Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate

Abstract

In this paper, based on SIR and SEIR epidemic models with a general nonlinear incidence rate, we incorporate time delays into the ordinary differential equation models. In particular, we consider two delay differential equation models in which delays are caused (i) by the latency of the infection in a vector, and (ii) by the latent period in an infected host. By constructing suitable Lyapunov functionals and using the Lyapunov-LaSalle invariance principle, we prove the global stability of the endemic equilibrium and the disease-free equilibrium for time delays of any length in each model. Our results show that the global properties of equilibria also only depend on the basic reproductive number and that the latent period in a vector does not affect the stability, but the latent period in an infected host plays a positive role to control disease development.

Traveling waves for a nonlocal dispersal SIR model with delay and external supplies

“…Huang et al [16] showed that the so-called basic reproductive number of (2.2) is R 0 ¼ b lþc . They further proved that F is global asymptotically stable if R 0 1, while if R 0 > 1; E is global asymptotically stable for any s P 0.…”

“…We use Lyapunov method motivated by [12,16,27,28]. We first claim that IðnÞ > 0 for any n 2 R. Indeed, suppose that there exists x 0 2 R such that Iðx 0 Þ ¼ 0, then I 0 ðx 0 Þ ¼ 0 and ð J Ã I À IÞðx 0 Þ P 0 with equality holding if and only if I 0.…”

“…However, motivated by [41,12], for R 0 > 1 and c > c à , we obtain the existence of traveling waves of (1.1) by Schauder's fixed point theorem with upper-lower solutions and a limiting argument. The boundary asymptotic behavior of traveling waves at þ1 is obtained by Lyapunov method which is motivated by [16,12,11]. Meanwhile, when c ¼ c à , by a limiting approach (see [3,37]), the existence of traveling waves with exact asymptotic boundary behavior is obtained.…”

Global Analysis for a Delay-Distributed SIR Epidemic Model

“…In [13], Korobeinikov established global stability for ordinary differential version of 2. In [9], Huang et al have considered two epidemiological models with the constant time delays and nonlinear incidence rate of the form f (S)g(I). In [8], Huang and Takeuchi established the global properties of epidemiological models with constant delays and nonlinear incidence rate f (S, I).…”

An SIR Epidemic Model with Time Delay and General Nonlinear Incidence Rate

“…In the context of epidemiology, delays can be caused by a variety of factors. The most common reasons for a delay are (i) the latency of the infection in a vector and (ii) the latency of the infection in an infected host [30]. In these cases, some time should elapse before the level of infection in the infected host or the vector reaching a sufficiently high level to transmit the infection further.…”