Epidemic characteristics of two classic models and the dependence on the initial conditions

Jianquan Li; Yiqun  Li; Yali  Yang

doi:10.3934/mbe.2016027

Article Contents

2016, Volume 13, Issue 5: 999-1010. Doi: 10.3934/mbe.2016027

This issue Previous Article Type-dependent stochastic Ising model describing the dynamics of a non-symmetric feedback module Next Article Modeling the role of healthcare access inequalities in epidemic outcomes

Epidemic characteristics of two classic models and the dependence on the initial conditions

1.
Science College, Air Force Engineering University, Xi'an 710051
2.
College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062

Received: November 30, 2015

Revised: April 30, 2016

Published: June 2016

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Abstract

The epidemic characteristics, including the epidemic final size, peak, and turning point, of two classical SIR models with disease-induced death are investigated when a small initial value of the infective population is released. The models have mass-action (i.e. bilinear), or density dependent (i.e. standard) incidence, respectively. For the two models, the conditions that determining whether the related epidemic characteristics of an epidemic outbreak appear are explicitly determine by rigorous mathematical analysis. The dependence of the epidemic final size on the initial values of the infective class is demonstrated. The peak, turning point (if it exists) and the corresponding time are found. The obtained results suggest that their basic reproduction numbers are one factor determining the epidemic characteristics, but not the only one. The characteristics of the two models depend on the initial values and proportions of various compartments as well. At last, the similarities and differences of the epidemic characteristics between the two models are discussed.

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Mathematics Subject Classification: Primary: 92D30.

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