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Backward bifurcation analysis for two continuous and discrete epidemiological models
Author(s) -
Anguelov Roumen,
Dukuza Kenneth,
Lubuma Jean M.S.
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5138
Subject(s) - mathematics , bifurcation , transcritical bifurcation , comparison theorem , dynamical systems theory , bifurcation diagram , bifurcation theory , nonlinear system , basic reproduction number , period doubling bifurcation , mathematical analysis , demography , population , physics , quantum mechanics , sociology
The bifurcation analysis of a continuous n ‐dimensional nonlinear dynamical system with a nonhyperbolic equilibrium point is done by using the main theorem in the work of Castillo‐Chavez and Song. We derive an analog of this theorem for discrete dynamical systems. We design nonstandard finite difference schemes for a susceptible‐infectious‐susceptible epidemiological model with vaccination and for a malaria model. For the latter model, we sharpen the interval of the values of the disease induced death rate for which backward bifurcation may occur. Applying the discrete theorem, it is shown that each nonstandard finite difference scheme replicates the property of the continuous model of having backward bifurcation at the value one of the basic reproduction number.