Abstract
A delayed epidemic model with nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals is considered. By analyzing the corresponding characteristic equations, the effects of time delay on the stability of the equilibria are studied. By choosing time delay as bifurcation parameter, the critical value of time delay at which a Hopf bifurcation occurs is obtained. In order to derive the normal form of the Hopf bifurcation, an extended method of multiple scales is developed and used. Then, the amplitude of bifurcating periodic solution and the conditions which determine the stability of the bifurcating periodic solution are obtained. The validity of analytical results is shown by their consistency with numerical simulations.
Highlights
Mathematical models describing the transmission of infectious diseases have played an important role in understanding the mechanism of disease transmission and controlling the spread of infectious diseases
Some SIR and SIRS models have been analyzed in detail [2,3,4,5,6,7]
The SEIR epidemic models, where E denotes the number of individuals who are infected but not yet infectious, are developed to investigate the role of the incubation period in disease transmission
Summary
Mathematical models describing the transmission of infectious diseases have played an important role in understanding the mechanism of disease transmission and controlling the spread of infectious diseases. The SEIR epidemic models, where E denotes the number of individuals who are infected but not yet infectious, are developed to investigate the role of the incubation period in disease transmission. Capasso and Serio [18] introduced a saturated incidence rate g(I)S into epidemic models, where g(I) = βI/(1 + αI), where βI measures the infection force of the disease and 1/(1 + αI) describes the psychological effect or inhibition effect from the behavioral change of the susceptible individuals with the increase of the infectious individuals. C1 and c2 are the removal rate constants from classes I and Q, respectively In this model, we assume that the susceptible individuals were infected before time delay τ which is the latent period.
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