Stability and phase portraits of susceptible-infective-removed epidemic models with vertical transmissions and linear treatment rates
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Date
2017-12-14
Authors
Hoti, Marvin
Huo, Xi
Lan, Kunquan
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We study stability and phase portraits of susceptible-infective-removed (SIR) epidemic models with horizontal and vertical transmission rates and linear treatment rates by studying the reduced dynamical planar systems under the assumption that the total population keeps unchanged. We find out all the ranges of the parameters involved in the models for the infection-free equilibrium and the epidemic equilibrium to be positive. The novelty of this paper lies in the demonstration and justification of the parameter conditions under which the positive equilibria are stable focuses or nodes. These phase portraits provide more detailed descriptions of behaviors and extra biological understandings of the epidemic diseases than local or global stability of the models. Previous results only discussed the stability of the SIR models with horizontal or vertical transmission rates and without treatment rates. Our results involving vertical transmission and treatment rates will exhibit the effect of the vertical transmissions and the linear treatment rates on the epidemic models.
Description
Keywords
SIR model, Vertical transmission, Treatment rate, Stability, Node, Focus, Saddle-node
Citation
Hoti, M., Huo, X., & Lan, K. (2017). Stability and phase portraits of susceptible-infective-removed epidemic models with vertical transmissions and linear treatment rates. <i>Electronic Journal of Differential Equations, 2017</i>(306), pp. 1-17.