Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models. (English) Zbl 1187.34121

Yang, Yu; Xiao, Dongmei

Discrete Contin. Dyn. Syst., Ser. B 13, No. 1, 195-211 (2010).

Summary: A disease transmission model of SIRS type with latent period and nonlinear incidence rate is considered. Latent period is assumed to be a constant \(\tau\), and the incidence rate is assumed to be of a specific nonlinear form, namely, \(\frac{kI(t-\tau)S(t)}{1+\alpha I^{h}(t-\tau)}\), where \(h\geq 1\). Stability of the disease-free equilibrium, and existence, uniqueness and stability of an endemic equilibrium for the model, are investigated. It is shown that, there exists the basic reproduction number \(R_0\) which is independent of the form of the nonlinear incidence rate, if \(R_0\leq 1\), then the disease-free equilibrium is globally asymptotically stable, whereas if \(R_0>1\), then the unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region for the model in which there is no latency, and periodic solutions can arise by Hopf bifurcation from the endemic equilibrium for the model at a critical latency. Some numerical simulations are provided to support our analytical conclusions.

MSC:

34K60	Qualitative investigation and simulation of models involving functional-differential equations
34K20	Stability theory of functional-differential equations
34K18	Bifurcation theory of functional-differential equations

Keywords:

SIRS model; latent period; nonlinear incidence rate; global stable; Hopf bifurcation