Marvin Hoti, Xi Huo, Kunquan Lan
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.
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 |
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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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