Disease Control of Delay SEIR Model with Nonlinear Incidence Rate and Vertical Transmission

The aim of this paper is to develop two delayed SEIR epidemic models with nonlinear incidence rate, continuous treatment, and impulsive vaccination for a class of epidemic with latent period and vertical transition. For continuous treatment, we obtain a basic reproductive number ℜ 0 and prove the global stability by using the Lyapunov functional method. We obtain two thresholds ℜ* and ℜ ∗ for impulsive vaccination and prove that if ℜ* < 1, then the disease-free periodic solution is globally attractive and if ℜ ∗ > 1, then the disease is permanent by using the comparison theorem of impulsive differential equation. Numerical simulations indicate that pulse vaccination strategy or a longer latent period will make the population size infected by a disease decrease.

[1]  Rui Xu,et al.  Global stability of a delayed SEIRS epidemic model with saturation incidence rate , 2010 .

[2]  Shigui Ruan,et al.  Global analysis of an epidemic model with nonmonotone incidence rate , 2006, Mathematical Biosciences.

[3]  Paul Bosch,et al.  Modeling the Spread of Tuberculosis in Semiclosed Communities , 2013, Comput. Math. Methods Medicine.

[4]  J. Andrus,et al.  Eradication of poliomyelitis: progress in the Americas. , 1991, The Pediatric infectious disease journal.

[5]  Mohammad A. Safi,et al.  Global Stability Analysis of SEIR Model with Holling Type II Incidence Function , 2012, Comput. Math. Methods Medicine.

[6]  R. Anderson,et al.  Sexual contact patterns between men and women and the spread of HIV-1 in urban centres in Africa. , 1991, IMA journal of mathematics applied in medicine and biology.

[7]  R. May,et al.  Regulation and Stability of Host-Parasite Population Interactions: I. Regulatory Processes , 1978 .

[8]  A. Sabin,et al.  Measles, killer of millions in developing countries: strategy for rapid elimination and continuing control , 2004, European Journal of Epidemiology.

[9]  S. Levin,et al.  Dynamical behavior of epidemiological models with nonlinear incidence rates , 1987, Journal of mathematical biology.

[10]  Zhen Jin,et al.  GLOBAL STABILITY OF A SEIR EPIDEMIC MODEL WITH INFECTIOUS FORCE IN LATENT, INFECTED AND IMMUNE PERIOD , 2005 .

[11]  Li Changguo,et al.  The effect of constant and pulse vaccination on an SIR epidemic model with infectious period , 2011 .

[12]  P. Kaye.  Infectious diseases of humans: Dynamics and control , 1993 .

[13]  Andrei Korobeinikov,et al.  Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission , 2006, Bulletin of mathematical biology.

[14]  Edward M. Lungu,et al.  THE EFFECTS OF VACCINATION AND TREATMENT ON THE SPREAD OF HIV/AIDS , 2004 .

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

[16]  Lansun Chen,et al.  The dynamics of a new SIR epidemic model concerning pulse vaccination strategy , 2008, Appl. Math. Comput..

[17]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[18]  H. Bedford,et al.  Measles , 1889, BMJ.

[19]  Zhidong Teng,et al.  The effects of pulse vaccination on SEIR model with two time delays , 2008, Appl. Math. Comput..

[20]  Y. Iwasa,et al.  Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models , 1986, Journal of mathematical biology.

[21]  Xuebin Chi,et al.  The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission , 2002 .

[22]  Zhidong Teng,et al.  Dynamic analysis of an SIR epidemic model with state dependent pulse vaccination , 2012 .

[23]  Zhidong Teng,et al.  Impulsive Vaccination of an SEIRS Model with Time Delay and Varying Total Population Size , 2007, Bulletin of mathematical biology.

[24]  M Rush,et al.  The epidemiology of measles in England and Wales: rationale for the 1994 national vaccination campaign. , 1994, Communicable disease report. CDR review.

[25]  M. E. Alexander,et al.  Periodicity in an epidemic model with a generalized non-linear incidence. , 2004, Mathematical biosciences.

[26]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[27]  Lansun Chen,et al.  Modeling and analysis of a predator-prey model with disease in the prey. , 2001, Mathematical biosciences.

[28]  Alberto d'Onofrio,et al.  On pulse vaccination strategy in the SIR epidemic model with vertical transmission , 2005, Appl. Math. Lett..

[29]  Roy M. Anderson,et al.  REGULATION AND STABILITY OF HOST-PARASITE POPULATION INTERACTIONS , 1978 .

[30]  Alberto d'Onofrio,et al.  Stability properties of pulse vaccination strategy in SEIR epidemic model. , 2002, Mathematical biosciences.