Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate

Abstract In this paper, the SEIR epidemic model with vertical transmission and the saturating contact rate is studied. It is proved that the global dynamics are completely determined by the basic reproduction number R0(p, q), where p and q are fractions of infected newborns from the exposed and infectious classes, respectively. If R0(p, q) ⩽ 1, the disease-free equilibrium is globally asymptotically stable and the disease always dies out. If R0(p, q) > 1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at the endemic equilibrium state if it initially exists.

[1]  F. V. Vleck,et al.  Stability and Asymptotic Behavior of Differential Equations , 1965 .

[2]  K. Dietz,et al.  Overall Population Patterns in the Transmission Cycle of Infectious Disease Agents , 1982 .

[3]  Morris W. Hirsch,et al.  Systems of differential equations that are competitive or cooperative. VI: A local Cr Closing Lemma for 3-dimensional systems , 1991 .

[4]  James S. Muldowney,et al.  On Bendixson′s Criterion , 1993 .

[5]  James S. Muldowney,et al.  Compound matrices and ordinary differential equations , 1990 .

[6]  James S. Muldowney,et al.  On R.A. Smith's Autonomous Convergence Theorem , 1995 .

[7]  James S. Muldowney,et al.  A Geometric Approach to Global-Stability Problems , 1996 .

[8]  M. Li,et al.  Global dynamics of a SEIR model with varying total population size. , 1999, Mathematical Biosciences.

[9]  R. A. Smith,et al.  Some applications of Hausdorff dimension inequalities for ordinary differential equations , 1986, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[10]  S. Busenberg,et al.  Delay differential equations and dynamical systems , 1991 .

[11]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[12]  Zhen Jin,et al.  Global stability of an SEI epidemic model with general contact rate , 2005 .

[13]  Paul Waltman,et al.  Persistence in dynamical systems , 1986 .

[14]  D Greenhalgh,et al.  Some results for an SEIR epidemic model with density dependence in the death rate. , 1992, IMA journal of mathematics applied in medicine and biology.

[15]  Zhien Ma,et al.  Global dynamics of an SEIR epidemic model with saturating contact rate. , 2003, Mathematical biosciences.

[16]  Liancheng Wang,et al.  Global Dynamics of an SEIR Epidemic Model with Vertical Transmission , 2001, SIAM J. Appl. Math..

[17]  K. L. Cooke,et al.  Analysis of an SEIRS epidemic model with two delays , 1996, Journal of mathematical biology.

[18]  K. Cooke,et al.  The population dynamics of two vertically transmitted infections. , 1988, Theoretical population biology.

[19]  Michael Y. Li,et al.  Global stability for the SEIR model in epidemiology. , 1995, Mathematical biosciences.

[20]  Charles Pugh,et al.  The C1 Closing Lemma, including Hamiltonians , 1983, Ergodic Theory and Dynamical Systems.

[21]  Shiwu Xiao,et al.  An SIRS model with a nonlinear incidence rate , 2007 .

[22]  Shigui Ruan,et al.  Uniform persistence and flows near a closed positively invariant set , 1994 .

[23]  Robert H. Martin.  Logarithmic norms and projections applied to linear differential systems , 1974 .

[24]  David Greenhalgh,et al.  Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity , 1997 .

[25]  R. May,et al.  Population biology of infectious diseases: Part II , 1979, Nature.

[26]  Jin Zhen,et al.  Global stability of an SEI epidemic model , 2004 .