Abstract
We analyze local asymptotic stability of an SIRS epidemic model with a distributed delay. The incidence rate is given by a general saturated function of the number of infective individuals.Our first aim is to find a class of nonmonotone incidence rates such that a unique endemic equilibrium is always asymptotically stable.We establish a characterization for the incidence rate, which shows that nonmonotonicity with delay in the incidence rate is necessary for destabilization of the endemic equilibrium. We further elaborate the stability analysis for a specific incidence rate. Here we improve a stability condition obtained in [Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B 13 (2010) 195-211], which is illustrated in a suitable parameter plane. Two-parameter plane analysis together with an application of the implicit function theorem facilitates us to obtain an exact stability condition. It is proven that as increasing a parameter, measuring saturation effect, the number of infective individuals at the endemic steady state decreases, while the equilibrium can be unstable via Hopf bifurcation. This can be interpreted as that reducing a contact rate may cause periodic oscillation of the number of infective individuals, thus disease can not be eradicated completely from the host population, though the level of the endemic equilibrium for the infective population decreases. Numerical simulations are performed to illustrate our theoretical results.
Highlights
In modeling of disease transmission dynamics an important ingredient is the incidence rate, describing the number of new infective individuals arising in a host population per unit of time
If the nonmonotone incidence rate can be interpreted as an inhibition effect due to an intervention policy [28, 29], one can successfully decrease the level of the endemic equilibrium by reducing the number of contact rate
In Proposition 3.1 we prove that the model has a unique endemic equilibrium if and only if the basic reproduction number is greater than one
Summary
In modeling of disease transmission dynamics an important ingredient is the incidence rate, describing the number of new infective individuals arising in a host population per unit of time. Force of infection is given by a nonlinear bounded function of the number of infective individuals, which can be interpreted as saturation or psychological effect in the disease transmission dynamics. We first aim to detect a class of nonmonotone incidence rates such that the endemic equilibrium is always asymptotically stable To this aim we consider an SIRS epidemic model with distributed delays. It is shown that a parameter, measuring saturation effect, is responsible for destabilization of the endemic equilibrium via Hopf bifurcation, the number of infective individuals would fluctuate periodically This can be interpreted as that, if the nonmonotonicity is due to an intervention policy during an epidemic outbreak as in [28], periodic oscillation is possible by reducing the contact rate (as increasing the saturation level), though the level of the endemic equilibrium decreases. Since the right hand side of equations (2.3) is locally Lipschitzian on C and a priori bound for solutions is given (see e.g. [7]), it can be shown that (2.3) has a unique positive solution defined on (0, ∞) for each initial function
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