which can determine the range for epidemic
outbreak.
is studied. Stability and bifurcation of all feasible equilibrium states are discussed. An endemic bubble is
observed through Hopf bifurcation in case of unique endemic equilibrium. Generalized Hopf bifurcation in
case of two endemic equilibrium is pointed out. In case of Bogdanov-Takens bifurcation there are two pairs
of feasible bifurcation thresholds. We also study the corresponding spatial model to understand the impact
of movement of individuals on the persistence and extinction of an infectious disease. It is evident that not
every susceptible person in a society is to blame for transmitting the disease. In such a situation, almost
all compartmental models, such as SI, SIS, SIR, SEIR, etc., are unable to predict the spread of the disease.
To overcome this unrealistic feature, in this study, we divide the susceptible population into unaware and
aware classes. This encourages us to extend the SIR model into an UAIR epidemic model with bilinear
incidence rate in the unaware class and a saturated type of incidence rate in the aware class. We discuss
the existence of Hopf bifurcation and Turing instability, which determines the Turing space in the spatial
domain. Extensive numerical simulations are performed to illustrate various type of pattern formation like
holes, strips, and holes-strips mixture inside the Turing space, which can determine the range for epidemic
outbreak.