Electron. J. Differential Equations, Vol. 2017 (2017), No. 306, pp. 1-17.

Stability and phase portraits of susceptible-infective-removed epidemic models with vertical transmissions and linear treatment rates

Marvin Hoti, Xi Huo, Kunquan Lan

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.

Submitted October 19, 2016. Published December 14, 2017.
Math Subject Classifications: 34C23, 92D25, 34D20, 34D23.
Key Words: SIR model; vertical transmission; treatment rate; stability; node; focus; saddle-node.

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Marvin Hoti
Faculty of Science
University of Ontario, Institute of Technology
Oshawa, Ontario, Canada L1H 7K4.
email: marvin.hoti@ryerson.ca
Xi Huo
Department of Mathematics
University of Miami
1365 Memorial Drive
Coral Gables, FL 33146, USA
email: huoxi@yorku.ca
  Kunquan Lan
Department of Mathematics
Ryerson University
Toronto, Ontario, Canada M5B 2K3
email: klan@ryerson.ca

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